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+{"text":"\\section{Introduction}\n\n\nObservations and numerical (radiation-)MHD models of the formation of massive protostars along with their accretion disks and collimated jets have made clear progress of the last decade. Even though there are several observational examples of massive protostellar jets \\citep[see e.g.][]{Purser2016}, direct observations and estimations of the radii of their associated accretion disks are rarer. The source for the HH 80-81 radio jet, for example, has an accretion disk with an estimated radius of $950\\text{--}1300\\unit{au}$, and an inner radius of $\\lesssim 170\\unit{au}$, with the mass of the protostar in the range of $4\\text{--}18\\unit{M_\\odot}$ \\citep{Girart2017, Carrasco-Gonzalez2010}.\n\n\n\n\nRecently, a new generation of multi-scale observations of the star-forming region IRAS 21078+5211 \\citep{Moscadelli2021multi, Moscadelli2022} have revealed the existence of a disk-jet system around a protostar of $5.6\\pm 2 \\unit{M_\\odot}$. Emission peaks from molecular rotational transitions of $\\mathrm{CH_3CN}$ and $\\mathrm{HC_3N}$ trace the kinematical footprint of a Keplerian-like accretion disk of around $400\\unit{au}$ in diameter surrounding the protostar. For the first time, the direct study of the launching mechanism of the jet has been possible, thanks to water maser observations performed with Very Long Base Interferometry (VLBI) \\citep{Moscadelli2022}, which traced individual streamlines in the jet and reached a resolution of $0.05\\unit{au}$.\n\nIn \\cite{Moscadelli2022}, we modeled the disk-jet system in IRAS 21078+5211 with a resistive-radiation-gravito-magnetohydrodynamical simulation of a forming massive star starting from the collapse of a cloud core. We followed the evolution of the protostar, the formation of the accretion disk, and the launching, acceleration, collimation, propagation and termination of the magnetically-driven outflows, which self-consistently form within the computational domain. Thanks to the use of an axisymmetrical grid, we were able to reach unprecedented resolutions in the regions surrounding the forming massive star (up to $0.03\\unit{au}$), allowing for the comparison with the VLBI data (which has a resolution of $0.05\\unit{au}$). We found good agreement between the simulated and observed disk size (given the protostellar mass) as well as the main features of the jet. The simulation was part of a larger study that examines the effects of the natal environment on the formation of a massive star. We present the full series of simulations in this article, with special focus on the dynamics of the accretion disk, while the dynamics of the magnetically-driven outflows are the focus of an upcoming article (hereafter Paper II).\n\n\n\n\nPreviously, the numerical simulations performed by \\cite{Anders2018} were able to show the self-consistent magneto-centrifugal launching of the jet and a tower flow driven by magnetic pressure in the context of massive star formation. The present study builds on their work by improving the treatment of the thermodynamics of the gas and dust with the inclusion of radiation transport with the gray flux-limited diffusion approximation \\citep{Kuiper2020}, and increasing the resolution of the grid. Those changes enable us to examine the vertical structure of the accretion disk. Previous numerical studies of the formation of a massive star in an environment with magnetic fields did not considered magnetic diffusion (e.g. \\citealt{Myers2013} and \\citealt{Seifried2011}) or approximated the thermodynamics of the system with the use of a barotropic \\citep[e.g.][]{Machida2020} or isothermal \\citep{Anders2018} equation of state instead of solving for radiation transport. Recent studies by \\cite{Mignon-Risse2021} and \\cite{Commercon2022}, which include ambipolar diffusion as the non-ideal MHD effect and do include radiation transport, find indications of a magneto-centrifugally launched jet (although their grid does not properly resolve the launching region), and smaller disks with stronger magnetic fields. The question of the size of the disk is explored in more detail in the present article, as we find an opposite trend in our results. For further literature review, we refer the reader to Sect. \\ref{S: previous}.\n\n\nBecause of computational costs, previous efforts have focused on only a handful of natal environmental conditions for the massive (proto)star. Our setup, in turn, allows us to explore a wide range of natal environmental conditions and their impact on the outcome of the formed massive star and its disk-jet system.  In section \\ref{S: model method} we describe the model, physical effects considered, and the parameter space of the simulations. We present an overview of the evolution of the system based on the fiducial case of the parameter space in \\ref{S: evolution}. In section \\ref{S: dynamics}, we focus on the dynamical processes observed in the accretion disk, i.e., the effect of the magnetic field, Ohmic dissipation and radiation transport. In section \\ref{s:parameter-study} we explore the dependence of the disk size and evolution with the initial conditions for the gravitational collapse, and finally in section \\ref{S: previous} we offer a comparison of our results with previous numerical studies.\n\n\n\n\n\\section{Model and method} \\label{S: model method}\n\\subsection{Description of the model}\nWe model axisymmetrically the gravitational collapse of a cloud core of mass $M_C$ and a radius of $R_C=0.1\\unit{pc}$ (cf. Fig. \\ref{overview_evol}), which has an initial density profile of the form\n\\begin{equation}\n\\rho(r,t=0) = \\rho_0 \\left(\\frac{r}{r_0}\\right)^{\\beta_\\rho}.\n\\end{equation}\nThe constant $r_0$ is chosen to be $1\\unit{au}$ and $\\rho_0$ is determined by computing the mass of the cloud with the integral of the density profile over $0 < r < R_C$. The cloud is in slow initial rotation, given by an initial angular velocity profile of the form\n\\begin{equation}\n\\Omega(R,t=0) = \\Omega_0 \\left( \\frac{R}{R_0} \\right)^{\\beta_\\Omega},\n\\end{equation}\nwhere $R$ is the cylindrical radius and $R_0$ is chosen to be $10\\unit{au}$. The constant $\\Omega_0$ is fixed by computing the ratio of rotational to gravitational energy $\\zeta \\equiv E_r\/E_g$ from the initial density and rotation profiles; $\\zeta$ is then a parameter of the simulation.\n\nThe cloud core is threaded by an initially uniform magnetic field directed parallel to the rotation axis. Its magnitude $B_0$ is determined by the normalized mass-to-flux ratio $\\bar \\mu$,\n\\begin{equation}\n\\bar \\mu \\equiv \\frac{M_C\/\\Phi_B}{\\left(M_C\/\\Phi_B\\right)_\\mathrm{crit}}.\n\\end{equation}\nThe denominator of the previous expression represents the critical value of the mass-to-flux ratio, i.e., the value for which the gravitational collapse is halted by the magnetic field under idealized conditions. We take the value from \\cite{MouschoviasSpitzer1976}. The magnetic flux $\\Phi_B$ is simply computed as the flux across the midplane, $B_0 \\cdot \\pi R_C^2$, which in magnitude is the same flux that enters or leaves the spherical cloud. \n\n\\subsection{Physics}\n\nThe simulations solve for the equations of magnetohydrodynamics, with the addition of Ohmic resistivity as a non-ideal effect, self-gravity, the gravitational force from the forming massive star, and radiation transport for the thermal emission from the gas and dust, on an axisymmetric grid. Next, we describe the set of equations solved.\n\nThe dynamics of the gas and dust are followed by numerically solving the following system of conservation laws, using the Pluto code \\citep{Mignone2007}:\n\\begin{equation}\n\\partial_t \\rho + \\vec \\nabla \\cdot (\\rho \\vec v) = 0,\n\\end{equation}\n\\begin{equation}\n{\\partial_t (\\rho \\vec v) } + \\vec \\nabla \\cdot \\left[  \\rho \\vec v \\otimes \\vec v -  \\tfrac{1}{4\\pi}{\\vec B \\otimes \\vec B} + P_t  \\mathsf{I} \\right] = \\rho \\vec{a}_\\text{ext},\n\\end{equation}\n\\begin{equation}\n{\\partial_t \\vec B} + \\vec \\nabla \\times (c \\vec{\\mathcal E}) = 0,\n\\end{equation}\n\\[ {\\partial_t (E^K+E^\\text{th}+E^B)} \\]\n\\begin{equation} \\label{e:eneq}\n\\quad\\quad + \\vec \\nabla \\cdot \\left[ (E^K+E^\\text{th} +  P)\\vec v + c \\vec{\\mathcal E} \\times \\vec B \\right] = \\rho \\vec v \\cdot \\vec a_\\text{ext},\n\\end{equation}\nwhich correspond to the continuity equation, momentum equation, induction equation and energy conservation equation, respectively. The gas is weakly ionized and has density $\\rho$, velocity $\\vec v$ and magnetic field $\\vec B$. The magnetic field must also satisfy the solenoidality condition $\\vec \\nabla \\cdot \\vec B = 0$. The total pressure $P_t = P + \\tfrac{1}{8\\pi}{B^2}$ is composed of the thermal and magnetic pressures. The electric field $\\vec{\\mathcal E}$ can be directly substituted by the expression\n\\begin{equation}\nc\\vec{\\mathcal E} = -\\vec v \\times \\vec B + \\eta \\vec \\nabla \\times \\vec B,\n\\end{equation}\nwhere $\\eta(\\rho,T)$ is the Ohmic resistivity. The energy density of Eq. \\ref{e:eneq} is separated into its components: kinetic ($E^K$), thermal or internal ($E^\\mathrm{th}$) and magnetic ($E^B$). The acceleration source term $\\vec a_\\text{ext}$ is a sum of the gravity of the forming star $\\vec a_{g\\star} = -GM_\\star\/r^2\\, \\vec e_r$, the self-gravity of the gas $\\vec a_\\text{sg}$ and the viscosity source term $\\vec a_\\nu$.\n\n\nThe self-gravity acceleration source term is given by $\\vec a_\\text{sg} = -\\nabla \\Phi_\\text{sg}$. The self-gravity potential $\\Phi_\\text{sg}$ is found by solving Poisson's equation $\\nabla^2 \\Phi_\\text{sg} = 4\\pi G\\rho$ with a diffusion Ansatz, according to \\cite{Kuiper2010circ}. Self-gravity is inherently a three-dimensional effect, and a cloud with the characteristics we consider here is expected to produce an accretion disk with spiral arms. Since we perform the simulations on a two-dimensional axisymmetric grid, we mimic the missing angular momentum transport by the gravitational torques with the addition of a shear viscosity source term $\\vec a_\\nu = \\vec \\nabla \\Pi\/\\rho$, where the viscosity tensor $\\Pi$ is fixed with the $\\alpha$-parametrization of \\cite{ShakuraSunyaev1973} and no bulk viscosity is considered. This approach was extensively studied in \\cite{Kuiper2011}, and we refer the reader to that article for more details on the computation of the viscosity tensor, as well as comparisons of the efficacy of angular momentum transport using this approach in comparison to non-viscous 3D models.\n\n\nWe treat radiation transport with the gray flux-limited diffusion approximation, by using the module Makemake described in ample detail in \\cite{Kuiper2020}. Specifically, we point  the reader to sections 2.3.2 and 2.3.4 in that reference. In a nutshell, two equations are solved simultaneously for treating radiation transport. The first one is the zeroth-moment of the radiation transport equation within a given volume, which essentially says that the net emitted energy density (emission minus absorption) that is not lost through the boundary as a radiative flux, has to be stored as a local radiative energy density. This equation is simplified by assuming  that the radiative flux can be written as a diffusion term in terms of the radiation energy density (hence flux-limited diffusion). The second equation describes the changes to $E^\\text{th}$: the net absorbed energy density (absorption minus emission) has to be stored as internal energy in the gas. Both energy fields are solved without previous assumption of equilibrium between them; this is the so-called two-temperature approach \\cite{Commercon2011rad}. In all but one of the simulations (see Sect. \\ref{S: disk var eta} for details), a constant value of the opacity for the dust and gas of $1\\unit{cm^2\\, g^{-1}}$ was used. The initial dust-to-gas mass ratio is set to $1\\%$. The irradiation from the forming massive star is not considered in this study.\n\n\nWe use the resistivity model by \\cite{Machida2007} (which is based on a numerical study by \\citealt{Nakano2002})\n\\begin{equation}\n\\eta = \\frac{740}{X_e} \\left({\\frac{T}{10\\unit{K}}}\\right)^{1\/2} \\left[  1 - \\tanh \\left( \\frac{n_H}{10^{15} \\unit{cm^{-3}}} \\right) \\right] \\unit{cm^2\\, s^{-1}},\n\\end{equation}\nwhere we use for simplicity $n_H=\\rho\/\\mu_H$ as the number density of hydrogen nuclei ($\\mu_H$ being the molecular weight of hydrogen) and the ionization fraction $X_e$ is given by\n\\begin{equation}\nX_e = 5.7\\cdot 10^{-4} \\left(\\frac{n_H}{\\unit{cm^{-3}}}\\right)^{-1}.\n\\end{equation}\n\n\nAt all moments in time, the properties of the formed massive star are computed using the evolutionary tracks for high-mass stars calculated by \\cite{HosokawaOmukai2009evol} and the instantaneous values of the stellar mass and accretion rate.\n\n\\subsection{Boundary conditions}\nThe computational domain assumes symmetry with respect to the rotation axis and equatorial symmetry with respect to the midplane. This means that scalar quantities and the velocity are simply reflected across the $z$-axis and the midplane. In the case of the magnetic field, it is reflected across the $z$-axis, but across the midplane, the parallel component switches sign instead of the normal component. The same symmetries imply that the boundaries across the azimuthal direction are periodic. Both the inner and outer boundaries (i.e., the sink cell and the outermost part of the cloud) impose a zero gradient condition for the magnetic field and the azimuthal and polar components of the velocity; for the radial component of the velocity and the density, only outflow but no inflow is allowed. Therefore, all matter that goes inside of the sink cell is considered as accreted.\n\n\\subsection{Initial conditions: parameter space and numerical configuration}\n\n\\begin{table}\n\\caption{Grid resolutions}\n\\label{t:grid}\n\\centering\n\\begin{tabular}{c c c c c}\n\\hline\\hline\nGrid & $N_r$ & $N_\\theta$ & $\\Delta x_\\text{min}\\ [\\unit{au}]$ & $\\Delta x_{1000}\\  [\\unit{au}]$ \\\\\n\\hline\nx1 & 56 & 10 & 0.51 & 160 \\\\\nx2 & 112 & 20 & 0.25 & 80 \\\\\nx4 & 224 & 40 & 0.12 & 40 \\\\\nx8 & 448 & 80 & 0.06 & 20\\\\\nx16 & 896 & 160 & 0.03 & 10\\\\\n\\hline \n\\end{tabular}\n\\end{table}\n\n\nWe present an extensive parameter study of the system, with the express aim of aiding in the understanding of the physical processes that intervene in the formation and evolution of the disk and the magnetically-driven outflows. In total, we present results from 31 different runs. Instead of describing all the parameters in multiple tables, we show a summary of the parameter space as a diagram in the right panel of Fig. \\ref{overview_evol}.\n\n\nWe used five different axisymmetrical grids with increasing resolution, that we name x1, x2, x4, x8 and x16. The grid for the radial coordinate $r$ increases logarithmically with distance, starting from the radius of the inner boundary ($3\\unit{au}$ in the fiducial case), to the radius of the cloud core ($0.1\\unit{pc}$). We often refer to the inner boundary as the sink cell. Using the assumed symmetries of the system, the colatitude $\\theta$ extends from $0$ to $\\pi\/2$ radians, and it is divided linearly into cells such that they have approximately the same dimensions in both the radial and polar directions for each distance to the center of the cloud. Table \\ref{t:grid} details the number of cells in each direction, the minimum cell size and a reference cell size at $1000\\unit{au}$ for all the grids used in this study, and assuming the sink cell size of the fiducial case.\n\n\nThe parameters with direct relation to physical quantities we considered are: the mass of the cloud core $M_C$, the resistivity model used, the normalized mass-to-flux ratio $\\bar \\mu$, the initial density profile exponent $\\beta_\\rho$, the initial rotation profile exponent $\\beta_\\Omega$, and the ratio of the rotational-to-gravitational energy $\\zeta$. The values for each parameter in the fiducial case are highlighted in blue in Fig. \\ref{overview_evol}. The fiducial case was run on all the grids. Modifications of the fiducial case are then investigated while keeping the rest of the parameters constant. Because of this, we refer in the paper to each simulation by the modified parameter followed by the grid it was run on; for example, $M_C = 150\\unit{M_\\odot}\\text{ [x8]}$ refers to a cloud of $150\\unit{M_\\odot}$, with the rest of the parameters as in the fiducial case for grid x8. The full set of values with direct relation to physical quantities was run on the base grid x4, and some selected values, which are underlined in Fig. \\ref{overview_evol}, were run on grid x8 as well. During the preparation of this study, we also ran a part of this physical parameter space on the lower resolution grids, however, we do not include the results here because they are redundant. Details of the choice of the parameters, as well as the comparison of results obtained, are presented in Sect. \\ref{s:parameter-study}.\n\n\nThe numerical convergence of the results was studied by changing the values of the sink cell radius, the strength of the $\\alpha$ shear viscosity and the Alfv\u00e9n limiter using the grid x2 as a base. A more detailed explanation of these latter  parameters, as well as the results of the convergence study can be found in Appendix \\ref{s:convergence}.\n\n\n\n\n\\section{Overview of the evolution of the system}\\label{S: evolution}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{imgraw\/overview_evol}\n\t\\caption{Initial conditions, overview of the evolution of the system, and summary of the parameter space presented in this study.}\n\t\\label{overview_evol}\n\\end{figure*}\n\nFigure \\ref{overview_evol} shows a schematic overview of the features observed during the evolution of the system, based on the results of the fiducial case but generally present in most of the simulations of the study. We describe these characteristic features and processes in the following subsections case-by-case.\n\n\\subsection{Gravitational infall epoch}\nAs soon as the simulation begins, the cloud starts collapsing under its own gravity. The flow quickly becomes super-Alfv\u00e9nic (the kinetic flow velocity is larger than the Alfv\u00e9n speed), and the magnetic field lines are dragged by the infall accordingly. The result of this collapse is that the configuration of magnetic field lines adopts an ``hourglass'' shape, as described in \\cite{Galli2006}, and that has been observed in \\cite{Beltran2019}.\n\n\\subsection{Protostar}\nA massive (proto)star is formed at the center of the cloud, which reaches a mass of $\\sim 20\\unit{M_\\odot}$ after $30\\unit{kyr}$. The luminosity of the (proto)star is dominated by the conversion of gravitational energy into radiation during the first $20\\unit{kyr}$, and only after that time the stellar luminosity becomes comparable to the accretion luminosity. We point out that both the stellar and accretion luminosities are not taken into account in this study, but we will report on these radiative feedback effects in future work.\n\n\\subsection{Magneto-centrifugal epoch}\n\nDue to the conservation of angular momentum, gas coming from large scales rotates faster as it reaches the center of the cloud. At $t\\sim 5\\unit{kyr}$, enough angular momentum is transported onto the center of the cloud to form a Keplerian-like accretion disk as a result. The disk grows in time due to the continued infall of material from large scales.\n\n\nThe accretion disk can be morphologically divided into a thin and a thick layer. The thin layer of the disk  is a region of the midplane consisting of material in rotation with speeds close to the Keplerian speed. The density of the thin layer increases towards the center of the cloud $10^{-16}$ to $10^{-11}\\unit{g\\, cm^{-3}}$. Enclosing the thin layer, there is a less dense thick layer (typical densities: $10^{-14}\\unit{g\\,cm^{-3}}$) which is also rotating.\n\n\nRoughly at the same time as the disk forms, magnetically-driven outflows are launched, which we classify according to their speeds and driving mechanisms. The jet is launched by the magneto-centrifugal mechanism (in a way similar to the model of \\citealt{BlandfordPayne1982}) and it reaches typical speeds higher than $100\\unit{km\\,s^{-1}}$. Rotation drags the magnetic field lines, as it has been observed in \\cite{Beuther2020} and \\cite{Girart2013}. This creates a magnetic pressure gradient that drives a slower and broader tower flow (cf. the model of \\citealt{Lynden-Bell2003}). Its typical speeds are of the order of $10\\unit{km\\,s^{-1}}$, and it broadens with time. A discussion of the dynamical processes present in the launching of outflows is offered in Paper~II.\n\n\nOn the interface between inflow along the thick layer of the disk and outflow along the cavity, an additional structure is identified, which we refer as the \\emph{cavity wall}. Densities at the cavity wall are higher than those in the cavity, but also higher than the surrounding infalling envelope and the thick layer of the disk. In terms of its dynamics, material from the cavity wall contributes episodically to both the inflow and the outflow.\n\n\\subsection{Magnetic braking epoch}\n\nAs magnetic field lines are dragged by rotation, the resulting magnetic tension exerts a torque on the gas that brakes it, infalling as a consequence of the angular momentum loss. The inclusion of Ohmic dissipation makes magnetic braking negligible at early times in the simulation, but at around $t\\sim 15 \\unit{kyr}$, magnetic braking starts to affect the innermost region of the disk and the cavity wall ($r\\lesssim 30\\unit{au}$). As a result, a magnetically-braked region is formed, where the material is mostly infalling but where the disk densities are kept because of constant replenishment through material from the cavity wall.\n\n\nThe extraction of angular momentum from the inner region of the cloud by magnetic braking modifies or even interrupts the magneto-centrifugal mechanism that drives the jet, but not the tower flow, which is mostly present at large scales. The tower flow broadens over time because the continuous dragging of magnetic field lines by rotation increases the magnetic pressure gradient on top of the disk, which is itself growing in time as well. This outflow broadening mechanism offers an explanation for the earliest outflow broadening observed and discussed in \\cite{Beuther2005}. An analysis and discussion of the effects of magnetic braking in outflows, as well as their propagation over time into the large scales of the cloud is offered in Paper~II, which focuses on the outflow physics of the same simulation data analyzed here.\n\n\n\\section{Dynamics of the disk} \\label{S: dynamics}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{imgraw\/thindisk_keplerianity.pdf}\n\t\\caption{Keplerianity on the midplane, computed with the fiducial simulation on grid x8.}\n\t\\label{thindisk_keplerianity}\n\\end{figure}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{imgraw\/thindisk_energies.pdf}\n\t\\caption{Contributions to the specific energy in the thin layer of the disk, calculated with the fiducial simulation on grid x8.}\n\t\\label{thindisk_energies}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{imgraw\/stratification_torus_1}\n\t\\caption{Dynamics of the thick layer of the disk. The panels of the figure are grouped in columns such that they show: the density structure of the thick layer (\\emph{a} and \\emph{e}), the offset from Keplerianity at three different heights in the thick layer (\\emph{b} and \\emph{f}), the contributions to the specific energy (\\emph{c} and \\emph{g}), and the vertical balance of forces (\\emph{d} and \\emph{h}). Data from the fiducial simulation on grid x8 was used for this figure.}\n\t\\label{thickdisk_dynamics}\n\\end{figure*}\n\nAfter giving an overview on the general evolution of the system, we aim at understanding the dynamical processes present in the disk in detail by taking the fiducial case and examining selected terms of the equations of motion. With the identification of these processes, we are able to perform a more effective and comprehensive comparison of results for the full parameter space, which we present in Sect. \\ref{s:parameter-study}.\n\n\\subsection{Radial dynamics in the disk}\n\nThe forming massive star accretes mass through the accretion disk, which implies that the velocity in the disk must have a component in the radially-inwards direction. The torque necessary for the angular momentum transport in the disk is provided by gravitational torques of forming spiral arms, which is modeled here through the use of the $\\alpha$ shear viscosity model. Nevertheless, $v_\\phi \\gg v_R$ in the disk, and so, it is useful to examine the effects of the different forces that act in the disk under the assumption of radial equilibrium.\n\n\nIn cylindrical coordinates $(R,\\phi,z)$, the radial component of the MHD momentum equation, setting $\\theta=\\pi\/2$ (midplane), assuming axisymmetry, is given by\n\\begin{equation}\\label{e:mhd-R}\n\\frac{\\partial (\\rho v_R)}{\\partial t} + \\nabla \\cdot \\left( \\rho v_R \\vec v - \\frac{B_R \\vec B}{4\\pi}  \\right) + \\frac{\\partial P_t}{\\partial R} = \\rho g_R + \\frac{\\rho v_\\phi^2 - B_\\phi^2\/(4\\pi)}{R},\n\\end{equation}\nwhere the divergence operator $\\nabla \\cdot$ acts according to\n\\begin{equation}\\label{e:nablaop}\n\\nabla \\cdot \\vec f(R,z) = \\frac{1}{R}\\frac{\\partial (R f_R)}{\\partial R} + \\frac{\\partial f_z}{\\partial z}\n\\end{equation}\nonto the vector fields $\\vec f(R,z) \\in \\{ \\rho v_R \\vec v, B_R\\vec B\\}$; $\\vec g$ is the total gravitational force from the star and self-gravity, and $P_t$ is the sum of the thermal and magnetic pressures.\n\n\n\nIn order to examine the forces that define radial equilibrium in the disk, we consider a parcel of gas moving in a circular orbit, for which we set $v_R = 0$ in eq. \\ref{e:mhd-R} and obtain\n\n\\begin{equation}\\label{e:mhd-R2}\n \\frac{\\rho v_\\phi^2 - B_\\phi^2\/(4\\pi)}{R} = -\\rho g_R + \\frac{\\partial P}{\\partial R} +  \\frac{\\partial}{\\partial R}\\left(\\frac{B^2}{8\\pi}\\right) -\\nabla \\cdot  (B_R \\vec B).\n\\end{equation}\n\nFor $\\vec B = \\vec 0$, $P = 0$, and considering the midplane, one recovers the well-known condition for gravito-centrifugal equilibrium (on the co-rotating frame), which defines the Keplerian velocity, $v_K = \\sqrt{GM(r)\/R}$; $M(r)$ being the enclosed mass (dominated by the mass of the protostar for small $r$). The use of the enclosed mass is an approximation to the use of the full potential given by the Poisson equation.\n\n\\subsubsection{Keplerianity}\nThe degree of gravito-centrifugal equilibrium in the whole disk can be measured by computing the ratio of the centrifugal force density to the gravitational force density:\n\\begin{equation}\n\\frac{a^\\mathrm{c}_R}{\\rho g_R} = \\frac{\\rho v_\\phi^2}{r\\sin\\theta} \\cdot \\frac{r^2}{\\rho G M(r) \\sin\\theta} = \\frac{v_\\phi^2}{v_K^2\\sin^3\\theta} .\n\\end{equation}\nThis motivates us to define the offset from Keplerianity as\n\\begin{equation}\n\\text{offset from Keplerianity} = \\left|\\frac{v_\\phi}{v_K\\sin^{3\/2}\\theta}\\right|-1,\n\\end{equation}\nwhich in the midplane corresponds to the fractional difference of the azimuthal velocity with respect to the Keplerian value.\n\n\nFigs. \\ref{thindisk_keplerianity} and \\ref{thickdisk_dynamics} show the offset from Keplerianity presented as a percentage, in such a way that the negative values indicate sub-Keplerianity, and the positive values, super-Keplerianity. In the midplane, there is a Keplerian region within $\\pm 25 \\%$ for all times, which corresponds to the thin accretion disk. The Keplerianity drops outside of the disk, which corresponds to the infalling envelope. At the exterior boundary of the disk, the mostly-rotating material encounters the mostly-infalling material from the region of the envelope, creating a centrifugal barrier that is seen as a slight super-Keplerianity in Fig. \\ref{thindisk_keplerianity}. Even though we ignore the effect of viscosity in the analysis of this section, we note that, as it transports angular momentum outwards, its effect is to increase positively the deviation from Keplerianity. The rest of the deviations from Keplerianity in the disk are discussed in the following sections.\n\nPanels \\emph{b} and \\emph{f} of Fig. \\ref{thickdisk_dynamics} contrast the values of the offset from Keplerianity in the midplane with values in planes parallel to it, which slice through the thick layer of the disk and reveal its dynamical structure. In general, the thick layer also shows Keplerianity within $\\pm 25\\%$, but it decreases with altitude because the cylindrical-radial component of gravity also decreases with altitude. The regions close to the cavity become increasingly super-Keplerian, while the regions close to the infalling envelope become increasingly sub-Keplerian.\n\n\\subsubsection{Specific energies}\nWe study the dynamics of the thin and thick layers of the disk by computing the terms of the Bernoulli equation, that is, the contributions to the specific energy of the material. This allows us to isolate the role of each term in the equations of magnetohydrodynamics and compare its relative importance. The results of the energy calculations are available in Fig. \\ref{thindisk_energies} for the midplane, and Fig. \\ref{thickdisk_dynamics} for the plane $z=z_1$. The contributions to the specific energy we considered are:\n\\begin{itemize}\n\\item the gravitational specific energy \\begin{equation}e^\\text{grav} = \\frac{GM(r)}{r},\\end{equation}\n\\item the thermal specific energy \\begin{equation} \\label{e:energies-thermal}\ne^\\text{th} = \\frac{P}{\\rho (\\Gamma - 1)},\n\\end{equation}\n\\item the contribution to the specific kinetic energy by the azimuthal component of velocity \\begin{equation}\ne^{K}_\\phi = \\frac{v_\\phi^2}{2},\n\\end{equation}\n\\item the contribution to the specific kinetic energy by the spherical-radial component of velocity\n\\begin{equation}\ne^K_r = \\frac{v_r^2}{2},\n\\end{equation}\n\\item the contribution to the specific magnetic energy by the toroidal component of the magnetic field\n\\begin{equation}\ne^B_\\text{tor} = \\frac{B_\\phi^2}{8\\pi \\rho},\n\\end{equation}\n\\item the contribution to the specific magnetic energy by the poloidal component of the magnetic field\n\\begin{equation}\ne^B_\\text{pol} = \\frac{B_r^2 + B_\\theta^2}{8\\pi \\rho}.\n\\end{equation}\n\\end{itemize}\n\n\n\nInitially, the specific gravitational energy increases as a function of $r$ because it is given by the enclosed mass (see Fig. \\ref{thindisk_energies}a). As the collapse progresses (Fig. \\ref{thindisk_energies}b), the gravity of the forming massive star becomes more important, which means that the curve for $e^\\text{grav}$ has a point-source-like behavior close to the center of the cloud. At large scales, the enclosed mass of the envelope becomes important. As expected, the infalling envelope has $e^K_r > e^K_\\phi$, but the advection of angular momentum from large to small scales causes $e^K_\\phi$ to increase until $e^K_\\phi > e^K_r$, forming the disk. When the magnetic field lines are dragged by rotation in the disk, the toroidal contribution to the magnetic energy increases.\n\n\\subsubsection{Thermal and magnetic pressure} \\label{S: dynamics - PB}\n\nThe following two sections examine the contributions of the remaining terms of eq. \\ref{e:mhd-R2} to the equilibrium of cylindrical radial forces in the disk and their impact to the values of Keplerianity shown in Fig. \\ref{thindisk_keplerianity}.\n\n\n\nFirst, we only consider the additional radial support that the thermal pressure gradient $\\partial P\/\\partial R$ gives to the material in the disk, which be inferred by using eq. \\ref{e:energies-thermal} and Figs. \\ref{thindisk_energies} and \\ref{thindisk_denstemp}. In agreement with the theory, we find that the thermal pressure gradient provides an additional small support against gravity and therefore makes material in radial equilibrium to appear to be slightly sub-Keplerian.\n\n\n\nIn Gaussian units, the magnetic pressure and magnetic energy density correspond to the same quantity, which means that we can use the specific energy density to study the magnetic pressure support in the disk. In the infalling envelope, the poloidal component of the magnetic field contributes the most to magnetic pressure (see, e.g., Fig. \\ref{thindisk_energies}b), given that the initial magnetic field distribution is parallel to the rotation axis. When the disk is formed, the toroidal component of the magnetic field increases in general because the magnetic field lines are dragged by rotation. The toroidal magnetic field becomes dominant over the poloidal component, which additionally allows us to neglect the magnetic tension term $\\nabla \\cdot (B_R \\vec B)$ from eq. \\ref{e:mhd-R2} in most parts of the disk.\n\n\nThe magnetic pressure $P_B = \\rho (e^B_\\text{tor} + e^B_\\text{pol})$ decreases with distance in the infalling envelope (compare Figs. \\ref{thindisk_energies}b and  \\ref{thindisk_denstemp}a), and overall in the disk as well, although the more complex structure of the magnetic field in the thin and thick layers of the disk means that there are regions of local increase with distance. Similarly to our argument with the thermal pressure gradient, $v_\\phi < v_K$ for a negative magnetic pressure gradient. Therefore, the radius of the disk should be slightly larger compared to the non-magnetic case, given that $v_\\phi$ decreases with distance and the angular momentum contained at larger $R$ becomes sufficient to support a circular orbit. This argument is revisited in Sect. \\ref{S: disk var B} after considering the effects of magnetic braking (Sect. \\ref{S: disk MB}) and magnetic diffusion (Sect. \\ref{S: disk m diff}).\n\n\nAdditionally to magnetic pressure, the second term of the left hand side of eq. \\ref{e:mhd-R2} reveals that the azimuthal component of the magnetic field reduces the effectiveness of the centripetal force required to keep a circular orbit. This can be measured by comparing the azimuthal component of the kinetic and magnetic energies:\n\\begin{equation}\n\\frac{B_\\phi^2}{4\\pi R} \\cdot \\frac{R}{\\rho v_\\phi^2}  = \\frac{e^B_\\text{tor}}{e^K_\\phi}.\n\\end{equation}\nIn the thin layer of the disk, the contribution of this term is negligible, but in the thick layer, they are comparable. In terms of Keplerianity, this term can cause the material to be slightly super-Keplerian; this can be observed when comparing panels \\emph{f} and \\emph{g} of Fig.  \\ref{thickdisk_dynamics}.\n\n\n\n\\subsection{Ohmic dissipation} \\label{S: disk m diff}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{imgraw\/Rem}\n\\caption{Magnetic Reynolds number, computed with the fiducial simulation on grid x8. For $t=10\\unit{kyr}$, the radius of the disk is around $80\\unit{au}$, while for $t=22.75\\unit{kyr}$, the exterior radius is $\\approx 700\\unit{au}$ and the interior radius is $\\approx 20\\unit{au}$.}\n\\label{Rem}\n\\end{figure}\n\n\n\nAccording to the model of Ohmic resistivity by \\cite{Machida2007}, which we use, the resistivity increases progressively with density for the number densities we obtain here ($n_H \\lesssim 10^{13}\\unit{cm^{-3}}$). We computed the ratio of the magnitude of the induction and diffusion terms in the induction equation, which is analogous to the magnetic Reynolds number, as\n\\begin{equation}\\label{e:Rem}\n\\mathrm{Re}_m = \\frac{|\\vec \\nabla \\times (\\vec v \\times \\vec B)|}{|\\eta \\vec \\nabla \\times  (\\vec \\nabla \\times \\vec B)|},\n\\end{equation}\nand show the results in Fig. \\ref{Rem}, which are computed for two instants of time from the fiducial case. At $t=10\\unit{kyr}$, the plasma is fully diffusive in most regions of the thin layer of the disk; the thick layer is dominated by advection but it experiences magnetic diffusion in around one part per hundred; and the cavity wall is also partially diffusive. Later in time, at $t=22.75\\unit{kyr}$, only the highest density regions close to the massive protostar are dominated by diffusion, the thin layer experiences magnetic diffusion of around one part per hundred, and the thick layer is almost completely dominated by advection.\n\n\nOhmic resistivity acts mostly on the toroidal component of the magnetic field, as it can be seen by comparing the energy curves in Figs. \\ref{thindisk_energies} and \\ref{thickdisk_dynamics}. In the thick layer of the disk at late times (Fig. \\ref{thickdisk_dynamics}g) $e_\\text{tor}^B$ roughly follows $e_\\phi^K$. This sets the picture for the case where magnetic diffusion is low. On the other hand, in the thin layer of the disk, at early times (Fig. \\ref{thindisk_energies}b), $e_\\text{tor}^B$ and $e_\\phi^K$ are clearly decoupled: the toroidal magnetic field component becomes low inside of the disk, corresponding to the region where diffusion dominates over advection. In the outer disk, where the density is lower, the resistivity is lower as well, and the toroidal component of the magnetic field is allowed to increase. \n\n\n\\subsection{Magnetic braking} \\label{S: disk MB}\nThe momentum equation in the azimuthal direction, considering axisymmetry and circular motion, reduces to\n\\begin{equation}\n\\frac{\\partial (\\rho v_\\phi)}{\\partial t}  =  \\frac{1}{R^2}\\frac{\\partial}{\\partial R}(R^2 B_\\phi B_R) + \\frac{\\partial}{\\partial z}(B_\\phi B_z).\n\\end{equation}\nThe right hand side of this equation is equal to zero if there are no magnetic fields, and it leads to the conservation of angular momentum. The terms on the right hand side constitute the azimuthal component of the magnetic tension force, which exerts a torque that reduces angular momentum locally. This process is known as magnetic braking, and it can be understood as a consequence of the dragging of magnetic field lines by rotation \\citep{Galli2006}.\n\nEven though Ohmic resistivity reduces the toroidal component of the magnetic field (cf. $e^B_\\text{tor}$ in Fig. \\ref{thindisk_energies}b), it does not completely suppress it. Over time, the magnetic field lines are wound enough so that magnetic braking happens in the innermost parts of the disk. In Fig. \\ref{thindisk_energies}c, which corresponds to $t=22.75\\unit{kyr}$,  $e^B_\\text{tor}$ is higher in the disk than when it is measured at $t=10\\unit{kyr}$, while $e^K_\\phi$ decreases over the same interval of time in the inner $20\\unit{au}$; those quantities show the reduction of angular momentum due to magnetic tension. In the face of the reduction of angular momentum, the material falls toward the protostar, as evidenced by the increase in $e^K_r$; this creates the magnetically-braked region. The cavity wall and the thick layer of the disk are also affected by magnetic braking; material from several directions is then delivered to the magnetically-braked region.\n\n\n\\subsection{Vertical support}\n\nIn the vertical direction (i.e., parallel to the rotation axis), the momentum equation reads\n\\begin{equation}\\label{e:mhd-z}\n\\frac{\\partial (\\rho v_z)}{\\partial t} + \\nabla \\cdot \\left(\\rho v_z \\vec v  - \\frac{B_z \\vec B}{4\\pi}\\right) + \\frac{\\partial P_t}{\\partial z} = \\rho g_z\n\\end{equation}\nwith the same definition of the divergence as given in eq. \\ref{e:nablaop}. Considering a parcel of gas moving in a circular orbit ($v_z=0$), and given that in the disk region $B_z \\ll B_\\phi$, eq. \\ref{e:mhd-z} becomes\n\\begin{equation}\n\\frac{\\partial P}{\\partial z} + \\frac{\\partial}{\\partial z} \\left( \\frac{B^2}{8\\pi} \\right) = \\rho g_z.\n\\end{equation}\n\n\nThe ratio of the  thermal to magnetic pressure is equivalent to the ratio of the specific thermal energy to the total magnetic energy. In the thin layer of the disk (Fig. \\ref{thindisk_energies}), thermal pressure dominates over magnetic pressure, from which we conclude that the thin layer is supported mainly by thermal pressure (at late times, magnetic pressure becomes comparable but not much higher than thermal pressure). On the contrary, the thick layer of the disk is supported by magnetic pressure, as evidenced by Figs. \\ref{thickdisk_dynamics}d and \\ref{thickdisk_dynamics}h, where we present a direct comparison of both pressure gradients to gravity in a vertical slice that crosses the disk. As the vertical component of gravity vanishes at the midplane, both curves increase towards $z=0$. In the thick layer of the disk, thermal pressure decreases, while the magnetic pressure gradient compensates gravity. At the cavity wall, thermal pressure increases because it has a higher density than the thick layer of the disk. Finally, in the cavity, magnetic pressure dominates over gravity in the vertical direction, which is expected in the case of a magnetically-driven outflow.\n\n\n\\subsection{Effects of other forms of magnetic diffusion} \\label{S: other magn diff}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{imgraw\/brhox16-700.png}\n\\caption{Distribution of the magnetic field strength as a function of density. This histogram was computed with the fiducial simulation run on grid x16, for time $t=7\\unit{kyr}$. The color scale indicates the polar angle: the dark purple colors correspond to the midplane (with the disk corresponding to $\\rho \\gtrsim 10^{-15} \\unit{g\\,cm^{-3}}$) and the light blue colors correspond to the rotation axis (where the jet is located).}\n\\label{brho}\n\\end{figure}\n\nIn this study, the effects of ambipolar diffusion and the Hall effect are not considered, and this formally constitutes a caveat of our approach which we plan to address in a future study. In this section, we discuss the expected effects that other forms of magnetic diffusivity might have on the results reported in the previous sections, with a focus on the fiducial case. We examine every region of interest for the discussion: the accretion disk, the infalling envelope and the outflow cavity. The simulation catalog also includes an investigation of the effects of the resistivity model; the results can be used to further assess the missing magnetic diffusivity and are presented in Sect. \\ref{S: disk var eta}.\n\nFirst, we consider only the accretion disk. The curves for ambipolar diffusivity and Ohmic resistivity given in e.g. \\cite{Marchand2022,Marchand2016} and \\cite{Tsukamoto2020} reveal that, at least for the range of number densities $10^8 \\lesssim n_H \\lesssim 10^{13}\\unit{cm^{-3}}$, which are found in the thin layer of the accretion disk, the values of the diffusivities are similar within the uncertainties of the dust models considered for their derivation, differing by about a factor of ten. The distribution of the magnetic field strength with density is presented in Fig. \\ref{brho}, where we use a color scale that highlights with darker colors the midplane. For the densities of the accretion disk ($\\rho > 10^{-14}\\unit{g\\,cm^{-3}}$), the magnetic field strength increases with density, until it reaches between $0.1$ and $1\\unit{G}$. When comparing the magnetic field distribution to the results of a recent study by \\cite{Commercon2022}, who considered the effects of ambipolar diffusion but not Ohmic dissipation, we find consistent results for the thin layer of the disk. In the thick layer, however, the addition of higher, more realistic magnetic diffusivities might reduce the vertical magnetic pressure support. This means that thinner accretion disks than those reported here would be expected in a more realistic setting.\n\nAs for the effects of magnetic braking, we expect that ambipolar diffusion and the Hall effect reduce the magnetic Reynolds number, that is, the material would become more diffusive. Contrary to Ohmic dissipation, ambipolar and Hall diffusivities depend on magnetic field, and so the reduction could be of particular importance at later times, when considerable magnetic flux has been transferred to the central regions of the cloud core. This would cause a delay in the formation of the magnetically-braked region in the inner disk (see also the discussion in Sect. \\ref{S: disk var eta}). On the other hand, magnetic braking could play an important role in determining the total angular momentum transferred to the forming massive star, as well as the growth of HII regions by terminating the magnetically-driven outflows (Martini et al., in prep.). An investigation of the role of all forms of magnetic diffusivity in the long-term evolution of the accretion disk and the jet is necessary. However, at least for early times ($t\\lesssim 20\\unit{kyr}$) in the formation of the massive star, our results fit observational constraints (see Sect. \\ref{S: obs}).\n\n\nAs a second region of interest, we consider the infalling envelope. According to the curves by \\cite{Marchand2022}, ambipolar diffusivities tend to be higher at lower densities ($n_H < 10^{8} \\unit{cm^{-3}}$). This suggests that we are missing key magnetic diffusion during the early phases of gravitational collapse, where densities are low. However, there are theoretical and observational results that suggest that the collapse phase is still reasonably modeled in our simulations. In this simulation series, we consider the case of weak magnetic fields (see the discussion on this assumption in Sect. \\ref{S: disk var B}). Because ambipolar diffusivities increase with magnetic field strength, we expect magnetic diffusion to play a stronger role in cases when the magnetic fields are initially strong. In the studies by \\cite{Commercon2022} and \\cite{Mignon-Risse2021}, who considered $\\bar \\mu = 5$ and ambipolar diffusion but no Ohmic dissipation, the magnetic field distribution for low densities coincides well with what is depicted in Fig. \\ref{brho}. Based on this, we would not expect a strong qualitative divergence of our results for the gravitational collapse epoch ($t \\lesssim 5 \\unit{kyr}$) upon the addition of ambipolar diffusion to our models. From the observational point of view, the role of ambipolar diffusion as a necessary enabler for gravitational collapse in a magnetized cloud core was recently questioned in \\cite{2022Natur.601...49C}. These latter authors found a supercritical low-mass prestellar core, which is not yet self-gravitating. This means that the magnetic field was diffused before the gravitational collapse takes place, that is, earlier than predicted by the classical picture of ambipolar diffusion turning a subcritical core supercritical and in a consistent way with our simplification.\n\n\nLastly, we take a look at the magnetic diffusivities in the outflow cavity, which is the focus of Paper II. Observations of protostellar jets have found evidence of ionized material \\citep[see e.g.,][]{Moscadelli2021multi, Rodriguezkamenetzky2017, Carrasco-Gonzalez2021, Guzman2010}, possibly from shock ionization. As discussed in \\cite{Marchand2016,Marchand2022}, ionization leads to a general decrease in the magnetic diffusivities. So far, the models of ambipolar diffusion and the Hall effect done in e.g., \\cite{Marchand2016,Marchand2022} and \\cite{Tsukamoto2020} do not consider the shock ionization in the outflow cavity. Material in the outflow cavity and close to the massive protostar is very low density and is in presence of strong magnetic fields, which means that neglecting shock ionization would overestimate magnetic diffusivities coming from the ambipolar diffusion and Hall effect terms, when comparing to the curves by \\cite{Marchand2022}. By only considering Ohmic dissipation (for which the resistivity is low for low densities and independent of magnetic field strength), the material in the cavity behaves closer to ideal MHD theory, in agreement to what has been found from observational evidence (see Sect. \\ref{S: obs} and \\cite{Moscadelli2022}). This can be seen in Fig. \\ref{brho}, where the magnetic field strengths in the cavity (light-colored dots) are able to increase while keeping the values constrained in the disk (dark-colored dots). A realistic treatment of the problem, however, would require the consideration of all forms of magnetic diffusivity (which would probably impact the disk dynamics) and the effect that thermal ionization and shock ionization have on the magnetic diffusivities corresponding to the outflow cavity.\n\n\n\n\\subsection{Density and temperature profiles}\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{imgraw\/x8-profiles}\n\t\\caption{(a) Density and (b) temperature profiles on the midplane, for the fiducial simulation on grid x8.}\n\t\\label{thindisk_denstemp}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{imgraw\/thickdisk_denstemp.pdf}\n\t\\caption{(a) Density and (b) temperature for two different altitudes in the thick layers of the disk (blue lines), and the corresponding profile for the thin layer (black). The data corresponds to the fiducial simulation on grid x8.}\n\t\\label{thickdisk_denstemp}\n\\end{figure}\n\n\nFigure \\ref{thindisk_denstemp} presents the density and temperature profiles for the midplane. The thin layer has densities over $10^{-16} \\unit{g\\,cm^{-3}}$, and they reach over $10^{-11}\\unit{g\\,cm^{-3}}$ at the inner boundary. For $t\\lesssim 15\\unit{kyr}$, the density profile of the thin layer follows an approximate power law. Later, the magnetically-braked region is formed, corresponding to the high density feature observed for $r \\lesssim 30 \\unit{au}$ in Fig. \\ref{thindisk_denstemp}a. Some of the material of the magnetically-braked region is drawn from the thin layer of the disk: the density profile of the thin layer reveals the existence of a plateau close to its magnetic braking radius, which coincides as well with the decrease in the azimuthal contribution to the specific kinetic energy due to magnetic braking (Fig. \\ref{thindisk_energies}c). However, another part of the material in the magnetically-braked region corresponds to the inflows from the cavity wall and thick layer of the disk, both of which also experience magnetic braking and continuously replenish it.\n\n\nThe temperature profile of the thin layer follows roughly a power law of the form $T\\propto r^{-1\/2}$, and it experiences a general increase over time. The transition between the disk and the envelope is accompanied by a change in the temperature gradient; the transition between the magnetically-braked region and the thin layer of the disk is more subtle. We remind the reader that the present simulations do not include the effect of irradiation from the star, for which we anticipate two additional features not present in Fig. \\ref{thindisk_denstemp}b, namely, an increase in temperature due to the flux from the forming massive star, and the existence of the dust evaporation front \\citep[see][]{Kuiper2010circ}. Those features are only relevant for $t\\gtrsim 20 \\unit{kyr}$.\n\n\n\nFig. \\ref{thickdisk_denstemp}a presents the density profile at two different altitudes crossing the thick layer of the disk at the time $t=10\\unit{kyr}$ (when $M_\\star = 3 \\unit{M_\\odot}$), \n accompanied by a comparison with the density in the midplane. Differently than the thin layer of the disk, the thick layer has a nearly uniform density of $\\sim 10^{-14}\\unit{g\\, cm^{-3}}$. The density drop for $r\\lesssim 10 \\unit{au}$ corresponds to the outflow cavity. The mean density in the thick layer decreases slightly with time, however, it roughly remains within the same order of magnitude throughout the simulation.\n\n\nBecause the density is nearly uniform, the temperature of the thick layer of the disk remains also relatively uniform, at around $200\\unit{K}$ in comparison to the power-law temperature of the thin layer. This means that it should be possible in principle to observe the vertical stratification of the disk by distinguishing the line emission from each layer.\n\n\n\\section{Dependence of the resulting disk properties on initial cloud properties} \\label{s:parameter-study}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{comp-img\/rdisk-norm}\n\t\\caption{Radius of the disk ($R_\\text{disk}$) and its magnetic braking radius ($R_\\text{mb}$) for different initial values of the mass of the cloud, magnetic field, density profile, rotational profile, rotational energy and resistivity models. The transparent lines in the panels that show $R_\\text{mb}$ indicate the full data, while the solid lines show the moving average. The colored dots in panel A4 indicate $t=30\\unit{kyr}$ in each simulation. For all the panels in rows B and C, $M_C=100\\unit{M_\\odot}$ was used.}\n\t\\label{rdisk}\n\\end{figure*}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{comp-img\/mstar-norm}\n\t\\caption{Mass of the protostar, corresponding to the mass in the sink cell, for several values of the (\\emph{a}--\\emph{b}) mass of the cloud, (\\emph{c}) resistivity model, (\\emph{d}) mass-to-flux ratio, (\\emph{e}) density profile, and (\\emph{f}) rotation profile and rotational energy. Panel \\emph{b} shows the same results as panel \\emph{a}, but normalized in such a way that scalability can be readily seen. For panels (c)--(f), $M_C = 100\\unit{M_\\odot}$ and $t_\\text{ff} = 53.73\\unit{kyr}$.}\n\t\\label{mstar}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{comp-img\/disk_morph}\n\t\\caption{Density structure (column 1) and specific energies (column 2) of the disk for selected values of the normalized values of the mass-to-flux ratio (rows B and C), and rotation profile (column D). All the snapshots were taken at $t=15\\unit{kyr}$ of evolution. All cases consider a rotational-to-gravitational energy of 4\\%.}\n\t\\label{disk_morph}\n\\end{figure}\n\nWe perform a series of simulations to explore the large parameter space of initial conditions. We vary the fiducial model model described above in terms of the initial mass of the cloud core, its density and rotation profiles, its rotational-to-gravitational energy ratio, its initial magnetic field strength and probe the effect of different models of Ohmic resistivity. In this section, we discuss how those changes affect the processes discussed in Sect. \\ref{S: dynamics}. As a probe, we use the radial extent of the Keplerian-like disk (between the magnetic braking radius and the [outer] radius), defined as the region where the material is within $\\pm 25\\%$ Keplerianity, and the results of which are presented in Fig. \\ref{rdisk} for the whole parameter space. Similarly, the mass of the forming massive star is presented in Fig. \\ref{mstar} for the whole parameter space. We also examine differences in the density structure of the disk for selected values of the parameters, and relate them to the dynamical information revealed by the specific energy; the results are shown in Fig. \\ref{disk_morph}.\n\n\\subsection{Dependence on cloud core mass} \\label{S: disk var M}\n\nCloud cores of masses of $50, 100, 150$ and $200 \\unit{M_\\odot}$ were considered, which form stars ranging masses from $6$ to $75\\unit{M_\\odot}$ after $30\\unit{kyr}$ of evolution (see Fig. \\ref{mstar}a). Further evolution and additional physical effects which are not considered here would be necessary to determine the final mass of the star. When changing the initial core mass of the fiducial model, we also change the normalization of the rotation profile in order to keep the rotational-to-gravitational energy ratio as in the fiducial model. The mass and size of the cloud core determines its free-fall timescale\n\\begin{equation}\n\tt_\\text{ff} = \\left(\\frac{\\pi^2 R_C^3}{8GM_C}\\right)^{1\/2},\n\\end{equation}\nwhich is $52.4\\unit{kyr}$ for the fiducial case of $M_C = 100\\unit{M_\\odot}$ and $R_C = 0.1\\unit{pc}$. Panels A1 and A2 of Fig. \\ref{rdisk} show that the disk and the magnetically-braked region are smaller for the lower-mass cases when plotted against time, however, when investigated as a function of the stellar mass as a fraction of the initial mass of the cloud (panels A3 and A4), both the size of the disk and the magnetic braking radius coincide well. We find that resulting curves roughly follow the empirical linear relation\n\\begin{equation}\n R_\\text{disk}(m) \\approx R_\\text{onset} + R_\\infty \\left( m - m_\\text{onset} \\right) ,\n\\end{equation}\nwhere we define\n\\begin{equation}\nm \\equiv \\frac{M_\\star}{M_C}.\n\\end{equation}\n$R_\\text{onset} \\equiv 3\\unit{au}$ is the minimum radius of the disk we can detect in the simulation data (the size of the sink cell), $m_\\text{onset}=0.016$ is the corresponding normalized mass of the protostar for $R_\\text{onset}$, and $R_\\infty=6380\\unit{au}$ can be interpreted as the approximate value of the radius of the disk at the hypothetical end of the collapse where $m\\sim 1$. This latter value will never be reached in reality because of stellar feedback: \\cite{KuiperHosokawa2018}, e.g., found that the disk lifetime is expected to be of only a few $10^5\\unit{yr}$ (see also  \\citealt{Kee2019} for the conditions for limiting accretion through the disk). A similar empirical disk size formula was previously offered in \\cite{Kuiper2011}.\n\nWe also verified that the mass of the protostar, expressed as a fraction of the mass of the cloud, scales with time presented as fraction of the free-fall timescale (cf. panels \\emph{a} and \\emph{b} of Fig. \\ref{mstar}). This result indicates that despite the differences in the scale-dependent microphysics, namely, the radiative transfer, resistivity and self-gravity, the disk scales well with the free-fall timescale, which allows our results to be used with a more ample range of cloud masses than what is presented here. Defining $\\tau \\equiv t\/t_\\text{ff}$, we also find an approximate fit for the mass of the protostar with a power law:\n\\begin{equation}\n  m(\\tau) \\approx k_1\\, \\tau^{k_2},\n\\end{equation}\nwhere $k_1 \\approx 0.56$ and $k_2 \\approx 1.8$.\n\n\\subsection{Ohmic resistivity} \\label{S: disk var eta}\n\nThe Ohmic resistivity model controls the diffusivity of the plasma, and therefore it has an impact on magnetic braking. We studied the effects of using the resistivity model of \\cite{Machida2007} in two versions: the full expression dependent on both density and temperature, and the same expression but with a fixed temperature of $\u00ba   T=10\\unit{K}$, as it was used in the isothermal study of \\cite{Anders2018}. We denote the first one as $\\eta(\\rho,T)$ and the second one as $\\eta (\\rho)$ given the near linear behavior of the resistivity in the range of densities for which we are interested. We estimate that the version that considers the temperature dependence differs at most by a factor of $100$ in regions close to the protostar with respect to the version with the fixed temperature. We compare our results as well with the ideal MHD case ($\\eta = 0$) and the non-magnetic (hydrodynamical) case ($\\eta\\to \\infty$). In order to mimic the magnetic diffusivity expected in the disk region if ambipolar diffusion were taking into account, we also include a simulation with an artificially high Ohmic resistivity (ten times the temperature-independent value, denoted as $10\\eta(\\rho)$). This factor is motivated by the difference in the curve for ambipolar diffusivity and Ohmic resistivity in \\cite{Marchand2022}, for disk densities. There is an additional difference between the simulations of this series: while most simulations of this study consider a constant dust and gas opacity of $1\\unit{cm^2\\, g^{-1}}$, the simulation with $\\eta(\\rho,T)$ uses the dust opacity table from \\cite{LaorDraine1993} and a constant gas opacity of $10^{-2}\\unit{cm^2\\, g^{-1}}$.\n\n\n\nFor the rest of Sect. \\ref{s:parameter-study}, we present the size of the disk as a function of the stellar mass as a fraction of the cloud mass, instead of time, because the calculation of the offset from Keplerianity (which defines the disk) is dependent on the gravity of the (proto)star, and by fixing it we are able to compare and assess the effects of the other terms that influence the radial equilibrium of the disk, namely, the angular momentum and pressure support. Additionally, our results can be scaled as a result of the discussion in Sect. \\ref{S: disk var M}.\n\n\nIn the panel B1 of Fig. \\ref{rdisk}, we observe that the magnetically-braked region develops earlier (when the mass of the star is lower, cf. Fig. \\ref{mstar}c) in the ideal MHD case, as compared to the corresponding curves for the simulations that consider resistivity. The case with $\\eta(\\rho)$ requires a more massive protostar (more time evolution) for the magnetic braking radius to appear, a trend that is confirmed with the simulations that consider higher resistivity: the temperature-dependent resistivity delays the appearance of magnetic braking, while the simulation with the artificial factor of ten in resistivity delays it even further. As expected, the disk in the case with no magnetic fields does not develop a magnetic braking radius. A disk is able to form at all in the ideal MHD case because the relatively weak magnetic field considered for this parameter scan.\n\n\nFrom this evidence, we infer that the presence of resistivity delays the action of magnetic braking compared to the results only considering ideal MHD. Higher resistivities delay magnetic braking more, however, as discussed in Sect. \\ref{S: disk MB}, because magnetic diffusion does not completely suppress the toroidal magnetic field, the Lorentz force continues to grow until enough magnetic tension force is built to decelerate the material. As a result, the innermost parts of the disk lose their centrifugal support.\n\n\nThe disk radius (Fig. \\ref{rdisk}B2) is in general larger in the magnetic cases in comparison to the non-magnetic case. Moreover, the disk radius of the run with no resistivity (ideal MHD) is marginally larger than the ones with resistivity. Magnetic braking removes angular momentum from the disk because of the torque produced by magnetic tension; therefore, a smaller disk would be expectable. However, magnetic tension is highest in the inner disk, where the magnetic field lines are wound the most, and not in the outer disk. As discussed in Sect. \\ref{S: dynamics - PB}, magnetic pressure can provide an additional radial support for the disk. This means that in the ideal MHD case, where the toroidal magnetic field is high in the disk, the additional magnetic pressure supports a larger disk. Magnetic diffusion lowers the toroidal magnetic field, and therefore the disk does not have radial magnetic pressure support: this can be seen at early times (where $M_\\star < 5\\unit{M_\\odot}$ or $m = 0.05$) where the simulations with $\\eta(\\rho,T)$ and $10\\eta(\\rho)$ yield a disk radius that almost matches the one obtained in the non-magnetic case. As the toroidal magnetic field increases over time due to the winding of magnetic field lines, the magnetic pressure support also increases and the disk becomes larger than the non-magnetic case.\n\n\nGiven that magnetic braking delivers mass onto the protostar, we investigate the role of resistivity in the mass of the forming massive star. The corresponding curves for the mass of the protostar as a function of time are shown in Fig. \\ref{mstar}c. The results for the resistive and non-resistive cases are very similar. The stellar mass, however, increases more in the non-magnetic case in comparison to the magnetic cases. This happens because the protostar in the non-magnetic case accretes material not only through the disk, but also through the infalling envelope along the bipolar directions. On the contrary, in the magnetic cases, there are magnetic outflows that impede accretion through the axis, resulting in a slightly lower mass.\n\n\n\\subsection{Initial magnetic field strength} \\label{S: disk var B}\n\n\nWe study values of the normalized mass-to-flux ratio ranging from $\\bar \\mu = 2$ (strongest magnetic field) to $\\bar\\mu = 40$ (weakest magnetic field).%\n\n\nThe results of the radial extent of the centrifugally-supported disk are shown in the panels B1 and B2 of Fig. \\ref{rdisk}. The curve for $\\bar \\mu = 2$ goes quickly to zero, which means that the structure formed is strongly sub-Keplerian and it is not classified as a Keplerian-like disk according to our criterion. A more detailed study of the specific energies revealed that the structure has a comparable radial and azimuthal velocities. Our result of no disk for $\\bar \\mu = 2$ should be taken with caution, because we do not include ambipolar diffusion in our calculations. As ambipolar diffusivities increase with magnetic field (contrary to Ohmic dissipation), the cases with strong magnetic fields are missing magnetic diffusivity that may allow the disk to form (see the discussions in Sects. \\ref{S: other magn diff} and \\ref{S: previous amb diff}).\n\n\nThe other values of the mass-to-flux ratio confirm that magnetic braking is responsible for the formation of the magnetically-braked region because higher initial magnetic fields build enough magnetic tension earlier. Additionally, the magnetically-braked region is larger for stronger magnetic fields. The radius of the disk, on the other hand, is larger for stronger magnetic fields, which supports the hypothesis that higher magnetic pressure provides additional support against gravity and allows material from larger radii (with lower angular momentum) to be part of the disk. As for the accretion onto the protostar, Fig. \\ref{mstar}d reveals that higher magnetic fields do tend to deliver more mass onto the protostar, due to stronger magnetic braking at late stages.\n\n\nThe rows A--C of Fig. \\ref{disk_morph} present a comparison of the morphological differences of the disk when it has evolved for $t=15\\unit{kyr}$ under different values of the mass-to-flux ratio. For $\\bar \\mu = 40$, the weaker magnetic field produces a disk with a thick layer that is smaller than the thin layer. Both regions of the disk are of the same size for $\\bar \\mu = 20$, which confirms magnetic pressure as the nature of the vertical support in the thick disk. However, the part of the thin layer that is not enveloped by the thick layer of the disk becomes inflated by magnetic pressure, as the curve for the toroidal contribution to the specific magnetic energy reveals in Fig. \\ref{disk_morph}B2: in the outer disk ($r\\sim 100\\unit{au}$), $e^B_\\text{tor}$ becomes higher than $e^\\text{th}$. Nevertheless, a similar analysis of the specific energies at later times reveals that the thick layer of the disk grows over time until it completely envelops the thin layer (which happens at around $t\\sim 19\\unit{kyr}$). At $t\\approx 25.5\\unit{kyr}$, the thin and thick disk layers of the disk of the case with $\\bar \\mu = 40$ become morphologically and dynamically similar to the results shown in panels A1 and A2 corresponding to the $\\bar \\mu = 20$ at $15\\unit{kyr}$; the difference being the size of the disk and the mass of the protostar. For a stronger initial magnetic field ($\\bar \\mu = 5$, row C of Fig. \\ref{disk_morph}), the thick layer becomes thicker because of the increased magnetic pressure.\n\n\\subsection{Initial density profile}\nA steeper initial density profile accelerates the accretion onto the protostar at early times because of the increased concentration of mass (and therefore gravity) in the center of the cloud. Apart from a density profile with $\\beta_\\rho = -1.5$, we also tried a profile with $\\beta_\\rho = -2$. The mass of the protostar (Fig. \\ref{mstar}e) increases rapidly at early times with $\\beta_\\rho = -2$, but the accretion rate decelerates over time, which is the opposite behavior observed for $\\beta_\\rho = -1.5$. This behavior can be readily seen from the power-law fits to those curves: the normalization constant is similar in both cases, but while the fiducial case is described by $m \\propto \\tau^{1.80}$, the steeper initial density profile exhibits $m \\propto \\tau^{0.94}$. The  radius of the disk (Fig. \\ref{rdisk}C2) is larger in the case of the shallower density profile when seen as a function of the stellar mass and its behavior is longer linear. However, given that the stellar mass increases more rapidly in the case of the steeper density profile, in this latter case, a disk of a given size is observed earlier in time. A similar comparison of stellar masses and time reveals that magnetic braking happens at roughly the same time for both simulations, because the magnetic field is wound by rotation in a similar way in both cases. While the magnetically-braked region appears at $M \\approx 3\\unit{M_\\odot}$ when $\\beta_\\rho = -1.5$, it appears at $M\\approx 17\\unit{M_\\odot}$ for $\\beta_\\rho = -2$; however, the protostar reaches both masses at the same time, $t \\approx 0.2 t_\\text{ff} \\approx 10.7\\unit{kyr}$. As a result, we conclude that a disk that does not exhibit a magnetically-braked region at a given time can be obtained for a range of protostellar masses by adjusting the initial density profile.\n\n\\subsection{Initial rotation profile} \\label{S: disk var rotprof}\n\n\nWe investigated the effect of two scenarios: initial solid-body rotation and a rotation profile that scales with the cylindrical radius at $\\Omega \\propto R^{-\\beta_\\Omega}$, with $\\beta_\\Omega = \\beta_\\rho\/2 = -0.75$. This choice of $\\beta_\\Omega$ keeps the initial ratio of rotational to gravitational energy independent of the radius of the cloud. Additionally, the particular value of $-0.75$ produces a cloud for which the offset from Keplerianity is uniformly negative. We complemented this parameter scan with an investigation of the effects of using a lower initial rotational-to-gravitational energy ratio $\\zeta$ of 1\\% instead of 4\\% as in the fiducial case.\n\n\nAn inspection between panels D1 and A1 of Fig. \\ref{disk_morph} reveals that the thick layer of the disk is flatter in the cloud with the steep rotation profile. The higher angular velocity at the center of the cloud allows for the formation of a larger accretion disk earlier in the simulation, in comparison with the cloud initially rotating as a solid-body. Panel C4 of Fig. \\ref{rdisk} confirms this: the curves for the disk radius for $\\beta_\\Omega = -0.75,\\,\\zeta = \\{0.01,0.04\\}$ are in general higher than the corresponding curves for $\\beta_\\Omega = 0$, and resemble a power law rather than a linear function as it is the case for solid-body rotation. For a fixed value of $\\beta_\\Omega$, the disk becomes larger with more content of initial rotational energy. Additionally, both curves for $\\beta_\\Omega = -0.75$ show that the disk forms at very early times during the simulation compared to $\\beta_\\Omega = 0$: for the solid-body case, angular momentum conservation during the gravitational collapse provides the necessary increase of angular momentum in the center of the cloud in order to build up a disk, while the steep rotation profile already provides some of this angular momentum from the start. Interestingly, the cases $\\beta_\\Omega = -0.75,\\,\\zeta = 0.01$ and $\\beta_\\Omega = 0,\\,\\zeta = 0.04$ produce a similar disk radius. The mass of the formed protostar (Fig. \\ref{mstar}f) is lower in the cases with a steep rotation profile and higher initial rotational energy. This is consequence of the earlier formation of a disk that is also larger (and therefore more massive), at it reduces the efficiency of accretion onto the protostar (in \\citealt{Oliva2020}, we utilized this fact to study disk fragmentation in the absence of magnetic fields). The magnetic braking radius of the disk is very similar for all cases when plotted as a function of the protostellar mass, however, given that the masses grow different in time, magnetic braking appears later in the case with $\\beta_\\Omega = 0,\\,\\zeta = 0.04$.\n\n\nFrom the dynamical point of view, a comparison between panels D2 and A2 of Fig. \\ref{disk_morph} reveals that in general, the different contributions to the specific energy have a more uniform radial gradient. Given that the offset from Keplerianity in the midplane can also be computed as $1-2e^{K}_\\phi\/e^\\text{grav}$, the disk formed with the steep rotation profile is more uniformly Keplerian. This is also manifested when comparing the curves for $e^{K}_r$ inside of the disk in both cases.\n\n\n\n\n\n\\section{Comparison to observations} \\label{S: obs}\n\n\\subsection{Natal environment of massive star formation}\nThe parameter space in our simulation set was motivated by observational typical values found in regions of massive star formation. Surveys of cloud cores in the early stages of massive star formation have found power-law density profiles with exponents ranging in $1.5\\lesssim -\\beta_\\rho \\lesssim 2.6$ \\citep[see, for example,][]{2022AA...657A...3G,Beuther2018, 2002ApJ...566..945B, 2002ApJS..143..469M, 2003A&A...409..589H}, with fragmenting cores having typical values of around $\\beta_\\rho \\approx -1.5$. We take values in the extremes of this interval.\n\nFor the initial angular momentum, we start from the results of the survey by \\citep{Goodman1993}, who found that the the ratio of rotational to gravitational energy of cloud cores ($\\zeta$) is of a few per cent, with typical values of around 2\\%. Those authors fitted linear gradients to the cloud cores that had signs of rotation, and found solid-body profiles. Our second choice for the initial rotation profile, $\\beta_\\Omega = \\beta_\\rho\/2$, is motivated by keeping $\\zeta$ independent of the radius of the cloud core, as explained in Sect. \\ref{S: disk var rotprof}.\n\nThe values of the mass-to-flux ratio are motivated by the supercriticality in star-forming cloud cores \\citep[e.g.][]{Beuther2020, Girart2013} and the following analysis. In low-mass prestellar cores, the magnitudes of the magnetic field have been found to be of the order of a few to tens of micro Gauss \\citep{2022Natur.601...49C, 2008ApJ...680..457T}. As the pre-collapse conditions in the massive star formation case are currently not known, we consider two scenarios (see also the discussion in \\citealt{Machida2020}):\n\\begin{enumerate}\n\\item That the normalized mass-to-flux ratio is independent of mass, which means that magnetic field strengths of 100 to $1000\\,\\mu\\mathrm{G}$ are required in the cloud prior to the gravitational collapse.\n\\item That the magnetic field strength of the high-mass star formation case is similar to the low-mass counterpart, which implies that the mass-to-flux ratio must be high.\n\\end{enumerate}\nThe first case is considered in the low mass-to-flux ratios in our parameter space ($\\bar \\mu \\sim 5$), while the second case corresponds to high mass-to-flux ratios, including our fiducial case ($\\bar \\mu = 20$).\n\n\\subsection{Size of the disk}\nThe analysis in Sect. \\ref{s:parameter-study} has shown that the radius of the disk is more strongly determined by the initial density and rotation (angular momentum) profiles of the large scale initial condition, as compared to the magnetic field and resistivity models, within the ranges of those values expected from observations. Panel B4 of Fig. \\ref{rdisk} shows that when $M_\\star\/M_C \\sim 0.15$, the average $R_\\text{disk}$ for the values of $\\bar \\mu$ we considered is around $850\\unit{au}$, with a variation of the order of $\\pm 200\\unit{au}$. However, the radius of the disk and its growth rate vary much more with the density profile and the rotation profile (panels C2 and C4 of Fig. \\ref{rdisk}). Smaller disks are produced with a lower and flatter initial distribution of angular momentum, and with shallower initial density profiles. This conclusion is crucial for the comparison of our results with previous studies, which is done in Sect. \\ref{S: previous}.\n\nWith regards to the available observational evidence of disk-jet systems around forming massive stars, we see that in principle it is possible to obtain a variety of disk sizes that are not only dependent on the age of the system but also on the initial distribution of matter and angular momentum, and not so strongly determined by magnetic braking in the outer regions of the disk. For example, assuming that the molecular cloud has the typical values of $M_C = 100\\unit{M_\\odot}$ in a sphere of radius $0.1\\unit{pc}$ of our fiducial case, we obtain a disk of radius $\\sim 1200 \\unit{au}$ when the mass of the protostar is $M_\\star \\sim 10 \\unit{M_\\odot}$, if we assume a steep and fast initial rotation profile ($\\beta_\\Omega = -0.75$, $\\zeta = 0.04$), which would provide a possible initial configuration for the observations of HH 80-81 \\citep{Girart2017}. \n\n\nIn \\cite{Moscadelli2022}, we utilized the fiducial simulation run on grid x16 to model the accretion disk and jet of IRAS 21078+5211. For that particular system, observations of molecular rotational transitions had revealed the existence of a Keplerian-like accretion disk of around $200 \\unit{au}$ in size \\citep{Moscadelli2021multi}. In order to find a model of our catalog that was consistent with the observations, in \\cite{Moscadelli2022} we used the mass of the protostar, the radius of the disk, and the morphology of the ejected material to constrain the parameter space and the time elapsed. In that study, we found strong agreement between the velocity field of the observed water masers and the velocity streamlines predicted by the chosen model from our catalog. This good agreement in turn constitutes evidence that the results for the size of the disk as a function of the pre-collapse conditions --which we present in Sect. \\ref{s:parameter-study}-- are in line with what is expected from observations. We note, however, that the wide streamline traced by the maser points (Fig. 2 of \\citealt{Moscadelli2022}) is slightly closer to the assumed disk plane compared to the reference wide streamline from the simulations. This can be caused by a number of factors, for example, a slightly different assumed perspective between the simulations and the observations, but a possible explanation is that the disk in IRAS 21078+5211 is thinner than what is predicted in the simulations.\n\n\n\n\n\\section{Comparison with previous numerical studies} \\label{S: previous}\n\nThe present study is a continuation of the work started by \\cite{Anders2018}. Their simulations correspond to our fiducial case but with a grid equivalent to our x2 grid, and used an isothermal equation of state. An isothermal equation of state yields a much smaller pressure scale height of the disk or thinner disk, respectively. In combination with the rather coarse grid resolution, the thickness of the disk could not be resolved on the numerical grid, making an in-depth convergence study problematic. In that sense, the addition of radiation transport and higher resolutions has enabled us to resolve the thin layer of the disk, and clearly differentiate it from the surrounding region that we call the thick layer of the disk. Also, they do not report about the existence of the magnetically-braked region, however, it is visible upon close examination of their Fig. 21 as a region where the vertical velocity is negative close to the center of the disk. This was originally thought to be a numerical effect caused by the additional mass created by the Alfv\u00e9n limiter, however, we performed a parameter scan with several values of the Alfv\u00e9n limiter, the highest values of which produce negligible artificial mass, and found a magnetically-braked region in all of them.\n\n\\subsection{Studies with ideal MHD}\n\\cite{Banerjee2007}, \\cite{Seifried2011}, \\cite{Myers2013} and \\cite{Rosen2020} conducted simulations of the formation of massive stars under the assumption of ideal MHD. Although \\cite{Banerjee2007} observed magnetically-driven outflows, their disk is always sub-Keplerian. \\cite{Myers2013} started from a cloud of $300 M_\\odot$, an initial velocity profile with supersonic turbulence and no rotation, and $\\bar \\mu = 2$ with $B_z \\propto R^{-1\/2}$. They included grey radiative transfer and radiative protostellar feedback, and used a 3D AMR grid with a maximum resolution of $10 \\unit{au}$, as well as an additional isothermal high resolution run ($\\Delta x = 1.25 \\unit{au}$). They only find a disk in their high resolution run; it has a radius of $40\\unit{au}$ when $t\\sim 0.2 t_\\mathrm{ff}$, for which $M_\\star \\sim 3.5 M_\\odot$. It is not trivial to compare their results to ours because of the difference in initial velocity fields, and the use of an isothermal equation of state. Using the scalability of our results with the mass of the cloud, we find the radius of the disk to be $128\\unit{au}$ for $t = 0.2 t_\\mathrm{ff}$ in the simulation for $\\bar \\mu = 2$. However, if we take $M_\\star\/M_C$ and compute the corresponding fraction of the free-fall timescale, we get $t\/t_\\mathrm{ff} \\approx 0.104$, for which our disk has a radius of $\\sim 30 \\unit{au}$, which is in line with the fact that supersonic turbulence delays the formation of an accretion disk as enough angular momentum has to assemble close to the forming star. \\cite{Rosen2020} also considered a cloud core with supersonic turbulence and radiation transport, however their coarse grid (minimum cell size of $20\\unit{au}$) only allows them to partially resolve a disk-like structure and therefore a direct comparison to our results is not possible.\n\n\\cite{Seifried2011} considered a setup similar to ours in terms of the mass and radius of the cloud, as well as its initial density and rotation profiles, but used a cooling function instead of radiation transport, and only considered the ideal MHD approximation. The code FLASH with an AMR grid of maximum resolution of $4.7\\unit{au}$ was used. The authors explored several values of $\\bar \\mu$ ranging from 2.6 to 26, but with a uniform plasma $\\beta$, and obtained a Keplerian-like accretion disk for $\\bar \\mu = 26$. They observe a drop in Keplerianity for the inner parts of the disk in the case of $\\bar \\mu = 10.4$, and credit it to magnetic braking, in agreement with our results; however, they do not observe a disk for $\\bar \\mu < 5.2$ because the high density of the thin layer of the disk decouples the gas from the magnetic field im resistive MHD runs.\n\n\\subsection{Studies including Ohmic resistivity}\n\\cite{Matsushita2017} and \\cite{Machida2020} use the same Ohmic resistivity model as we do \\citep{Machida2007}, but both consider a barotropic equation of state instead of solving for radiation transport. In \\cite{Matsushita2017}, a disk was formed, however, their analyses are strongly focused on the magnetic outflows. \\cite{Machida2020} consider cloud cores of radius $0.2\\unit{pc}$ with masses ranging from $11$ to $545$ solar masses, an enhanced Bonnor-Ebert density profile, and slow initial solid-body rotation. If we expand our cloud to $0.2 \\unit{pc}$, keeping the same density profile as in the fiducial case, the mass of the cloud becomes $283 \\unit{M_\\odot}$, which means that our setup is similar to their simulation series E. The authors also make a parameter scan with the  normalized mass-to-flux ratio with values ranging from 2 to 20, and use a three-dimensional AMR grid with a maximum resolution of $0.62\\unit{au}$. After $10\\unit{kyr}$ of evolution, they find disk radii of a few hundred astronomical units, which develops spiral arms and fragments for some configurations.\n\n\n\\subsection{Studies including ambipolar diffusion} \\label{S: previous amb diff}\n\n\\cite{Mignon-Risse2021} and \\cite{Commercon2022} studied the formation of massive stars from $100 \\unit{M_\\odot}$ cloud cores of radius $0.2\\unit{pc}$, with a centrally-condensed initial density profile; which roughly correspond to our configuration $M_C = 50\\unit{M_\\odot}$. The cloud cores are initially in solid body rotation and are threaded by a uniform magnetic field determined from the normalized mass-to-flux ratio $\\bar \\mu = 5 \\text{ and } 2$. They treated radiation transport with the flux-limited diffusion approach and gray stellar irradiation; for magnetic diffusivity, they considered ambipolar diffusion using the model from \\cite{Marchand2016} but no Ohmic resistivity, and ran their simulations on a three-dimensional AMR grid with the code RAMSES down to a maximum resolution of $5\\unit{au}$. In a nutshell, those studies constitute the ones which include the most similar physical ingredients but to our setup but with ambipolar diffusion instead of Ohmic dissipation as the resistive effect and utilizing a different grid method.\n\nBoth studies report on thin accretion disks which are vertically supported by thermal pressure, in agreement with our results; however, they do not observe a surrounding thick layer of the disk supported by magnetic pressure. %\nOpposite to our findings, the authors find smaller disks with stronger magnetic fields, however, they do find a bigger disk for their ideal MHD run compared to the runs with ambipolar diffusion. In that case, the disk is not only bigger, but also inflated and therefore less dense, in agreement with our results. They attribute the finding of smaller disks to higher ionization in the outer disk, and therefore more magnetic braking. In turn, our results indicate that magnetic braking is highest in the inner disk, where the fluid and the magnetic field lines are dragged faster by rotation, although resistivity causes the dragging to be partial and delayed because of the dominance of magnetic diffusion. Moreover, in our results for the ideal MHD case we observe simultaneously the largest disk and the strongest magnetic braking, because it mostly affects the inner disk.\n\nA comparison of the radius of the disk that we report here and the disk size reported in \\cite{Commercon2022} can only be made qualitatively and with several considerations in mind, apart from the difference in the non-ideal MHD effect considered. First, the density profiles in both studies are different because of the presence of an initial density plateau in the inner $\\sim 4125\\unit{au}$ in \\cite{Commercon2022} and \\cite{Mignon-Risse2021}, in contrast to our continuous power-law slope down to the smallest scales. This density plateau affects the gravitational collapse and therefore, the elapsed time and mass of the protostar cannot be directly compared. Second, the initial rotational to gravitational energy ratio of the cloud is not the same in the fiducial cases of both studies, but we can roughly compare our $\\zeta = 0.04$ case to the ``fast'' case studied in \\cite{Commercon2022}, which has $\\zeta = 0.05$. This is of special importance because of what is shown in Fig. \\ref{rdisk}C4: the radius of the disk is strongly influenced by the initial rotation of the cloud, even more than the initial magnetic field (cf. Fig. \\ref{rdisk}B4) or resistivity (Fig. \\ref{rdisk}B2). Finally, the disk identification criterion is not the same in both studies: while \\cite{Commercon2022} use a set of criteria that involves the radial and vertical dynamics of the disk as well as a density threshold, we consider a purely dynamical criterion (radial equilibrium between gravity and centrifugal force) as it is more usual in observational studies. With this in mind, we note that in our simulation $M_C = 50\\, \\unit{M_\\odot}\\ \\text{[x4]}$, the  radius of the disk is around $470\\unit{au}$ after $27\\unit{kyr}$, while their disk is around the same value after $51\\unit{kyr}$. However, the mass of the protostar at that time is lower in our case ($4.6\\unit{M_\\odot}$) than theirs ($9.1\\unit{M_\\odot}$); both the difference in time and mass can be reasonably attributed to the initial density profile. Additionally, while we find that the disk grows in size with time, most of the nonideal MHD simulations by \\cite{Commercon2022} find disks that stop growing at around $100\\unit{au}$.\n\n\n\\cite{Commercon2022} and \\cite{Mignon-Risse2021} do not report the existence of a magnetically-braked region, for which we offer the following explanations, apart from the different criteria for disk identification we already discussed. First, ambipolar diffusion adds magnetic diffusivity, which may have the effect of delaying the formation of the magnetically-braked region. In more detail: we report on the fact that higher Ohmic resistivities delay the onset of magnetic braking and with it, the formation of the magnetically-braked region. Ambipolar diffusion also decouples the magnetic field from the flow (even though it may act in a different direction than Ohmic resistivity), which leads us to assume that a similar delaying effect on magnetic braking could be obtained upon the inclusion of ambipolar diffusion in our calculations. We note, however, that if this delay in the onset of magnetic braking is longer than the timescale of gravitational collapse of the cloud, the massive star and the disk may form without ever developing a magnetically-braked region. %\nSecond, spatial resolution: their 3D AMR grid has a minimum cell size (maximum spatial resolution) of $5\\unit{au}$, while our grid has minimum cell sizes of the order of $10^{-1}\\unit{au}$ close to the sink cell for the simulation series x1, x2 and x4, and $10^{-2}\\unit{au}$ for the simulation series x8 and x16. Third, their sink particle algorithm requires the definition of an accretion radius, set to four times the minimum cell size, i.e., $20\\unit{au}$, which is the size of the magnetically-braked region for most of the time in our fiducial case. As a consequence, gas orbiting the massive protostar within the accretion radius or in the forming magnetically-braked region cannot be resolved on their finest AMR level while it is resolved on our spherical grid.\n\nFinally, we mention the study by \\cite{Masson2016}, who carried out simulations of low-mass star formation considering both ideal MHD and ambipolar diffusion. Although we cannot compare our results quantitatively with theirs, there are several qualitative similarities. The authors observe an accretion disk vertically supported by thermal pressure, with magnetic pressure dominating above the disk. Their figure 14 shows a structure with relatively high angular momentum enveloping the disk for both the diffusive and ideal cases, which is reminiscent of the thick layer of the disk we observe, however, due to the different definitions of the disk used in both studies, we cannot establish a direct correspondence. The same figure also shows that in the ideal MHD case, the inner parts of the disk are destroyed by magnetic braking at late times. Nevertheless, their corresponding run that considers ambipolar diffusion also exhibits for late times a region of low angular momentum (probably due to magnetic braking) in a conical region around the inner disk and that could deliver material to it, in a way that is reminiscent of the early stages of magnetic braking we observe in our simulations. Those results seem to support the idea that increasing the diffusivity leads to a delay but not complete suppression of magnetic braking in a magnetized disk embedded in a collapsing cloud.\n\n\\section{Summary and conclusions}\n\nWe have modeled the formation of a massive star with a series of 30 magnetohydrodynamical simulations including Ohmic resistivity, radiation transport from thermal emission of the dust and gas, and self gravity. The series of simulations covered a wide range of cloud masses, magnetic field strengths, density profiles, rotation profiles, and ratios of rotational to gravitational energy, in line with currently estimated values from observations. We also performed a convergence study to test the robustness of our results.\n\nAfter analyzing the fiducial case of our parameter study in depth, we found the following general features of the system:\n\n\\begin{itemize}\n\t\\item After the initial gravitational collapse, a Keplerian-like accretion disk is formed.\n\t\\item The accretion disk is divided into two layers: a thin layer supported vertically by thermal pressure, and a surrounding thick layer supported by magnetic pressure. The thin layer of the disk appears only in simulations with sufficiently high resolution.\n\t\\item At early times magnetic diffusion due to Ohmic resistivity is strong in the inner parts of the disk and it greatly reduces magnetic braking there during the magneto-centrifugal epoch ($5 \\unit{kyr} \\lesssim t \\lesssim 15\\unit{kyr}$). As time progresses, and the magnetic field lines are continuously dragged by rotation, magnetic braking is observed in the innermost $\\sim 50\\unit{au}$ of the disk for the fiducial case in our parameter space.\n\t\\item Magnetic pressure can increase the size of the accretion disk.\n\\end{itemize}\n\nWhen examining the full parameter space of initial conditions for the onset of gravitational collapse, we find that:\n\\begin{itemize}\n\t\\item Our results for the size of the disk and the mass gain of the protostar scale with the initial mass of the cloud, despite the non-scalability of self-gravity and the thermodynamics considered.\n\t\\item The thickness of the thick layer of the disk is controlled by the initial magnetic field strength.\n\t\\item For a cloud with an initial density profile $\\rho \\propto r^{-1.5}$ and in solid-body rotation, the disk grows roughly linearly in size as $R_\\text{disk} \\approx [ 6380 M_\\star\/M_C - 98 ]\\unit{au} $. The stellar mass grows approximately like $M_\\star \\propto (t\/t_\\text{ff})^{1.5..1.9}$.\n\t\\item The size of the disk is more strongly determined by the initial distribution of density and rotational energy in the cloud than by the strength of the magnetic field.\n\t\\item Multiple initial configurations of the cloud can produce a given disk size and (proto)stellar mass. This means that observations of disk-jet systems constrain (as opposed to determine) the possible conditions for the onset of gravitational collapse, and more measurements (distribution and strength of magnetic fields, for example) are needed to break the degeneracies.\n\\end{itemize}\n\nIn a follow-up article in preparation, we will perform a dynamical analysis of the magnetically-driven outflows of the same dataset we present here.\n\n\\begin{acknowledgements}\nWe thank Richard Nies for his contributions to the analysis of part of the dataset at the early stages of the project. GAO acknowledges financial support from the Deutscher Akademischer Austauschdienst (DAAD), under the program Research Grants - Doctoral Projects in Germany, and complementary financial support for the completion of the Doctoral degree by the University of Costa Rica, as part of their scholarship program for postgraduate studies in foreign institutions. RK acknowledges financial support via the Emmy Noether and Heisenberg Research Grants funded by the German Research Foundation (DFG) under grant no.~KU 2849\/3 and 2849\/9.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCoupling molecules to the quantized radiation field inside an optical cavity creates a set of new photon-matter hybrid states, which are commonly referred to as polaritons,\\cite{Ebbesen2016ACR,kowalewski_manipulating_2017,Flick2017pnas,Ribeiro2018,Feist2018,Mandal2019JPCL} which have been shown to facilitate new chemical reactivities.\\cite{Hutchison2012ACIE,Ebbesen2016ACR,Thomas2019S,Mandal2019JPCL,Mandal2020JPCB} Theoretical investigations play a crucial role in understanding the fundamental limit and basic principles in this emerging field,\\cite{Mandal2019JPCL,Feist2018,Luk2017jctc,Groenhof2018jpcl,Groenhof2019jpcl,Groenhof2019jpcl,Groenhof2021JCP} as these polariton chemical reactions often involve a rich dynamical interplay among the electronic, nuclear, and photonic degrees of freedom (DOFs). Accurately simulating polaritonic quantum dynamics remains a challenging task and is beyond the scope of photochemistry or quantum optics.\\cite{kowalewski_manipulating_2017}\n\nThe trajectory-based non-adiabatic dynamics approaches\\cite{Tully2012jcp,BarbattiSH,Mai2015} play an important role in simulating the non-adiabatic dynamics of the coupled electronic-nuclear DOFs. Two of the most commonly used mixed quantum-classical (MQC) methods are the Ehrenfest and fewest switches surface hopping (FSSH) approaches.\\cite{Tully,tully94jcp} Both approaches describe the electronic subsystem quantum mechanically, and treat the nuclear DOFs classically. It is thus a natural idea for the theoretical chemistry community to extend these two approaches to investigate polariton chemistry by treating the electronic-photonic DOFs (or so-called polariton subsystem) quantum mechanically and the nuclear DOFs classically. Incorporating the description of the photon field into the MQC methods has become a basic strategy to simulate polariton chemistry.\\cite{Luk2017jctc,Groenhof2018jpcl,Groenhof2019jpcl,Groenhof2021JCP,Fregoni2018,Fregoni2020,fregoni_strong_2020,Zhang2019jcp} The key ingredient in the MQC simulations of polariton dynamics is the expression of the nuclear gradient. Recently, we derived a rigorous expression of the nuclear gradient using the quantum electrodynamics (QED) Hamiltonian without making the usual approximations,\\cite{Zhou2022} such as the rotating wave approximation. These gradient expressions, together with the corresponding MQC approaches (Ehrenfest and FSSH approaches), is valid for any any number of electronic states or Fock states at any light-matter coupling strength. However, the inherent semi-classical approximation in these approaches can lead to the break-down of detailed balance\\cite{ParandekarJCTC2006} (incorrect long time population) in Ehrenfest dynamics and the creation of artificial electronic coherence\\cite{subotnik2016arpc} or incorrect\nchemical kinetics\\cite{subotnik2016arpc} for the FSSH dynamics without invoking \\textit{ad hoc} decoherence corrections.\n\nIn response to these theoretical challenges, a wide range of non-adiabatic dynamics approaches have been developed in the diabatic representation. Many of them belong to the family of non-adiabatic mapping dynamics which are based on the Meyer-Miller-Stock-Thoss (MMST) mapping formalism.\\cite{MeyerJCP1979,StockPRL1997,ThossPRA1999} These methods include partial linearized density matrix\\cite{HuoJCP2011,HuoMP2012} (PLDM), symmetrical quasi-classical\\cite{CottonJCP2013,CottonJPCA2013} (SQC), the quantum-classical Liouville equation (QCLE) dynamics.\\cite{HsiehJCP2012,HsiehJCP2013} In particular, the recently developed $\\gamma$-SQC has been shown\\cite{CottonJCP2019_2} to provide impressively accurate non-adiabatic photo-dissociation quantum dynamics with coupled Morse potentials through the adjusted zero-point energy (ZPE) parameter of the mapping variables, thus appearing to be a promising method to simulate on-the-fly quantum dynamics of complex molecular systems. In addition, the spin-mapping Linearized Semi-Classical approach (spin-LSC)\\cite{richardson2019,richardson2020,Duncan2022JCP}, which uses generalized spin mapping representation\\cite{richardson2020} for the electronic DOF as well as the Linearization approximation\\cite{MillerJCP98,shi2004} for the nuclear DOF, has also shown a significant improvement of the population dynamics in the system-bath model problems (such as in spin-boson systems\\cite{richardson2019} and many-state exciton Hamiltonians of light-harvesting complexes.\\cite{richardson2020}) The $\\gamma$-SQC approach has already demonstrated\\cite{WeightJCP2021} its ability to outperform MQC approaches (Ehrenfest and FSSH) in describing the electronic non-adiabatic dynamics for {\\it ab initio} on-the-fly simulations. These new mapping approaches should, in principle, also outperform the MQC methods in simulating the polaritonic non-adiabatic dynamics that happens in the electron-photon subspace coupling to the motion of the nuclei. Unfortunately, to the best of our knowledge, there are only limited studies of using mapping dynamics to investigate polariton chemistry for model systems with strict diabatic states.\\cite{Mandal2019JPCL,Chowdhury2021jcp,Mandal2020JPCB}\n\nRecently, we have developed the quasi-diabatic (QD) propagation scheme\\cite{MandalJCTC2018,MandalJCP2018,MandalJPCA2019,SandovalJCP2018,ZhouJCPL2019, WeightJCP2021} as a general framework to seamlessly combine a diabatic quantum dynamics approach, such as the mapping based methods,\\cite{CottonJCP2019_2,richardson2020} with the adiabatic outputs of an electronic structure method. The QD propagation scheme uses the adiabatic states at a reference nuclear geometry (the so-called ``crude adiabatic\" states) as the {\\it locally well-defined} diabatic states during a short-time propagation and then dynamically updates the QD basis at each consecutive nuclear propagation step. In this propagation scheme, one does not construct a {\\it global diabatic representation} but instead, uses a sequence of locally diabatic representations (one for each short-time segment) to propagate the dynamics. We have both analytically\\cite{MandalJCTC2018} and numerically\\cite{MandalJPCA2019,SandovalJCP2018} demonstrated that the QD scheme provides exactly the same results compared to the direct diabatic quantum dynamics at the single trajectory level. \n\nIn this work, we generalize the QD propagation scheme to simulate polariton non-adiabatic dynamics in a molecule-cavity hybrid system. In particular, we use the adiabatic-Fock state at a reference nuclear geometry as the locally well-defined diabatic basis to propagate the polariton dynamics, and dynamically update the definition of these local diabatic states between two consecutive propagation steps. These adiabatic-Fock states are tensor products of the electronic adiabatic states for the molecular system and the Fock states of the photon field inside an optical cavity. We use the Shin-Metiu (SM) model\\cite{Shin1995jcp,Hoffmann2020} as the ``\\textit{ab initio}\" model molecular system to investigate strong and ultra-strong light-matter interactions between a molecule and an optical cavity. Through numerical simulations, we demonstrate the accuracy of using both $\\gamma$-SQC\\cite{CottonJCP2019_2} and spin-LSC\\cite{richardson2019,richardson2020} to obtain non-adiabatic polariton dynamics, which outperforms widely used MQC approaches.\n\n\\section{Theory and Methods} \\label{methods}\n\\subsection{The Pauli-Fierz QED Hamiltonian}\nThe Pauli-Fierz (PF) QED Hamiltonian for one molecule coupled to quantized radiation field inside an optical cavity can be written as \n\\begin{equation}\\label{total_PF_H}\n\\hat{H}=\\hat{T}_\\mathrm{n}+\\hat{H}_\\mathrm{en}+\\hat{H}_\\mathrm{p}+\\hat{H}_\\mathrm{enp}+\\hat{H}_\\mathrm{d},\n\\end{equation}\nwhere $\\hat{T}_\\mathrm{n}$ represents the nuclear kinetic energy operator, $\\hat{H}_\\mathrm{en}$ is the electronic Hamiltonian that describes electron-nucleus interactions. Further, $\\hat{H}_\\mathrm{p}$, $\\hat{H}_\\mathrm{enp}$, and $\\hat{H}_\\mathrm{d}$ represent the photonic Hamiltonian, electronic-nuclear-photonic interactions, and the dipole self-energy (DSE) term, respectively. A full derivation of this Hamiltonian, as well as its connection with the various atomic cavity QED models can be found in the Appendix of Ref.~\\citenum{Chowdhury2021jcp}.\n\nThe electronic-nuclear potential $\\hat{H}_\\mathrm{en}$, which describes the common molecular Hamiltonian excluding the nuclear kinetic energy is described as follows\n\\begin{equation}\\label{eqn:Hen}\n\\hat{H}_\\mathrm{en} = \\hat{T}_\\mathrm{e} + \\hat{V}_\\mathrm{ee} + \\hat{V}_\\mathrm{en} + \\hat{V}_\\mathrm{nn}.\n\\end{equation}\nThe above expression includes electronic kinetic energy, electron-electron interaction, electron-nucleus interaction and nucleus-nucleus interaction. The expressions of these four terms can be found in previous work.\\cite{Doltsinis2002jctc,Schafer2018pra,Marxbook} Modern electronic structure theory have been developed around solving the eigenvalue problem of $\\hat{H}_\\mathrm{en}$, providing the following electronically adiabatic energy and its corresponding state\n\\begin{equation}\\label{HenPsi}\n\\hat{H}_\\mathrm{en}|\\phi_\\alpha(\\mathbf{R}) \\rangle= E_{\\alpha}({\\mathbf R})|\\phi_\\alpha(\\mathbf{R}) \\rangle.\n\\end{equation}\nHere, $|\\phi_\\alpha(\\mathbf{R}) \\rangle$ represents the $\\alpha_\\mathrm{th}$ many-electron adiabatic state for a given molecular system, with the adiabatic energy $E_{\\alpha}({\\mathbf R})$.\n\nFor clarity, we restrict our discussions to the cavity with only one photonic mode, and all the formulas presented here can be easily generalized into a more realistic, many-mode cavity. The photonic Hamiltonian is written as \n\\begin{equation}\\label{eqn:Hp}\n\\hat{H}_\\mathrm{p}=\\frac12 \\left(\\hat{p}_\\mathrm{c}^2+\\omega_\\mathrm{c}^2\\hat{q}_\\mathrm{c}^2\\right)=\\hbar\\omega_\\mathrm{c} \\Big(\\hat{a}^{\\dagger}\\hat{a}+\\frac12 \\Big),\n\\end{equation}\nwhere $\\hat{q}_\\mathrm{c}=\\sqrt{\\hbar\/2 \\omega_\\mathrm{c}}(\\hat{a}^{\\dagger}+\\hat{a})$ and $\\hat{p}_\\mathrm{c}=i\\sqrt{\\hbar\\omega_\\mathrm{c}\/2}(\\hat{a}^{\\dagger}-\\hat{a})$ are photon field operators, $\\hat{a}^{\\dagger}$ and $\\hat{a}$ are the photonic creation and annihilation operators, respectively and $\\omega_\\mathrm{c}$ is the photon frequency.\n\nThe light-matter coupling term (electronic-nuclear-photonic interactions) under the dipole gauge is expressed as \n\\begin{equation}\\label{eq:enp}\n\\hat{H}_\\mathrm{enp}=\\omega_\\mathrm{c}\\hat{q}_\\mathrm{c} ( \\boldsymbol{\\lambda} \\cdot {\\hat{\\boldsymbol \\mu}})=g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot  {\\hat{\\boldsymbol \\mu}} (\\hat{a}^{\\dagger} + \\hat{a}).\n\\end{equation}\nwhere $\\boldsymbol{\\lambda}=\\lambda \\cdot{\\boldsymbol\\epsilon}$ characterizes the cavity photon field strength, ${\\boldsymbol\\epsilon}$ is the direction of the field polarization. The photon field strength is determined by the volume of the cavity as $\\lambda=\\sqrt{1\/\\epsilon_{0}\\mathcal{V}_{0}}$, where $\\epsilon_{0}$ is the permittivity inside the cavity and $\\mathcal{V}_{0}$ is the effective quantization volume inside the cavity. Another way to characterize the light-matter coupling strength is using $g_\\mathrm{c}=\\sqrt{\\hbar\\omega_\\mathrm{c}\/2}\\lambda$. Note that the common notation used in the literature,\\cite{Kowalewski2016JPCL,Mandal2019JPCL} the definition of $g_\\mathrm{c}$ also includes $\\boldsymbol{\\lambda} \\cdot {\\hat{\\boldsymbol \\mu}}$. Further, the total dipole operator of both electrons and nuclei is defined as\n\\begin{equation}\\label{dipole}\n{\\hat{\\boldsymbol \\mu}} = -\\sum_ie \\hat{{\\bf r}}_i +\\sum_{j} Z_{j} e \\hat{\\bf{R}}_{j}, \n\\end{equation}\nwhere $-e$ is the charge of the electron and $Z_{j} e $ is the charge of the $j_\\mathrm{th}$ nucleus.\n\nFinally, the DSE term is expressed as\n\\begin{equation}\\label{eqn:Hd}\n\\hat{H}_\\mathrm{d}=\\frac12 (\\boldsymbol{\\lambda} \\cdot \\hat{\\boldsymbol \\mu})^2=\\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega_\\mathrm{c}} (\\boldsymbol{\\epsilon} \\cdot \\hat{\\boldsymbol \\mu})^2.\n\\end{equation}\nThis is a necessary term in the PF Hamiltonian and ensures both gauge invariance of the Hamiltonian\\cite{Taylor2020PRL,Mandal2020JPCB} and a bounded ground state.\\cite{Rokaj2018JPB,Mandal2020JPCB,Schaefer2020AP} In this work, we do not consider the cavity loss. The cavity loss can be effectively incorporated by using Lindblad dynamics approaches with the MQC simulations.\\cite{Koessler2022}\n\nFor the molecule-cavity hybrid system, a convenient basis for quantum dynamics simulations could be the photon-dressed electronic adiabatic states \n\\begin{equation}\\label{adia-fock}\n|\\psi_i({\\mathbf R}) \\rangle=|\\phi_\\alpha({\\mathbf R}) \\rangle \\otimes |n\\rangle \\equiv|\\phi_{\\alpha}({\\bf R}),n\\rangle,\n\\end{equation}\nwhere quantum number $i\\equiv\\{\\alpha,n\\}$ indicates both the adiabatic electronic state of the molecule and the Fock state. Note that we have introduced a shorthand notation in Eq.~\\ref{adia-fock}, which will be used throughout the rest of this paper. This is one of the most straightforward choices of basis for the hybrid system because of the readily available adiabatic electronic information (\\textit{e.g.,} wavefunctions, energies, and the dipole matrix) from electronic structure calculations that we need to construct the elements of Hamiltonian. \n\nIn the MQC simulation, such as the Ehrenfest or FSSH approach, or the recently developed mapping non-adiabatic approaches, the total molecular Hamiltonian is expressed as\n\\begin{equation}\\label{totalH}\n\\hat{H}=\\hat{T}_\\mathrm{n}+\\hat{V},\n\\end{equation}\nwhere $\\hat{T}_\\mathrm{n}$ represents the nuclear kinetic energy operator, and $\\hat{V}$ represents the rest of the Hamiltonian. For a bare molecular system, $\\hat{V}=\\hat{H}_\\mathrm{en}$ expressed in Eq.~\\ref{eqn:Hen}.\nFor a molecule-cavity hybrid system, \n\\begin{equation}\\label{eqn:Henp}\n\\hat{V}=\\hat{H}_\\mathrm{en}+\\hat{H}_\\mathrm{p}+\\hat{H}_\\mathrm{enp}+\\hat{H}_\\mathrm{d}\\equiv \\hat{H}_\\mathrm{pl},\n\\end{equation}\nwhich is commonly referred to as the polariton Hamiltonian,\\cite{Flick2017jctc,Flick2017pnas} also denoted as $\\hat{H}_\\mathrm{pl}$. In a similar way that electronic adiabatic states are defined in Eq.~\\ref{HenPsi}, one can further define the polaritonic state\\cite{Flick2017jctc,Flick2017pnas} as the eigenstate of $\\hat{V}=\\hat{H}_\\mathrm{pl}$ (see definition in Eq.~\\ref{eqn:Henp}) through the following eigenequation\n\\begin{equation}\\label{polariton}\n\\hat{H}_\\mathrm{pl}|\\mathcal{E}_{J}({\\bf R})\\rangle=\\mathcal{E}_{J}({\\bf R})|\\mathcal{E}_{J}({\\bf R})\\rangle,\n\\end{equation}\nwhere $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is the polariton state with polariton energy $\\mathcal{E}_{J}({\\bf R})$. The polariton eigenstate can be expressed as \n\\begin{equation}\\label{pol-exp}\n|\\mathcal{E}_{J}({\\bf R})\\rangle=\\sum_{\\alpha,n}c^{J}_{\\alpha, n}({\\mathbf R})|\\phi_{\\alpha}({\\bf R}),n\\rangle,\n\\end{equation}\nwhere $c^{J}_{\\alpha, n}({\\mathbf R})=\\langle \\phi_{\\alpha}({\\bf R}),n|\\mathcal{E}_{J}({\\bf R})\\rangle$ and $\\mathcal{E}_{J}({\\bf R})$ can be obtained by diagonalizing the matrix of $\\hat{V}=\\hat{H}_\\mathrm{pl}$ (constructed from the adiabatic-Fock state basis in Eq.~\\ref{adia-fock}) as\n\\begin{equation}\\label{eqn:diag}\n\\mathbf{U}^{\\dagger}[V({\\mathbf R})]\\mathbf{U} = [\\mathcal{E}({\\mathbf R})],\n\\end{equation}\nwhere\n\\begin{equation}\n[V({\\mathbf R})]_{ij} = \\langle\\psi_i({\\mathbf R})| \\hat{V} |\\psi_j({\\mathbf R}) \\rangle.\n\\end{equation}\nNote that the ${\\bf R}$-dependence of $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is entirely coming from the ${\\bf R}$-dependence of the adiabatic states $|\\phi_\\alpha({\\mathbf R}) \\rangle$, and the Fock state $|n\\rangle$ is completely ${\\bf R}$-independent. Meanwhile, the $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is the eigenstate of $\\hat{V}$, whereas the adiabatic state $|\\phi_\\alpha({\\mathbf R}) \\rangle$ is only the eigenstate of $\\hat{H}_\\mathrm{en}$, and not for $\\hat{V}$.\n\n\\subsection{Quasi-Diabatic Propagation Scheme for Molecular Cavity QED}\\label{sec:QD}\nThe QD propagation scheme explicitly addresses the discrepancy between accurate quantum dynamics methods in the diabatic representation and the electronic structure methods in the adiabatic representation. The essential idea of the QD scheme is to use the electronic adiabatic states associated with a reference geometry as the local diabatic states during a short-time quantum propagation and dynamically updates the definition of the QD states along the time-dependent nuclear trajectory.\\cite{MandalJCTC2018,MandalJCP2018,MandalJPCA2019,SandovalJCP2018,ZhouJCPL2019, WeightJCP2021}\n\nIn this work, we apply the QD propagation scheme to the case of molecular cavity QED. This requires the use of a convenient basis with a reference nuclear geometry as the $\\it locally$ well-defined diabatic basis, in the sense that its character is fixed (which is automatically guaranteed because of the fixed reference geometry by construction) as well as it is a complete basis (which is only true when the geometry is close to this reference geometry). The potential candidate for this basis is the adiabatic-Fock state $|\\psi_i({\\mathbf R}) \\rangle=|\\phi_\\alpha({\\mathbf R}),n\\rangle$ (Eq.~\\ref{adia-fock}), which is not the same as the polariton states $|\\mathcal{E}_{J}({\\bf R})\\rangle$ (Eq.~\\ref{pol-exp}) except for the zero-coupling limit. In this work, we use the adiabatic-Fock state as the convenient choice due to its simplicity in terms of the polariton coupling and nuclear gradient expressions in the QD propagation framework.\n\nConsider a short-time propagation of the nuclear DOFs during $t\\in[t_0, t_1]$, where the nuclear positions evolve from ${\\bf R}(t_0)$ to ${\\bf R}(t_1)$, and the corresponding adiabatic-Fock basis (defined in Eq.~\\ref{adia-fock}) are $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\}$ and $\\{|\\psi_{j}{\\bf R}(t_1))\\rangle\\}$. We uses the basis $\\{|\\psi_{i}({\\bf R}_0)\\rangle\\equiv|\\phi_\\alpha({\\mathbf R_0}),n\\rangle\\}$ at the reference nuclear geometry ${\\bf R}(t_0)$ as the {\\it diabatic} basis during this short-time propagation, such that\n\\begin{equation}\\label{eqn:qdidea}\n|\\psi_{i}({\\bf R}_{0})\\rangle\\equiv|\\psi_{i}({\\bf R}(t_0))\\rangle,~~\\mathrm{for}~t\\in[t_0,t_1].\n\\end{equation}\nWith the above QD basis defined independently of ${\\bf R}(t)$ within each propagation segment, the electronic derivative couplings vanish while $\\hat{V}({\\bf R}(t))$ in the QD basis becomes off-diagonal. With this local diabatic basis, all of the necessary diabatic quantities can be evaluated and used to propagate quantum dynamics during $t\\in[t_0,t_1]$. \n\nDuring this propagation step, the matrix element of $\\hat{V}$ in the QD basis is evaluated as\n\\begin{equation}\\label{eqn:vijt} \nV_{\\alpha\\beta, m n}({\\bf R}(t))  = \\langle \\phi_\\alpha ({\\bf R}_0), m| \\hat V ({\\bf R}(t))| \\phi_\\beta ({\\bf R}_0), n\\rangle.\n\\end{equation}\nFor on-the-fly simulations, this quantity is obtained from a linear interpolation\\cite{Rossky-Webster} between $V_{{\\alpha}{\\beta},m n} ({\\bf R}_{0})$ and $V_{\\alpha\\beta,m n}({\\bf R}(t_1))$ as follows\n\\begin{align}\\label{eqn:interpolation}\n&V_{\\alpha\\beta,m n}({\\bf R}(t))= V_{\\alpha\\beta,m n}({\\bf R}_{0}) \\\\\n&~~~~~+ \\frac {(t - t_{0})}{(t_{1} - t_{0})}\\big[V_{\\alpha\\beta,mn}({\\bf R}(t_{1})) - V_{\\alpha\\beta,mn}({\\bf R}_{0})\\big].\\nonumber\n\\end{align} \nThe above linear interpolation scheme can be further improved in future work and one potential choice is the recently developed norm-preserving interpolation scheme.\\cite{meek2014evaluation,jain2016efficient}\n\nIt is straightforward to evaluate $V_{{\\alpha}{\\beta},m n} ({\\bf R}_{0})$ and $V_{\\alpha\\beta,m n}({\\bf R}(t_1))$ separately for the molecule-cavity hybrid system, as discussed below. Using electronic {\\it ab initio} calculation, as well as the properties of $\\hat{a}^{\\dagger}$ and $\\hat{a}$ for the photonic DOF, we can explicitly evaluate each term of $V_{\\alpha\\beta,m n}({\\bf R}_{0})$ (see Eq.~\\ref{eqn:Henp}) as follows\n\\begin{subequations}\n\\begin{align}\n&H^\\mathrm{en}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{en} ({\\bf R}_{0})|\\phi_\\beta({\\bf R}_{0}), n\\rangle  \\nonumber \\\\ \n&~~~= E_{\\alpha}({\\bf R}_{0}) \\delta_{{\\alpha},{\\beta}} \\delta_{m,n} \\\\\n&H^\\mathrm{p}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{p} | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber \\\\ \n&~~~= \\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\\\\n&H^\\mathrm{enp}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{enp} ({\\bf R}_{0}) | \\phi_\\beta({\\bf R_{0}}), n \\rangle \\nonumber \\\\\n&~~~= g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\beta}({\\bf R}_{0}) \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1} \\big) \\\\\n&H^\\mathrm{d}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{d} ({\\bf R}_{0}) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\\n&~~~= \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega_\\mathrm{c}} \\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\gamma} ({\\bf R}_{0})) (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\gamma\\beta} ({\\bf R}_{0})) \\delta_{m,n} \\nonumber\\\\ &\n~~~\\equiv D^2_{\\alpha\\beta}({\\bf R}_{0}) \\delta_{m,n},\n\\end{align}\n\\end{subequations}\nwhere $\\hat{H}_\\mathrm{p}$ (see its definition in Eq.~\\ref{eqn:Hp}) is an ${\\bf R}$-independent operator, the sum $\\sum_{\\gamma}$ in the matrix element of $\\hat{H}_\\mathrm{d}$ runs over the diabatic states, and $D^{2}_{\\alpha\\beta}$ denotes the elements of DSE. Further, the matrix element of the dipole operator under the diabatic representation is expressed as\n\\begin{equation}\\label{dipole-mat}\n{\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}_{0})\\equiv\\langle \\phi_\\alpha ({\\bf R}_{0}) |\\hat{\\boldsymbol\\mu} ({\\bf R}_{0})|\\phi_\\beta ({\\bf R}_{0}) \\rangle.\n\\end{equation}\nSimilarly, at time $t_1$, the matrix element $V_{\\alpha\\beta, m n} ({\\bf R}(t_1))=\\langle \\phi_\\alpha ({\\bf R}_{0}), m| \\hat V ({\\bf R}(t_1))| \\phi_\\beta ({\\bf R}_{0}), n\\rangle$ can also be written explicitly, with each term expressed as follows\n\\begin{subequations}\n\\begin{align}\n&H^\\mathrm{en}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle\\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{en} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\\n&~~~= H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1)) \\delta_{m,n} \\\\\n&H^\\mathrm{p}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{p} | \\phi_\\beta({\\bf R}_{0}),n \\rangle \\nonumber\\\\ \n&~~~= \\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\\\\n&H^\\mathrm{enp}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{enp} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\ \n&~~~= g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\beta}({\\bf R}(t_1)) \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1} \\big)\\\\\n&H^\\mathrm{d}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{d} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R_{0}}), n \\rangle \\nonumber\\\\ \n&~~~=\\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\gamma} ({\\bf R}(t_1))) (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\gamma\\beta} ({\\bf R}(t_1))) \\delta_{m,n} \\nonumber\\\\ \n&~~~\\equiv D^2_{\\alpha\\beta}({\\bf R}(t_1)) \\delta_{m,n}, \n\\end{align}\n\\end{subequations}\nwhere $H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1))\\equiv\\langle\\phi_\\alpha({\\bf R}_{0})| \\hat{H}_\\mathrm{en} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0})\\rangle$, and ${\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1))\\equiv\\langle \\phi_\\alpha ({\\bf R}_{0}) |\\hat{\\boldsymbol\\mu} ({\\bf R}(t_1))|\\phi_\\beta ({\\bf R}_{0}) \\rangle$.\\\\\n\nTo conveniently calculate $H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1))$ and ${\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1))$, we use the following relations\n\\begin{subequations}\n\\begin{align}\nH^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1)) & =\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\tilde{H}^{\\mathrm{en}}_{\\lambda\\nu}({\\bf R}(t_1)) S^{\\dagger}_{\\beta\\nu} \\label{eqn:elect2} \\\\\n{\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1)) & =\\sum_{\\lambda\\nu}S_{\\alpha\\lambda}\\tilde{{\\boldsymbol\\mu}}_{\\lambda\\nu} ({\\bf R}(t_1)) S^{\\dagger}_{\\beta\\nu}, \\label{eqn:mu2}\n\\end{align}\n\\end{subequations}\nwhere the matrix elements at ${\\bf R}(t_1)$ are expressed as\n\\begin{subequations}\n\\begin{align}\n\\tilde{H}^{\\mathrm{en}}_{\\lambda\\nu}({\\bf R}(t_1)) & =\\langle \\phi_{\\lambda}({\\bf R}(t_{1}))| \\hat{H}_\\mathrm{en} ({\\bf R}(t_1))|\\phi_{\\nu}({\\bf R}(t_{1})) \\rangle \\\\\n&=E_{\\lambda}({\\bf R}(t_1))\\delta_{\\lambda\\nu} \\nonumber \\\\\n\\tilde{\\boldsymbol\\mu}_{\\lambda\\nu} ({\\bf R}(t_1)) & =\\langle \\phi_{\\lambda}({\\bf R}(t_{1}))| \\hat{\\boldsymbol\\mu} ({\\bf R}(t_1)) |\\phi_{\\nu}({\\bf R}(t_{1})) \\rangle,\n\\end{align}\n\\end{subequations}\nand the overlap matrix between two electronic adiabatic states (with two different nuclear geometries) are  \n\\begin{subequations}\n\\begin{align}\nS_{\\alpha\\lambda}&= \\langle \\phi_{\\alpha}({\\bf R}_0)|\\phi_{\\lambda}({\\bf R}(t_{1}))\\rangle\\label{eqn:wf_overlap}\\\\\nS^{\\dagger}_{\\beta\\nu}&= \\langle \\phi_{\\nu}({\\bf R}(t_{1}))|\\phi_{\\beta}({\\bf R}_0)\\rangle.\\label{eqn:wf_overlap_diag}\n\\end{align}\n\\end{subequations}\nUsing the above information, as well as Eq.~\\ref{eqn:interpolation}, we can obtain each tern of $V_{\\alpha\\beta,m n}({\\bf R}(t))$ for propagating the dynamics of the quantum subsystem that contains both electronic and photonic DOFs.\n\nNext, we need to evaluate the nuclear gradients to propagate the dynamics of the classical subsystem, which contains the nuclear DOFs. In particular, we need to evaluate the nuclear gradients on each term of $\\nabla V_{\\alpha\\beta,mn}({\\bf R}(t_1))$. First, let us focus on the gradient term from the electron-nuclear coupling term\\cite{ZhouJCPL2019} as follows\n\\begin{align}\\label{dHen}\n&{\\nabla} H^\\mathrm{en}_{\\alpha\\beta,m n}({\\bf R}(t_1))\\\\\n&= {\\nabla} \\langle \\phi_{\\alpha} ({\\bf R}(t_0)),m|\\hat{H}_\\mathrm{en}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0)),n\\rangle \\nonumber\\\\\n&=\\langle \\phi_{\\alpha} ({\\bf R}(t_0))|{\\nabla}\\hat{H}_\\mathrm{en}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0))\\rangle\\cdot\\langle m|n\\rangle \\nonumber\\\\\n&=\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\langle \\phi_\\lambda({{\\bf R}({t_1})}) |\\nabla{\\hat{H}}_{\\mathrm{en}} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle S^{\\dagger}_{\\beta\\nu}\\cdot \\delta_{mn}. \\nonumber\n\\end{align}\nHere, from the first to the second line, we have used the fact that neither $\\langle\\phi_{\\alpha}({\\bf R}_0)|$ nor $\\langle m|$ are ${\\bf R}$-dependent, which allows ${\\nabla}$ to bypass both and directly act on $\\hat{V}({\\bf R}(t_{1}))$. We have also used the fact that $\\hat{H}_\\mathrm{en}$ does not contain any photonic operators. The gradient term $\\langle \\phi_\\lambda({{\\bf R}({t_1})}) |\\nabla{\\hat{H}}_{\\mathrm{en}} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle$ can be evaluated using the following well-known equality\\cite{tully94jcp}\n\\begin{equation}\\label{eqn:HF}\n\\langle \\phi_{\\lambda} ({\\bf R}) |\\nabla \\hat{H}_\\mathrm{en}({\\bf R}) | \\phi_{\\nu} ({\\bf R}) \\rangle = \n\\begin{cases}\n     \\nabla E_{\\lambda} \\quad & (\\lambda=\\nu) \\\\\n      {\\bf d}_{\\lambda\\nu} \\left( E_{\\nu} - E_{\\lambda} \\right) \\quad & (\\lambda \\neq \\nu), \n\\end{cases}\n\\end{equation}\nwhere the non-adiabatic coupling (NAC) vector (or so-called  derivative coupling) is \n\\begin{equation}\\label{elec-nac}\n{\\bf d}_{\\lambda\\nu}=\\langle \\phi_{\\lambda} ({\\bf R}) |\\nabla|\\phi_{\\nu} ({\\bf R}) \\rangle.\n\\end{equation}\n\nFor the gradient on the matrix $H^\\mathrm{p}_{\\alpha\\beta,m n}$, because there is no nuclear DOF in $\\hat{H}_\\mathrm{p}$, thus \n\\begin{equation}\n\\nabla H^\\mathrm{p}_{\\alpha\\beta,m n}({\\bf R}({t_1}))= \\nabla \\big[\\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\big] = 0.\n\\end{equation}\n\nFor the gradient on the light-matter interaction term $H^\\mathrm{enp}_{\\alpha\\beta,m n}$, we have\n\n\\begin{widetext}\n\\begin{align}\\label{eqn:gradient_en}\n&\\nabla H^\\mathrm{enp}_{\\alpha\\beta,m n}({\\bf R}(t_1))= {\\nabla}\\langle \\phi_{\\alpha} ({\\bf R}(t_0)),m|\\hat{H}_\\mathrm{enp}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0)),n\\rangle \\\\\n& = \\langle \\phi_{\\alpha} ({\\bf R}(t_0))|{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0))\\rangle\\cdot \\boldsymbol{\\epsilon}\\cdot  g_\\mathrm{c} \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1}\\big)\\nonumber\\\\\n&=\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\langle \\phi_\\lambda({{\\bf R}({t_1})})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle S^{\\dagger}_{\\beta\\nu}\\cdot \\boldsymbol{\\epsilon}\\cdot g_\\mathrm{c} \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1}\\big),\\nonumber\n\\end{align}\n\\end{widetext}\n\nwhere $S_{\\alpha\\lambda}$ and $S^{\\dagger}_{\\beta\\nu}$ are defined in Eq.~\\ref{eqn:wf_overlap} and Eq.~\\ref{eqn:wf_overlap_diag}, respectively. To evaluate the term $\\langle\\phi_\\lambda({\\bf R})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R})| \\phi_\\nu({\\bf R)}\\rangle$ that appears in Eq.~\\ref{eqn:gradient_en}, we use a simple relation based on the chain rule as follows\n\\begin{widetext}\n\\begin{align}\\label{eq:muderive}\n&\\langle\\phi_\\lambda({\\bf R})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R})| \\phi_\\nu({\\bf R})\\rangle=\\nabla\\langle\\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle - \\langle\\nabla \\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle-\\langle \\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\nabla\\phi_{\\nu}({\\bf R})\\rangle \\nonumber \\\\\n&=\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R}) -\\sum_{\\gamma}\\langle\\nabla \\phi_{\\lambda}({\\bf R})|\\phi_{\\gamma}({\\bf R})\\rangle {\\boldsymbol\\mu}_{\\gamma \\nu}({\\bf R})-\\sum_{\\gamma}{\\boldsymbol\\mu}_{\\lambda\\gamma} ({\\bf R})\\langle\\phi_{\\gamma} ({\\bf R}) |\\nabla\\phi_{\\nu}({\\bf R})\\rangle\\nonumber\\\\\n&=\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})+ \\sum_{\\gamma}\\big[{\\bf d}_{\\lambda \\gamma}({\\bf R}) {\\boldsymbol\\mu}_{\\gamma \\nu}({\\bf R}) -{\\boldsymbol\\mu}_{\\lambda\\gamma}({\\bf R}){\\bf d}_{\\gamma\\nu}({\\bf R})\\big], \n\\end{align}\n\\end{widetext}\nwhere ${\\bf d}_{\\lambda\\gamma}({\\bf R})=\\langle\\phi_{\\lambda} ({\\bf R})|\\nabla\\phi_{\\gamma}({\\bf R})\\rangle=-\\langle\\nabla\\phi_{\\lambda}({\\bf R})|\\phi_{\\gamma}({\\bf R})\\rangle$ and ${\\bf d}_{\\gamma\\nu}({\\bf R})=\\langle\\phi_{\\gamma} ({\\bf R})|\\nabla\\phi_{\\nu}({\\bf R})\\rangle=-\\langle\\nabla\\phi_{\\gamma}({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle$ are the electronic derivative couplings (defined in Eq.~\\ref{elec-nac}). Note that from the first line to the second line, we have inserted $\\hat{\\mathcal P}=\\sum_{\\gamma}|\\phi_{\\gamma}({\\bf R})\\rangle\\langle\\phi_{\\gamma}({\\bf R})|$, which is the resolution of identity in the electronic subspace. As one can clearly see, this term requires the evaluation of derivative coupling ${\\bf d}_{\\lambda\\gamma}({\\bf R})$ and ${\\bf d}_{\\gamma\\nu}({\\bf R})$, and the derivative on dipole matrix element $\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})$.\nFor most of the electronic structure methods, the derivatives on dipole matrix elements $\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})$ are not implemented. Nevertheless, recent theoretical development has made these quantities available.\\cite{Zhang2019jcp}\n\nFinally, the gradient from the DSE term is expressed as follows\n\\begin{widetext}\n\\begin{align} \\label{eqn:gradient_d}\n&\\nabla H^\\mathrm{d}_{\\alpha\\beta,m n}({\\bf R}(t_1))= {\\nabla} \\langle \\phi_{\\alpha} ({\\bf R}_{0}),m|\\hat{H}_\\mathrm{d}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0}), n\\rangle \\\\\n=& {\\nabla} \\Big[\\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\Big] \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\delta_{m,n} \\nonumber \\\\\n=& \\Big[\\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\nonumber  \\\\\n& + (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) | \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\Big] \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\delta_{m,n},\\nonumber\n\\end{align}\n\\end{widetext}\nwhere the term $\\langle \\phi_{\\alpha} ({\\bf R}(t_0)) | \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}(t_0))\\rangle$ can be evaluated using Eq.~\\ref{eqn:mu2}, and the $\\langle \\phi_{\\alpha} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle$ type of derivative can be computed in the same fashion as elaborated in Eq.~\\ref{eqn:gradient_en}.\n\nUsing the matrix elements $V_{\\alpha\\beta,m n}({\\bf R}(t))$ (Eq.~\\ref{eqn:interpolation}) and the nuclear gradient $\\nabla V_{\\alpha\\beta,mn}({\\bf R})$ (as outlined in Eqs.~\\ref{dHen} -~\\ref{eqn:gradient_d}), one can in principle use any trajectory-based approaches or wavepacket approaches with guiding trajectories\\cite{IzmaylovJCP2018,Ben-Nun1998jcp,mcEhrenfest} to propagate the quantum dynamics in the time step of $t\\in[t_0,t_1]$. During the next short-time propagation segment $t\\in[t_1,t_2]$, the QD scheme adopts a new reference geometry ${{\\bf R}'_{0}}\\equiv {\\bf R}(t_1)$ and new {\\it diabatic} basis $|\\psi_{j}({{\\bf R}'_{0}})\\rangle\\equiv|\\psi_{j}({\\bf R}(t_1))\\rangle=|\\phi_{\\beta}({\\bf R}(t_{1})),m\\rangle$. Between the $t\\in [t_0,t_1]$ propagation and the $t \\in [t_1,t_2]$ propagation segments, all of the necessary quantities will be transformed from $\\{|\\psi_{i}({\\bf R}_{0})\\rangle\\}$ to the $\\{|\\psi_{j}({\\bf R}'_{0})\\rangle\\}$ basis, using the relation \n\\begin{equation}\\label{eqn:basis}\n    |\\psi_{j}({\\bf R}(t_{1}))\\rangle=\\sum_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle|\\psi_{i}({\\bf R}(t_{0}))\\rangle.\n\\end{equation}\n\nFor the model calculations in this work, these overlap integrals are evaluated as \n\\begin{equation}\\label{overlap}\n\\langle\\psi_{i}({\\bf R}(t_{0}))|\\psi_{j}({\\bf R}(t_{1}))\\rangle=\\langle\\phi_{\\alpha}({\\bf R}(t_{0}))|\\phi_{\\beta}({\\bf R}(t_{1}))\\rangle\\cdot\\langle n|m\\rangle,\n\\end{equation}\nwhere the electronic adiabatic state overlaps $\\langle\\phi_{\\alpha}({\\bf R}(t_{0}))|\\phi_{\\beta}({\\bf R}(t_{1}))\\rangle$ are directly calculated using the discrete variable representation (DVR) basis, and the Fock states are orthonormal to each other $\\langle n|m\\rangle=\\delta_{n,m}$. \nWhen performing the transformation in Eq.~\\ref{eqn:basis} (as well as in  Eq.~\\ref{map-trans2} for the non-adiabatic mapping methods), the eigenvectors maintain their mutual orthogonality subject to a very small error when they are expressed in terms of the previous basis due to the incompleteness of the basis.\\cite{GranucciJCP2001,PlasserJCP2012} Nevertheless, the orthogonality remains to be well satisfied among  $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\}$ or $\\{|\\psi_{j}({\\bf R}(t_1))\\rangle\\}$. This small numerical error generated from each step can, however, accumulate over many steps and cause a significant error at longer times, leading to non-unitary dynamics.\\cite{GranucciJCP2001,PlasserJCP2012} This potential issue can be easily resolved by using orthonormalization procedure among the vectors of the overlap matrix composed by $\\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle$, as been done in our previous work\\cite{MandalJCP2018} for simulating photo-induced charge transfer dynamics. Here, we perform the L\\\"owdin orthogonalization procedure\\cite{LowdinJCP1950} as commonly used in the local diabatization approach\\cite{GranucciJCP2001} to ensure unitary propagation.\n\nAs the nuclear geometry closely follows the reference geometry throughout the propagation, the QD basis forms a convenient and compact basis. Note that, in principle, one needs an infinite set of crude adiabatic states $\\{|\\psi_{i}({\\bf R}_0)\\rangle\\}$ to represent the time-dependent electronic wavefunction because the electronic wavefunction could change rapidly with the motion of the nuclei, and the crude adiabatic basis is only convenient when the reference geometry ${\\bf R}_{0}$ is close to the nuclear geometry ${\\bf R}$. By dynamically updating the basis in the QD scheme, the time-dependent electronic wavefunction is expanded with the ``moving crude adiabatic basis\"\\cite{IzmaylovJCP2018} that explores the most relevant and important parts of the Hilbert space, thus requiring only a few states for quantum dynamics propagation. \n\n\\subsection{Non-adiabatic Mapping Dynamics Methods}\\label{subsec:mapping_method}\nThe Meyer-Miller-Stock-Thoss (MMST) formalism\\cite{MeyerJCP1979,StockPRL1997,ThossPRA1999} maps the discrete electronic DOFs onto continuous phase space variables. In the strict diabatic basis $\\{|i\\rangle\\}$ (in the sense that $\\langle i|\\nabla|j\\rangle=0$ for all $|i\\rangle$ and $|j\\rangle$), the total Hamiltonian in Eq.~\\ref{totalH} is expressed as\n\\begin{equation}\\label{Diabatic-Hamiltonian}\n\\hat{H}={\\frac {{\\bf P}^2} {2{\\bf M}}}+\\sum_{i}V_{ii}(\\hat{\\bf R})|i\\rangle\\langle i|+\\sum_{i\\neq j}V_{ij}(\\hat{\\bf R})|i\\rangle\\langle j|,\n\\end{equation}\nwhere $V_{ij}(\\hat{\\bf R})=\\langle i|\\hat{V}(\\hat{\\bf r},\\hat {\\bf R})|j\\rangle$ are the matrix elements of the electronic Hamiltonian. Note that here $|i\\rangle$ is used to represent the strict diabatic basis, and not to be confused with the adiabatic-Fock state $|\\psi_i({\\mathbf R}) \\rangle=|\\phi_{\\alpha}({\\bf R}),n\\rangle$ introduced in Eq.~\\ref{adia-fock}. Nevertheless, based on the QD scheme, these adiabatic-Fock states with a reference geometry ${\\bf R}_{0}$ will be used as the diabatic state in the neighborhood of the reference geometries, as indicated in Eq.~\\ref{eqn:qdidea}.\n \nIn the non-adiabatic mapping approach, the Hamiltonian operator in Eq.~\\ref{Diabatic-Hamiltonian} is transformed into the following MMST Hamiltonian\n\\begin{equation} \\label{eq:mapham} \n\\mathcal{H}_\\mathrm{m}={\\frac {{\\bf P}^2} {2{\\bf M}}}+{\\frac{1}{2}}\\sum_{ij}V_{ij}({\\bf R})\\left(p_{i}p_{j}+q_{i}q_{j}-2\\gamma_{j}\\delta_{ij}\\right),\n\\end{equation}\nwhere $2\\gamma_{j}$ is viewed as a parameter\\cite{MillerFD2016} which specifies the ZPE of the mapping oscillators.\\cite{UweJCP1999,MillerFD2016,richardson2019,richardson2020} In principle, $2\\gamma_{j}$ is state-specific and trajectory-specific.\\cite{CottonJCP2019_2} The MMST mapping Hamiltonian has been historically justified by Stock and Thoss using harmonic oscillator's raising and lowering operators as the mapping operator.\\cite{StockPRL1997,ThossPRA1999} Recently, it has been derived using the $SU(N)$ Lie group theory or so-called generalized spin mapping approach.\\cite{richardson2020}\n\nClassical trajectories are generated based on Hamilton's equations of motion (EOM)\n\\begin{subequations}\\label{eq:mapeqn}\n\\begin{eqnarray}  \n    \\dot q_{j} &=& \\partial \\mathcal{H}_\\mathrm{m}\/ \\partial p_{j};~~\\dot p_{i} = -\\partial \\mathcal{H}_\\mathrm{m} \/ \\partial q_{i}\\\\\n    \\dot {\\bf R} &=& \\partial \\mathcal{H}_\\mathrm{m}\/ \\partial {\\bf P};~~ \\dot {\\bf P}=-\\partial \\mathcal{H}_\\mathrm{m} \/ \\partial {\\bf R}= {\\bf F}, \n\\end{eqnarray}\n\\end{subequations}\nwith the nuclear force expressed as\n\\begin{equation}\n\\label{eq:force}\n{\\bf F}=-{\\frac {1} {2}}\\sum_{ij}\\nabla V_{ij}({\\bf R})\\big(p_{i}p_{j}+q_{i}q_{j}-2\\gamma_{j}\\delta_{ij}\\big).\n\\end{equation}\nOverall, the MMST mapping provides a consistent classical footing for both electronic and nuclear DOFs, and the non-adiabatic transitions between electronic states are captured through the classical motion of the fictitious harmonic oscillators. The non-adiabatic dynamics obtained from this formalism have shown good performance in the {\\it ab initio} on-the-fly dynamics.\\cite{ZhouJCPL2019,HuJCTC2021,WeightJCP2021}\n\nTo sample the initial electronic condition and estimate the population, it is also convenient to use the action-angle variables, $\\{\\varepsilon_{j}, \\theta_{j}\\}$, which are related to the canonical mapping variables $\\{p_{j}, q_{j}\\}$ through \n\\begin{equation}\\label{action-angle}\n    \\varepsilon_j = \\frac{1}{2}\\left(p_j^2 + q_j^2 \\right);~~~\\theta_j =-\\tan^{-1}\\left( \\frac{p_j}{q_j}\\right),\n\\end{equation}\nand the inverse relations\n\\begin{equation}\\label{eqn:action_to_mapping}\n    q_j = \\sqrt{2 \\varepsilon_j }\\cos(\\theta_j);~~p_j =-\\sqrt{2 \\varepsilon_j}\\sin(\\theta_j),\n\\end{equation}\nwhere $\\varepsilon_j$ is a positive-definite action variable that is directly proportional to the mapping variables' radius in action-space.\\cite{CottonJCP2019_2} \n\nThe SQC approach calculates the population of electronic state $|j\\rangle$, which is to be evaluated as\\cite{MillerFD2016}\n\\begin{align}\\label{eqn:wignersqc}\n\\rho_{jj}(t)&=\\mathrm{Tr}_{\\bf R}\\left[\\hat{\\rho}(0)e^{i\\hat{H}t\/\\hbar}|j\\rangle\\langle j|e^{-i\\hat{H}t\/\\hbar}\\right]\\\\\n&\\approx\\int d\\boldsymbol{\\tau} \\rho_\\mathrm{W}({\\bf P},{\\bf R})W_i({\\boldsymbol \\varepsilon}(0))W_j({\\boldsymbol \\varepsilon}(t)),\\nonumber\n\\end{align}\nwhere $\\hat{\\rho}(0)=\\hat{\\rho}_{\\bf R} \\otimes |i\\rangle \\langle i|$ is the initial density operator, $\\rho_\\mathrm{W}({\\bf P},{\\bf R})$ is the Wigner transform of $\\hat{\\rho}_{\\bf R}$ operator for the nuclear DOFs, ${\\boldsymbol \\varepsilon} = \\{\\varepsilon_1,\\varepsilon_2,...,\\varepsilon_\\mathcal{N}\\}$ is the positive-definite action variable vector for $\\mathcal{N}$ electronic states,\\cite{CottonJCP2019_2} $W_i({\\boldsymbol \\varepsilon})= \\delta (\\varepsilon_i - (1+\\gamma_{i}))\\prod_{i \\neq j} \\delta(\\varepsilon_j-\\gamma_{j})$ is the Wigner transformed action variables,\\cite{CottonJCP2016} and $d\\boldsymbol{\\tau}\\equiv d{\\bf P}\\cdot d{\\bf R}\\cdot d{\\boldsymbol \\varepsilon} \\cdot d\\boldsymbol{\\theta}$. For practical reasons, the above delta functions in $W_i({\\boldsymbol \\varepsilon})$ are broadened using a distribution function (so-called window function) that can be used to bin the resulting electronic action variables in action-space.\\cite{MillerFD2016} Further, we use the $\\gamma$-SQC approach,\\cite{CottonJCP2019_2} which uses a {\\it state-specific} and {\\it trajectory-specific} $\\gamma_{j}$ parameter in Eq.~\\ref{eq:mapham} to correct the initial force according to the initially populated state. This method has been proven to provide very accurate non-adiabatic dynamics in model photo-dissociation problems (coupled Morse potential), as well as outperform FSSH (with decoherence correction) in {\\it ab initio} on-the-fly simulations.\\cite{HuJCTC2021, WeightJCP2021} The details of $\\gamma$-SQC are provided in Appendix~\\ref{apsec:details_of_sqc}.\n\nFor the spin-LSC approach,\\cite{richardson2019,richardson2020} one chooses a universal ZPE parameter $2\\gamma_{j}=\\Gamma$ for all states and trajectories. The spin-LSC population dynamics is calculated as\n\\begin{align}\\label{eqn:wignerlsc}\n\\rho_{jj}(t)&=\\mathrm{Tr}_{\\bf R}\\left[\\hat{\\rho}_{R}\\otimes|i\\rangle\\langle i|e^{i\\hat{H}t\/\\hbar}|j\\rangle\\langle j|e^{-i\\hat{H}t\/\\hbar}\\right]\\\\\n&\\approx\\int d\\boldsymbol{\\tau} \\rho_\\mathrm{W}({\\bf P},{\\bf R}) [|i\\rangle\\langle i|]_\\mathrm{s}(0)\\cdot[|j\\rangle\\langle j|]_\\mathrm{\\bar{s}}(t),\\nonumber\n\\end{align}\nwhere the population estimators are obtained from the Stratonovich-Weyl transformed electronic projection operators, with the expressions as follows\\cite{richardson2020}\n\\begin{subequations}\\label{sw-estimator}\n\\begin{align}\n&\\big[|i\\rangle\\langle i|\\big]_\\mathrm{s}=\\frac{1}{2}(q^2_i+p^2_i-\\Gamma)\\\\\n&\\big[|j\\rangle\\langle j|\\big]_\\mathrm{\\bar s}\n=\\frac{\\mathcal{N}+1}{2(1+\\frac{\\mathcal{N}\\Gamma}{2})^2}\\cdot (q^2_{j}+p^2_{j})-\\frac{1-\\frac{\\Gamma}{2}}{1+\\frac{\\mathcal{N}\\Gamma}{2}}.\n\\end{align}\n\\end{subequations}\nThe parameter $\\Gamma$ is related to the radius of the generalized Bloch sphere $r_\\mathrm{s}$ through $\\Gamma=\\frac{2}{\\mathcal N}(r_\\mathrm{s}-1)$, where $\\mathrm{s}$ and $\\mathrm{\\bar s}$ are complementary indices in the Stratonovich-Weyl transform. Among the vast parameter space, one of the best performing choices\\cite{richardson2019,richardson2020} is when $r_\\mathrm{s}=r_\\mathrm{\\bar s}=\\sqrt{\\mathcal{N}+1}$, which is referred to as $\\mathrm{s}=\\mathrm{W}$, leading to a ZPE parameter\n\\begin{equation}\\label{Gamma}\n\\Gamma=\\frac{2}{\\mathcal{N}}(\\sqrt{\\mathcal{N}+1}-1),\n\\end{equation}\nas well as the identical expression of $[|i\\rangle\\langle i|]_\\mathrm{s}$ and $[|j\\rangle\\langle j|]_\\mathrm{\\bar s}$ in Eq.~\\ref{sw-estimator}. We further use the focused initial condition\\cite{richardson2019,richardson2020} that replaces the sampling of the mapping variables in the $d\\boldsymbol{\\tau}$ integral of Eq.~\\ref{eqn:wignerlsc} with specific values of the mapping variables, such that $\\frac{1}{2}(q^2_i+p^2_i-\\Gamma)=1$ for initially occupied state $|i\\rangle$ and $\\frac{1}{2}(q^2_j+p^2_j-\\Gamma)=0$ for the initially unoccupied states $|j\\rangle$. The angle variables $\\{\\theta_{j}\\}$ (Eq.~\\ref{action-angle}) are randomly sampled\\cite{richardson2020} in the range of $[0,2\\pi)$.\n\nUsing the QD propagation scheme, one can directly perform non-adiabatic using both $\\gamma$-SQC and spin-LSC in their original diabatic formalism, with the information from the ``ab initio\" polaritonic calculations of the molecule-cavity hybrid system. Using the schemes outlined in Sec.~\\ref{sec:QD}, one can obtain the polariton coupling $\\langle\\psi_{i}({\\bf R}_{0})|\\hat{V}({\\bf R})|\\psi_{j}({\\bf R}_{0})\\rangle$ (see Eq.~\\ref{eqn:vijt} and Eq.~\\ref{eqn:interpolation}) and nuclear gradient $\\nabla\\langle\\psi_{i}({\\bf R}_{0})|\\hat{V}({\\bf R})|\\psi_{j}({\\bf R}_{0})\\rangle$ (see Eq.~\\ref{dHen}-Eq.~\\ref{eqn:gradient_d}), which are the necessary ingredients to solve the MMST mapping EOMs in Eq.~\\ref{eq:mapeqn} and Eq.~\\ref{eq:force}. Between two propagation steps, the QD basis is transformed from $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\equiv|\\phi_\\alpha({\\mathbf R}(t_0)),n\\rangle\\}$ to $\\{|\\psi_{j}({\\bf R}(t_1))\\rangle\\equiv|\\phi_\\beta({\\mathbf R}(t_1)),m\\rangle\\}$. This leads to the corresponding transform of mapping variables between the two consecutive QD bases as follows\\cite{MandalJCTC2018,ZhouJCPL2019}\n\\begin{subequations}\\label{map-trans2}\n\\begin{align}\n    &\\sum_{i} q_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))|\\psi_{j}({\\bf R}(t_{1}))\\rangle \\rightarrow q_{j}\\\\\n    &\\sum_{i} p_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle \\rightarrow p_{j},\n\\end{align}\n\\end{subequations}\nwhere the overlaps between the two steps are evaluated using Eq.~\\ref{overlap} (see discussion under that equation). More computational details for the $\\gamma$-SQC and spin-LSC are provided in section \\ref{sec:comp-detail}.\n\n\\section{Computational Details}\n\\subsection{The Model System}\nIn this work, we use the asymmetrical Shin-Metiu model\\cite{Shin1995jcp,Hoffmann2020} as the ``\\textit{ab initio}\" model molecular system to investigate strong light-matter interactions between a molecule and an optical cavity. The model contains a transferring proton (nucleus) and an electron, as well as two fixed ions labeled as donor (D) and acceptor (A), as shown in Fig.~\\ref{fig:adiabatic_PES}a. This model is usually used to describe the proton-coupled electron transfer (PCET) reaction and has been studied recently using the exact factorization approach to investigate how cavity can influence chemical reactivities.\\cite{Maitra2019,Hoffmann2020,Maitra2021} The electron-nuclear interaction potential operator $\\hat{H}_\\mathrm{en}$ (c.f. Eq.~\\ref{eqn:Hen}) is expressed as\n\\begin{equation}\\label{Hen}\n\\hat{H}_\\mathrm{en}=\\sum_{\\sigma=\\pm 1} \\left( \\frac{1}{|R+\\frac{\\sigma L}{2}|}-\\frac{\\mathrm{erf}\\Big( \\frac{|r+\\frac{\\sigma L}{2}|}{a_{\\sigma}} \\Big)}{|r+\\frac{\\sigma L}{2}|} \\right) - \\frac{\\mathrm{erf}\\Big(\\frac{|R-r|}{a_f}\\Big)}{|R-r|},\n\\end{equation}\nwhere the first term represents the potential of the transferring proton, the second term represents the potential of the transferring electron, and the third term represents the electron-proton coupling. We choose the same parameters used in Ref.~\\citenum{Hoffmann2020}, which is $L=19 $ a.u., $a_+=3.1$ a.u., $a_{-}=4.0$ a.u., $a_f=5.0$ a.u. and the proton mass is $M=1836$ a.u. To calculate the electronic properties of the SM model, we use the Sinc discrete variable representation (DVR) basis\\cite{Colbert1992} to represent the electronic adiabatic states. These adiabatic states $|\\phi_{\\alpha}(R)\\rangle$ are computed on-the-fly for a given nuclear configuration $R$ by solving Eq.~\\ref{HenPsi}. The details are provided in Appendix~\\ref{Exact}.\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_1.pdf}\n  \\end{minipage}%\n   \\caption{(a) The schematic illustration of the asymmetrical SM model, where one proton and one electron can transfer between two fixed ions (donor and acceptor). The distance between the donor and acceptor is 19 a.u. Here $V_p$ and $V_e$ are the potential of the transferring proton and electron, respectively. (b) The first two adiabatic electronic states, (c) NAC between these two electronic states and (d) their permanent and transition dipole moments of the SM model. The PESs of the polaritonic states inside the cavity are obtained with the light-matter coupling strength (e) $g_\\mathrm{c}=0.001$ and (f) $g_\\mathrm{c}=0.005$. The color used in (e) and (f) is coded according to $\\langle \\hat{a}^{\\dagger}\\hat{a}\\rangle$, as shown in the upper position of panel (e).}\n\\label{fig:adiabatic_PES}\n\\end{figure}\n\nFig.~\\ref{fig:adiabatic_PES}a also depicts the model potential in Eq.~\\ref{Hen}, with the black curve depicting the proton potential (the first term in Eq.~\\ref{Hen}) and the green curve depicting the electron potential (the second term in Eq.~\\ref{Hen}). Fig.~\\ref{fig:adiabatic_PES}b presents the two lowest adiabatic electronic states of the SM model (red and blue curves). There is an avoided crossing between the ground and the first excited state potential energy surfaces (PESs) near $R=2.0$ a.u.. Fig.~\\ref{fig:adiabatic_PES}c presents the NAC between them (the green curve), which shows a strong coupling near the avoided crossing region. The matrix elements of the dipole operator under the adiabatic representation (Eq.~\\ref{dipole-mat}) of the SM model are presented in Fig.~\\ref{fig:adiabatic_PES}d. \n\nWhen coupling the SM molecular model with the cavity, the photon frequency of the cavity mode is chosen as $\\hbar \\omega_\\mathrm{c}=2.721$ eV ($\\approx \\, 0.1 $ a.u.). Further, we assume that the cavity field polarization direction ${\\boldsymbol{\\epsilon}}$ is always aligned with the direction of the dipole operator $\\hat{\\boldsymbol\\mu}$, such that ${\\boldsymbol{\\epsilon}}\\cdot{\\boldsymbol\\mu}_{\\gamma\\nu}={\\mu}_{\\gamma\\nu}$ (for $\\{\\nu,\\gamma\\}=\\{e,g\\}$) where ${\\mu}_{\\gamma\\nu}$ is the magnitude of $\\hat{\\boldsymbol\\mu}$. Explicitly considering the angle between ${\\boldsymbol{\\epsilon}}$ and $\\hat{\\boldsymbol\\mu}$ will generate a polariton induced conical intersection (even for a diatomic molecule), which will induce geometric phase effects.\\cite{Farag2021PCCP} We consider two different light-matter coupling strengths $g_\\mathrm{c}=0.001$ a.u. and $g_\\mathrm{c}=0.005$ a.u. in this work. The normalized coupling strength is often defined as\\cite{Kockum2019rev} $\\eta\\equiv g_\\mathrm{c}\\cdot|\\boldsymbol{\\epsilon} \\cdot  {{\\boldsymbol \\mu}}_{eg}|\/\\omega_\\mathrm{c}$ where $|\\boldsymbol{\\epsilon} \\cdot {{\\boldsymbol \\mu}}_{eg}|$ is the typical magnitude of the transition dipole projected along the field polarization direction. For the coupling strength considered above (and taking $\\hbar\\omega_\\mathrm{c}\\approx 2.721$ eV for the model calculation), the normalized coupling strength is $\\eta = 0.06$ (for $g_\\mathrm{c}=0.001$ a.u.) and $\\eta = 0.3$ (for $g_\\mathrm{c}=0.005$ a.u.). When $0.1<\\eta<1$, the light and matter interaction achieves the ultra-strong coupling regime,\\cite{Kockum2019rev,Nori2019natphys} which is difficult to achieve but still within the reach of the current experimental setup.\\cite{Shegai_NC_2020,Haran2016} Thus, besides the pure theoretical value to derive the exact nuclear gradient expression, our computational results are also within the reach of the near future experimental setup. \n\n\nThe polaritonic PESs $\\mathcal{E}_{J}(R)$ associated with polariton states $|\\mathcal{E}_{J}(R)\\rangle$ (see their definition in Eq.~\\ref{polariton}) are presented in Fig.~\\ref{fig:adiabatic_PES}e with the light-matter coupling strength $g_\\mathrm{c}=0.001$ a.u. and in Fig.~\\ref{fig:adiabatic_PES}f with the light-matter coupling strength $g_\\mathrm{c}=0.005$. These polariton potentials are color coded (as shown in the inset of panel (e)) based on the expectation value of $\\langle \\hat{a}^{\\dagger}\\hat{a}\\rangle$ indicated on top of this panel. Note that this should not be viewed as a ``photon number\" operator under the dipole gauge used in the PF Hamiltonian\\cite{Mandal2020JPCL,Schaefer2020AP} because the rigorous photon number operator should be obtained by applying the Power-Zienau-Woolley (PZW) Gauge transformation\\cite{PZW,Cohen-Tannoudji,Taylor2020PRL} on the photon number operator $\\hat{a}^{\\dagger}\\hat{a}$. Nevertheless, it can be viewed as an approximate estimation of the photon number when the light-matter couplings are not in the ultra-strong coupling regime.\n\\cite{Forn-Diaz2019rmp}\n\nThe initial state (for $t=0$) of the molecule-cavity hybrid system is\n\\begin{equation}\\label{eq:initial-state}\n|\\Phi(t=0)\\rangle=|e,0\\rangle\\otimes|\\chi\\rangle,\n\\end{equation}\nwhich corresponds to a Franck-Condon excitation of the hybrid system to the $|e,0\\rangle$ state, with $|\\chi\\rangle$ as the initial nuclear wavefunction. For the SM model in this work, we use $\\chi(R)=\\langle R|\\chi\\rangle\\sim \\exp[-M\\omega_{0}(R-R_{0})^2\/2\\hbar]$, where $M$ is the mass of the proton (nucleus in the SM model), $R_0$ is the position with a minimum potential energy of the ground electronic state. Here, $\\chi(R)$ is the vibrational ground state wavefunction on the ground electronic states, centered at $R_0$ under the harmonic approximation, with the harmonic oscillation frequency being $\\omega_{0}$. We use the parameters in the original reference\\cite{Hoffmann2020} for $R_{0}=-4$ and $\\omega_{0}=0.000382$ a.u. To solve the exact quantum dynamics, we use the DVR basis for the nuclear DOF and the adiabatic-Fock state for the electronic-photonic subsystem. The details of the exact quantum dynamics are provided in Appendix~\\ref{Exact}. \n\n\\subsection{Details of $\\gamma$-SQC and spin-LSC Dynamics}\\label{sec:comp-detail}\n\nTo perform the $\\gamma$-SQC dynamics, we need to sample the initial condition for the quantum subsystem. In this work, we first sample the action-angle variables $\\{\\varepsilon_{j}, \\theta_{j}\\}$ then transform them to the mapping variables$\\{p_{j},q_{j}\\}$ using Eq.~\\ref{eqn:action_to_mapping}. Among them, the action variables $\\{\\varepsilon_{j}\\}$ are sampled according to the window function in Eq.~\\ref{EQ:TriangleWindow}, and the angle variables $\\{\\theta_{j}\\}$ are randomly sampled from $[0,2\\pi)$. The triangle window is used in this work, although the square window generates similar results. \n\nFor the spin-LSC dynamics, we use the focused initial conditions\\cite{richardson2020} as described in section \\ref{subsec:mapping_method}, where the action variable $\\varepsilon_{i}$ is set to be $1+\\Gamma\/2$ for the initially occupied state and $\\Gamma\/2$ for the initially unoccupied state, with $\\Gamma$ expressed in Eq.~\\ref{Gamma}. The angle variables $\\{\\theta_{j}\\}$ are randomly generated between $[0,2\\pi)$ as in the $\\gamma$-SQC method. The canonical mapping variables are obtained from Eq.~\\ref{eqn:action_to_mapping}.\n\nThe initial nuclear distribution of all trajectory-based simulations (Ehrenfest, FSSH, $\\gamma$-SQC and spin-LSC) are generated by sampling the Wigner density \n\\begin{equation}\\label{wigner}\n[\\langle R|\\chi\\rangle]_\\mathrm{w}=\\frac{1}{\\hbar\\pi}e^{-M(P^2+\\omega^2(R-R_{0})^2)\/\\omega\\hbar},\n\\end{equation}\nwhich is the Wigner transformation of the nuclear wavefunction $\\chi(R)=\\langle R|\\chi\\rangle$ in the initial state (see Eq.~\\ref{eq:initial-state}). Here, $R$ and $P$ are the nuclear coordinate and momentum, respectively. The initial state for the electronic-photonic subsystem is set to $|e0\\rangle$. The nuclear time step used in the QD-$\\gamma$-SQC and QD-spin-LSC is $dt=0.1$ fs, with $100$ equally spaced electronic time steps for the mapping variables' integration during each nuclear time step. The equation of motion in Eq.~\\ref{eq:mapeqn}-Eq.~\\ref{eq:force} are integrated using a second-order symplectic integrator for the MMST variables\\cite{kelly2012mapping,church2018} for a given nuclear time step, and these mapping variables are transformed based on Eq.~\\ref{map-trans2} between two adjacent nuclear time steps due to the change of the QD basis. The population dynamics using all MQC and mapping methods were averaged over $5000$ trajectories, although $3000$ trajectories were enough to produce the basic trend of the polariton dynamics, see Fig. S3 in the Supplementary Material. The light-matter coupling strength $g_\\mathrm{c}$ was chosen to be $0.001$ and $0.005$, according to our previous work.\\cite{Zhou2022}\n\nWe also benchmark the results of non-adiabatic mapping dynamics approaches with commonly used MQC approaches, including the Ehrenfest dynamics and the FSSH method. The details of these two MQC approaches are provided in Appendix~\\ref{apsec:Ehrenfest_FSSH_Details}. In particular, the Ehrenfest dynamics is equivalent to choosing $\\gamma=0$ in the mapping theory (see Eq.~\\ref{eq:mapham}) and an initial action-angle variables condition (see Eq.~\\ref{action-angle} and Eq.~\\ref{eqn:action_to_mapping}) of $\\varepsilon_j=\\delta_{ij}$ (for the initially occupied state $|i\\rangle$) and $\\theta_{j}=0$ (for all state $|j\\rangle$). One can thus use the same QD scheme and the mapping equation to obtain the results of the Ehrenfest dynamics.\\cite{Subotnik2016JCP} \n\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_2.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states in Shin-Metiu-cavity model obtained from (a) Ehrenfest dynamics, (b) FSSH approach, (c) $\\gamma$-SQC method, and (d) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Two electronic states and two Fock states are considered in the simulation, and the light-matter coupling strength $g_\\mathrm{c}=0.001$ a.u.}\n\\label{fig:gc001_results}\n\\end{figure}\n\n\\section{Results}\nFig.~\\ref{fig:gc001_results} presents the population dynamics of the adiabatic-Fock states simulated using the approximate methods (open circles), including the MQC approaches (Ehrenfest and FSSH) and the mapping dynamics methods ($\\gamma$-SQC and spin-LSC), compared to the numerically exact results (solid lines). The light-matter coupling strength is chosen to be $g_\\mathrm{c}=0.001$ a.u. The system is initially prepared in the $|e0\\rangle$ state and decays quickly into the $|g1\\rangle$ state during the first $\\sim12$ fs due to the large light-matter coupling strength from the large transition dipole moment ($\\boldsymbol{\\mu}_{eg}$) between adiabatic electronic state $|g\\rangle$ and $|e\\rangle$ (as shown in Fig.~\\ref{fig:adiabatic_PES}d). Then, the system starts to oscillate between $|e0\\rangle$ and $|g1\\rangle$ until about $20$ fs. All of the MQC and mapping dynamics methods can describe the above process reasonably well compared to the exact results. After that, all the dynamics results (including the exact one) show a fast population increase of the $|g0\\rangle$ state, which is due to the electronic NAC $d_{eg}$ that directly couples the $|e0\\rangle$ state to $|g0\\rangle$ state (gold lines). All of the approximate methods can qualitatively describe such a trend, but the MQC methods (panels a-b) are less accurate compared to the mapping-based methods (panels c-d), in terms of the rising of the $|g0\\rangle$ population as well as its long time plateau. Moreover, both Ehrenfest and FSSH dynamics predict a significant population transfer from $|g1\\rangle$ to $|e1\\rangle$ state (panels a-b) as an {\\it artifact} that is not shown in the exact dynamics results. In contrast, the $\\gamma$-SQC and spin-LSC methods perform much better, where the population transfer process from $|g1\\rangle$ to $|e1\\rangle$ state is largely suppressed (panels c-d). Overall, the mapping methods outperform the MQC methods in this small light-matter coupling case. It is worth mentioning that the population dynamics results obtained with the FSSH method can be significantly improved if one uses the proper estimator.\\cite{Landry2013JCP} We have provided details of this approach and numerical results in Appendix~\\ref{apsec:Ehrenfest_FSSH_Details}. Even so, the FSSH method is still facing many challenges from the improper treatment of the quantum coherence and frustrated hop problems, which have been widely discussed.\\cite{StockPRL1997,subotnik2016arpc,PrezdoSH,Granucci2007} \n\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_3.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states in Shin-Metiu-cavity model obtained from (a) Ehrenfest dynamics, (b) FSSH approach, (c) $\\gamma$-SQC method, and (d) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Two electronic states and two Fock states are considered in the simulation, and the light-matter coupling strength $g_\\mathrm{c}=0.005$ a.u.}\n\\label{fig:gc005_results}\n\\end{figure}\n\nFig.~\\ref{fig:gc005_results} presents the polariton population dynamics with the coupling strength $g_\\mathrm{c}=0.005$ a.u. The oscillation between $|e0\\rangle$ and $|g1\\rangle$ state population appears much earlier and faster compared to the $g_\\mathrm{c}=0.001$ results due to the larger light-matter coupling between $|e0\\rangle$ and $|g1\\rangle$ states (see Eq.~\\ref{eq:enp}). Further, the $|g0\\rangle$ and $|e1\\rangle$ states are also getting populated at an earlier time, due to the permanent dipole $\\mu_{gg}$ and $\\mu_{ee}$ that couples $|g1\\rangle$ state to $|g0\\rangle$ state and $|e0\\rangle$ state to $|e1\\rangle$ state, respectively. Similar to the $g_\\mathrm{c}=0.001$ case, all the MQC and mapping dynamics provide a reasonable accuracy for the population dynamics at a short time, while the mapping methods perform much better than the MQC methods at a longer time. In addition, the spin-LSC method outperforms $\\gamma$-SQC method in the description of $|g0\\rangle$ and $|e0\\rangle$ state population after $t=20$ fs, as shown in Fig.~\\ref{fig:gc005_results}c-d.\n\nUntil now, all of our simulations are restricted in the Hilbert subspace formed by two electronic states ($|g\\rangle$ and $|e\\rangle$) and two photonic Fock states ($|0\\rangle$ and $|1\\rangle$). The system could explore a larger Hilbert space due to the increasing light-matter coupling strength. Thus, we systematically check the polariton dynamics using the exact wavepacket dynamics method with a larger number of electronic adiabatic states and Fock states, as shown in Fig. S1 and S2 in the Supplementary Material. The results show that, for the small light-matter coupling strength case ($g_\\mathrm{c}=0.001$ a.u.), truncation to the Hilbert subspace formed by two electronic states and two Fock states is enough to give an accurate description of the population dynamics for the SM model studied in this work. However, for the larger coupling strength case ($g_\\mathrm{c}=0.005$ a.u.), the polariton dynamics will converge when including four adiabatic electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$ with energies in ascending order) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$ with photon number in ascending order). \n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_4.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states with (a,b) Ehrenfest dynamics, (c,d) FSSH approach, (e,f) $\\gamma$-SQC method, and (g,h) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Four adiabatic electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$ with energies in ascending order) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$ with photon number in ascending order) are considered in the simulations and the light-matter coupling strength is  $g_\\mathrm{c}=0.005$ a.u. Only the adiabatic-Fock states with observable populations of more than $0.01$ are plotted.}\n\\label{fig:more-states}\n\\end{figure}\n\nTo further test the performance of the mapping methods ($\\gamma$-SQC and spin-LSC) as well as the MQC methods (Ehrenfest and FSSH) in such a large Hilbert subspace, which includes sixteen states formed by the tensor product of four electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$). Fig.~\\ref{fig:more-states} presents the results of using Ehrenfest dynamics (a-b), FSSH (c-d), $\\gamma$-SQC (e-f) and spin-LSC (g-h). Besides the adiabatic-Fock states already appear in the four-states subspace ($|g0\\rangle$, $|e0\\rangle$, $|g1\\rangle$ and $|e1\\rangle$), we can also see some other states ($|h0\\rangle$, $|f0\\rangle$, $|g2\\rangle$ and $|f1\\rangle$) are populated due to the increasing light-matter coupling strength. All of the MQC (Ehrenfest, FSSH) and mapping ($\\gamma$-SQC and spin-LSC) dynamics results provide accurate agreement with the exact one in the short time ($< 20$ fs). After that, all of the methods start to generate less accurate results (especially for the $|g0\\rangle$ population). Note that both $\\gamma$-SQC and spin-LSC perform less accurately compared to the situation in a smaller Hilbert subspace (Fig.~\\ref{fig:gc005_results}). This is because both $\\gamma$-SQC and spin-LSC are sensitive to the number of states of the system. For $\\gamma$-SQC, more states means less trajectory landed in the population action window.\\cite{Cotton_JCP_2019_many_states} For spin-LSC, the ZPE correction $\\Gamma$ (Eq.~\\ref{Gamma}) explicitly depends on the number of states $\\mathcal{N}$. This suggests a need for the future development of more accurate dynamics approaches. The current work, nevertheless, paves the way for those future methods to be directly used for simulating on-the-fly polariton quantum dynamics.\n\n\\section{Conclusions}\nIn this work, we generalize the quasi-diabatic (QD) propagation scheme\\cite{MandalJCP2018,ZhouJCPL2019,MandalJPCA2019} to simulate the non-adiabatic polariton dynamics in molecule-cavity hybrid systems. The adiabatic-Fock states, which are the tensor product states of the adiabatic electronic states of the molecule and photon Fock states, are used as the {\\it locally} well-defined {\\it diabatic} states for the dynamics propagation.\\cite{ZhouJCPL2019,MandalJPCA2019} These locally well-defined diabatic states allow using any diabatic quantum dynamics methods for dynamics propagation, and the definition of these states will be updated at every nuclear time step. The benefit of using such adiabatic-Fock states is that one can conveniently obtain the electronic adiabatic states energies, the nuclear gradient, the dipole moments and NACs between these states, which are necessary ingredients in molecular cavity QED simulations. \n\nWe use the recently developed non-adiabatic mapping dynamics approaches, $\\gamma$-SQC\\cite{CottonJCP2019_2} and spin-LSC\\cite{richardson2020} to investigate polariton dynamics of a Shin-Metiu model coupled to an optical cavity.\\cite{Hoffmann2020,Zhou2022} To benchmark the results of the obtained polariton dynamics, we performed simulations using the Ehrenfest dynamics and the FSSH approaches, as well as the numerically exact polariton wavepacket propagation. The results show that the mapping methods can accurately describe the population dynamics of the molecule-cavity system both at a short time and a longer time when compared to exact dynamics results. In addition, the mapping methods outperform the Ehrenfest and FSSH approaches at a long time dynamics. The numerical results also demonstrate that the performance of the mapping methods ($\\gamma$-SQC and spin-LSC) becomes less accurate with an increased number of states in the simulation, indicating the need for future theoretical development.\n\nWe envision that the theoretical development in this work will provide the emerging polariton chemistry field with a general theoretical tool that enables direct {\\it ab initio} on-the-fly simulations of polariton photochemical processes. We also anticipate that the theoretical developments in this work will enable many recently developed diabatic quantum dynamics approaches to directly simulate polariton quantum dynamics.\n\n\\section*{Acknowledgments}\nThis work was supported by the National Science Foundation CAREER Award under Grant No. CHE-1845747 and by a Cottrell Scholar award (a program by Research Corporation for Science Advancement). Computing resources were provided by the Center for Integrated Research Computing (CIRC) at the University of Rochester. D.H. appreciates valuable discussions with Wanghuai Zhou. We also appreciate valuable suggestions by Prof. Yihan Shao.\n\n\\section*{Conflict of Interest}\nThe authors have no conflicts to disclose.\n\n\\section*{Supplementary Material}\nSee Supplementary Material for additional results of the convergence test for the number of trajectories and the number of adiabatic electronic states and Fock states.\n\n\\section*{Availability of Data}\t\nThe data that support the findings of this study are available from the corresponding author upon a reasonable request.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\subsection{Background}\n\\label{subsec:background} Let $P = \\{p_1,\\ldots,p_n\\}$ be a set of\n$n$ points in $\\mathbb{R}^3$. The \\emph{2-center problem} for $P$ is to\nfind two congruent balls of minimum radius whose union covers $P$.\nThis is a special case of the general $p$-center problem in\n$\\mathbb{R}^d$, which calls for covering a set $P$ of $n$ points in\n$\\mathbb{R}^d$ by $p$ congruent balls of minimum radius. If $p$ is\npart of the input, the problem is known to be NP-complete~\\cite{MK}\neven for $d=2$, so the complexity of algorithms for solving the\n$p$-center problem, for any fixed $p$, is expected to increase more\nthan polynomially in $p$. Agarwal and Procopiuc showed that the\n$p$-center problem in $\\mathbb{R}^d$ can be solved in $n^{O(p^{1-1\/d})}$\ntime~\\cite{AP}, improving upon a naive $n^{O(p)}$-solution. At the\nother extreme end, the 1-center problem (also known as the\n\\emph{smallest enclosing ball} problem) is known to be an LP-Type\nproblem, and can thus be solved in $O(n)$ randomized expected time\nin any fixed dimension, and also in deterministic linear time\n~\\cite{CM, NML, NMLT}. Faster approximate solutions to the general\n$p$-center problem have also been proposed~\\cite{AP, BHI, BE}.\n\nIf $d$ is not fixed, the 2-center problem in $\\mathbb{R}^d$ is\nNP-Complete~\\cite{MK2}. The 2-center problem in $\\mathbb{R}^2$ has a\nrelatively rich history, mostly in the past two decades. Hershberger\nand Suri~\\cite{HS} showed that the decision problem of determining\nwhether $P$ can be covered by two disks of a given radius $r$ can be\nsolved in $O(n^2 \\log n)$ time. This has led to several\nnearly-quadratic algorithms~\\cite{ASP, DE, JK} that solve the\noptimization problem, the best of which, due to Jaromczyk and\nKowaluk~\\cite{JK}, runs in $O(n^2 \\log n)$ deterministic time.\nSharir~\\cite{MS} considerably improved these bounds and obtained a\ndeterministic algorithm with $O(n \\log^9 n)$ running time. His\nalgorithm combines several geometric techniques, including\nparametric searching, searching in monotone matrices, and dynamic\nmaintenance of planar configurations. Chan~\\cite{TC} (following an\nimprovement by Eppstein~\\cite{DEF}) improved the running time to\n$O(n \\log^2 n \\log^2 \\log n)$.\n\nThe only earlier work on the 2-center problem in $\\mathbb{R}^3$ we are aware of is by Agarwal~\\textsl{et~al.}~\\cite{AES},\nwhich presents an algorithm with $O(n^{3+\\varepsilon})$ running time, for any $\\varepsilon > 0$. It uses\na rather complicated data structure for dynamically maintaining upper and lower envelopes of bivariate functions.\n\n\\subsection{Our results}\n\\label{subsec:results} We present two randomized algorithms for the\n2-center problem in $\\mathbb{R}^3$. We first present an algorithm whose\nexpected running time is $O(n^3 \\log^5 n)$. It is conceptually a\nnatural generalization of the earlier algorithms for the planar\n2-center problem~\\cite{ASP, DE, JK}; its implementation however is\nconsiderably more involved. The second algorithm runs in $O((n^2\n\\log^5 n) \/(1-r^*\/r_0)^3)$ expected time, where $r^*$ is the common\nradius of the 2-center balls and $r_0$ is the radius of the smallest\nenclosing ball of $P$.  This is based on some of the ideas in\nSharir's planar algorithm~\\cite{MS}, but requires several new\ntechniques. As in the previous algorithms, we first present\nalgorithms for the decision problem: given $r > 0$, determine\nwhether $P$ can be covered by two balls of radius $r$. We then\ncombine it with an adaptation of Chan's randomized optimization\ntechnique~\\cite{TCG} to obtain a solution for the optimization\nproblem. In both cases, the asymptotic expected running time of the\noptimization algorithm is the same as that of the decision procedure\n(which itself is deterministic).\n\nThe paper is organized as follows. Section~\\ref{sec:sketches}\nbriefly sketches our two solutions. Section~\\ref{sec:cubic_alg}\npresents the near-cubic algorithm, and\nSection~\\ref{sec:improved_alg} presents the improved algorithm. A\nkey ingredient of both algorithms is a dynamic procedure for testing\nwhether the intersection of a collection of balls in $\\mathbb{R}^3$\nis nonempty. We present the somewhat technical details of this\nprocedure in Section~\\ref{sec:spherical_polytopes}, and conclude in\nSection~\\ref{sec:discussion} with a few open problems.\n\n\\section{Sketches of the Solutions}\n\\label{sec:sketches}\n\n\\subsection{The near-cubic algorithm}\n\\label{subsec:n^3_sketch} To solve the decision problem, in the less\nefficient but conceptually simpler manner, we use a standard\npoint-plane duality, and replace each point $p \\in P$ by a dual\nplane $p^*$, and each plane $h$ by a dual point $h^*$, such that the\nabove-below relations between points and planes are preserved. We\nnote that if $P$ can be covered by two balls $B_1, B_2$ (not\nnecessarily congruent), then there exists a plane $h$ (containing\nthe circle ${\\partial}{B_1} \\cap {\\partial}{B_2}$, if they intersect at all, or\nseparating $B_1$ and $B_2$ otherwise) separating $P$ into two\nsubsets $P_1, P_2$, such that $P_1 \\subset B_1$ and $P_2 \\subset\nB_2$. We therefore construct the arrangement ${\\cal A}$ of the set $\\{p^*\n\\mid p \\in P\\}$ of dual planes. It has $O(n^3)$ cells, and each cell\n$\\tau$ has the property that, for any point $w \\in \\tau$, its primal\nplane $w^*$ separates $P$ into two subsets of points, $P_\\tau^+$ and\n$P_\\tau^-$, which are the same for every $w \\in \\tau$, and depend\nonly on $\\tau$. We thus perform a traversal of ${\\cal A}$, which proceeds\nfrom each visited cell to a neighbor cell. When we visit a cell\n$\\tau$, we check whether the subsets $P_\\tau^+$ and $P_\\tau^-$ can\nbe covered by two balls of radius $r$, respectively. To do so, we\nmaintain dynamically the intersection of the sets $\\{B_r(p)\\mid p\n\\in P_\\tau^+\\}$, $\\{B_r(p)\\mid p \\in P_\\tau^-\\}$, where $B_r(p)$ is\nthe ball of radius $r$ centered at $p$, and observe that (a) any\npoint in the first (resp., second) intersection can serve as the\ncenter of a ball of radius $r$ which contains $P_\\tau^+$ (resp.,\n$P_\\tau^-$), and (b) no ball of radius $r$ can cover $P_\\tau^+$\n(resp., $P_\\tau^-$) if the corresponding intersection is empty.\nMoreover, when we cross from a cell $\\tau$ to a neighbor cell\n$\\tau'$, $P_\\tau^+$ changes by the insertion or deletion of a single\npoint, and $P_\\tau^-$ undergoes the opposite change, so each of the\nsets of balls $\\{B_r(p)\\mid p \\in P_\\tau^+\\}$, $\\{B_r(p)\\mid p \\in\nP_\\tau^-\\}$ changes by the deletion or insertion of a single ball.\nAs we know the sequence of updates in advance, maintaining\ndynamically the intersection of either of these sets of balls can be\ndone in an offline manner. Still, the actual implementation is\nfairly complicated. It is performed using a variant of the\nmulti-dimensional parametric searching technique of\nMatou\\v{s}ek~\\cite{JM} (see also~\\cite{TCA, CMS, NPT}). The same\nprocedure is also used by the second improved algorithm. For the\nsake of readability, we describe this procedure towards the end of\nthe paper, in Section~\\ref{sec:spherical_polytopes}.\n\n\nThe main algorithm uses a segment tree to represent the sets\n$P_\\tau^+$ (and another segment tree for the sets $P_\\tau^-$).\nRoughly, viewing the traversal of ${\\cal A}$ as a sequence $\\Sigma$ of\ncells, each ball $B_r(p)$ has a \\emph{life-span} (in $P_\\tau^+$),\nwhich is a union of contiguous maximal subsequences of cells $\\tau$,\nin which $p \\in P_\\tau^+$, and a complementary life-span in\n$P_\\tau^-$. We store these (connected portions of the) life-spans as\nsegments in the segment tree. Each leaf of the tree represents a\ncell $\\tau$ of ${\\cal A}$, and the balls stored at the nodes on the path\nto the leaf from the root are exactly those whose centers belong to\nthe set $P_\\tau^+$ (or $P_\\tau^-$). By precomputing the intersection\nof the balls stored at each node of the tree, we can express each of\nthe intersections $\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^+\\}$ and\n$\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^-\\}$, for each cell $\\tau$, as\nthe intersection of a logarithmic number of precomputed\nintersections (see also~\\cite{DE}). We show that such an\nintersection can be tested for emptiness in $O(\\log^5 n)$ time. This\nin turn allows us to execute the decision procedure with a total\ncost of $O(n^3 \\log^5 n)$. We then return to the original\noptimization problem and apply a variant of Chan's randomization\ntechnique~\\cite{TCG} to solve the optimization problem by a small\nnumber of calls to the decision problem, obtaining an overall\nalgorithm with $O(n^3 \\log^5 n)$ \\emph{expected} running\ntime.\\footnote{\\small The earlier algorithm in~\\cite{AES} follows\nthe same general approach, but uses an even more complicated, and\nslightly less efficient machinery for dynamic emptiness testing of\nthe intersection of congruent balls.}\n\n\\subsection{The improved solution}\n\\label{subsec:n^2_sketch} The above algorithm runs in nearly cubic\ntime because it has to traverse the entire arrangement ${\\cal A}$, whose\ncomplexity is $O(n^3)$. In Section~\\ref{sec:improved_alg} we improve\nthis bound by traversing only portions of ${\\cal A}$, adapting some of the\nideas in Sharir's improved solution for the planar\nproblem~\\cite{MS}. Specifically, Sharir's algorithm solves the\ndecision problem (for a given radius $r$) in three steps, treating\nseparately three subcases, in which the centers $c_1, c_2$ of the\ntwo covering balls are, respectively, far apart ($|c_1 c_2| > 3r$),\nat medium distance apart ($r < |c_1 c_2| \\leq 3r$) and near each\nother ($|c_1 c_2| \\leq r$). We base our solution on the techniques\nused in the first two cases, which, for simplicity, we merge into a\nsingle case (as done in~\\cite{DEF} for the planar case), and extend\nit so that we only need to assume that $|c_1c_2| \\geq \\beta r$, for\nany fixed $\\beta > 0$. In more detail, letting $B_r(p)$ denote the\ndisk of radius $r$ centered at a point $p$, Sharir's algorithm\nguesses a constant number of lines $l$, one of which separates the\ncenters $c_1, c_2$ of the respective solution disks $D_1,D_2$, so\nthat the set $P_L$ of the points to the left of $l$ is contained in\n$D_1$. We then compute the intersection $K(P_L) = \\bigcap_{p \\in\nP_L} B_r(p)$, and intersect each ${\\partial}{B_r(p)}$, for $p \\in P_R = P\n\\setminus P_L$ (the subset of points to the right of $l$), with\n${\\partial}{K(P_L)}$. It is easily seen that ${\\partial}{K(P_L)}$ has linear\ncomplexity and that each circle ${\\partial}{B_r(p)}$, for $p \\in P_R$,\nintersects it at two points (at most). This produces $O(n)$ critical\npoints (vertices and intersection points) on ${\\partial}{K(P_L)}$ and\n$O(n)$ arcs in between. As argued in~\\cite{MS}, it suffices to\nsearch these points and arcs for possible locations of the center of\n$D_1$ (and dynamically test whether the balls centered at the\nuncovered points have nonempty intersection).\n\nGeneralizing this approach to $\\mathbb{R}^3$, we need to guess a separating plane $\\lambda$, to retrieve the subset $P_L \\subseteq P$ of points to the left of $\\lambda$, to compute ${\\partial}{K(P_L)}$ (which, fortunately, still has only linear complexity), to intersect ${\\partial}{B_r(p)}$, for each $p \\in P_R$, with ${\\partial}{K(P_L)}$, and to form the arrangement of the resulting intersection curves. Each cell of this arrangement is a candidate for the location of the center of the left covering ball $B_1$, and for each placement in $\\tau$, $B_1$ contains the same fixed subset of $P$ (which depends only on $\\tau$).\n\nHowever, the complexity of the resulting arrangement $M_K$ on\n${\\partial}{K(P_L)}$ might potentially be cubic. We therefore compute only\na portion $M$ of $M_K$, which suffices for our purposes, and prove\nthat its complexity is only $O(n^2)$. This is the main geometric\ninsight in the improved algorithm, and is highlighted in\nLemma~\\ref{lemma:quadratic_K_P_L}. We show that if there is a\nsolution then $O(1\/\\beta^3)$ guesses suffice to find a separating\nplane. This implies that the running time of the improved decision\nprocedure is $O((1\/\\beta^3) n^2\\log^5 n)$. Thus, it is nearly\nquadratic for any fixed value of $\\beta$. We show that one can take\n$\\beta = 2(r_0\/r -1)$, where $r_0$ is the radius of the smallest\nenclosing ball of $P$.\n\nTo solve the optimization problem, we conduct a search on the\noptimal radius $r^*$, using our decision procedure, starting from\nsmall values of $r$ and going up, halving the gap between $r$ and\n$r_0$ at each step\\footnote{\\small We have to act in this manner to\nmake sure that we do not call the decision procedure with values of\n$r$ which are too close to $r_0$, thereby losing control over the\nrunning time.}, until the first time we reach a value $r > r^*$.\nThen we use a variant of Chan's technique~\\cite{TCG}, combined with\nour decision procedure, to find the exact value of $r^*$. The way\nthe search is conducted guarantees that its cost does not exceed the\nbound $O((1\/\\beta^3) n^2 \\log^5 n)$, for the separation parameter\n$\\beta = 2(r_0\/r^* -1)$ for $r^*$. Hence, we obtain a randomized\nalgorithm that solves the 2-center problem for any positive\nseparation of $c_1$ and $c_2$, and runs in $O((n^2 \\log^5 n)\n\/(1-r^*\/r_0)^3)$ expected time.\n\n\n\n\\section{A Nearly Cubic Algorithm}\n\\label{sec:cubic_alg}\n\n\\subsection{The decision procedure}\n\\label{subsec:decision_procedure} In this section we give details of\nthe implementation of our less efficient solution, some of which are\nalso applicable for the improved solution. Recall from the\ndescription in Section~\\ref{sec:sketches} that the decision\nprocedure, on a given radius $r$, constructs two segment trees $T^+,\nT^-$, on the life-spans of the balls $B_r(p)$, for $p \\in P$ (with\nrespect to the tour of the dual plane arrangement ${\\cal A}$). Each leaf\nis a cell $\\tau$ of ${\\cal A}$, and the balls, whose centers belong to\n$P_\\tau^+$ (resp., $P_\\tau^-$), are those stored at nodes on the\npath from the root to $\\tau$ in $T^+$ (resp., $T^-$).\n\nFor each node $u$ of $T^+$, let $S_u$ denote the intersection of all\nthe balls (of radius $r$) stored at $u$. We refer to each $S_u$ as a\n\\emph{spherical polytope}; see~\\cite{BCT, BLNP, BN} for (unrelated)\nstudies of spherical polytopes. We compute each $S_u$ in $O(|S_u|\n\\log |S_u|)$ deterministic time, using the algorithm by\nBr\\\"{o}nnimann et al.~\\cite{BCM} (see also~\\cite{CS, ER} for\nalternative algorithms). Since the arrangement ${\\cal A}$ consists of\n$O(n^3)$ cells, standard properties of segment trees imply that the\ntwo trees require $O(n^3 \\log n)$ storage and $O(n^3 \\log^2 n)$\npreproccessing time.\n\nClearly, the intersection $K(P_\\tau^+)$ (resp., $K(P_\\tau^-)$) of\nthe balls whose centers belong to $P_\\tau^+$ (resp., $P_\\tau^-$) is\nthe intersection of all the spherical polytopes $S_u$, over the\nnodes $u$ on the path from the root to $\\tau$ in $T^+$ (resp.,\n$T^-$).\n\n\\paragraph{Intersection of spherical polytopes.}\nLet ${\\cal S} = \\{S_1,\\ldots,S_t\\}$ be the set of $t = O(\\log n)$\nspherical polytopes stored at the nodes of a path from the root to a\nleaf of $T^+$ or of $T^-$, where, as above, a spherical polytope is\nthe intersection of a finite set of balls, all having the common\nradius $r$. Each $S_i$ is the intersection of some $n_i$ balls, and\n$\\sum_{i=1}^t n_i \\leq n$. Our current goal is to determine, in\npolylogarithmic time, whether the intersection $K$ of the spherical\npolytopes in ${\\cal S}$ is nonempty. If this is the case for at least one\npath of $T^+$ and for the same path in $T^-$ then $r^* \\leq r$, and\notherwise $r^* >r$. Moreover, if there exist a pair of such paths\nfor which both intersections have nonempty interior, then $r^* < r$\n(because we can then slightly shrink the balls and still get a\nnonempty intersection). If no such pair of paths have this property,\nbut there exist pairs with nonempty intersections (with at least one\nof them being degenerate) then $r^* = r$.\n\nThe algorithm for testing emptiness of $K$ is technical and fairly\ninvolved. For the sake of readability, we delegate its description\nto Section~\\ref{sec:spherical_polytopes}. It uses a variant of\nmultidimensional parametric searching which somewhat resembles\nsimilar techniques used in ~\\cite{TCA, CMS, JM, NPT}. It is\nessentially independent of the rest of the algorithm (with some\nexceptions, noted later). We summarize it in the following\nproposition.\n\n\\begin{proposition}\n\\label{prop:intersection_test} Let ${\\cal S}$ be a collection of spherical\npolytopes, each defined as the intersection of at most $n$ balls of\na fixed radius $r$. Let $N$ denote the sum, over the polytopes of\n${\\cal S}$, of the number of balls defining each polytope. After a\npreprocessing stage, which takes $O(N \\log n)$ time and uses $O(N)$\nstorage, we can test whether any $t \\leq \\log n$ polytopes of ${\\cal S}$\nhave a nonempty intersection in $O(\\log^5 n)$ time, and also\ndetermine whether the intersection has nonempty interior.\n\\end{proposition}\n\nHence, we check, for each cell $\\tau$, whether each of $K(P_\\tau^+)$\nand $K(P_\\tau^-)$ are nonempty and non-degenerate. To this end, we\ngo over each path of $T^+$, and over the same path of $T^-$, and\ncheck, using the procedure described in\nProposition~\\ref{prop:intersection_test}, whether the spherical\npolytopes along the tested paths (of $T^+$ and of $T^-$) have a\nnonempty intersection (and whether these intersections have nonempty\ninteriors). We stop when a solution for which both $K(P_\\tau^+)$ and\n$K(P_\\tau^-)$ are nonempty and non-degenerate is obtained, and\nreport that $r^* < r$. Otherwise, we continue to test all cells\n$\\tau$. If at least one degenerate solution is found (i.e., a\nsolution where both $K(P_\\tau^+), K(P_\\tau^-)$ are nonempty, and at\nleast one of them has nonempty interior), we report that $r^* = r$,\nand otherwise $r^* > r$.\n\nBy proposition~\\ref{prop:intersection_test}, the cost of this\nprocedure is $O(n^3 \\log^5 n)$. This subsumes the cost of all the\nother steps, such as constructing the arrangement ${\\cal A}$ and the\nsegment trees $T^+, T^-$. We therefore get a decision procedure\nwhich runs in $O(n^3 \\log^5 n)$ (deterministic) time.\n\n\\subsection{Solving the optimization problem}\n\\label{subsec:optimization_problem}\nWe now combine our decision procedure with the randomized\noptimization technique of Chan~\\cite{TCG}, to obtain an algorithm\nfor the optimization problem, which runs in $O(n^3 \\log^5 n)$\n\\emph{expected} time. Our application of Chan's technique, described\nnext, is somewhat non-standard, because each recursive step has also\nto handle global data, which it inherits from its ancestors.\n\nChan's technique, in its ``purely recursive'' form, takes an optimization problem that has to compute an optimum value $w(P)$ on an input set $P$. The technique replaces $P$ by several subsets $P_1,\\ldots,P_s$, such that $w(P) = \\min\\{w(P_1),\\ldots,w(P_s)\\}$, and $|P_i| \\leq \\alpha|P|$ for each $i$ (here $\\alpha < 1$ and $s$ are constants). It then processes the subproblems $P_i$ in a \\emph{random} order, and computes $\\displaystyle\\min_i w(P_i)$ by comparing each $w(P_i)$ to the minimum $w$ collected so far, and by replacing $w$ by $w(P_i)$ if the latter is smaller.\\footnote{\\small So the value of $w$ keeps shrinking.} Comparisons are performed by the decision procedure, and updates of $w$ are computed recursively. The crux of this technique is that the expected number of recursive calls (in a single recursive step) is only $O(\\log s)$, and this (combined with some additional enhancements, which we omit here) suffices to make the expected cost of the whole procedure asymptotically the same as the cost of the decision procedure, for \\emph{any} values of $s$ and $\\alpha$. Technically, if the cost $D(n)$ of the decision procedure is\n$\\Omega(n^\\gamma)$, where $\\gamma$ is some fixed positive constant, the expected running time is $O(D(n))$ provided that\n\\begin{equation}\n\\label{equation:lnr}\n(\\ln s + 1) \\alpha^\\gamma < 1.\n\\end{equation}\nHowever, even when (\\ref{equation:lnr}) does not hold ``as\nis'', Chan's technique enforces it by compressing $l$ levels of the\nrecursion into a single level, for $l$ sufficiently large, so its expected cost is still $O(D(n))$. See~\\cite{TCG} for details.\n\n\nTo apply Chan's technique to our decision procedure,\nwe pass to the dual space, where each point $p \\in P$ is mapped to a\nplane $p^*$, as done in the decision procedure. We obtain the set $P^* =\n\\{p^* \\mid p \\in P\\}$ of dual planes, and we consider its arrangement\n${\\cal A} = {\\cal A}(P^*)$, where each cell $\\tau$ in ${\\cal A}$ represents an\nequivalence class of planes in the original space, which separate $P$\ninto the same two subsets of points $P_\\tau^+, P_\\tau^-$.\n\nTo decompose the optimization problem into subproblems, as required\nby Chan's technique, we construct a $(1\/\\varrho)$-\\emph{cutting} of\nthe dual space. We recall that, given a collection $\\it{H}$ of $n$\nhyperplanes in $\\mathbb{R}^d$ and a parameter $1 \\leq \\varrho \\leq\nn$, a $(1\/\\varrho)$-\\emph{cutting} of ${\\cal A}(\\it{H})$ of size $q$ is a\npartition of space into $q$ (possibly unbounded) openly disjoint\n$d$-dimensional simplices $\\Delta_1,\\ldots,\\Delta_q$, such that the\ninterior of each simplex $\\Delta_i$ is intersected by at most\n$n\/\\varrho$ of the hyperplanes of $\\it{H}$. See~\\cite{JMC} for more\ndetails.  We use the following well known result~\\cite{BC, CF}:\n\n\\begin{lemma}\n\\label{lemma:cutting}\nGiven a set $\\it{H}$ of $n$ hyperplanes in $\\mathbb{R}^d$, a\n$(1\/\\varrho)$-cutting of ${\\cal A}(\\it{H})$ of size $O(\\varrho^d)$ can be constructed\nin time $O(n\\varrho^{d-1})$, for any $\\varrho \\leq n$.\n\\end{lemma}\n\nReturning to our setup, we construct a $(1\/\\varrho)$-cutting for\n${\\cal A}(P^*)$, for a specific constant value of $\\varrho$, that we will\nfix later, and obtain $O(\\varrho^3)$ simplices, such that the\ninterior of each of them is intersected by at most $n\/\\varrho$\nplanes of $P^*$. Each simplex $\\Delta_i$ corresponds to one\nsubproblem and contains some (possibly only portions of) cells\n$\\tau_1,\\ldots,\\tau_k$ of the arrangement ${\\cal A}$. We recall that each\ncell $\\tau_j$ represents an equivalence class of planes which\nseparate $P$ into two subsets of points $P_{\\tau_j}^+$ and\n$P_{\\tau_j}^-$. Hence, $\\Delta_i$ represents a collection of such\nequivalence classes. All these subproblems have in common the sets\n$(P^*)_{\\Delta_i}^+$, $(P^*)_{\\Delta_i}^-$, consisting,\nrespectively, of all the planes that pass fully above $\\Delta_i$ and\nthose that pass fully below $\\Delta_i$. (These sets are dual to\nrespective subsets $P_{\\Delta_i}^+, P_{\\Delta_i}^-$ of $P$, where\n$P_{\\Delta_i}^+$ is contained in all the sets $P_{\\tau_j}^+$, for\nthe cells $\\tau_j$, that meet $\\Delta_i$, and symmetrically for\n$P_{\\Delta_i}^-$.) Note that most of the dual planes belong to\n$(P^*)_{\\Delta_i}^+ \\cup (P^*)_{\\Delta_i}^-$; the ``undecided''\nplanes are those that cross the interior of $\\Delta_i$, and their\nnumber is at most $n\/\\varrho$. We denote the set of these planes as\n$(P^*)_{\\Delta_i}^0$ (and the set of their primal points as\n$P_{\\Delta_i}^0$).\n\n\nTo apply Chan's technique, we construct two segment trees on the\narrangement of $(P^*)_{\\Delta_i}^0$, as described in Section~\\ref{subsec:decision_procedure}. Consider one of these segment trees, $T^+$,\nthat maintains the set of balls ${\\cal B}^+ = \\{B_r(p) \\mid p \\in\nP_{\\tau_j}^+\\}$. Each cell $\\tau_j$ in $\\Delta_i$ is represented by a\nleaf of $T^+$. Each ball is represented as a collection of disjoint\nlife-spans, with respect to a fixed tour of the cells of ${\\cal A}((P^*)_{\\Delta_i}^0)$,\nwhich are stored as segments in $T^+$, as described earlier.\nIn addition, we compute the intersection of the balls centered\nat the points of $P_{\\Delta_i}^+$, in $O(n \\log n)$ time, and store it at\nthe root of $T^+$. Note that, as we go down the recursion, we keep\nadding planes to $(P^*)_{\\Delta_i}^+$, that is, points to $P_{\\Delta_i}^+$, and the\nactual set $P_{\\Delta_i}^+$ of points dual to the planes above the\ncurrent $\\Delta_i$ is the union of logarithmically many subsets, each\nobtained at one of the ancestor levels of the recursion, including the current step.\nHowever, we cannot inherit the precomputed intersections of the balls\nin these subsets of $P_{\\Delta_i}^+$ from the previous levels, since, as we go down the\nrecursion, Chan's technique keeps `shrinking' the radius of the balls. Hence, each\ntime we have to solve a decision subproblem, we compute the\nintersection of the balls centered at the points of $P_{\\Delta_i}^+$\n(collected over all the higher levels of the recursion)\nfrom scratch. (See below for details on the additional cost incurred\nby this step.) We build a second segment tree $T^-$\nthat maintains the balls of ${\\cal B}^- = \\{B_r(p) \\mid p \\in\nP_{\\tau_j}^-\\}$, in a fully analogous manner. The running time so far (of the decision procedure)\nis $O(n \\log n + m^3 \\log^2 m)$, where $m$ is the\nnumber of planes in $(P^*)_{\\Delta_i}^0$ and $n$ is the size of the\ninitial input set $P$.\n\nTo solve the decision procedure for a given subproblem associated\nwith a simplex $\\Delta_i$, we test, by going over all the\nroot-to-leaf paths in $T^+$ and $T^-$, whether there exists a cell\n$\\tau$ (overlapping $\\Delta_i$), for which the intersections of the\nspherical polytopes on the two respective paths in $T^+$ and $T^-$\nare nonempty (and, if nonempty, whether they both have nonempty\ninteriors). The overall cost of this step, iterating over the\n$O(m^3)$ cells of ${\\cal A}((P^*)_{\\Delta_i}^0)$ and applying the\nprocedure from Section~\\ref{subsec:decision_procedure} for\nintersecting spherical polytopes, is $O(m^3 \\log^5 n)$.\n\nWhen the recursion bottoms out, we have two subsets\n$P_{\\Delta_i}^+$, and $P_{\\Delta_i}^-$ of $O(n)$ points, and a\nconstant number of points in $P_{\\Delta_i}^0$. Hence, we try the\nconstant number of possible separations of $P_{\\Delta_i}^0$ into an\nordered pair of subsets $P_1$ and $P_2$, and, for each of these separations, we compute the two\nsmallest enclosing balls of the sets $P_{\\Delta_i}^+ \\cup P_1$ and\n$P_{\\Delta_i}^- \\cup P_2$ in linear time. If both $P_{\\Delta_i}^+\n\\cup P_1$ and $P_{\\Delta_i}^- \\cup P_2$ can be covered by balls of\nradius $r$, for at least one of the possible separations of\n$P_{\\Delta_i}^0$ into two subsets, then we have found a solution for\nthe 2-center problem. (Discriminating between $r^* = r$ or $r^* < r$\nis done as in Section~\\ref{subsec:decision_procedure}.)\n\nWe now apply Chan's technique to this decision procedure. Note that\nthis application is not standard because the recursive subproblems are\nnot ``pure'', as they also involve the ``global'' parameter $n$. We\ntherefore need to exercise some care in the analysis of the expected\nperformance of the technique.\n\nSpecifically, denote by $T(m,n)$ an upper bound on the expected\nrunning time of the algorithm, for preprocessing\na recursive subproblem involving $m$ points, where the initial input\nconsists of $n$ points. Then $T(m,n)$ satisfies the following recurrence.\n\n\\begin{equation}\n\\label{eqn:T(m,n)}\nT(m,n) \\leq \\left\\{ \\begin{array}{ll}\n\\ln(c\\varrho^3)T(m\/\\varrho,n) + O(m^3 \\log^5 n + n\\log n),   & \\mbox{for $m \\geq \\varrho$,}\\\\\nO(n),                                        & \\mbox{for $m <\n\\varrho$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $c$ is an appropriate absolute constant (so that $c\\varrho^3$\nbounds the number of cells of the cutting), and $\\varrho$ is chosen\nto be a sufficiently large constant so that (\\ref{equation:lnr})\nholds (with $s = c\\varrho^3, \\alpha = 1\/\\varrho$, and $\\gamma = 3$).\nIt is fairly routine (and we omit the details) to show that the\nrecurrence (\\ref{eqn:T(m,n)}) yields the overall bound $O(n^3 \\log^5\nn)$ on the expected cost of the initial problem; i.e., $T(n,n) =\nO(n^3 \\log^5 n)$. We thus obtain the following intermediate result.\n\n\\begin{theorem}\nLet $P$ be a set of $n$ points in $\\mathbb{R}^3$. A 2-center for $P$\ncan be computed in $O(n^3 \\log^5 n)$ randomized expected time.\n\\end{theorem}\n\n\\section{An Improved Algorithm}\n\\label{sec:improved_alg}\n\n\\subsection{An improved decision procedure $\\Gamma$}\n\\label{subsec:improved_decision_procedure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{beta_a_s.pstex_t}\n\\caption{\\small \\sf The points $q_1, q_1', q_2, q_2'$ prevent $|c_1\nc_2|$ from getting smaller.} \\label{figure:beta}\n\\end{center}\n\\end{figure}\n\nConsider the decision problem, where we are given a radius $r$ and a parameter $\\beta > 0$, and\nhave to determine whether $P$ can be covered by two balls of radius\n$r$, such that the distance between their centers $c_1, c_2$ is at\nleast $\\beta r$. (Details about supplying a good lower bound for $\\beta$ will be given in Section~\\ref{sec:separated_centers_optimization}.)\nBy this we mean that there is no placement of two balls of radius $r$, which cover $P$, such that the distance between their centers is smaller than $\\beta r$; see Figure~\\ref{figure:beta}.\n\nThis assumption is easily seen to imply the following property: Let\n$C_{12}$ denote the intersection circle of ${\\partial}{B_1}$ and ${\\partial}{B_2}$\n(assuming that $B_1 \\cap B_2 \\neq \\emptyset$). Then any hemisphere\n$\\nu$ of ${\\partial}{B_1}$, such that (a) the plane $\\pi$ through $c_1$\ndelimiting $\\nu$ is disjoint from $C_{12}$, and (b) $\\nu$ and\n$C_{12}$ lie on different sides of $\\pi$, must contain a point $q$\nof $P$, for otherwise we could have brought $B_1$ and $B_2$ closer\ntogether by moving $c_1$ in the normal direction of $\\pi$, into the\nhalfspace containing $c_2$ (and $C_{12}$). See\nFigure~\\ref{figure:case2_assumption}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{case2_assumption_s.pstex_t}\n\\caption{\\small \\sf The plane $\\pi$ passes through $c_1$ and is\ndisjoint from $C_{12}$. The hemisphere $\\nu$ delimited by $\\pi$,\nwhich lies on the side of $\\pi$ not containing $C_{12}$, must\ncontain a point $q$ of $P$.} \\label{figure:case2_assumption}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{new_case2_a_s.pstex_t}\n\\caption{\\small \\sf $v_1$ is the leftmost point of the intersection\ncircle $C_{12}$.} \\label{figure:new_case2_1}\n\\end{center}\n\\end{figure}\n\n\\smallskip\n\\noindent{\\bf Guessing orientations and separating planes.} We\nchoose a set $D$ of canonical orientations, so that the maximum\nangular deviation of any direction $u$ from its closest direction in\n$D$ is an appropriate multiple $\\alpha$ of $\\beta$. The connection\nbetween $\\alpha$ and $\\beta$ is given by the following reasoning.\nFix a direction $v \\in D$ so that the angle between the orientation\nof $c_1 c_2$ and $v$ is at most $\\alpha$. Rotate the coordinate\nframe so that $v$ becomes the $x$-axis. As above, let $C_{12}$\ndenote the intersection circle of ${\\partial}{B_1}$ and ${\\partial}{B_2}$\n(assuming that the balls intersect). Let $v_1$ be the leftmost point\nof $C_{12}$ (in the $x$-direction); see\nFigure~\\ref{figure:new_case2_1}. If $B_1$ and $B_2$ are disjoint\n(which only happens when $|c_1 c_2| > 2r$) we define $v_1$ to be the\nleftmost point of $B_2$. To determine the value of $\\alpha$, we note\nthat (in complete analogy with Sharir's algorithm in the\nplane~\\cite{MS}) our procedure will try to find a $yz$-parallel\nplane, which separates $c_1$ from $v_1$. For this, we want to ensure\nthat $x(v_1) - x(c_1) > \\beta r\/4$, say, to leave enough room for\nguessing such a separating plane. Let $\\theta$ denote the angle\n$\\varangle v_1c_1c_2$ (see Figure~\\ref{figure:new_case2_2}). Using\nthe triangle inequality on angles, the angle between\n$\\overrightarrow{c_1 v_1}$ and the $x$-axis is at most $\\theta +\n\\alpha$, so $x(v_1) - x(c_1) \\geq r \\cos(\\theta + \\alpha)$. Hence,\nto ensure the above separation, we need to choose $\\alpha$, such\nthat $\\cos(\\theta + \\alpha) > \\beta\/4$. Since $|c_1 c_2| \\geq \\beta\nr$, we have $\\cos \\theta \\geq \\beta\/2$. Hence, it suffices to choose\n$\\alpha$, such that\n$$\\alpha \\leq \\cos^{-1} \\frac{\\beta}{4} - \\cos^{-1} \\frac{\\beta}{2} =\n    \\sin^{-1} \\frac{\\beta}{2} - \\sin^{-1} \\frac{\\beta}{4}\n    = \\Theta(\\beta).$$\nWith this constraint on $\\alpha$, the size of $D$ is\n$\\Theta\\left(1\/\\alpha^2\\right) = \\Theta\\left(1\/\\beta^2\\right)$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{choosing_alpha_s2.pstex_t}\n\\caption{\\small \\sf $x(v_1) - x(c_1) \\geq r \\cos(\\theta + \\alpha)$.}\n\\label{figure:new_case2_2}\n\\end{center}\n\\end{figure}\n\nWe draw $O(1\/\\beta)$ $yz$-parallel planes, with horizontal\nseparation of $\\beta r\/4$, starting at the leftmost point of $P$\n(with respect to the guessed orientation). One of these planes will\nseparate $v_1$ from $c_1$. Thus, the total number of guesses that we\nmake (an orientation in $D$ and a separating plane) is\n$O(1\/\\beta^3)$. The following description pertains to a correct\nguess, in which the properties that we require are satisfied. (If\nall guesses fail, the decision procedure has a negative answer.)\n\n\\smallskip\n\\noindent{\\bf Reducing to a 2-dimensional search.}\nBy the property noted above, the left hemisphere $\\nu_{\\lambda_0}$ of ${\\partial}{B_1}$, delimited by\nthe $yz$-parallel plane $\\lambda_0$ through $c_1$,\nmust pass through at least one point $q$ of $P$ (see Figure~\\ref{figure:new_case2_left}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{leftsemiball_s2.pstex_t}\n\\caption{\\small \\sf The separating plane $\\lambda$ and its parallel\ncopy $\\lambda_0$ through $c_1$. The hemisphere $\\nu_{\\lambda_0}$ of\n${\\partial}{B_1}$ to the left of $\\lambda_0$ must contain a point $q$ of\n$P$.} \\label{figure:new_case2_left}\n\\end{center}\n\\end{figure}\n\n\nLet $P_L$ denote the subset\nof points of $P$ lying to the left of $\\lambda$. Then $P_L$\nmust be fully\ncontained in $B_1$ and contain $q$.\nWe compute the intersection $K(P_L) = \\bigcap \\{B_r(p) \\mid p \\in P_L\\}$\nin $O(n \\log n)$ time~\\cite{BCM}. If $K(P_L)$ is empty, then $P_L$ cannot be covered\nby a ball of radius $r$ and we determine that the currently assumed\nconfiguration does not yield a positive solution for the decision problem.\nOtherwise, since $P_L \\subseteq B_1$, $c_1$\nmust lie in\n$K(P_L)$. Moreover, since $q \\in P_L$ lies on the \\emph{left} portion\nof ${\\partial}{B_1}$, $c_1$ must\nlie on the \\emph{right} portion of the boundary of $K(P_L)$. Finally, since $c_1$ lies to the\nleft of $\\lambda$, only the portion $\\sigma_L$ of the right part of ${\\partial}{K(P_L)}$ to the left of $\\lambda$ has to be considered.\nIf $K(P_L)$ is disjoint from $\\lambda$ then $\\sigma_L$ is just the right\nportion of ${\\partial}{K(P_L)}$. Otherwise, $\\sigma_L$ has a ``hole'', bounded by ${\\partial}{K(P_L)} \\cap \\lambda$, which is a convex piecewise-circular curve, being the boundary of the intersection of the disks $B_r(p) \\cap \\lambda$, for $p \\in P_L$.\n\nWe partition $\\sigma_L$ into \\emph{quadratically many} cells, such\nthat if we place the center $c_1$ of the left solution ball $B_1$ in\na cell $\\tau$, then, no matter where we place it within $\\tau$,\n$B_1$ will cover the same subset of points from $P$. To construct\nthis partition, we intersect, for each $p \\in P_R = P \\setminus\nP_L$, the sphere ${\\partial}{B_r(p)}$ with $\\sigma_L$ and obtain a curve\n$\\gamma_p$ on $\\sigma_L$; this curve bounds the portion of the\nunique face of ${\\partial}{K(P_L \\cup \\{p\\})}$ within $\\sigma_L$. Hence,\nwithin $K(P_L)$, it is a closed connected curve (it may be\ndisconnected within $\\sigma_L$, though). Let $M$ denote the\narrangement formed on $\\sigma_L$ by the curves $\\gamma_p$, for $p\n\\in P_R$, and by the arcs of $\\sigma_L$. Apriori, $M$ might have\ncubic complexity, if many of the $O(n^2)$ pairs of curves $\\gamma_a,\n\\gamma_b$, for $a,b \\in P_R$, traverse a linear number of common\nfaces of $\\sigma_L$, and intersect each other on many of these\nfaces, in an overall linear number of points. Equivalently, the\n``danger'' is that the intersection circle $C_{ab}$ of a\ncorresponding pair of spheres ${\\partial}{B_r(a)}, {\\partial}{B_r(b)}$, for $a,b\n\\in P_R$, could intersect a linear number of faces of $\\sigma_L$\n(and each of these intersections is also an intersection point of\n$\\gamma_a$ and $\\gamma_b$). See Figure~\\ref{figure:new_case2_cubic}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\n\\input{new_case2_cubic.pstex_t}\n\n\\caption{\\small \\sf In a general setup (different than ours), an intersection circle of two balls (the dotted circle) may\nintersect a linear number of faces of ${\\partial}{K(P_L)}$.}\n\\label{figure:new_case2_cubic}\n\\end{center}\n\\end{figure}\n\n\\smallskip\n\\noindent{\\bf Complexity of $M$.} Fortunately, in the assumed\nconfiguration, this cubic behavior is impossible --- $C_{ab}$ can\nmeet only a constant number of faces of $\\sigma_L$. Consequently,\nthe overall complexity of $M$ is only quadratic. This crucial claim\nfollows from the observation that, for $C_{ab}$ to intersect many\nfaces of $\\sigma_L$, it must have many short arcs, each delimited by\ntwo points on $\\sigma_L$ and lying outside $K(P_L)$. The main\ngeometric insight, which rules out this possibility, and leads to\nour improved algorithm, is given in the following lemma.\n\n\\begin{lemma}\n\\label{lemma:quadratic_K_P_L}\nLet $\\lambda$ be a $yz$-parallel plane, which separates $v_1$ from $c_1$.\nLet $P_L \\subseteq P$ be the subset of points of $P$ to the left of $\\lambda$, and let $P_R = P \\setminus P_L$.\nLet $C_{ab}$ denote the intersection circle of ${\\partial}{B_r(a)}, {\\partial}{B_r(b)}$, for some pair of points $a, b \\in P_R$, and let $q \\in P_L$. If the arc $\\omega = C_{ab} \\setminus B_r(q)$ is smaller than a semicircle of $C_{ab}$, then at least one of its endpoints must lie to the right of $\\lambda$.\n\\end{lemma}\n\n\\begin{proof}The situation and its analysis are depicted in Figure~\\ref{figure:lemma_setup}. To slightly simplify the analysis, and without loss of generality, assume that $r = 1$. Let $h$ be the plane passing through $a$, $b$ and $q$. Let $c_{ab}$\ndenote the midpoint of $ab$, and let $w$ denote the center of the\ncircumscribing circle $Q$ of $\\triangle qab$. Denote the distance $|ab|$\nby $2x$, and the radius of $Q$ by $y$ (so $|wp_1|=|wp_2|=|wq|=y$).\nNote that $c_{ab}$ and $w$ lie in $h$ and that $y \\ge x$.\nObserve that $c_{ab}$ is the center of the intersection circle $C_{ab}$ of\n${\\partial}{B_r(a)}$ and ${\\partial}{B_r(b)}$. See Figure~\\ref{figure:lemma_setup}(a).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular*}{0.8\\textwidth}{c c c}\n            \\hspace{-35pt} {\\input{h_setup_s2.pstex_t} } &  & {\\input{C_ab_s2.pstex_t} } \\\\\n    \\small (a) & \\hspace{15pt} & \\small (b) \\\\\n    \\hspace{45pt} & {\\input{abq_s2.pstex_t} } & \\hspace{45pt}\\\\\n     \\hspace{-35pt} & \\small (c) &\n          \n          \n           \\end{tabular*}\n\n       \\vspace{20pt}\n\\caption{\\small \\sf The setup in Lemma~\\ref{lemma:quadratic_K_P_L}:\n           (a) the setup within the plane $h$; (b) the setup within\n    $C_{ab}$; (c) $ww'$ lies on the bisector of $ab$ in\n           the direction that gets away from $q$.}\n\n\\label{figure:lemma_setup}\n\\end{center}\n\\end{figure}\n\nThe intersection points $z,z'$ of $C_{ab}$ and ${\\partial}{B_r(q)}$ are the\nintersection points of the three spheres\n${\\partial}{B_r(a)}$, ${\\partial}{B_r(b)}$, and ${\\partial}{B_r(q)}$.\nThey lie on the line $\\ell$ passing through $w$ and orthogonal\nto $h$, at equal distances $\\sqrt{1-y^2}$ from $w$. See Figure~\\ref{figure:lemma_setup}(b).\n(If $y>1$ then $z$ and $z'$ do not exist, in which case $C_{ab}$ does not intersect ${\\partial}{B_r(q)}$; in what follows we assume\nthat $y\\le 1$.)\nHence, within $C_{ab}$, $zz'$ is a chord of length $2\\sqrt{1-y^2}$.\nIn the assumed setup, $z$ and $z'$\ndelimit a short arc $\\omega$ of $C_{ab}$, which lies outside $B_r(q)$,\nso points on the arc are (equally) closer to $a$ and $b$ than to\n$q$.\n\n\nHence, the projection of the arc $\\omega$ onto $h$ is a small\ninterval $ww'$, which lies on the bisector of $ab$ in the\ndirection that gets away from $q$; that is, it lies on the Voronoi\nedge of $ab$ in the diagram ${\\rm Vor}(\\{a,b,q\\})$ within $h$. See Figure~\\ref{figure:lemma_setup}(c).\nMoreover, $c_{ab}$ also lies on the bisector, but it has to lie on the\nother side of $w$, or else the smaller arc $\\omega$ would have to lie\ninside $B_r(q)$. That is, $c_{ab}$ has to be closer to $q$ than to $a$\nand $b$. Since $\\lambda$ separates $a$ and $b$ from $q$, it also separates\n $c_{ab}$ from $q$. Moreover, the preceding arguments are easily seen to imply that $wq$ crosses $ab$ (as in Figure~\\ref{figure:lemma_setup}(a)), which implies that $\\lambda$ also separates $q$ and $w$, so\n$w$ has to lie to the right of $\\lambda$. Since $z$ and $z'$ lie on\ntwo sides of $w$ on the line $\\ell$, at least one of them has to lie on\nthe same side of $\\lambda$ as $w$ (i.e., to the right of\n$\\lambda$). This completes the proof.\n\\end{proof}\n\nLet $a,b \\in P_R$ and consider those arcs of $C_{ab}$ which lie outside $K(P_L)$ but their endpoints lie on $\\sigma_L$. Clearly, all these arcs are pairwise disjoint. At most one such arc can be larger than a semicircle. Let $\\omega$ be an arc of this kind which is smaller than a semicircle, and let $q \\in P_L$ be such that one endpoint of $\\omega$ lies on ${\\partial} B_r(q)$. Then $\\omega' = C_{ab} \\setminus B_r(q)$ is contained in $\\omega$ and therefore is also smaller than a semicircle. By Lemma~\\ref{lemma:quadratic_K_P_L}, exactly one endpoint of $\\omega'$ lies to the right of $\\lambda$ (the other endpoint lies on $\\sigma_L$).\nNote that $C_{ab}$ cannot have more than two such short arcs lying outside $K(P_L)$, since, due to the convexity of $C_{ab}$, only two arcs of $C_{ab}$ can have their two endpoints lying on opposite sides of $\\lambda$. Hence the number of arcs of $C_{ab}$ under consideration is at most 3, implying that $\\gamma_a$ and $\\gamma_b$ intersect at most three times, and thus the complexity of $M$ is $O(n^2)$, as asserted.\n\n\\smallskip\n\\noindent{\\bf Constructing and searching $M$.} The next step of the\nalgorithm is to compute $M$. We have already constructed\n${\\partial}{K(P_L)}$, in $O(n \\log n)$ time, and, in additional linear\ntime, we can compute its portion $\\sigma_L$ to the left of $\\lambda$\n(we omit the straightforward details). We compute the intersection\ncurve $\\gamma_p$ of $B_r(p)$ and $\\sigma_L$, for each $p \\in P_R$,\nin $O(n \\log n)$ time, by computing the intersection $K(P_L \\cup\n\\{p\\})$, and obtaining the curve which bounds the portion of the\nunique face of ${\\partial}{K(P_L \\cup \\{p\\})}$ within $\\sigma_L$. If\nnecessary, we also split $\\gamma_p$ into portions, such that each\nportion is contained in a different face of $\\sigma_L$. The total\ncost of computing all curves $\\{\\gamma_p \\mid p \\in P_R\\}$, and\nspreading them along the faces of $\\sigma_L$, is $O(n^2 \\log n)$.\nThen, for each face $f$ of $\\sigma_L$, we consider the portions of\nall the arcs $\\gamma_p$, for $p \\in P_R$, within $f$, and compute\ntheir arrangement (which is the portion of $M$ which lies in $f$).\nTo this end, we use standard line-sweeping~\\cite{book}, to report\nall the intersections of $n$ curves in the plane in $O((n+k) \\log\nn)$ time, where $k = k_f$ is the complexity of the resulting\narrangement on $f$. Hence, the total cost of computing the portion\nof $M$ on all the faces of $\\sigma_L$ is $\\sum_{f \\in \\sigma_L}\nO((n+k_f) \\log n) = O(n^2 \\log n) + O(\\log n) \\cdot \\sum_{f \\in\n\\sigma_L} k_f = O(n^2 \\log n)$, since the complexity of $M$ is\n$O(n^2)$.\n\nWe next perform a traversal of the cells of $M$ in a manner similar to the one used in Section~\\ref{sec:cubic_alg}, via a tour, which proceeds from each visited cell to an adjacent one. For each cell $\\tau$ that we visit, we place\nthe center $c_1$ of $B_1$ in $\\tau$, and maintain dynamically the subset\n$P_\\tau^+$ of points of $P$ not covered by $B_1$. (Here, unlike the algorithm of Section~\\ref{sec:cubic_alg}, the complementary set  $P_\\tau^-$ is automatically covered by $B_1$ and there is no need to test it.) As before,\nwhen we move\nfrom one cell $\\tau$ to an adjacent cell $\\tau_1$, $P_{\\tau_1}^+$ gains\none point or loses one point. This implies that this tour generates only $O(n^2)$ connected life-spans of the points of $P$, where a life-span of a point $p$ is a maximal connected interval of the tour, in which $p$ belongs to $P_\\tau^+$. We can thus use a segment tree $T_M$ to store these life-spans, as before. Each\nleaf $u$ of $T_M$ represents a cell $\\tau$ of $M$, and the balls not containing $\\tau$ are those with life-spans that are stored at the nodes on the path from the root to $u$. Since $M$ has a quadratic number of\ncells, $T_M$ has a total\nof $O(n^2)$ leaves. Arguing exactly as in Section~\\ref{subsec:decision_procedure}, we can compute $T_M$ in overall $O(n^2 \\log^2 n)$ time, and the total storage used by $T_M$ is $O(n^2 \\log n)$.\n\nAs in Section~\\ref{subsec:decision_procedure}, we next test, for\neach leaf $u$ of $T_M$, whether the spherical polytopes along the\npath from the root to $u$ have non-empty intersection. We do this\nusing the parametric search technique described in\nProposition~\\ref{prop:intersection_test}, which takes $O(\\log^5 n)$\ntime for each path, for a total of $O(n^2 \\log^5 n)$. More\nprecisely, as above, we also need to distinguish between $r = r^*$\nand $r > r^*$. We therefore stop only when both the intersection\nalong the path and the cell of $\\sigma_L$ corresponding to $u$ are\nnon-degenerate, and then report that $r^* < r$. Otherwise, we\ncontinue running the above procedure over all paths of $T_M$, and\nrepeat it for each of the $O(1\/\\beta^3)$ combinations of an\norientation $v$ and a separating plane $\\lambda$. If we find at\nleast one (degenerate\\footnote{Note that $\\bigcap\\{B_r(p) \\mid p \\in\nP_\\tau^-\\}$ is non-degenerate if $\\tau$ is a 2-face or an edge. If\n$\\tau$ is a vertex we test for degeneracy as in the procedure in\nSection~\\ref{subsec:decision_procedure}. Determining whether\n$\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^+\\}$ is degenerate is also\nperformed using that procedure.}) solution, we report that $r^* =\nr$, and otherwise conclude that $r^*\n> r$. Hence, the cost of handling Case~2, and thus also the overall\ncost of the decision procedure, is $O((1\/\\beta^3) n^2 \\log^5 n)$.\n\n\\subsection{Solving the optimization problem}\n\\label{sec:separated_centers_optimization}\n\nWe now combine the decision procedure $\\Gamma$ described in Section~\\ref{subsec:improved_decision_procedure} with the randomized optimization\ntechnique of Chan~\\cite{TCG} (as briefly described in Section~\\ref{subsec:optimization_problem}),\nto obtain a solution for the optimization problem.\n\nThe decision procedure $\\Gamma$, on a specified radius $r$, relies on an apriori knowledge of a lower bound $\\beta$ for the separation ratio\n$|c_1c_2|\/r$. To supply such a $\\beta$, let $r_0$ denote the radius of the smallest\nenclosing ball of $P$, and observe that if there exist two balls $B_1, B_2$ of radius $r$ covering $P$ then the smallest ball $B^*$ enclosing $B_1 \\cup B_2$ must be at least as large as the smallest enclosing ball of $P$, so its radius must be at least $r_0$. Since this radius is\n$(1+\\beta\/2)r$ (see Figure~\\ref{figure:new_case2_SEB}), we have\n$(1+ \\beta\/2)r \\geq r_0$ or $\\beta \\geq 2(r_0\/r -1)$.\nIt follows that the running time of the decision procedure $\\Gamma$ is\n$$O\\left(\\frac{1}{\\beta^3} n^2 \\log^5 n\\right) =\nO\\left(\\frac{1}{\\left(1-r\/r_0 \\right)^3} n^2 \\log^5 n\\right).$$\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{B_s2.pstex_t}\n\\caption{\\small \\sf The smallest enclosing ball $B^*$ of $B_1 \\cup\nB_2$.} \\label{figure:new_case2_SEB}\n\\end{center}\n\\end{figure}\n\nChan's technique starts with a very big $r$ (for all practical\npurposes we can start with $r = r_0$) and shrinks it as it iterates\nover the subproblems. Therefore, running Chan's technique in a\nstraightforward manner, starting with $r = r_0$, will make it\npotentially very inefficient, because the initial executions of\n$\\Gamma$, when $r$ is still close to $r_0$, may be too expensive due\nto the large constant of proportionality (not to mention the run at\n$r_0$ itself, which the algorithm cannot handle at all). We need to\nfine-tune Chan's technique, to ensure that we do not consider values\nof $r$ which are too close to $r_0$. To do so, we consider the\ninterval $(0, r_0)$ which contains $r^*$, and run an ``exponential\nsearch'' through it, calling $\\Gamma$ with the values $r_i = r_0\n\\left(1-1\/2^i\\right)$, for $i = 1,2, \\ldots$, in order, until the\nfirst time we reach a value $r' = r_i \\geq r^*$. Note that $1-\nr'\/r_0 = 1\/2^i$ and $1\/2^i < 1- r^*\/r_0 < 1\/2^{i-1}$, so our lower\nbound estimates for the separation ratio $\\beta$ at $r'$ and at\n$r^*$ differ by at most a factor of $2$, so the cost of running\n$\\Gamma$ at $r'$ is asymptotically the same as at $r^*$. Moreover,\nsince the (constants of proportionality in the) running time bounds\non the executions of $\\Gamma$ at $r_1,\\ldots,r_i$ form a geometric\nsequence, the overall cost of the exponential search is also\nasymptotically the same as the cost of running $\\Gamma$ at $r^*$. We\nthen run Chan's technique, with $r'$ as the initial minimum radius\nobtained so far. Hence, from now on, each call to $\\Gamma$ made by\nChan's technique will cost asymptotically no more than the cost of\ncalling $\\Gamma$ with $r'$ (which is asymptotically the same as\ncalling $\\Gamma$ with $r^*$).\n\n\\paragraph{Combining Chan's technique with the decision procedure $\\Gamma$.}\nTo apply Chan's technique with our decision procedure, we use the\nsame cutting-based decomposition as in\nSection~\\ref{subsec:optimization_problem}. That is, we replace each\npoint $p \\in P$ by its dual plane $p^* \\in P^*$, and construct a\n$(1\/\\varrho)$-cutting of ${\\cal A}(P^*)$, for some sufficiently large\nconstant parameter $\\varrho>0$. We then apply Chan's technique to\nthe resulting subproblems (where each subproblem corresponds to a\nsimplex $\\Delta_i$ of the cutting), using the improved decision\nprocedure $\\Gamma$ on each of them, and recursing into some of them,\nas required by the technique. As in Section~\\ref{sec:cubic_alg}, the\nrecursion and the application of the decision procedure are not\n``pure'', because they need to consider also those planes that miss\nthe current simplex. (Note that in the problem decomposition we use,\nfor simplicity, the full 3-dimensional arrangement ${\\cal A}(P^*)$, of\ncubic size. This, however, does not affect the asymptotic running\ntime, because we have only a constant number of subproblems, and\nChan's technique recurses into only an expected logarithmic number\nof them.) Given a radius $r$, we compute the lower bound $\\beta =\n2\\left(\\frac{r_0}{r} -1\\right)$ for the separation ratio\n$\\frac{|c_1c_2|}{r}$, where $c_1, c_2$ are the centers of the two\ncovering balls, as above. Consider the application of $\\Gamma$ to a\nsubproblem represented by a simplex $\\Delta_i$ of the cutting. The\npresence of ``global'' points (those dual to planes passing above or\nbelow $\\Delta_i$) forces us, as in\nSection~\\ref{subsec:optimization_problem}, to modify the ``pure''\nversion of $\\Gamma$ described above. We use the same notations as in\nSection~\\ref{sec:cubic_alg}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{h_lambda_s2.pstex_t}\n\n\\caption{\\small \\sf $h_\\lambda$ does not contain any point of\n$P_{\\Delta_i}^+$.} \\label{figure:h_lambda}\n\\end{center}\n\\end{figure}\n\nWe again rotate the coordinate axes, in $O\\left(1\/\\beta^2\\right)$\nways (in the same manner as in the ``pure'' decision procedure), and\ndraw $O(1\/\\beta)$ $yz$-parallel planes, such that, at the correct\norientation, one of these planes, $\\lambda$, separates $c_1$ from\n$v_1$ (if there is a solution for $r$). As in the pure case, we may\nassume that the $x$-span of $P$ is at most $5r$; a larger span is\nhandled earlier. We assume, without loss of generality, that\n$P_{\\Delta_i}^- \\subseteq B_1$, and that $P_{\\Delta_i}^+ \\subseteq\nB_2$. Recall also that the points in the left halfspace $h_\\lambda$\nbounded by $\\lambda$ are all contained in $B_1$. Moreover, the plane\n$\\pi$ containing the intersection circle $C_{12}$ is dual to a point\n$\\pi^*$, which has to separate $(P^*)_{\\Delta_i}^+$ from\n$(P^*)_{\\Delta_i}^-$. Hence, all the points of $P_{\\Delta_i}^+$ have\nto lie on the other side of $\\pi$, and in $B_2$, which is easily\nseen to imply that none of them can lie in $h_\\lambda$. See\nFigure~\\ref{figure:h_lambda}. We thus verify that $P_{\\Delta_i}^+\n\\cap h_\\lambda = \\emptyset$, aborting otherwise the guess of\n$\\lambda$. (Note that, in contrast, points of $P_{\\Delta_i}^-$ can\nalso lie to the right of $\\lambda$.)\n\nWe now have a subset $P_L \\subseteq P_{\\Delta_i}^0$ of $O(m)$ points\nto the left of $\\lambda$, which are assumed, together with the\npoints of $P_{\\Delta_i}^-$, to be contained in $B_1$. Note however\nthat, for Lemma~\\ref{lemma:quadratic_K_P_L} to hold, we have to\ndefine $\\sigma_L$ only in terms of the points to the left of\n$\\lambda$. Therefore, we compute the surface $\\sigma_L' = {\\partial}{K(P_L\n\\cup (P_{\\Delta_i}^- \\cap h_\\lambda)) \\cap h_\\lambda}$ and search on\nit for a placement of the center $c_1$ of $B_1$. However, since the\nremaining points of $P_{\\Delta_i}^-$ are also assumed to belong to\n$B_1$, we need to consider only the portion of $\\sigma_L'$ inside\n$\\bigcap\\{B_r(p) \\mid p \\in P_{\\Delta_i}^- \\setminus h_\\lambda\\}$.\nLet $\\sigma_L''$ denote this portion. It is easy to compute\n$\\sigma_L''$ in $O(n \\log n)$ time. It is easily checked that $c_1$\nmust lie on $\\sigma_L''$ (if there is a solution for the current\nsituation). So far, the cost of the decision procedure also depends\n(cheaply --- see below) on the initial input size $n$, but the\nsaving in this setup comes from the fact that it suffices to\nintersect the $O(m)$ spheres ${\\partial}{B_r(p)}$, for $p \\in\nP_{\\Delta_i}^0 \\setminus h_\\lambda$, with $\\sigma_L''$ to obtain the\nmap $M$, since only the points of $P_{\\Delta_i}^0$ are\n``undecided''. (The points of $P_{\\Delta_i}^+$ are always placed in\n$B_2$ as already discussed.)\n\nNote that $\\sigma_L''$ need not to be connected, so it may seem\nimpossible to visit all the cells of $M$ in a single connected tour.\nNevertheless, we will be able to do it, in a manner detailed below.\nWe thus build a segment tree $T_M$ to maintain the subset $P'(c_1)$\nof points of $P$ not covered by $B_1$. We build and query $T_M$ as\nis done in Section~\\ref{subsec:decision_procedure}, except for the\nfollowing modifications. First, note that the points of\n$P_{\\Delta_i}^+$ are assumed to be contained in $B_2$. Thus, the\npoints of $P_{\\Delta_i}^+$, that in the decision procedure were\nconsidered in building $M$, do not need to be considered as part of\n$M$ now, rather it is enough to build the spherical polytope\n$\\bigcap \\{B_r(p) \\mid p \\in P_{\\Delta_i}^+ \\}$ and place it at the\nroot of $T_M$. Second, we claim that $M$ is of complexity $O(m n)$.\nTo see this, let ${\\cal C}^0$ denote the set of curves $\\{{\\partial}{B_r(p)} \\cap\n\\sigma_L'' \\mid p \\in P_{\\Delta_i}^0\\}$. Each pair of curves of\n${\\cal C}^0$ can intersect each other in only a constant number of points,\nas proved in Section~\\ref{subsec:improved_decision_procedure}.\nHence, the complexity of the arrangement of the $O(m)$ curves in\n${\\cal C}^0$, formed on $\\sigma_L''$, is $O(m^2)$. However, $\\sigma_L''$\nitself is of complexity $O(n)$, and each edge of $\\sigma_L''$ may\nintersect the curves of ${\\cal C}^0$ at $O(m)$ points. Hence, the\ncomplexity of the map $M$ is $O(m n)$, but the number of its\nvertices that lie in the interior of the faces of $M$ is only\n$O(m^2)$.\n\nTo overcome the possible disconnectedness of $\\sigma_L''$, we\nproceed as follows. We consider the (connected) network of the\n$O(n)$ edges of $\\sigma_L'$, and intersect each of these edges with\nthe $m$ balls $B_r(p)$, for $p \\in P_{\\Delta_i}^0$. We construct a\ntour of this network, which visits $O(mn)$ arcs along the edges of\n$\\sigma_L'$, and append to this ``master tour'' separate tours of\neach face of $\\sigma_L''$. We get in this way a single grand tour of\nthe cells of $M$ (which also traverses some superfluous arcs of\n$\\sigma_L' \\setminus \\sigma_L''$), of length $O(mn)$, which has the\nincremental property that we need: Moving from any cell or arc of\nthe tour to a neighbor cell or arc incurs an insertion or a deletion\nof a single point into\/from $P'(c_1)$.\n\n\n\\paragraph{Running time.}\nFor each cell of $M$ we run the procedure described in\nProposition~\\ref{prop:intersection_test} for determining whether the\nintersection of the corresponding spherical polytopes is nonempty\n(and whether it has nonempty interior). Therefore, solving each\nsubproblem requires $O(m n \\log^5 n)$ time. The $O(m n \\log n)$ time\nrequired to build $M$, and the $O(n \\log n)$ time required to\nconstruct the intersection of the balls in $\\{B_r(p) \\mid p \\in\nP_{\\Delta_i}^+\\}$, are all subsumed in that cost. Repeating this for\neach of the $O(1\/\\beta^3)$ guesses of an orientation and a\nseparating plane, results in $O\\left((1\/\\beta^3) mn \\log^5 n\\right)$\nrnning time. When the recursion bottoms out, we handle it the same\nway as in Section~\\ref{subsec:optimization_problem}.\n\nArguing similarly to the less efficient solution, we obtain the\nfollowing recurrence for the maximum expected cost $T(m, n)$ of\nsolving a recursive subproblem involving $m$ ``local'' points, where\n$n$ is the number of initial input points in $P$.\n\n\\begin{equation}\n\\label{eqn:T(m,n)_2} T(m,n) \\leq \\left\\{ \\begin{array}{ll}\n\\ln(c\\varrho^3)T(m\/\\varrho,n) + O\\left((1\/\\beta^3) m n \\log^5 n\\right),   & \\mbox{for $m \\geq \\varrho$,}\\\\\nO(n),                                        & \\mbox{for $m <\n\\varrho$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $c$ is an appropriate absolute constant (as in\nSection~\\ref{subsec:optimization_problem}), $\\varrho$ is the\nparameter of the cutting, chosen to be a sufficiently large constant\n(to satisfy (\\ref{equation:lnr}), as above, with $\\gamma = 2$), and\n$\\beta = 2\\left(r_0\/r'-1\\right)$, where $r'$ is the value of $r$ at\nwhich the initial exponential search is terminated.\n\nIt can be shown rather easily (and we omit the details, as we did in\nthe preceding section), that the recurrence~(\\ref{eqn:T(m,n)_2})\nyields the overall bound $O\\left((1\/\\beta^3) n^2 \\log^5 n\\right)$ on\nthe expected cost of the initial problem; i.e.,\n$$T(n,n) = O\\left((1\/\\beta^3) n^2 \\log^5 n\\right).$$ We thus finally obtain\nour main result:\n\\begin{theorem}\nLet $P$ be a set of $n$ points in $\\mathbb{R}^3$. A 2-center for $P$\ncan be computed in $O((n^2 \\log^5 n)\/(1-r^*\/r_0)^3 )$ randomized\nexpected time, where $r^*$ is the radius of the balls of the\n2-center for $P$ and $r_0$ is the radius of the smallest enclosing\nball of $P$.\n\\end{theorem}\n\n\\section{Efficient Emptiness Detection of Intersection of Spherical\nPolytopes} \\label{sec:spherical_polytopes} In this section we\ndescribe an efficient procedure for testing emptiness (and\nnon-degeneracy) of the intersection of spherical polytopes, as\nprescribed in Proposition~\\ref{prop:intersection_test}. Let ${\\cal S}$ be\na collection of spherical polytopes, each defined as the\nintersection of at most $n$ balls of a fixed radius $r$. Fix a\nspherical polytope $S \\in {\\cal S}$. To simplify the forthcoming analysis,\nwe assume that the centers of the balls involved in the polytopes of\n${\\cal S}$ are in general position, meaning that no five of them are\nco-spherical, and that there exists at most one quadruple of centers\nlying on a common sphere of radius $r$. As is well known, each ball\n$b$ participating in the intersection $S$ contributes at most one\n(connected) face to ${\\partial}{S}$ (see~\\cite{ER}). The vertices and edges\nof $S$ are the intersections of two or three bounding spheres,\nrespectively (at most one vertex might be incident to four spheres).\nHence ${\\partial}{S}$ is a planar (or, rather, spherical) map with at most\n$|S|$ faces, which implies that the complexity of ${\\partial}{S}$ is\n$O(|S|)$.\n\nWe preprocess $S$ into a point-location structure. We first\npartition ${\\partial}{S}$ into its upper portion ${\\partial}{S}^+$ and lower\nportion ${\\partial}{S}^-$. We project vertically each of ${\\partial}{S}^+$ and\n${\\partial}{S}^-$ onto the $xy$-plane and obtain two respective planar maps\n$M^+$ and $M^-$ (see Figure~\\ref{figure:projection}). For each face\n$\\zeta$ of each map we store the ball $b$ that created it; that is,\n$\\zeta$ is the projection of the (unique) face of ${\\partial}{S}$ that lies\non ${\\partial}{b}$. The $xy$-projection $S^*$ of $S$ is equal to both\nprojections of ${\\partial}{S^+}$, ${\\partial}{S^-}$, and is bounded by a convex\ncurve $E^*$ that is the concatenation of the $xy$-projections of\ncertain edges of $S$ and of portions of horizontal equators of some\nof its balls.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{projection_log8.pstex_t}\n\\caption{\\small \\sf Projecting ${\\partial}{S_i}^-$ vertically onto the\n$xy$-plane (left), and the point location structure for the\nresulting map $M_i^-$ (right).} \\label{figure:projection}\n\\end{center}\n\\end{figure}\n\nWe apply the standard point-location algorithm of Sarnak and\nTarjan~\\cite{ST} to each of the maps $M^+, M^-$. That is, we divide\neach planar map into slabs by parallel lines (to the $y$-axis)\nthrough each of the endpoints (and locally $x$-extremal points) of\nthe arcs obtained by projecting the edges of ${\\partial}{S}$, including the\nnew equatorial arcs. Using the persistent search structure\nof~\\cite{ST}, the total storage is linear in $|S|$ and the\npreprocessing cost is $O(|S| \\log |S|)$, where $|S|$ is the number\nof balls forming $S$. To locate a point $q_0$ in $M^+$ (or in\n$M^-$), we first find the slab in the $x$-structure that contains\n$q_0$, and then find the two curves between which $q_0$ lies in the\n$y$-structure.\\footnote{\\small All these standard details are\npresented to make more precise the infrastructure used by the\nhigher-dimensional routines $\\Pi_1$ and $\\Pi_2$.}\n\nTo determine whether $q \\in S^*$, we locate the face $\\zeta^+$\n(resp., $\\zeta^-$) of the map $M^+$ (resp., $M^-$) that contains\n$q$, as just described. Each of these faces can be a 2-face, an edge\nor a vertex. We therefore retrieve a set ${\\cal B}^+$ (resp., ${\\cal B}^-$) of\nthe one, two, or three or four balls associated (respectively) with\nthe 2-face, edge or vertex containing $q$. (We omit here the easy\nconstruction of witness balls when the faces $\\zeta^+$ and $\\zeta^-$\nare not associated with any ball, that is, $q \\notin S^*$.)\n\nLet ${\\cal B}$ denote the set ${\\cal B}^+ \\cup {\\cal B}^-$. We observe that $q \\in\nS^*$ if and only if the $z$-vertical line $\\lambda_q$ through $q$\nintersects $S$. Moreover, we have, by construction, $\\lambda_q \\cap\nS = \\lambda_q \\cap (\\bigcap {\\cal B})$. Hence $q \\in S^*$ if and only if\n$s \\coloneqq \\lambda_q \\cap (\\bigcap {\\cal B}) \\neq \\emptyset$. Clearly,\nif we put $N = \\sum_{S \\in {\\cal S}} |S|$, then the preprocessing stage\ntakes a total of $O(N \\log n)$ time and requires $O(N)$ storage.\n\nNext, let $S_1,\\ldots,S_t$ be $t \\leq \\log n$ spherical polytopes of\n${\\cal S}$, for which we want to determine whether $K = \\bigcap_{i=1}^t\nS_i$ is nonempty (and, if so, whether it has nonempty interior). We\nsolve this problem by employing a technique similar to the\nmulti-dimensional parametric searching technique of\nMatou\\v{s}ek~\\cite{JM} (see also~\\cite{AES, TCA, CMS, NPT}). We\nsolve in succession the following three subproblems, $\\Pi_0(q)$,\nwhere $q$ is a point in the $xy$-plane, $\\Pi_1(l)$, where $l$ is a\n$y$-parallel line in the $xy$-plane, and $\\Pi_2$, over the entire\n$xy$-plane. In the latter problem we wish to to determine whether\nthe $xy$-projection $K^*$ of $K$ is nonempty. During the execution\nof the algorithm for solving $\\Pi_2$, we call recursively the\nalgorithm for solving $\\Pi_1(l)$, for certain $y$-parallel lines $l\n\\subset \\mathbb{R}^2$, and we wish to determine whether $K^*$ meets\n$l$. If so, then $\\Pi_2$ is solved directly (with a positive\nanswer). Otherwise, we wish to determine which side of $l$, within\n$\\mathbb{R}^2$, can meet $K^*$ (since $K^*$ is convex, there can\nexist at most one such side). The recursion bottoms out at certain\npoints $q \\in l$, on which we run $\\Pi_0(q)$ to determine whether\n$K^*$ contains $q$. If so, then $\\Pi_1(l)$ is solved directly (with\na positive answer). Otherwise, we determine which side of $q$,\nwithin $l$, can meet $K^*$, and continue the search accordingly.\n\nOur solutions to the subproblems $\\Pi_k$, $0\\leq k\\leq 2$, are based\non generic simulations of the standard point-location machinery of\nSarnak and Tarjan~\\cite{ST} mentioned above. In each of the\nsubproblems, if we find a point in $f \\cap K^*$, for the respective\npoint, line, or the entire $xy$-plane $f$, we know that $K \\neq\n\\emptyset$ and stop right away. If $f \\cap K^* = \\emptyset$, we want\nto ``prove'' it, by returning a small set of \\emph{witness balls}\n$b_1,\\ldots,b_y$, where, for each $j$, $b_j$ is one of the balls\nthat participates in some spherical polytope $S_i$ (so $b_j\n\\supseteq S_i$), so that their intersection $K_0 = \\bigcap_{j=1}^y\nb_j$ satisfies $f \\cap K_0^* = \\emptyset$ (where, as above, $K_0^*$\nis the $xy$-projection of $K_0$). If $K_0 = \\emptyset$ then $K =\n\\emptyset$ too and we stop. Otherwise (when $f$ is a line or a\npoint), $K_0$ determines the side of $f$ (within $\\mathbb{R}^2$ if\n$f$ is a line, or within the containing line $l$ if $f$ is a point)\nthat might meet $K^*$; the opposite side is asserted at this point\nto be disjoint from $K^*$. We use this information to perform binary\nsearch (or, more precisely, parametric search) to locate $K^*$\nwithin the flat, from which we have recursed into $f$.  The\nexecution of the algorithm for solving $\\Pi_2$ will therefore either\nfind a point in $K$ or determine that $K = \\emptyset$, because it\nhas collected a small (as we will show, polylogarithmic) number of\nwitness balls, whose intersection, which has to contain $K$, is\nfound to be empty.\n\n\\paragraph{Solving $\\Pi _0(q)$ for a point $q$.}\n\\label{subsec:Pi_0}\nHere we have a point $q \\in \\mathbb{R}^2$ and we wish to determine\nwhether $q \\in K^*$. To do so, we locate $q$ in each of the maps\n$M_i^+$ (the $xy$-projection of ${\\partial} S_i^+$) and $M_i^-$ (the\n$xy$-projection of ${\\partial} S_i^-$), for each $i=1,\\ldots,t$. If $q$\nlies outside the projection of at least one polytope $S_i$ then $q\n\\notin K^*$, and we return the witness balls that prove that $q\n\\notin S_i^*$. Otherwise, as explained above, each point location\nreturns a set ${\\cal B}_i$ of $O(1)$ witness balls for $S_i$. We compute\nthe $t$ line segments $s_i = \\lambda_q \\cap (\\bigcap{\\cal B}_i)$, for each\n$i = 1,\\ldots,t$, where $\\lambda_q$ is, as above, the $z$-vertical\nline through $q$. We then have $K_0 \\coloneqq \\lambda_q \\cap K =\n\\bigcap_{i=1}^t s_i$, so it suffices to compute this intersection\n(in $O(t)$ time) and test whether it is nonempty. If $K_0$ is\nnonempty, then we have found a point $q'$ in $K$. Otherwise, we\nreturn the set ${\\cal B}_0 = \\bigcup \\{{\\cal B}_i \\mid 1 \\leq i \\leq t\\}$ of up\nto $5 \\log n$ balls as witness balls for the higher-dimensional step\n(involving the $y$-parallel line containing $q$).\n\nThe time complexity for solving $\\Pi_0(q)$ is $O(\\log^2 n)$, since\nit takes $O(\\log n)$ time to compute, for each of the $O(\\log n)$\nspherical polytopes $S_i$, the intersection $\\lambda_q \\cap S_i$.\n\n\\paragraph{\\bf Solving $\\Pi _1(l)$ for a line $l$.}\n\\label{subsec:Pi_1}\nHere we have a $y$-parallel line $l \\subset \\mathbb{R}^2$ and we\nwish to determine whether $K^*$ meets $l$. We first locate $l$ in\neach of the planar maps $M_i^+$ and $M_i^-$ of each $S_i$, and find\nthe slabs $\\psi_i^+$ and $\\psi_i^-$, which contain $l$ (in some\ncases $l$ is the common bounding line of two adjacent slabs\n$\\psi_i'$ and $\\psi_i''$ of $M_i^+$ or of $M_i^-$, so we retrieve\nboth slabs). We then run a binary search through the $y$-structure\nof each of the obtained slabs to find a point in $K^* \\cap l$, if\none exist. In each step of the search, within some fixed slab\n$\\psi_0$, we consider an arc $\\gamma$ of the $y$-structure, and\ndetermine whether $K^*$ meets $l$ above or below $\\gamma$ (within\n$\\mathbb{R}^2$), assuming $K^* \\cap l \\neq \\emptyset$. To this end,\nwe find the intersection point $q_0 = l \\cap \\gamma$, and run the\nalgorithm for solving $\\Pi_0(q_0)$ (see Figure~\\ref{figure:pi_1}).\nIf $q_0 \\in K^*$, then we have found a point $q'$ in $K$, and we\nimmediately stop. Otherwise, we have a set ${\\cal B}_0$ of up to $5 \\log\nn$ balls returned by the algorithm for solving $\\Pi_0(q_0)$. We test\nwhether the $xy$-projection $K_0^*$ of $\\bigcap {\\cal B}_0$ intersects\n$l$. If $K_0^* \\cap l = \\emptyset$, then (due to the convexity of\n$K$) we know which side of $l$ (within $\\mathbb{R}^2$) meets $K^*$,\nand we return ${\\cal B}_0$ as a set of witness balls for the\nhigher-dimensional (planar) step. Otherwise (again due to the\nconvexity of $K$), we know which side of $\\gamma$, within $l$, meets\n$K^*$, and we continue the search through the $y$-structure of\n$\\psi_0$ on this side. We continue the search in this manner, until,\nfor each $S_i$, we obtain an interval $\\xi_i$ of $l$ between two\nconsecutive arcs of the $y$-structure of $\\psi_0$, which meets $K^*$\n(assuming $K^* \\cap l \\neq \\emptyset$). Let $\\Xi$ denote the\ncollection of all these intervals. Clearly, $K^* \\cap l \\subseteq\n\\bigcap \\Xi$. We find the lowest endpoint $E^-$ among the top\nendpoints of the intervals in $\\Xi$ and the highest endpoint $E^+$\namong the bottom endpoints of the intervals in $\\Xi$, and test\nwhether $E^-$ is above $E^+$. If so, we consider the set ${\\cal B}_1$ of\nup to $10 \\log n$ witness balls returned by the algorithms for\nsolving $\\Pi_0(E^-)$ and $\\Pi_0(E^+)$. If the $xy$-projection\n$K_1^*$ of $\\bigcap {\\cal B}_1$ intersects $l$, then $K^*$ meets $l$ and\nwe stop immediately, for we have found that $K$ is nonempty.\nOtherwise, we know which side of $l$ (within $\\mathbb{R}^2$) can\nmeet $K^*$, and we return ${\\cal B}_1$ as a set of witness balls for the\nhigher (planar) recursive level. If $E^-$ is not above $E^+$, then\n$K^* \\cap l = \\emptyset$ and we return ${\\cal B}_1$ as a set of witness\nballs for the higher (planar) recursive level as well.\\footnote{With\nsome care, the number of witness balls can be significantly reduced.\nWe do not go into this improvement, because handling the witness\nballs is an inexpensive step, whose cost is subsumed by the cost of\nthe other steps of the algorithm.}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{pi_1_log8.pstex_t}\n\\caption{\\small \\sf The line $l$ on which we run $\\Pi_1(l)$. The\npoint $q_0$ on which we run $\\Pi_0(q_0)$ is the intersection point\nof $l$ with some arc $\\gamma$.} \\label{figure:pi_1}\n\\end{center}\n\\end{figure}\n\nA naive implementation of the above procedure takes $O(\\log^4 n)$\ntime, since for each of the $O(\\log n)$ spherical polytopes $S_i$ we\nrun a binary search through the $y$-structure of at most two slabs\nof each of the maps $M_i^+$ and $M_i^-$, and in each of the binary\nsearch steps, we run the algorithm for solving $\\Pi_0(q_0)$ for some\npoint $q_0$. The other substeps take less time. However, we can\nimprove the running time by implementing it in a parallel manner and\nsimulating the parallel version sequentially with a smaller number\nof calls to $\\Pi_0$.\n\nWe only parallelize the binary searches through the $y$-structure of\neach $M_i^+$ and $M_i^-$, since the other substeps take less time.\nTo this end, we use $O(\\log n)$ processors, one for each of the\nplanar maps $M_i^+$ and $M_i^-$, and we run in parallel the binary\nsearch through the $y$-structure of each planar map using $O(\\log\nn)$ parallel steps. In each parallel step we need to ``compare''\n$O(\\log n)$ arcs with $K^*$ (one arc for each of the planar maps\n$M_i^+$, $M_i^-$). We therefore intersect each such arc with $l$ and\nobtain a set $Q$ of $O(\\log n)$ intersection points. We then run a\nbinary search through the points of $Q$ (to locate $K^*$) using\n$\\Pi_0$. This determines the outcome of the comparisons of each of\nthe arcs with $K^*$, and the parallel execution can proceed to the\nnext step. Applying this approach to each of the $O(\\log n)$\nparallel steps results in an $O(\\log^3 n \\log \\log n)$-time\nalgorithm for solving $\\Pi_1(l)$. However, we can slightly improve\nthis bound further using a simple variant of Cole's\ntechnique~\\cite{RC}. More precisely, in each parallel step we have a\ncollection $Q$ of $O(\\log n)$ weighted points, one for each map,\nwhich we need to compare with $K^*$. We select the (weighted) median\npoint $q_0$ of $Q$ and run $\\Pi_0(q_0)$. This determines the\noutcomes of the comparisons between $K^*$ and each of the points in\n$Q$ which lie to the opposite side of $q_0$ to the side containing\n$K^*$. Points in $Q$ which lie in the same side of $q_0$ as $K^*$,\nin level $j$ of the parallel implementation, are given weight\n$1\/4^{j-1}$ and we try to resolve their comparison to $K^*$ in the\nnext step. An easy calculation (simpler than the one used by Cole)\nshows that this method adds only $O(\\log n)$ steps to the $O(\\log\nn)$ parallel steps of the searches, and now in each parallel step we\nperform only one call to $\\Pi_0$ (see~\\cite{RC} for more details).\nTherefore, the total running time of $\\Pi_1(l)$ is $O(\\log^3 n)$.\n\n\\paragraph{\\bf Solving $\\Pi _2$.}\n\\label{subsec:Pi_2}\nWe next consider the main problem $\\Pi _2$, where we want to\ndetermine whether $K^* \\neq \\emptyset$ (i.e., whether $K \\neq\n\\emptyset$). We use parametric searching, in which we run the point\nlocation algorithm that we used for solving $\\Pi_0$, in the\nfollowing generic manner.\n\nIn the first stage of the generic point location, we run a binary\nsearch through the slabs of each of the planar maps $M_i^+$ and\n$M_i^-$, for $i=1,\\ldots,t$. In each step of the search through any\nof the maps, we take a line $l_0$ delimiting two consecutive slabs\nof the map, and run the algorithm for solving $\\Pi _1(l_0)$, thereby\ndeciding on which side of $l_0$ to continue the search. At the end\nof this stage, unless we have already found a point in $K$ or\ndetermined that $K$ is empty, we obtain a single slab in each map\nthat contains $K^*$.  Let $\\psi$ denote the intersection of these\nslabs, which must therefore contain $K^*$ (unless $K$ is empty). The\ncost of this part of the procedure is $O(\\log^5 n)$.\n\nIn the next stage of the generic point location, we consider each\nmap $M_i^+$ or $M_i^-$ (for simplicity we refer to it just as $M_i$)\nseparately, and run a binary search through the $y$-structure of its\nslab $\\psi_i$ that contains $\\psi$. In each step of the search we\nconsider an arc $\\gamma$ of the $y$-structure, and determine which\nside of $\\gamma$ (within the slab $\\psi$), can meet $K^*$, assuming\nthat $\\psi \\cap K^* \\neq \\emptyset$; if $\\gamma \\cap K^* \\neq\n\\emptyset$ we will detect it and stop right away. Before describing\nin detail how to resolve each comparison with an arc $\\gamma$, we\nnote that this results in $O(\\log n)$ comparisons of arcs $\\gamma$\nto $K^*$ for each of the $O(\\log n)$ planar maps $M_i^+$ and\n$M_i^-$. However, we can reduce the number of comparisons to $O(\\log\nn)$ in total, by simulating (sequentially) a parallel implementation\nof this step, as follows. There are $O(\\log n)$ parallel steps, and\nin each step we execute a single step of the binary search in each\nof the maps $M_i^+, M_i^-$. In each parallel step we need to compare\n$K^*$ to a set $G$ of $O(\\log n)$ arcs, one of each of the planar\nmaps $M_i^+, M_i^-$. Consider the portion ${\\cal A}'(G)$ of the\narrangement ${\\cal A}(G)$ of the arcs in $G$ which lies in $\\psi$. Let\n$L(G)$ denote the set of $O(\\log^2 n)$ $y$-parallel lines which pass\nthrough the vertices of ${\\cal A}'(G)$. We run a binary search through the\nlines of $L(G)$, using calls to the algorithm for $\\Pi_1$ to guide\nthe search, to locate $K^*$ amid these lines, in a total of\n$O(\\log^3 n \\log \\log n)$ running time. This step (if it did not\nfind a line crossing $K^*$) may trim $\\psi$ to a narrower slab\n$\\psi'$ in which $K^*$ must lie if $K^* \\neq \\emptyset$. Put $G' =\n\\{\\gamma \\cap \\psi' \\mid \\gamma \\in G\\}$, and observe that the arcs\nof $G'$ are pairwise disjoint and form a sorted sequence in the\n$y$-direction. We then perform a binary search through the arcs in\n$G'$, using $O(\\log \\log n)$ comparisons to $K^*$. Each comparison\nis carried out in $O(\\log^4 n)$ time, in a manner detailed below.\nOnce the binary search is terminated, we can determine the outcomes\nof the comparisons between $K^*$ and each of the arcs in $G'$ and\nproceed to the next parallel step. Applying this approach to each of\nthe $O(\\log n)$ parallel steps results in an $O(\\log^5 n \\log \\log\nn)$-algorithm for solving $\\Pi_2$. We again use an appropriate\nvariant of Cole's technique to improve the running time by a $\\log\n\\log n$ factor, in a manner similar to the one described in the\nsolution of $\\Pi_1$.\n\nTo carry out a comparison between an arc $\\gamma \\in G'$ and $K^*$,\nwe act under the assumption that $\\gamma \\cap K^* \\neq \\emptyset$,\nand try to locate a point of $\\gamma \\cap K^*$ in each of the other\nmaps. Suppose, to simplify the description, that we managed to\nlocate the entire $\\gamma$ in a single face of each of the other\nmaps $M_j^+$, $M_j^-$. This yields a set ${\\cal B}$ of $O(t)$ balls, so\nthat a point $v \\in \\gamma$ lies in $K^*$ if and only if it lies in\nthe $xy$-projection $K_0^*$ of $\\bigcap {\\cal B}$. We then test whether\n$\\gamma$ intersects $K_0^*$. If so, we have found a point in $K$ and\nstop right away. Suppose then that $K_0^* \\cap \\gamma = \\emptyset$.\nIf $K_0^* \\cap \\psi' = \\emptyset$ then $K$ must be empty, because we\nalready know that $K^* \\subset \\psi'$. If $K_0^* \\cap \\psi' \\neq\n\\emptyset$, then we know on which side of $\\gamma$ to continue the\nbinary search in (the portion within $\\psi'$ of) $\\psi_i$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{pi_2_log8.pstex_t}\n\\caption{\\small \\sf Comparing $\\gamma \\cap K^*$ with $\\delta$. The\noutcome of $\\Pi_1(l_0)$ determines (a) the side of $\\delta$ in which\nthe search in $\\psi_j$ should continue, and (b) the portion of\n$\\gamma$ which can still meet $K^*$. The subslab $\\psi'$ is drawn\nshaded.} \\label{figure:pi_2}\n\\end{center}\n\\end{figure}\n\n\n\n\nIn general, though, $\\gamma$ might split between several cells of a\nmap $M_j$, where $M_j$ denotes, as above, one of the maps $M_j^+$ or\n$M_j^-$. This forces us to narrow the search to a subarc of\n$\\gamma$, in the following manner. We run a binary search through\nthe $y$-structure of the corresponding slab $\\psi_j$ of $M_j$, which\ncontains $\\psi'$, and repeat it for each of the maps $M_j$. In each\nstep of the search, we need to compare $\\gamma$ (or, more precisely,\nsome point in $\\gamma \\cap K^*$) with some arc $\\delta$ of $\\psi_j$,\nwhich we do as follows. If $\\gamma$ lies, within $\\psi'$, completely\non one side of $\\delta$, we continue the binary search in $\\psi_j$\non that side of $\\delta$. If $\\gamma$ intersects $\\delta$, we pick\nan intersection point $v$ of $\\gamma$ and $\\delta$, pass a\n$y$-parallel line $l_0 \\subset \\mathbb{R}^2$ through $v$, and run\nthe non-generic version of the algorithm to solve $\\Pi_1(l_0)$. (See\nFigure~\\ref{figure:pi_2}.) As before, if $l_0 \\cap K^* \\neq\n\\emptyset$ we detect this and stop. Otherwise, we know which of the\ntwo portions of $\\gamma$, delimited by $v$, can intersect $K^*$. We\nrepeat this step for each of the at most four intersection points of\n$\\gamma$ and $\\delta$ (observing that these are elliptic arcs), and\nobtain a connected portion $\\gamma'$ of $\\gamma$, delimited by two\nconsecutive intersection points, whose relative interior lies\ncompletely above or below $\\delta$, so that $\\gamma \\cap K^*$, if\nnonempty, lies in $\\gamma'$. This allows us to resolve the generic\ncomparison with $\\delta$, and continue the binary search through\n$\\psi_j$. (On the fly, each comparison with a line $l_0$ narrows\n$\\psi'$ still further.)\n\nTo make this procedure more efficient, we perform the binary\nsearches through the slabs $\\psi_j$ in parallel, as follows. As\nbefore, we run in parallel the binary searches through each of the\nslabs $\\psi_j$ using $O(\\log n)$ parallel steps. In each parallel\nstep we need to compare a set $D$ of $O(\\log n)$ arcs to $\\gamma$,\none arc $\\delta$ from each planar map $M_j$. We intersect each of\nthe arcs in $D$ with $\\gamma$ and obtain a set $Z$ of $O(\\log n)$\nintersection points. Let $L_Z$ denote the set of the $O(\\log n)$\n$y$-parallel lines which pass through the points of $Z$. We run a\nbinary search through the lines of $L_Z$, using calls to the\nalgorithm for $\\Pi_1$ to guide the search, in a total of $O(\\log^3 n\n\\log \\log n)$ running time. We obtain a connected portion $\\gamma'$\nof $\\gamma$, delimited by two consecutive intersection points of\n$Z$, whose relative interior lies completely above or below each\n$\\delta \\in D$, so that $\\gamma \\cap K^*$, if nonempty, lies in\n$\\gamma'$. This allows us to resolve each comparison between $K^*$\nand an arc $\\delta \\in D$, assuming that $\\gamma \\cap K^* \\neq\n\\emptyset$, and we continue the binary search through each $M_j$ in\nthe same manner.\n\nWe again use a variant of Cole's technique~\\cite{RC} to slightly\nimprove this bound further. In each parallel step we have a\ncollection $Z$ of $O(\\log n)$ weighted points, each of which is an\nintersection point of $\\gamma$ with some arc $\\delta$ from one of\nthe planar maps $M_j$, and we need to compare each of the points of\n$Z$ with $K^*$. Let $D$ denote the set of these active arcs.\n\nNote that each arc $\\delta$ participating in this step\ncontributes (at most) four points to $Z$, for a total of at most\n$4|D|$ points. We perform three steps of a (weighted) binary search on\nthe points of $Z$, where each step takes the weighted median $z_0$\nof an appropriate portion of $Z$, and calls $\\Pi_1(l_0)$, where $l_0$\nis the vertical line through $z_0$. These $\\Pi_1$-steps resolve the\ncomparisons with $K^*$ of all but $1\/8$ of the points of $Z$, that is,\nat most $(1\/8)\\cdot 4|D| = |D|\/2$ points of $Z$ are still unresolved.\n\nIn other words, after the three calls\nto the algorithm for solving $\\Pi_1$ (in the first parallel\nstep of the execution), we can determine the outcomes of the\ncomparisons of at least half of the arcs in $D$ with $K^*$. We can\nthen proceed in this manner and apply Cole's technique (as before),\nby using only a constant number of calls to $\\Pi_1$ in each of the\n$O(\\log n)$ parallel steps of searching in all the maps.\nThis reduces a $\\log \\log n$ factor from the bound of the\nrunning time, so it is only $O(\\log^5n)$ time.\n\nWhen these searches terminate, we end up with a 2-face in each\n$M_j$, in which $\\gamma \\cap K^*$ lies (if nonempty), and we reach\nthe scenario described in a preceding paragraph. As explained there,\nwe can now either determine that $K \\neq \\emptyset$, or that $K =\n\\emptyset$, or else we know which side of $\\gamma$, within $\\psi_i$\n(or, rather, within $\\psi'$) can contain $K^*$, and we continue the\nbinary search through $\\psi_i$ on that side.\n\nWhen the binary search through $\\psi_i$ terminates, we have a 2-face\n$\\zeta_i$ of $M_i$, where $K^*$ must lie, and we retrieve the ball\n$b_i$ corresponding to $\\zeta_i$. We repeat this step to each of the\nmaps $M_i^+$ and $M_i^-$ of each of the $t$ spherical polytopes\n$S_i$, and obtain a set ${\\cal B}_1$ of $2t$ balls. In addition, the\nsearches through the maps $M_i^+$ and $M_i^-$ may have trimmed\n$\\psi'$ to a narrower strip $\\psi''$, and have produced a set ${\\cal B}_2$\nof witness balls, so that the $xy$-projection of their intersection\nlies inside $\\psi''$. ${\\cal B}_2$ may consist of a total of $O(t^3 \\log^2\nn)$ witness balls, as is easy to verify. In addition, the\nsecond-level searches produce an additional collection ${\\cal B}_2'$,\nconsisting of balls corresponding to faces of the maps $M_j^+$ and\n$M_j^-$, in which the second-level searches have ended; their\noverall number is $O(t^2 \\log n)$. Put $K_2 = \\bigcap ({\\cal B}_1 \\cup\n{\\cal B}_2 \\cup {\\cal B}_2')$. Hence $K \\neq \\emptyset$ if and only if $K_2 \\neq\n\\emptyset$.\n\nAs already noted, the overall running time of the emptiness\ndetection is $O(\\log^5 n)$.\n\nSo far, we have only determined whether $K$ is empty or not.\nHowever, to enable the decision procedure to discriminate between\nthe cases $r^* = r$ and $r^* < r$ we need to refine the algorithm,\nso that it can also determine whether $K$ has nonempty interior (we\nrefer to an intersection $K$ with this property as\n\\emph{non-degenerate}).  To do so, we make the following\nmodifications to the algorithm described above. Each step in the\nemptiness testing procedure which detects that $K \\neq \\emptyset$\nobtains a specific point $w$ that belongs to $K$. Moreover, $w$\nbelongs to the intersection $K_1$ of polylogarithmically many\nwitness balls, and does not lie on the boundary of any other ball.\nThis is because each of the procedures $\\Pi_0, \\Pi_1$, or $\\Pi_2$\nlocates the $xy$-projection $w^*$ of $w$ (which, for $\\Pi_1$ and\n$\\Pi_2$ is a generic, unknown point in $K$) in each of the maps\n$M_i^+, M_i^-, i = 1,\\ldots,t$, and the collection of the witness\nballs gathered during the various steps of the searches contains all\nthe balls that participate in the corresponding spherical polytopes\n$S_i$ on whose boundary $w$ can lie. Thus, when we terminate with a\npoint $w \\in K$, we find, among the polylogarithmically many witness\nballs, the at most four balls whose boundaries contain $w$ (recall\nour general position assumption), and test whether their\nintersection is the singleton $\\{w\\}$. It is easily checked that\nthis is equivalent to the condition that $K$ is degenerate.\n\nThis completes the description of the algorithm, and concludes the\nproof of Proposition~\\ref{prop:intersection_test}.\n\n\n\\section{Discussion and Open Problems.}\n\\label{sec:discussion} In this paper we presented two algorithms for\ncomputing the 2-center of a set of points in $\\mathbb{R}^3$. The first\nalgorithm takes near-cubic time, and the second one takes\nnear-quadratic time provided that the two centers are not too close\nto each other. Note that our second algorithm may be slightly\nrevised, so that it receives, in addition to $P$, a parameter\n$\\epsilon > 0$ as input, and returns a solution for the 2-center\nproblem for $P$, if $\\epsilon \\leq 1-\\frac{r^*}{r_0}$. To this end,\nwe run the exponential search until we reach a value of $r$ with\n$1-\\frac{r}{r_0} \\leq \\epsilon$. If along the search we have found a\nvalue of $r$ such that $r \\geq r^*$, we stop the search and run\nChan's technique with the constraint that $r^* \\leq r$, as above.\nOtherwise, we have $r^* > r_0(1-\\epsilon)$ and we may return the\nsmallest enclosing ball of $P$ as an $\\epsilon$-approximate solution\nfor the 2-center problem. This way, we ensure that the running time\nof our algorithm is $O(\\epsilon^{-3} n^2\\log^5 n)$.\n\nAn obvious open problem is to design an algorithm for the 2-center\nproblem that runs in near-quadratic time on all point sets in\n$\\mathbb{R}^3$.  Another interesting question is whether the 2-center\nproblem in $\\mathbb{R}^3$ is \\emph{{\\sc 3sum}-hard} (see~\\cite{GO}\nfor details), which would suggest that a near-quadratic algorithm is\n(almost) the best possible for this problem.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:Introduction}}\n\n\nThe organic-inorganic interfaces formed between peptides and surfaces \nare of interest to diverse fields like medicine, optoelectronics and energy storage \\cite{WangGiovanni2013, GuoCahen2016, Khatayevich, Mannoor, Guy, Zhao, Sarikaya:2003-biomim, Costa:2015:IMPORTANT, Heinz:2016:Review, Walsh:2017ci}.\nAmino acids and their oligomers -- i.e. peptides -- are particularly interesting materials because they are naturally biocompatible and offer a rich functional space already at the amino acid level, that can be extended by the combinatorial increase of molecular motifs available through the peptide bond formation. In these setups, the inorganic component offers a platform to immobilize and template the bioorganic counterpart, as well as to record electronic signals from interactions or reactions.\nHowever, emergent behavior makes it impossible to base the description of such interfaces solely on the study of its independent components. \n\nOn the experimental front, progress on the fundamental understanding of these interfaces has been achieved with recent soft landing techniques \\cite{Rauschenbach2017} that allow amino acids and peptides of a few tens of building blocks to be deposited on solid surfaces and subsequently imaged in ultra-high vacuum conditions employing scanning tunneling microscopy (STM) \\cite{Rauschenbach2016}. \nThese experiments have shown that small sequence modifications, changes in the protonation of a peptide, and charge transfer at the interface can yield drastic changes in the two-dimensional structure and self-assembly~\\cite{Rauschenbach2017, Abb2016, Humblot2016}.  \n\nExplaining the physical mechanisms that govern adsorption and pattern formation calls for systematic theoretical investigations with atomistic resolution of such interfaces. \nHowever, such studies are challenging as they require (i) accurate energetics for a system build-up of elements across the periodic table and where considerable charge rearrangement and chemical reactions can occur (ii) sampling and representing a large conformational space, and (iii) dealing with structure motifs that can only be represented by unit-cells containing hundreds of atoms. Pioneering work that used density-functional theory (DFT) to study amino-acid at inorganic substrates has focused on small or rigid amino acids, and a limited portion of their conformational space \\cite{DiFelice2003, DiFelice2004, Ghiringhelli2006, Arrouvel2007, Iori2008}. \nOne of the first studies that was dedicated to larger amino acids was performed by Hong and coworkers \\cite{Hong2009} and illustrates the challenge of properly sampling the large structure space of flexible biomolecules. These studies have clarified that an accurate energy function is only one of the ingredients needed to properly predict the structure of peptides at surfaces, with the sampling of structure space being just as important.\n\n\nWe propose to analyze how the structure space of a complex building block with multiple functional groups is affected by changes in its (non-biological) environment. To achieve this goal, we analyze a database of the arginine (Arg) amino-acid and its protonated counterpart (Arg-H$^+$) in the gas-phase and interfaced with Cu(111), Ag(111) and Au(111). Arginine is a good test bed because it is a relatively small molecule, but yet contains a very flexible side-chain and allows for different stable protonation states. We show that it is possible to perform an exhaustive conformational space exploration while still treating the potential-energy surface (PES) with DFT, thus capturing the charge rearrangements at the interface.\nThe outcome of such a computational structure-search is a large number of stationary-states on the respective PES, where not only the global minimum is of interest. Also the relative positioning of other local minima with respect to the structural degrees of freedom, as well as with respect to the energy scale of the PES, are relevant because they can reveal different basins and structures of interest under different conditions~\\cite{ropo2016trends, Rossi_2013, Rossi_2014, Schubert_2015, Baldauf_2015}.\nIn order to handle such high-dimensional data, we make use of recent advances in machine-learning methods that can help to visualize conformational preferences of adsorbed molecules~\\cite{Ceriotti2011, Tribello2012, Ceriotti2013, De2016, Bartok2017, De2017}.\n\nIn the following, we discuss the impact of the protonation state and the presence or absence of the surface on the accessible conformational space of arginine. We start by describing the procedure we followed to build the database of Arg and Arg-H$^+$ on Cu(111), Ag(111), and Au(111), based on thousands of first-principles structure optimizations which we make available to the community~\\cite{DOI_of_the_data_set_at_NOMAD}.\nWe then analyze this database and show how different patterns of bonding and charge transfer induce fundamental changes in the accessible conformational space. We also provide an analysis of property trends across the different metallic surfaces, including protonation-dependent stereoselective binding to the surfaces and deprotonation propensities.\n\n\n\\section{Computational methods \\label{sec:CompMethods}}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{F1_Protonation_states.jpg}\n\\caption{a) Pictorial representation of the arginine amino acid, including labels of chemical groups and atoms. b) Protomers of Arg that are addressed in this work. c) Protomers of Arg-H$^+$ that are addressed in this work. }\n\\label{fig:protonationsketch}\n\\end{figure}\n\nArginine is an amino acid with a flexible side-chain of three (aliphatic) CH$_2$ groups and a guanidino group. \nA depiction of the Arg molecule including the labeling of the different chemical groups and specific atoms we will refer to in the manuscript is shown in Fig. \\ref{fig:protonationsketch}(a). \nIn the context of this publication we use the term \\textit{protonation state} to distinguish between Arg and its singly-protonated form Arg-H$^+$. We use the word \\textit{protomers} to distinguish different arrangements of protons within molecules of the same sum formula, for example the protomers \\textbf{P1} to \\textbf{P5} of Arg or the protomers \\textbf{P6} and \\textbf{P7} of Arg-H$^+$, shown in Fig. \\ref{fig:protonationsketch}(b) and (c). In this section, we describe the computational setup, convergence tests, and sampling strategies we employed in order to build the database of Arg and Arg-H$^+$ adsorbed on different metal surfaces.\n\n\nAll the electronic structure calculations were carried out using the numeric atom-centered basis set all-electron code FHI-aims \\cite{Blum2009, Havu2009}. We used the standard \\textit{light} settings of FHI-aims for all species, except when stated otherwise. For modeling the adsorbed molecules, a $5\\times6$ surface unit cell with $4\\times4\\times1$ $k$-point sampling was employed. The slab contains 4 layers and we added a 50 \\AA~ vacuum in the $z$ direction in order to separate periodic images of the system. \nA surface unit cell of this size does not completely isolate neighboring molecules on the surface plane.\nIn order to estimate the magnitude of this spurious interaction, we calculated binding energies for three  Arg and three Arg-H$^+$ structures adsorbed on Cu(111) using different surface unit cell sizes. As shown in the SI (Tables S3 and S4), the relative binding energies change by no more than 50 meV when reaching a $10\\times12$ cell. \n\nAn accurate description of the dispersion interactions that play a role in weakly-bonded adsorbates on metallic surfaces can be achieved with newly-proposed methods that take into account electronic screening and the the many-body nature of the dispersion term \\cite{Hermann2019}; however, such methods are not yet computationally feasible for large-scale studies.\nConsidering dispersion interactions is, however, crucial to model such interfaces \\cite{Ruiz2016, Ruiz-thesis, Liu2013, Liu2012, Al-Saidi2012, VanRuitenbeek2012, Wagner2012, Carrasco2014}. We thus employ the PBE exchange-correlation functional augmented by the TS-vdW$^{\\text{surf}}$ \\cite{Ruiz2016} dispersion correction between pairs of atoms that involve the adsorbed molecule (molecule-molecule and molecule-surface dispersion). \nAdding pairwise vdW corrections for pairs of metal atoms did not result in a systematic improvement of the lattice constants due to the complex many-body nature of the electronic screening within the metallic bulk.  \nBecause the PBE lattice constants for Cu, Ag, and Au are in good agreement with experimental data (see Table S1 of the SI) and the electronic structure itself is not changed by the inclusion of these types of vdW interactions, we chose to use the simplest setup. \nWe optimized all structures \nuntil all forces in the system were below 0.01\\,eV\/\\AA\\,. \nWe also fixed the two bottom layers of the slabs in all optimizations. \nA dipole correction was applied in $z$ direction to compensate for the dipole formed by the asymmetric surface configurations. \n\n\\subsection{Database Generation \\label{sec:ModelGeenration}}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{F2_Relative_Energies.jpg}\n\\caption{(a-d) Correlation plots of relative energies of Arg or Arg-H$^+$ conformers on Cu, Ag, and Au (111) surfaces. Each dot corresponds to the same conformer optimized on the two surfaces addressed in each panel, color coded with respect to the RMSD (heavy atoms only) between the superimposed optimized structures without taking surface atoms into consideration. (e) Binding energies of Arg and Arg-H$^+$ on Cu(111), Ag(111) and Au(111) surfaces.}\n\\label{fig:relrel}\n\\end{figure}\n\n\nThe sampling of the structure space of the amino acid in two protonation states on metallic surfaces was performed by starting from a previously published data-set comprising stationary points of isolated amino acids and dipeptides \\cite{Ropo2016,ropo2016trends}. \nFor Arg, 1206 structures were present in the database.\nIn order to reduce the number of possibilities, but keeping a representative share of the structures, we considered the 300 lowest energy conformers, the 27 highest energy conformers, and 125 conformers uniformly spanning the energy range in between. \nFor the Arg-H$^+$ amino acid, all 215 structures present in the gas-phase data set were used in this study.\n\nWe distinguish \\textit{upstanding} positions of the molecules where the largest principal axis of the molecule is approximately perpendicular to the surface plane, from \\textit{flat lying} positions with an arrangement parallel to the surface.\nFor Arg, 3 lying configurations per structure were generated by randomly placing the molecule flat on the Cu(111) surface and then rotating it by $120^{\\circ}$ around the principal axis. \nTwo upstanding configurations were generated for the 25 lowest-energy gas-phase structures by first placing the molecule in a random upright orientation and then flipping it. \nFor Arg-H$^+$ a similar procedure was adopted: flat lying positions were created by $90^{\\circ}$ rotations around the principal axis and upstanding configurations were created for 27 structures that were uniformly selected from the whole energy range. \nIn summary, we considered a total of 1156 conformers of Arg@Cu(111) and 914 conformers of Arg-H$^+$@Cu(111).\n\nAll optimized structures that fell within a range of 0.5 eV from the global minimum on Cu(111) were transferred to Ag(111) and Au(111) and further optimized. In addition, we randomly picked 105 Arg-H$^+$ structures representing the higher energy range on Cu(111) to be further optimized on Ag(111) and Au(111). Moreover, for Arg, 180 randomly picked structures representing the higher energy range were considered on Ag(111) and 61 on Au(111). The total amount of calculated structures for each case is summarized in Table \\ref{tbl:totalnmbrofstructs}. \n\n\\begin{table}\n\\center\n \\caption{Number of calculated Arg and Arg-H structures in isolation and adsorbed on Cu(111), Ag(111) and Au(111).}\n \\label{tbl:totalnmbrofstructs}\n\\begin{tabular}{|r|c|c|c|c|}\n\\hline\n     & Gas phase & Cu(111) & Ag(111) & Au(111) \\\\\n\\hline\nArg  & 1206      & 1156    & 327     & 209  \\\\\n\\hline\nArg-H$^+$ & 215       & 914      & 718       & 721         \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe checked that this strategy ensured a sufficient sampling of the low-energy range of both Arg and Arg-H$^+$ on Ag(111) and Au(111) by analyzing the alterations in relative energy hierarchies on the different surfaces. In Fig. \\ref{fig:relrel}(a-d) each dot corresponds to a conformer that was optimized first on the Cu(111) surface and then post-relaxed on Ag(111) or Au(111). Within the lowest 0.5 eV range, we do not observe any significant rearrangement of the energy hierarchy with respect to the Cu(111) surface. The energy hierarchies of both Arg and Arg-H$^+$ on the Ag(111) and Au(111) surfaces are almost identical. The most pronounced outliers in all plots correlate with a higher root mean square displacement (RMSD) of the molecular atoms (i.e. disregarding the surface-adsorption site), thus pointing to a structural rearrangement of the molecule.\n\n\\subsection{Structure space representation and analysis \\label{sec:soap-map}}\n\nWe analyse the structure space of all systems considered, as well as its alterations by employing a dimensionality reduction procedure that makes it more intuitive to understand the high-dimensional space. Following Ref. \\cite{De2017}, we represent the local atom-centered environments of the structures through the smooth overlap of atomic positions (SOAP)\\cite{Bartok2017} descriptors. We then obtain the similarity matrix between different conformers with the regularized entropy match kernel (REMatch) \\cite{De2016}. We used SOAP descriptors with a cutoff of 5.0 \\AA, a Gaussian broadening of $\\sigma=0.5$ \\AA~ and an intermediate regularization parameter $\\gamma$=0.01. SOAP kernels were calculated only for heavy atoms in the molecule (disregarding metal and hydrogen atoms) and were obtained using the GLOSIM package \\cite{De2016, GLOSIM}. \n\nFor the dimensionality-reduced representation, we here chose to use the multi-dimensional scaling (MDS) algorithm as implemented in the \\texttt{scikit-learn} package\\cite{scikit-learn}. This algorithm is similar to the Sketchmap algorithm previously employed in Ref. \\cite{De2017}, but we found it more suitable for the data at hand, which is composed of decorrelated local stationary-points, instead of structures generated from molecular dynamics trajectories. In short, the low-dimensional map was obtained considering all calculated structures of Arg in the gas-phase and through minimizing the stress function\n\\begin{equation}\n\\sigma = \\sum_{i\\neq j} (D_{ij}-d_{ij})^2,  \n\\end{equation}\nwhere $D_{ij}$ is the similarity between structures $i$ and $j$ in the high-dimensional space and $d_{ij}$ the Cartesian distance in the low-dimensional (2D) space. \nWe then projected structures in different environments onto the pre-computed map of gas-phase Arg by fixing the parameters of the map and finding the low-dimensional coordinates $x$ that minimize  \n\\begin{equation}\n\\sigma_P(x)= \\sum_{i=1}^{N_{\\text{Arg}}} \\left(|X-X_i|-|x-x_i|\\right)^2 \n\\label{eq:proj}\n\\end{equation}\nwhere the sum runs over all structures in the optimized gas-phase Arg map and $X$ represents the high-dimensional representation of the new structure to be projected. In order to classify structural patterns, we employ the following notations:\n\\begin{itemize}\n    \\item We represent the protomers by the labels shown in Fig. \\ref{fig:protonationsketch}(b) and (c). \n    \\item We identify the presence of strong intramolecular hydrogen bonds (H-bonds) whenever the distances between the hydrogen connecting donor and acceptor are below 2.5 \\AA. We label the H-bond pattern between two atoms in the molecules according to the nomenclature shown in Fig. S3 in the SI.\n    \\item We further classify the structures according to the longest distance between two heavy atoms in the molecule.\n\\end{itemize}\n\n\\subsection{Binding energies and charge rearrangement \\label{sec:BindingEnergies}}\n\nThe binding energies for each structure were calculated as follows. For neutral Arg we computed \n$\nE_{\\textsf{b}} =  E_{\\textsf{mol@surf}} - E_{\\textsf{surf}} - E_{\\textsf{mol}}, \n$\nwhere $E_{\\textsf{mol@surf}}$ corresponds to the total energy of the interface, $E_{\\textsf{surf}}$ is the total energy of the pristine metallic slab and $E_{\\textsf{mol}}$ the total energy of the lowest energy gas-phase conformer.\n\nFor charged Arg-H$^+$, we considered the binding energy of a two-step reaction. \\textit{First}, the interface is formed between the charged molecule and the clean surface:\n$\nE_{\\textsf{b1}} = E_{\\textsf{mol}^+\\textsf{@surf}} - E_{\\textsf{surf}} - E_{\\textsf{mol}^+},\n$\nwhere $E_{\\textsf{mol}^+}$ the total energy of the most stable gas-phase conformer of the isolated charged molecule. \\textit{Second}, an electron from the metal neutralizes the unit cell where the adsorbed molecule is located, yielding\n$\nE_{\\textsf{b2}} = E_{\\textsf{mol@surf}} - E_{\\textsf{mol}^+\\textsf{@surf}} - E_{f},\n$\nwhere $E_f$ corresponds to the Fermi energy of the metallic surface. The final binding energy is thus considered to be\n\\begin{equation}\nE_{\\textsf{b}}^+ = E_{\\textsf{b1}} + E_{\\textsf{b2}} = E_{\\textsf{mol@surf}} - E_{\\textsf{surf}} - E_{f}- E_{\\textsf{mol}^+}.\n\\end{equation}\nFor reference, we report the values we used for $E_f$ at each surface in Table S5. These binding energies are shown in Fig. \\ref{fig:relrel}d, and will be discussed in detail in Section \\ref{sec:Results}.\n\n\nIn addition,  for conformers within 0.1 eV of the respective global minimum of each system, we have calculated free energies at finite temperatures within the harmonic approximation \\cite{McQuarrie, Fultz2010}, in order to address question about thermal stability of such structures. We have calculated\n$\nF_{\\textsf{harm}}(T) = E_{\\textsf{PES}} + F_{\\textsf{vib}}(T),\n$\nwhere $E_{\\textsf{PES}}$ is the total energy obtained from DFT, and we have used textbook expressions for the harmonic vibrational Helmholtz free energy $F_{\\textsf{vib}}(T)$. We have calculated the Hessian matrix only taking into account displacements of the adsorbate. The surface was only considered as an external field, which is a good approximation for physisorbed systems. \n\nTo address charge rearrangements after adsorption on the surface, we calculated electron density differences for selected  structures on each surface by computing\n$\n\\Delta\\rho = \\rho_{\\textsf{mol@surf}} - \\rho_{\\textsf{surf}} - \\rho_{\\textsf{mol}},  \n$\nand in the case of Arg and \n$\n\\Delta\\rho^{(+)} = \\rho_{\\textsf{mol@surf}} - \\rho_{\\textsf{surf}} - \\rho_{\\textsf{mol}^{(+)}},\n$\nin the case of Arg-H$^+$. In these expressions, $\\rho_{\\textsf{mol@surf}}$ is the total electron density of the interface, $\\rho_{\\textsf{surf}}$ is the electron density of the slab without molecule, and $\\rho_{\\textsf{mol}}$ and $\\rho_{\\textsf{mol}^+}$ are electron densities of neutral Arg and charged Arg-H$^+$ molecules with the same geometries as in interface. The $+$ sign denotes that the final density difference integrates to +1 electron in the case of Arg-H$^+$.\n\n\\section{Results and discussion \\label{sec:Results}}\n\n\\subsection{The unconstrained structure space: Arg in isolation \\label{sec:Isolation}}\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=0.87\\linewidth]{F3_Arg_isolated.jpg}\n\\caption{Low-dimensional map of Arg stationary points on the PES. Only points linked to structures with a relative energy of 0.5 eV or lower are colored. Representative structures of all conformer families are visualized as well as their H-bond distances (in turquoise) and longest distance between two heavy atoms (in red) of the molecule. The maps are colored with respect to a) relative energy, b) longest distance, and c) H-bond pattern. The size of the dots also reflect their relative energy, with larger dots corresponding to lower energy structures.}\n\\label{fig:Arg_Sketchmap}\n\\end{figure}\n\nWe start by analysing the unconstrained conformational space of Arg in isolation, which is formed by more than 1200 local stationary states \\cite{Ropo2016,ropo2016trends}. In order to rationalize different structural arrangements in this space, we utilize the dimensionality-reduction algorithm discussed in Section \\ref{sec:soap-map} and build a two-dimensional map. On this map, shown in Figure \\ref{fig:Arg_Sketchmap}(a), each dot represents one structure. A close proximity of the dots implies similarities of the heavy-atom arrangement between the conformations. This is the low-dimensional map that is taken as a reference for comparison throughout this manuscript. \n\nWe proceed to color-code the dots on the map according to different properties. In Fig. \\ref{fig:Arg_Sketchmap}(a) we show the map colored by the relative energy of each structure with respect to the global minimum $\\Delta E_{\\text{rel}}$. We only color structures with $\\Delta E_{\\text{rel}} < 0.5$ eV. The region with $\\Delta E_{\\text{rel}} < 0.1$ eV is colored red and is represented by 32 different structures that occupy different parts of the map. The dominant protomer (98.6\\%) among these conformers is the one labeled \\textbf{P1} in Fig. \\ref{fig:protonationsketch}, i.e. non-zwitterionic. \nHowever, the lowest energy structure, labeled \\textit{a} in panel (a) of Fig. \\ref{fig:Arg_Sketchmap}, is protomer \\textbf{P3}, with a shared proton between the carboxylic and the guanidino group. \nThis structure is compact, with the longest distance within the molecule of only 5.01 \\AA~ and presenting two strong intramolecular H-bonds.\nZwitterionic protomers, denoted as \\textbf{P4} and \\textbf{P5} in Fig. \\ref{fig:protonationsketch}, do not appear in the gas-phase.\n\nInspecting the map in Fig. \\ref{fig:Arg_Sketchmap}(a), it is clear that low-energy conformers are almost exclusively present in the upper hemisphere of the plot. \nThis can be rationalized in terms of the structural motifs that occupy these two halves of conformational space:\nIn Fig. \\ref{fig:Arg_Sketchmap}(b), we color-code the dots in terms of the longest extension of the conformers. \nWhile the upper hemisphere features compact structures, the lower hemisphere of the map is populated by extended conformers (with longest extensions between 7.5 \\AA~ and 10.0 \\AA).\nMany of them do not contain any H-bond or contain only one H-bond between the carboxyl and amino group.\nExtended conformers of Arg are energetically unfavoured in the gas-phase as the formation of strong H-bonds is crucial for the stabilization of Arg in isolation. \nComparing the different plots in Figure \\ref{fig:Arg_Sketchmap}, we see that all low-energy structures with $\\Delta E_{\\text{rel}} < 0.1$ eV are indeed compact with one or two H-bonds. \n\nIn Fig. \\ref{fig:Arg_Sketchmap}(c), we identify in total 13 different families with respect to the number and character of H-bonds in the molecule, with $\\Delta E_{\\text{rel}} < 0.5$ eV. Representative structures of all families are shown in panel (a). \nThis family classification  helps us understand why in Fig. \\ref{fig:Arg_Sketchmap}(a) there are structures of higher energies at similar regions as structures with lower energies. Even though these structures are typically in the same protomeric state and have a similar arrangement of the heavy-atoms, the carboxyl group can rotate, giving rise to different H-bond patterns.\nThese different arrangements can give rise to energy differences of up to 0.2 eV, as exemplified in Fig. S4 in the SI. Including hydrogens in the SOAP descriptors used to build the 2D map could provide a better energy separation, but it would prevent us from comparing different protonation states, as shown in the next section.\n\n\\subsection{Adding a proton: Arg-H$^+$ in isolation \\label{sec:Proton}}\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=0.9\\linewidth]{F4_ArgH_isolated.jpg}\n\\caption{Representative conformers of the populated structure families within 0.5 eV of the global minimum of isolated Arg-H$^+$ and low-dimensional projections of all populated conformers onto the Arg map. Grey dots represent all structures from the original map of isolated Arg in Figure 3, and serve as a guide to the eye. The maps are colored with respect to a) relative energy, b) longest distance within the molecule, and c) H-bond pattern.}\n\\label{fig:arghmaps}\n\\end{figure}\n\nArg-H$^+$ is of particular interest as it is the most abundant form of this amino acid under physiological pH conditions \\cite{BolgerBook1995}, and we thus investigate changes of the conformational space introduced by the addition of a proton to the Arg amino-acid. \nTo that end, we plot a projection of all stationary points of the Arg-H$^+$ PES with $\\Delta E_{\\text{rel}} < 0.5$ eV (referenced to its own global minimum) onto the map that was previously created for Arg. \nIn Fig. \\ref{fig:arghmaps}(a), we color the dots in the map according to $\\Delta E_{\\text{rel}}$, in Fig. \\ref{fig:arghmaps}(b) according to the longest distance between heavy atoms in the molecule, and in Fig. \\ref{fig:arghmaps}(c) according to the H-bond pattern. The grey dots in the maps represent all points in the Arg map of Figure \\ref{fig:Arg_Sketchmap} and are shown for ease of comparison.\n\nThe unique conformation types of Arg-H$^+$ can be grouped into 8 different families in this energy range, which are represented in Fig. \\ref{fig:arghmaps}(a). Most families only have one H-bond and there are no zwitterionic protomers. \nThis means that in isolation only the protomer \\textbf{P6} is populated.\nIt is worth noting that under physiological conditions (in solution), the zwitterionic protomer \\textbf{P7} is preferred. \n\n\nThere are only two very similar structures (same family) with $\\Delta E_{\\text{rel}} < 0.1$ eV in this case. The global minimum, labeled \\textit{a} in Fig. \\ref{fig:arghmaps}(a), contains two H-bonds within the molecule, between atoms N-N$\\mathrm{\\varepsilon}$ and O1-N$\\eta$ (see Fig. \\ref{fig:protonationsketch}). \nThis particular structure resembles the lowest-energy structure of Arg with a proton added to the carboxyl group.\nThis protonation results in an extension of the molecule by around 1 \\AA. \nThat correlates with the location of the lowest-energy structure being slightly shifted on the map towards the region containing more extended structures.\n\nWe note that the structure space of Arg-H$^+$ is contained within the conformational space of Arg and also drastically reduced in numbers if compared to Arg: \nThere are only 108 structures with $\\Delta E_{\\text{rel}} < 0.5$ eV, compared to 1179 structures in the Arg case. \nIn this energy range, regions of the map with very compact and very extended structures are not populated in this protonation state. \nThis can be traced to the constraint imposed by the addition of the proton, that make extended structures less stable due to the strong driving force to neutralize the charge imbalance created by the proton on the guanidino group.\nTo rationalize why the most compact conformers are also less populated, we show in Fig. \\ref{fig:ArghArg_electrondensitydifferencce} the electron-density differences between the lowest energy Arg-H$^+$ conformer and an Arg conformer created by fixing the same Arg-H$^+$ structure, but neutralizing the charge and removing the hydrogen connected to the carboxyl group. This modification yields the same covalent connectivity observed in the global minimum of Arg. \nWe show isosurfaces corresponding to electron accumulation in Arg-H$^+$ in red and electron depletion in Arg-H$^+$ (accumulation in Arg) in blue. \nWe observe a density surplus between the O1 and N$\\eta$ atoms in Arg, favoring the formation of a stronger H-bond leading to a more compact structure.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{diffelectron.jpg}\n\\caption{Electron density difference between Arg-H$^+$ and Arg calculated by neutralizing the charge and removing the hydrogen connected to the carboxyl group (marked in green) from the lowest energy structure of Arg-H$^+$. The isosurfaces of electron density with value $\\pm0.005$ e\/Bohr$^3$ corresponding to the a) regions of electron accumulation on Arg-H$^+$ and b) where the electron depletion on Arg-H$^+$, both compared to Arg.}\n\\label{fig:ArghArg_electrondensitydifferencce}\n\\end{figure}\n\n\\subsection{Adsorption of Arg on Cu, Ag, Au (111) surfaces}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{F6_Arg_surfaces.jpg}\n\\caption{\nLow-dimensional projections of conformers of Arg adsorbed on a) Cu(111), b) Ag(111), and c) Au(111), onto the gas-phase Arg map of Fig. 3. Only conformers within 0.5 eV from the respective global minimum are colored. Grey dots represent all structures from the original map of gas-phase Arg, and serve as a guide to the eye.\nIn each panel, representative structures are shown from two perspectives: a side view where molecule and surface are shown (bottom), and the corresponding top view (top) where only the molecule is shown. The longest distance within each visualized conformer is reported in red and H-bond lengths are reported in turquoise.}\n\\label{fig:argsurf_sketchmap}\n\\end{figure}\n\nWe now turn to the analysis of the conformational space of Arg when in contact with metal surfaces, namely Cu(111), Ag(111), and Au(111). \nIn Figure \\ref{fig:argsurf_sketchmap}, we show map-projections of the stationary points with $\\Delta E_{\\text{rel}} < 0.5$ eV (referenced to each respective global minimum) of Arg adsorbed on the three surfaces.\nThe conformational space of Arg upon adsorption is reduced and the conformers occupy similar regions of the map as the ones from Arg-H$^+$. \nWe will learn in the following that this is mainly due to the formation of strong bonds with the surface that result in steric constraints of the space and also partially due to electron donation from the molecule to the metallic surfaces.\n\nThe lowest energy structure lies on the same part of the map on all surfaces, which is different from the area where the gas-phase global minimum of Arg was located. These conformers, labeled \\textit{a} in Figure \\ref{fig:argsurf_sketchmap}(a), (b) and (c) form a strong H-bond between atoms O1 and N$\\epsilon$. The longest distance within molecule lies between 7.20-7.35 \\AA. This structure binds most strongly to all three surfaces through its amino and carboxyl groups. \n\nOther low-energy structures in all surfaces form strong bonds to the surfaces only through the carboxyl group, as exemplified by the structure labeled \\textit{b} in all panels of Fig. \\ref{fig:argsurf_sketchmap}. \nThese bonds are formed most favorably on \\textit{top} positions, i.e. vertically on top of a surface metal atom.\nIn particular for Cu(111), the atomic spacing of the Cu atoms on the surface favors both oxygens to bind on \\textit{top} positions simultaneously.\n\nThe favorable formation of these bonds is connected with the fact that all conformers with $\\Delta E_{\\text{rel}} < 0.2$ eV are in the protomeric state \\textbf{P3}, in which the carboxyl group is deprotonated.\nThe bonds to the surface and a favorable vdW attraction effectively flatten the molecular conformation, thus energetically favoring more elongated structures. \nProtomers of type \\textbf{P1}, which was dominant in the gas-phase, only appear  with $\\Delta E_{\\text{rel}} > 0.3$ eV on Cu and Ag, and with $\\Delta E_{\\text{rel}} > 0.2$ eV on Au.\nProtomers \\textbf{P4} and \\textbf{P5} are again not observed. \nRegarding the intramolecular H-bond patterns, within 0.5 eV from the global minimum\nwe can identify 7 different families on Cu(111), \nand 6 families on both Ag(111) and Au(111). \nThese families contain H-bonds where the carboxyl group predominantly participates. All families are represented in Table S4 in the SI.\n\n\n\n\\subsection{Adsorption of Arg-H$^+$ on Cu, Ag, Au (111) surfaces}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.83\\linewidth]{F7_ArgH_surfaces.jpg}\n\\caption{Low-dimensional projections of conformers of Arg-H$^+$ adsorbed on a) Cu(111), b) Ag(111), and c) Au(111), onto the gas-phase Arg map of Fig. 3. Only conformers within 0.5 eV from the respective global minimum are colored. Grey dots represent all structures from the original map of gas-phase Arg, and serve as a guide to the eye.\nIn each panel, representative structures are shown from two perspectives: a side view where molecule and surface are shown (bottom), and the corresponding top view (top) where only the molecule is shown. The longest distance within each visualized conformer is reported in red and H-bond lengths are reported in turquoise.}\n\\label{fig:arghsurf_map}\n\\end{figure}\n\nFinally, we characterize the conformational-space changes arising from the simultaneous addition of a proton and the adsorption onto metallic surfaces.\nIn Figure \\ref{fig:arghsurf_map}, we show the projection of Arg-H$^+$ conformers adsorbed on Cu, Ag, and Au(111) onto the map of isolated Arg. \nThese projections, in particular the comparison of the plots in Figs. \\ref{fig:argsurf_sketchmap} and \\ref{fig:arghsurf_map}, reveal that \nthe conformational space of adsorbed Arg-H$^+$ is larger than the one of adsorbed Arg. \nWhile Arg-H$^+$ features more than 500 conformers within $ \\Delta E_{\\text{rel}} < 0.5$ eV, Arg only counts about 150 conformers in the same energy range.\nInterestingly, the adsorption of Arg-H$^+$ to a metal surface also results in an increase of the occupied structure space in comparison to isolated Arg-H$^+$ (108 structures with $ \\Delta E_{\\text{rel}} < 0.5$ eV), shown in Fig. \\ref{fig:arghmaps}. \nIn fact, the structures occupy similar regions of the map as the ones occupied by Arg-H$^+$, with the addition of extended structures that are located in the bottom of the map.\n\n\nWe identify 4 different families on Cu(111) and 3 on Ag(111) and Au(111) with $ \\Delta E_{\\text{rel}} < 0.1$ eV. \nRepresentative conformers of these families are shown in Fig. \\ref{fig:arghsurf_map}. \nThe lowest energy conformer, labeled \\textit{a} in Fig. \\ref{fig:arghsurf_map}(a)-(c), appears on all surfaces at the same region of the map as for adsorbed Arg.  The largest distance within the molecule lies around 7 \\AA~ and it also has a strong H-bond linking the carboxyl-O and the N$\\varepsilon$ atoms. \nThe structure, however, does not present the same orientation to the surface as compared to the lowest energy conformer of Arg, and does not form strong bonds with the surface.\nWith the exception of the extended structure on Cu(111), labeled \\textit{d} in  Fig. \\ref{fig:arghsurf_map}(a), all conformers with $ \\Delta E_{\\text{rel}} < 0.1$ eV on all surfaces contain one intramolecular H-bond involving either \ncarboxyl-O and N$\\varepsilon$ atom (labeled a), \nbackbone N and N$\\varepsilon$ atoms (labeled b) or \ncarboxyl O and a N$\\eta$ atom (labeled c). \nCompared to adsorbed Arg, adsorbed Arg-H$^+$ structures become on average 1.0 \\AA~ more extended (see SI). The guanidino and carboxyl groups often lie parallel to the surface. \nThe protomer \\textbf{P6}, the only one present in the gas-phase, is dominantly populated also on the surfaces. \nHowever, we do observe very few conformers in the zwitterionic \\textbf{P7} state. These structures are at least 0.2 eV higher in energy than than the global minimum. \n\nWith respect to the number of bonds that Arg-H$^+$ forms with the surface, the picture is very different from adsorbed Arg. \nWithin the lower 0.15 eV, we do not observe short (strong) bonds of O or N atoms to the surfaces. \nThis lack of constraint by the surface contributes to the increased structure space of adsorbed Arg-H$^+$. \nIn addition, the molecule accepts electrons from the surface, becoming less positively charged, as we discuss in detail in the next section. \nWe conclude that Arg-H$^+$ interacts with the metallic surfaces mostly through van der Waals and electrostatic interactions.\n\n\\subsection{Electronic structure and trends across surfaces}\n\n\nIn the previous sections we focused on structural aspects of the adsorbed molecules and the most prominent bonds the molecules make with the metallic surfaces. In the following, we will discuss different aspects of the molecule-surface interactions with the goal of identifying trends across these systems.\n\nThe binding energies for all surfaces were calculated as discussed in Section \\ref{sec:BindingEnergies}. \nThe larger negative values in Fig. \\ref{fig:relrel}(d) correspond to stronger binding of the molecule to the surface. \nIn the case of adsorbed Arg, many conformers bind to Cu more strongly than to Ag and Au.\nAs mentioned previously, Arg forms strong bonds to the surfaces, but the binding of the deprotonated carboxyl group of Arg to the Cu(111) surface is geometrically favored as discussed above.\nIn the case of adsorbed Arg-H$^+$, there is no pronounced difference in binding strengths to the surfaces and the values are comparable to binding energies obtained for Arg adsorbed on Cu(111). \nThis correlates with the observation that the interaction of Arg-H$^+$ to the surfaces happens mostly through dispersion and electrostatic interactions. Despite the strong binding to the surface, it is also visible in Fig. \\ref{fig:relrel}(a) that the interaction of Arg-H$^+$ with the surface does not strongly template the conformations of this molecule, implying a low corrugation (i.e. homogeneity) of the molecule-surface interaction and allowing for a larger variety of conformers with similiar energies. This is in contrast to the molecule-surface interation of Arg, that is more inhomogeneous due to the formation of bonds through specific chemical groups.\nAdditionally, we have estimated harmonic vibrational free energies for all conformers with $\\Delta E_{\\text{rel}} < 0.1$ eV in each surface. In contrast to what has been reported for longer helical peptides \\cite{Rossi_2013,Schubert_2015}, the global minimum remains the same in all cases, as reported in Fig. S6 in the SI. For Arg-H$^+$ we observe relative energy rearrangements of up to 50 meV at 300 K, which changes the relative energy hierarchy of conformers less stable than the global minimum.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Closest_sketch_com_v1.jpg}\n\\caption{Low dimensional projections of adsorbed Arg and Arg-H$^+$ on Cu(111), Ag(111) and Au(111) color-coded with respect to the distance of the center of mass of the molecule with respect to the surface. Grey dots represent all structures from the original map of isolated Arg where the projection was made, and serve as a guide to the eye. }\n\\label{fig:Closest_map}\n\\end{figure}\n\nWe then focus on the distance between the molecule and the surfaces. We define this quantity by measuring the distance of the center of mass (COM) of the molecule with respect to the  \nsurface plane defined by the top layer of surface atoms.\nThese distances are collected in Fig. \\ref{fig:Closest_map}. \nThe COM is closer to Cu(111) than to Ag(111) and Au(111) for both Arg and Arg-H$^+$. \nIn addition, in all surfaces, Arg lies closer than Arg-H$^+$, in agreement with the observation that Arg forms stronger bonds to the surface. \nThe extended structures of Arg-H$^+$, at the bottom of the maps, tend to be closer to the surface than those that have H-bonds within the molecule, likely due to the stronger vdW attraction to the surface by extended conformations. \n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{F10_UpDown_representation.jpg}\n\\caption{Orientation of the C$_\\alpha$H group in a) \\textit{up} orientation (hydrogen pointing towards vacuum) and b) \\textit{down} orientation (hydrogen pointing towards the surfaces). c) The amount of structures with \\textit{up} and \\textit{down} orientation within 0.1\/0.5 eV from the global minimum of each surface.}\n\\label{fig:UpDown}\n\\end{figure}\n\n\nThe differences in COM distance to the surfaces between Arg and Arg-H$^+$ are apparently related to the preferred orientation of the chiral center of the molecule to the surface. \nThe chiral C$_\\alpha$ carbon can point its bonded hydrogen towards the surface (labeled \\textit{down} in the following), or towards the vacuum region (labeled \\textit{up} in the following).\nExamples of different molecular orientation are shown in Fig. \\ref{fig:UpDown}(a).\nThe dominant orientation with respect to the surface is different in the cases of Arg and Arg-H$^+$, as evidenced by the numbers presented in Fig. \\ref{fig:UpDown}(b). \nThe lower energy structures are mostly in the \\textit{up} orientation for Arg and mostly in the \\textit{down} orientation for Arg-H$^+$ (see also map in Fig. S7).\nAs discussed in the previous sections, despite showing different orientations of their C$_\\alpha$H groups, the lowest energy structures for both molecules adsorbed on each surface have very similar conformations. \nSince the addition or removal of a proton can apparently alter the preference of the chiral-center orientation, we propose that it could template different chiralities of  self-assembled super-structures on the surface \\cite{LingenfelderThesis}.\n\n\\begin{figure}[hb!]\n\\includegraphics[width=\\textwidth]{F13_El_difference_Z.jpg}\n\\caption{Electronic-density difference averaged over the directions parallel to the surface for the lowest energy conformers of Arg adsorbed on Cu(111) (a), Ag(111) (b), and Au(111) (c), as well as of Arg-H$^+$ adsorbed on Cu(111) (d), Ag(111) (e), and Au(111) (f). Positive values (red) correspond to electron density accumulation and negative values (blue) correspond to electron density depletion.\nIn each panel, we also show a side and top view of the 3D electronic density rearrangement.  Blue isosurfaces correspond to an electron density of +0.05 e\/Bohr$^3$ and red isosurfaces to -0.05 e\/Bohr$^3$. \\label{fig:charge-re}}\n\\end{figure}\n\nWe then investigated the rearrangement of the electronic density upon binding of the molecules to the different surfaces. \nIn Fig. \\ref{fig:charge-re} we show the electronic density rearrangement created by the lowest energy conformer at each surface, integrated over the axis parallel to the surface, overlaid on the side-view of the 3D density rearrangement. In addition, we show a top view of the density rearrangement in each case. Examples of further conformers are summarized in the SI, Figs. S8-S13. The data shows that Arg donates electrons to the surface, while Arg-H$^+$ accepts electrons from the surface. We have checked this propensity for selected conformers by integration of the electronic density rearrangement around the molecule and by calculating the Hirshfeld charge remaining on the molecule for the full database (see SI, Table S6). When comparing Hirshfeld charges on the molecule and those obtained from the electronic density rearrangement, we observe that Hirshfeld charges are always 0.3-0.5 e underestimated. \nIn addition, we observe that the depletion and accumulation of charge is not uniform through the lateral extension of the molecule. \nThis behavior is consistent with the level alignment predicted by the PBE Kohn-Sham energy levels, as shown in Fig. S14 in the SI. However, we note that quantitative values of charge transfer are often inaccurate at this level of theory, as characterized in Refs. \\cite{Egger_2015, Liu_2017}. Optimally tuned range-separated hybrid functionals would yield more accurate values, but their computational cost is prohibitive for the use in this whole database. Nevertheless, hybrid-functional calculations of selected conformers (see SI, Fig. S15) confirm the qualitative trend. Therefore, we conclude that the protonation state again critically impacts these systems, in this case by qualitatively changing the redistribution of electronic charge.   \n\n\n\\begin{figure}[ht!]\n \\centering\n \\includegraphics[width=0.65\\textwidth]{Figure_DehydroEnergies.jpg}\n \\caption{Energy differences upon hydrogen dissociation for selected conformers (see text and SI) of Arg and Arg-H$^+$ on all metallic surfaces. $\\Delta E = E_{\\textbf{dep}}-E$, where $E_{\\textbf{dep}}$ is the total energy of the dissociated structure after optimization (including the adsorbed hydrogen) and $E$ the energy of the optimized intact structure. A negative $\\Delta E$ indicates that deprotonation is favored.}\n \\label{fig:deprotonation}\n \\end{figure}\n\n\nIt was observed experimentally that amino acids can undergo deprotonation on reactive surfaces \\cite{BARLOW1998322, BarlowRaval2004, METHIVIER201588, MATEOMARTI2002191, MARTI200485, EralpTrolle2010}. \nHere we also investigated whether deprotonation of Arg and Arg-H$^+$  was favorable on any of the surfaces studied here, the details of the procedure can be found in the SI. The results are summarized in Fig. \\ref{fig:deprotonation}. They show that deprotonation of Arg-H$^+$ is favorable in Cu(111), such that Arg-H$^+$ would be predominantely deprotonated. The population of deprotonated Arg on Cu(111) could reach about 0.1\\% of the molecules at room temperature, from a simple Arrhenius estimation. Given that we have not observed any spontaneous dissociation upon optimization of Arg-H$^+$ on Cu(111), we conclude that, although favorable, this dissociation of H does not occur without a barrier. In all other surfaces, the barrier for dissociation would be rather high for both molecules.\n\n\n\\section{Conclusions}\n\nIn this paper, we have characterized the conformational space of the arginine amino acid in its neutral and protonated form in different non-biological environments, i.e. in isolation and in contact with metallic surfaces. In particular, we have analyzed how and why different parts of the conformational space become accessible or excluded depending on the protonation state and the environment, showing the importance of bond formation and charge rearrangement in these systems. \n\nThis study included the construction of a database based on thousands of structures optimized by density-functional theory including dispersion interactions. The construction of this database is a result in itself. We found, for example, that it is advantageous to start from a comprehensive sampling of the conformational space of the least-constrained molecular form, which in our case was the neutral Arg amino acid in the gas-phase. This is evidenced by the fact that in our low-dimensional projections, all low-energy conformers we observe on the surfaces, for both Arg and Arg-H$^+$ lie among structural conformations that were already present in the gas-phase sampling of Arg, albeit often with high relative energies. In addition, we find that within Cu, Ag, and Au surfaces, relative energies between different conformers are largely preserved when changing the substrate.\n\nIn recent years much effort has been invested in developing and improving classical force fields (FF) for simulation of organic-inorganic interfaces, including peptides on surfaces \\cite{Iori2009, Wright2013, Heinz2013}. However, as we discuss in the SI, the existing force fields are not accurate for these scenarios where amino acids are in contact with metal surfaces and under vacuum conditions. Nevertheless, these situations are extremely relevant for scanning tunneling microscopy experiments \\cite{Rauschenbach2017, Rauschenbach2016, Abb2016, KoslowskiThesis} and other technological applications \\cite{WangGiovanni2013, GuoCahen2016}. As we illustrate in the SI, even though these force-fields can sample the relevant areas of conformational space, they are not able to capture consistent energy hierarchies. Additionally, the molecular chemical groups show a preference to adsorb on different surface sites, which could have considerable impact on self-assembly studies. Databases such as this one will serve as an important source of data for further parametrization and improvement of these potentials. \n\nRegarding the structural space of Arg and Arg-H$^+$ adsorbed on (111) surfaces of Cu, Ag and Au, we have learned the following:\nThe adsorption of Arg leads to the formation of strong bonds to the surface that involve mostly the carboxyl and amino groups. This stabilizes the protomer that we label \\textbf{P3} in this work, where the carboxyl group is deprotonated and the side chain is protonated. This is different to the dominant protomer in the gas phase \\textbf{P1}. The bonds to the surface sterically constrain the conformations of this molecule, thus decreasing the amount of structures\nwith respect to the numbers observed in the gas phase. \nThe molecule also donates electrons to the surface, becoming slightly positively charged. \nWe do not observe fully extended structures lying on the surface and most conformers exhibit intramolecular H-bonds. The majority of conformers of Arg in the low-energy region adsorbs with the C$_{\\alpha}$H chiral center pointing the hydrogen atom away from the surfaces.\n\nArginine in its protonated form, i.e. Arg-H$^+$ is the most abundant form of this amino-acid in biological environments, where it typically adopts the zwitterionic protomer \\textbf{P7}. \nIn the gas-phase, we observe that the non-zwitterionic state \\textbf{P6} is dominant and that the addition of the proton decreases the number of allowed conformations with respect to isolated Arg due to the added electrostatic interactions, and the passivation of the carboxyl group that would otherwise be involved in intramolecular H-bonds. \nUpon adsorption to the metallic surfaces, we observe that the protomer \\textbf{P6} is still dominant and that there are no strong bonds formed to the surface. In addition, this molecule receives electrons from the surface, thus becoming less positively charged. Both effects conspire to yield a homogeneous (flat) molecule-surface interaction and a relatively high population of different structures in the low-energy range. Contrary to Arg, most low-energy conformers of Arg-H$^+$ adsorb with the C$_{\\alpha}$-H chiral center pointing the hydrogen atom towards to the surfaces. Finally, through the calculation of dissociation energies, we also conclude that the deprotonation of Arg-H$^+$ is energetically favorable only on Cu(111). \n\nOur observations regarding the preferred protomers and deprotonation propensities discussed above are consistent with the observations in the literature that the adsorption of amino acids in their anionic and deprotonated form is common on reactive metals like Cu(111) \\cite{Costa:2015:IMPORTANT}. \nOne pronounced difference that we find among surfaces is the average adsorption height of the molecules: They follow the trend Cu(111) $<$ Ag(111) $<$ Au(111), and Arg is always closer than Arg-H$^+$ to the same respective surface.\n\nThe set of electronic-structure calculations presented here show that a flexible amino-acid like Arginine presents a rich conformational space involving different protomeric states and molecule orientations with respect to the surface, allied to a complex charge rearrangement. Going forward, it is clear that the likes of this study based solely on DFT cannot become a routine method due to the elevated computational cost. Addressing the whole breadth of amino acids as well as self assembly of these structures on surfaces will profit from this study as a benchmark and a means to develop models, possibly based on different machine-learning techniques \\cite{Todorovic2019, Oganov2009, Dieterich2010, fafoom, genarris, HORMANN2019, Olsson8265}, that can bypass the cost of thousands of DFT structure optimizations.\n\n\n\\section*{Availability of the data}\nThe data presented here is has been uploaded to the NOMAD repository \\cite{DOI_of_the_data_set_at_NOMAD}.\n\n\n\\section*{Acknowledgements}\n This work was supported by the Max Planck-EPFL Center for Molecular Nanoscience and Technology. We acknowledge Michele Ceriotti for insightful discussions and critical comments to the manuscript. We also acknowledge fruitful discussions with Yair Litman, Stephan Rauschenbach and Sabine Abb. MR also acknowledges support from the Max Planck Research Network on Big-Data-Driven Science (BigMAX).\n\n\n\n\\section{Details of the calculations}\n\nFor Cu, Ag and Au, the bulk lattice constants were determined by optimizing the fcc unit cell. The convergence criteria were set to 0.001 eV\/\\AA~ for the final forces, 10$^{-4}$ e\/Bohr$^{3}$ for the charge density, and 10$^{-5}$ eV for the total energy of the system. A 30$\\times$30$\\times$30 k-grid mesh  was used for the sampling of the Brillouin zone. Relativistic effects were considered by the zeroth order regular approximation (ZORA) \\cite{VanLenthe2000, VanWullen1998}. The values obtained with the PBE functional\\cite{Perdew1997} are in good agreement with previous works \\cite{Liu2013, Haas2009} and are shown in Table \\ref{tbl:Bulk_constants}. In that Table, we compare these values with lattice constants obtained when including pairwise van der Waals dispersion from the original Tkatchenko-Scheffler scheme (+vdW)\\cite{Tkatchenko2009} and from the one that includes an effective electronic screening optimized for metallic surfaces (+vdW$^{\\text{surf}}$)\\cite{Ruiz2016}\\footnote{We here used the original parameters published in Ref. \\cite{Ruiz2016}.}. \n\n\\begin{table}[h!]\n\\center\n \\caption{Lattice constants (in \\AA) of bulk metals determined with the PBE, PBE+vdW and PBE+vdW$^\\text{surf}$ functionals (\\textit{light} settings). }\n \\label{tbl:Bulk_constants}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nMethod     & Cu & Ag & Au \\\\\n\\hline\nPBE                   & 3.633     & 4.156   & 4.157   \\\\\n\\hline\nPBE+vdW               & 3.545     & 4.077     & 4.114               \\\\\n\\hline\nPBE+vdW$^{\\text{surf}}$ & 3.604     & 4.022   & 4.173    \\\\\n\\hline\nExp              & 3.603      & 4.069     & 4.065      \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\center\n\\caption{Fermi energies calculated with the PBE functional for the 4-layer slabs with (111) surface orientation used in our calculations of the binding energies of Arg-H$^+$ to the different surfaces. All values in eV.}\n\\label{tbl:fermi}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n& Cu & Ag  & Au \\\\ \\hline\nSlab $E_f$ & -4.73     & -4.30     & -5.02   \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\center\n\\caption{Relative binding energies (in eV) of relaxed Arg@Cu for different surface unit cell sizes with a 8$\\times$8$\\times$1 k-grid for the cell sizes less than 10$\\times$12 and 4$\\times$4$\\times$1 for the 10$\\times$12 unit cell. All numbers are reported with respect to the binding energy for the structure A modelled with a $5\\times6$ surface unit cell.}\n\\label{tbl:surf_unit_cell_ArgCu}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nsize     & A & B & C \\\\\n\\hline\n5$\\times$6                   & 0.000     &  0.011   & 0.216  \\\\\n\\hline\n6$\\times$6                   & -0.011     &   -0.013    & 0.190              \\\\\n\\hline\n6$\\times$7                   & -0.021     &  -0.030     & 0.174    \\\\\n\\hline\n10$\\times$12                 &   -0.048     &  -0.053  &       0.151  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{Conv_unit_cell_ArgCu.jpg}\n\\caption{Structures that were used for the surface unit cell size convergence test of Arg@Cu. Image unit cell size is $5 \\times 6$.}\n\\label{fig:surfaceunitcellArgCu}\n\\end{figure}\n\n\\begin{table}[h!]\n\\center\n \\caption{Relative binding energies (in eV) of relaxed Arg-H$^+$@Cu for different surface unit cell sizes with a 8$\\times$8$\\times$1 k-grid for the cell sizes less than 10$\\times$12 and 4$\\times$4$\\times$1 for the 10$\\times$12 unit cell. All numbers are reported with respect to the binding energy for the structure A modelled with a $5\\times6$ surface unit cell.}\n\\label{tbl:surf_unit_cell_ArgHCu}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nsize     & A & B & C \\\\\n\\hline\n5$\\times$6                   & 0.000     &  0.080   & 0.035  \\\\\n\\hline\n6$\\times$6                   & -0.050     &   0.041    & -0.017              \\\\\n\\hline\n6$\\times$7                   & -0.055     &  0.029     & -0.033    \\\\\n\\hline\n10$\\times$12                 & -0.044      & -0.007  &   -0.057     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{Conv_unit_cell_ArgHCu.jpg}\n\\caption{Structures that were used for surface unit cell size convergence test for Arg-H$^+$@Cu. Image unit cell size is $5 \\times 6$.}\n\\label{fig:surfaceunitcellArgHCu}\n\\end{figure}\n\\clearpage\n\n\\section{Family Classification According to Hydrogen Bond Patterns}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{F4_Hbond_Notations.jpg}\n\\caption{Labeling of all H-bond patterns considered in this manuscript.}\n\\label{fig:hbonds}\n\\end{figure}\n\\clearpage\n\\section{Structure space representation}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.99\\linewidth]{Many_conf_similar.jpg}\n\\caption{Representative conformers with similar backbone structure but different H-bonds within the molecule. The different H-bond pattern can cause energy differences of up to 0.2 eV for similar structures, as discussed in the main text.}\n\\label{fig:hbonds}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.99\\linewidth]{Families_on_surfaces.jpg}\n\\caption{Projection of Arg and Arg-H$^+$ conformers adsorbed on the different metalic surfaces on the low-dimensional map of gas-phase Arg, colored according to the H-bond pattern.}\n\\label{fig:hbonds}\n\\end{figure}\n\n\\begin{table}[]\n\\caption{Number of different families within 0.1\/0.5 eV energy range for different systems.}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n           & Arg    & Arg-H$^+$ & \\multicolumn{3}{c|}{Arg} & \\multicolumn{3}{c|}{Arg-H$^+$} \\\\ \\hline\nAtomnames    &        &      & Cu     & Ag     & Au     & Cu      & Ag     & Au     \\\\ \\hline\nNO         & 0 \/599 & 0\/2  & 0\/2    & 0\/0    & 0\/0    & 1 \/162  & 0\/124  & 0\/98   \\\\ \\hline\nN-N$\\varepsilon$        & 0 \/70  & 0\/11 & 5\/10   & 0\/9    & 1\/9    & 4 \/80   & 3\/79   & 3\/77   \\\\ \\hline\nN-N$\\eta1$        & 7 \/87  & 0\/20 & 0\/11   & 0\/12   & 0\/13   & 0 \/78   & 0\/78   & 0\/73   \\\\ \\hline\nN-N$\\varepsilon$, O1-N$\\eta1$   & 2 \/11  & 2\/4  & 1\/2    & 2\/6    & 4\/6    & 0 \/4    & 0\/4    & 0\/5    \\\\ \\hline\nO1-N$\\varepsilon$        & 2 \/115 & 0\/31 & 6\/56   & 8\/62   & 9\/56   & 11\/152  & 6\/146  & 6\/140  \\\\ \\hline\nO1-N$\\eta1$        & 16\/237 & 0\/37 & 0\/66   & 0\/70   & 0\/71   & 5 \/135  & 4\/115  & 5\/109  \\\\ \\hline\nO1-N$\\varepsilon$, O1-N$\\eta1$   & 5 \/27  & 0\/2  & 0\/1    & 0\/1    & 0\/2    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\varepsilon$   & 0 \/5   & 0\/1  & 0\/0    & 0\/0    & 0\/0    & 0 \/1    & 0\/1    & 0\/1    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\eta1$   & 0 \/8   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\varepsilon$, N-N$\\eta1$   & 0 \/8   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\varepsilon$, O1-N$\\varepsilon$   & 0 \/7   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\eta1$ & 0 \/2   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nO1-N$\\eta1$, O2-N$\\eta2$   & 0 \/3   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\\section{Harmonic free energies}\nFree energies were calculated in the harmonic approximation \\cite{McQuarrie, Fultz2010} for selected molecules adsorbed on surfaces within 0.1 eV range.\n\n$$\nF(\\textrm{T}) = E_{\\textrm{PES}} + F_\\textrm{vib}(\\textrm{T}) + F_\\textrm{rot}(\\textrm{T}),\n$$\n\nwhere $E_{\\textrm{PES}}$ is the total energy obtained with DFT (PBE+vdW$^{\\textrm{surf}}$ functional), and\n\n$$\nF_{\\mathrm{vib}}(\\mathrm{T})=\\sum_{i}^{3 N-6}\\left[\\frac{\\hbar \\omega_{i}}{2}+k_{\\mathrm{B}} T \\ln \\left(1-e^{-\\beta \\hbar \\omega_{i}}\\right)\\right],\n$$\n\nwhere N is the total number of atoms in the molecule (metal atoms were not displaced and were taken into account in external field), $k_\\textrm{B}$ is Boltzmann constant, $T$ is the temperature, $\\omega_i$ are vibrational frequencies obtained by diagonalization of Hessian matrix with use of developing version of phonopy-FHI-aims \\cite{phonopy, phonopy-fidaynyan} and\n \n$$\nF_\\textrm{rot}(\\textrm{T}) =-k_\\textrm{B} T \\textrm{ln} \\left[ \\frac{\\sqrt{\\pi}}{\\sigma} \\left(  \\frac{8\\pi^2 I k_\\textrm{B} T}{h^2}  \\right) \\right],\n$$\nwhere $I$ is the moment of inertia of the molecule obtained after diagonalization of the inertia tensor of the molecule. For the adsorbed conformers, rotational contributions are completely neglected since rotation around all principal axes of the molecule become internal vibrational modes of the system.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.9\\textwidth]{Free_energies_300.jpg}\n\\caption{Harmonic free energies calculated for adsorbed structures within the lowest 0.1 eV total-energy range. E$_{\\textbf{PES}}$ corresponds to the total energy of the system obtained at DFT level and F$_{\\textbf{harm}}$ corresponds to the free energy of the system at 300 K calculated as described above.}\n\\label{sup:Harmonic_free_energies_300}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{F11_UpDown_sketch.jpg}\n\\caption{Low dimensional maps of Arg and Arg-H$^+$ adsorbed on Cu(111), Ag(111) and Au(111) color-coded with respect to the orientation of the C$_\\alpha$H group. Blue correspond to \\textit{up} orientation and red correspond to \\textit{down} orientation of the C$_\\alpha$H group.}\n\\label{fig:UpDown_map}\n\\end{figure}\n\n\\clearpage\n\\section{Charge rearrangement while on the surface}\n\n\n\\begin{table}[ht]\n\\caption{Calculated charge on the molecule with use of Hirshfeld partial charge analysis and by integration of the electron density difference in the molecular region. Values are in electrons. Conformations are depicted in the following Figures S8-13.}\n\\center\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\nConformer & Hirshfeld & Integral & Conformer & Hirshfeld & Integral \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Cu}     & \\multicolumn{3}{c|}{Arg-H$^+$@Cu}     \\\\ \\hline\na     & 0.11      & 0.19     & a     & 0.29      & 0.85     \\\\ \\hline\nb     & 0.03      & 0.30     & b     & 0.30      & 0.85     \\\\ \\hline\nc     & 0.04      & 0.31     & c     & 0.31      & 0.84     \\\\ \\hline\nd     & 0.08      & 0.26     & d     & 0.43      & 0.88     \\\\ \\hline\ne     & 0.01      & 0.24     & e     & 0.46      & 0.85     \\\\ \\hline\nf     & 0.11      & 0.30     & f     & 0.38      & 0.82     \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Ag}     & \\multicolumn{3}{c|}{Arg-H$^+$@Ag}     \\\\ \\hline\na     & 0.04      & 0.15     & a     & 0.28      & 0.83     \\\\ \\hline\nb     & -0.08     & 0.23     & b     & 0.30      & 0.83     \\\\ \\hline\nc     & -0.03     & 0.24     & c     & 0.31      & 0.82     \\\\ \\hline\nd     & -0.06     & 0.21     & d     & 0.43      & 0.86     \\\\ \\hline\ne     & -0.13     & 0.16     & e     & 0.46      & 0.85     \\\\ \\hline\nf     & 0.05      & 0.14     & f     & 0.36      & 0.86     \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Au}     & \\multicolumn{3}{c|}{Arg-H$^+$@Au}     \\\\ \\hline\na     & 0.06      & 0.05     & a     & 0.32      & 0.86     \\\\ \\hline\nb     & -0.01     & 0.29     & b     & 0.29      & 0.86     \\\\ \\hline\nc     & 0.00      & 0.30     & c     & 0.34      & 0.85     \\\\ \\hline\nd     & -0.10     & 0.25     & d     & 0.48      & 0.91     \\\\ \\hline\ne     & 0.01      & 0.23     & e     & 0.49      & 0.90     \\\\ \\hline\nf     & 0.06      & 0.31     & f     & 0.43      & 0.92     \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgCu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Cu(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgCu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgAg.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Ag(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgAg}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgAu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Au(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgAu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHCu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Cu(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHCu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHAg.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Ag(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHAg}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHAu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Au(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHAu}\n\\end{figure}\n\nIn order to take a look in electronic level alignments of interface system after adsorption the projected, angular-momentum resolved partial density of states (pDOS) averaged over all atoms of each species  were calculated and normalized per molecule and per surface respectively. For corresponding isolated molecular geometry HOMO and LUMO levels were calculated and plotted together with interface pDOS. These calculations were performed with higher number of k-grid points: 6x6x1. Gaussian broadening was chosen to be 0.05. \n\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=0.8\\textwidth]{DOS_PDOS.jpg}\n\\caption{Projected densities of states of the lowest energy structures on each surface. Filled area corresponds to the occupied states below highest occupied state (VBM) of the whole system. HOMO (black solid line) and LUMO (black dashed line) are the states of the corresponding gas-phase molecular conformer calculated with the same geometry as it adopts when adsorbed. The Fermi energy of the pristine slab is depicted with blue dashed line.} \n\\label{sup:DOS_PDOS}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=1.0\\textwidth]{PBE0.jpg}\n\\caption{Side and top views of the adsorbed structures of a) Arg on Cu(111) (conformer c in Fig. S8) and b) Arg-H$^+$ on Cu(111) (conformer a in Fig. S11). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3 with PBE0 functional.} \n\\label{sup:DOS_PDOS}\n\\end{figure}\n\n\\clearpage\n\\section{Deprotonation on the surfaces}\n\nIn Arg, we found it most favorable to detach the proton from the guanidino group, while for Arg-H$^+$, it was most favorable to detach the proton from the carboxyl group. In both cases we note that the final adsorbed species is a hydrogen, i.e., it does not carry a positive charge. We chose three representative conformers  at each surface: the lowest energy structure and other two with different H-bonds within the molecule. We placed the detached proton at a distance of at least 2.5 \\AA~ from the molecule and fully optimized the dissociated structures. Comparing the energy difference between the final and initial states gives a lower limit for the dissociation barrier.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.9\\textwidth]{Deprotonation.jpg}\n\\caption{Representative structures that were analyzed for the calculation of the deprotonation energies. Colored structures represent the deprotonated relaxed structure. The green translucent structures represent the initial structure from which hydrogen was removed and placed on surface. The hydrogen that was removed is highlighted in bright green. $\\Delta E$ (see main text) is also reported in each panel.}\n\\label{sup:Deprotonation}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\includegraphics[width=0.9\\textwidth]{Deprotonation_ver_0.jpg}\n\\caption{All structures that were analyzed for the calculation of the deprotonation energies. $\\Delta E$ (see main text) is also reported in each panel.}\n\\label{supfig:Deprotonation_v0}\n\\end{figure}\n\n\\clearpage\n\n\\section{Comparison of DFT with INTERFACE-FF}\n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=1.05\\textwidth]{Energies_dft_vs_FF.jpg}\n  \\caption{Comparison of the relative energies obtained from DFT optimized structures and the same structures after post-relaxation in with the INTERFACE force field. Dots on the diagonal line represent an optimal correlation. The red area marks structures that lie in the lower 0.5 eV energy range in DFT but above the 0.5 eV energy range in INTERFACE-FF. The green area marks the structures that are in the lower 0.5 eV energy range regardless of the level of theory. The grey area marks the structures that are above the 0.5 eV energy range in DFT but below the 0.5 eV energy range in INTERFACE-FF.}\n  \\label{sup:RelEn_ArgHCu_DFTvsFF}\n\\end{figure}\n\n\n\nSelected local minima obtained at DFT level of theory were optimized with the  INTERFACE-FF\\cite{Heinz:2016:Review} \nusing the NAMD package \\cite{Phillips2005}. Calculations were performed with periodic boundary conditions with the same cell size and number of Cu atoms as in the DFT calculations. \nWe obtained parameters for certain protonation states as described in the following.\nFor the calculation of Arg, two protomers \\textbf{P1} and \\textbf{P3} (as denoted in the main text) had to be prepared.\nThey are called ``ARN'' (P1)  and  ``ARZ'' (P3) in the topology file that is provided. \n\nThe parametrization of\n``ARZ'' proceeded by taking the C-terminus in the deprotonated form (PRES CTER) and the N-terminus in the neutral form (PRES NNEU) from \\texttt{top-all36-prot.rtf}, such that the protonation is COO-NH2 with total charge 0. The partial charge of the guanidino group is +1. Other parameters were taken from ARG (\\texttt{top-all22-prot-metals.inp}) which is in the protonated form, by default, in CHARMM force field.\n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=0.95\\textwidth]{FF_DFT_sketchmaps.jpg}\n  \\caption{Low-dimensional map of the conformational space of the Arg and Arg-H$^+$ molecules adsorbed on the Cu(111) surface. The map was optimized considering all DFT and INTERFACE-FF structures. Green dots represent conformations obtained at DFT level of theory and red dots represent conformations obtained after geometry optimization with INTERFACE-FF. Close proximity of the dots reflects their structural similarity.}\n  \\label{sup:ArgCu_moleculeDFTvsFF_Sketchmap}\n\\end{figure}\n\nThe parametrization of ``ARN'' proceeded by taking the parameters for neutral C-terminus and N-terminus (PRES CNEU and PRES NNEU from \\texttt{top-all36-prot.rtf}) and the deprotonated methyl-guanidinium group (RESI MGU2) from \\texttt{par-all36-cgenff.rtf}. Atom types and parameters for MGU2 were copied from \\texttt{par-all36-cgenff.rtf}. The missing parameters related to joining MGU2 with the rest of Arg were manually added to the topology file. They were obtained from the corresponding values appearing in the protonated Arg, assuming that CG331 == CT2, NG311 == NC2, HGPAM1 == HC, NG2D1 == NC2, CG2N1 == C, where needed. The partial charge of CD atom was manually adjusted (decreased by 0.1) in order to have a neutral molecule with neutral guanidino group. Parameters for the neutral C-terminus and N-terminus were taken from the  \\texttt{top-all36-prot.rtf} file (PRES CNEU and PRES NNEU) and manually added to the customized file of topology. \n\n\n\nArgH named as ``ARX'' has total charge +1 (partial charge of guanidino group is also +1) and COOH-NH2 termini which is neutral. All the other parameters were directly taken from the INTERFACE-FF \\texttt{all22-prot-metals} topology and parameter files. \n\nWe conclude from Fig. \\ref{sup:RelEn_ArgHCu_DFTvsFF} that DFT (PBE+vdW$^{\\text{surf}}$) and the INTERFACE-FF yield very different energy hierarchies. However, from Fig. \\ref{sup:ArgCu_moleculeDFTvsFF_Sketchmap}, we conclude that both levels of theory represent a similar conformational space. However, Table \\ref{sup:table-ff-dft-adsorption-sites} shows that DFT and the FF yield different adsorption site preferences for the amino and carboxyl groups. In particular, DFT predicts that O will adsorb almost exclusively on top sites, consistent with the accepted adsorption site preference of CO groups on the pristine Cu(111) surface. The FF predicts a larger population of other adsorption sites, in particular hollow sites, compared to DFT. \n\n\\begin{table}[htbp]\n\\center\n\\caption{Surface site adsorption preferences of chosen chemical groups in Arg and Arg-H$^+$. All numbers are reported as a percentage of the total number of conformers optimized with DFT (PBE+vdW$^{\\text{surf}}$) and the INTERFACE force field. \\label{sup:table-ff-dft-adsorption-sites}}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n\\hline\n                & \\multicolumn{4}{c|}{Arg@Cu(111)}               & \\multicolumn{4}{c|}{Arg-H$^+$@Cu(111)}                     \\\\ \\hline\n                & \\multicolumn{2}{c|}{Amino} &  \\multicolumn{2}{c|}{Carboxyl}    & \\multicolumn{2}{c|}{Amino} & \\multicolumn{2}{c|}{Carboxyl} \\\\ \\hline\nAdsorption site & DFT          & FF          & DFT        & FF   & DFT          & FF          & DFT            & FF           \\\\ \\hline\nTop             & 80           & 53          & 76         & 48   & 59           & 50          & 70             & 45           \\\\ \\hline\nBridge          & 9            & 18          & 14         & 18   & 18           & 20          & 15             & 22           \\\\ \\hline\nHollow-FCC      & 5            & 13          & 4          & 17   & 13           & 15          & 7              & 16           \\\\ \\hline\nHollow-HCP      & 6            & 16          & 5          & 17   & 10           & 15          & 9              & 18           \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect1}\n\nIn recent years there has been a revived interest in 2D shell models because of unconventional materials and extremely small aspect to thickness ratio, such as for instance thin polymeric films or biological membranes. For classical engineering materials and for non-extreme aspect to thickness ratios, available 3D FEM-Codes may readily be used such that the need for a truly 2D shell model does not arise anymore. However, for ultrathin specimens the application of a 3D  constitutive law is not  clear at all.  In these extreme cases one is led to employ a 2D shell model.\nThis paper is concerned with one such model, the geometrically non-linear resultant theory of shells. We consider the 6-parameter model of shells which involves two independent kinematic fields: the translation vector field and the rotation tensor field (in total 6 independent scalar kinematic variables). This theory of shells is one of the most general, and it is also very effective in the treatment of complex shell problems, as can be seen from the works \\cite{Pietraszkiewicz10,Eremeyev11,Pietraszkiewicz11}, among others. The resultant  6-parameter theory of shells was originally proposed by Reissner \\cite{Reissner74} and it has been considerably developed subsequently. An account of these developments and main achievements have been presented in the books of Libai and Simmonds \\cite{Libai98} and    Chr\\'o\\'scielewski, Makowski and  Pietraszkiewicz \\cite{Pietraszkiewicz-book04}.\nIn this approach,\nthe 2D equilibrium equations and static boundary conditions of the shell are derived exactly\nby direct through-the-thickness integration of the stresses in the 3D balance laws of linear and angular momentum. The kinematic fields are then constructed on the 2D level using the integral identity of the virtual work principle. Following this procedure, the 2D model is expressed in terms of stress resultants and work--averaged deformation fields defined on the shell base surface. It is interesting  that the kinematical structure of 6-parameter shells (involving the translation vector and rotation tensor) is identical to the kinematical structure of Cosserat shells (defined as material surfaces endowed with a triad of rigid directors describing the orientation of points). From this point of view, the 6-parameter theory of shells is related to the shell model proposed initially by the Cosserat brothers \\cite{Cosserat09neu} and developed by many authors, such as  Zhilin \\cite{Zhilin76}, Zubov \\cite{Zubov97}, Altenbach and Zhilin \\cite{Altenbach04}, Eremeyev and Zubov \\cite{Eremeyev08}, B\\^{\\i}rsan and Altenbach \\cite{Birsan10}. Using the so--called derivation approach, Neff \\cite{Neff_plate04_cmt,Neff_plate07_m3as} has established independently a Cosserat--type model for initially planar shells (plates) which is very similar to the  6-parameter resultant shell model. A   comparison between these two models has been presented in the paper \\cite{Birsan-Neff-Plates}, in the case of plates.\n\nOn the other hand, we should mention that the kinematic structure of the 6-parameter shell model is different from the kinematic structure of the so--called \\emph{Cosserat surfaces}, which are defined as material surfaces with one or more deformable directors attached to every point, see \\cite{Naghdi72,Antman95,Rubin00,Rubin04,Altenbach-Erem12,Altenbach-Erem-Review}. For instance, the kinematics of Cosserat surfaces with one deformable director  is characterized also by 6 degrees of freedom (3 for the position of material points and 3 for the orientation and stretch of the material line element through the thickness), which differ essentially from the 6 degrees of freedom in the 6-parameter resultant shell model.\n\nThe topic of existence of solutions for the 2D equations of linear and non-linear elastic shells has been treated in many works.   The results that can be found in the literature refer to various types of shell models and they employ different techniques, see e.g. \\cite{Koiter60,Simo89.1,Simo92,Steigmann08,Steigmann12,Sansour92,Aganovic06,Aganovic07,Tiba02,Davini75,Birsan08,Wisniewski10,Wisniewski12,Ibrahim94,Badur-Pietrasz86}. The method of formal asymptotic expansions is one method of investigation which allows   for the derivation and justification of plate and shell models. The existence theory for linear or nonlinear shells is presented in details in the books of Ciarlet \\cite{Ciarlet97,Ciarlet98,Ciarlet00}, together with many historical remarks and bibliographic references. Another fruitful approach to the existence theory of 2D plate and shell models (obtained as limit cases of 3D models) is the $\\Gamma$-convergence analysis of thin structures, see e.g. \\cite{Neff_Chelminski_ifb07,Neff_Danzig05,Neff_Hong_Reissner08,Paroni06,Paroni06b}. Concerning the geometrically  non-linear 6-parameter theory of elastic shells, there is no existence theorem published in the literature yet, as far as we are aware of. In the case of \\emph{linear} micropolar shells, the existence of weak solutions has been recently proved in \\cite{EremeyevLebedev11}. Existence results for the related (very similar) Cosserat--type model of initially planar shells have been established by Neff \\cite{Neff_plate04_cmt,Neff_plate07_m3as}. In \\cite{Neff_Chelminski_ifb07,Neff_Danzig05,Neff_Hong_Reissner08} the linearized version of this model has been analyzed and compared with the\nclassical membrane and bending plate models given by the Reissner--Mindlin or Kirchhoff--Love theories.\n\nIn the present work, we prove the existence of minimizers for the minimization problem of the total potential energy associated to the deformation of geometrically non-linear 6-parameter elastic shells. We emphasize that our work is not concerned with the derivation of the 2D shell model, but it presents existence results for the well-established 2D theory of 6-parameter elastic shells. It should be mentioned from the beginning that this model refers to shells made of a simple (classical) elastic material, not a generalized (Cosserat or micropolar) continuum. However, the rotation tensor field appears naturally in this theory, in the course of the exact through-the-thickness reduction of the 3D formulation of the problem to the 2D one \\cite{Libai98,Pietraszkiewicz-book04,Eremeyev06}. Thus, in spite of the above mentioned similarity with the kinematics of Cosserat shells, the material of the shell in the resultant 6-parameter model is described as a simple continuum (without any specific microstructure or material length scale). On the other hand, in the case of dimensional reduction of the 3D equations of micropolar shell-like bodies one can obtain the same 6-parameter theory with modified 2D constitutive equations, see e.g. \\cite{Altenbach-Erem09} for the linear case and \\cite{Neff_plate04_cmt,Neff_plate07_m3as,Zubov09} for the nonlinear case, or one can obtain more complex theories as in \\cite{Eringen67}.\n\nFor the proof of existence, we employ the direct methods of the calculus of variations and extend the techniques presented in  \\cite{Neff_plate04_cmt,Neff_plate07_m3as} to the case of general shells (with non-vanishing curvature in the reference configuration). In Section 2 we present briefly the kinematics of general  6-parameter shells and the equations of equilibrium. In Section 3 we give some alternative formulas for the strain tensor and  curvature tensor, which are written in the direct tensorial notation as well as in the component (matrix) notation. These expressions are needed subsequently in the proof of our main result.\nIn Section 4 we formulate the two-field minimization problem for general elastic shells, corresponding to mixed--type boundary conditions. Under the assumptions of convexity and coercivity of the quadratic strain energy function (physically linear material response), we prove the existence of minimizers over a large  set of admissible pairs. Thus, the minimizing solution pair is of class $\\boldsymbol{H}^1(\\omega, \\mathbb{R}^3)$ for the translation vector and   $\\boldsymbol{H}^1(\\omega, SO(3))$ for the rotation tensor. The existence result is valid for general anisotropic elastic shells having arbitrary geometry of the reference configuration.\nSection 5 includes some applications of the existence theorem and discussions of special cases. We present a convenient way to choose the initial directors and the parametrization of the reference surface. Then, we consider separately the cases of isotropic shells, orthotropic shells, and composite layered shells and we present   the respective forms of the strain energy densities. Applying the theorem stated previously, we establish the conditions on the constitutive coefficients that ensure the existence of minimizers in each situation. This analysis shows the usefulness of our theoretical result in the treatment of practical  problems for elastic shells.\n\n\n\n\n\\section{General 6-parameter resultant shells}\\label{sect2}\n\n\n\nConsider a general 6-parameter shell and denote with $S^0$ the base surface of the shell in the reference (initial) configuration and with $S$ the base surface in the deformed configuration. Let $O$ be a fixed point in the Euclidean space and $\\{\\boldsymbol{e}_1,\\boldsymbol{e}_2,\\boldsymbol{e}_3\\}$ the fixed orthonormal vector basis. The reference configuration is represented by the position vector $\\boldsymbol{y}^0$ (relative to the point $O$) of the base surface $S^0$ \\emph{plus} the structure tensor $\\boldsymbol{Q}^{0}$. The structure tensor is a second order proper orthogonal tensor which can be described by an orthonormal triad of directors  $\\{\\boldsymbol{d}^0_1,\\boldsymbol{d}^0_2,\\boldsymbol{d}^0_3\\}$  attached to every point \\cite{Pietraszkiewicz-book04,Eremeyev06}. Thus the reference (initial) configuration is characterized by the functions\n\\begin{equation}\\label{1}\n\\begin{array}{l}\n    \\,\\boldsymbol{y}^0:\\omega\\subset \\mathbb{R}^2\\rightarrow\\mathbb{R}^3,\\qquad\\qquad\\,\\,\\boldsymbol{y}^0=\\boldsymbol{y}^0(x_1,x_2) ,\\\\\n    \\boldsymbol{Q}^{0}:\\omega\\subset \\mathbb{R}^2\\rightarrow SO(3), \\qquad\\,\\,\\, \\boldsymbol{Q}^{0} = \\boldsymbol{d}_i^0 (x_1,x_2)\\otimes \\boldsymbol{e}_i\\,,\n    \\end{array}\n\\end{equation}\nwhere thus $(x_1,x_2)$ are material curvilinear coordinates on the surface $S^0\\,$. Throughout the paper Latin indexes $i,j,...$ take the values $\\{1,2,3\\}$, while Greek indexes $\\alpha,\\beta,...$ the values $\\{1,2\\}$. The usual Einstein summation convention over repeated indexes is employed. We assume that the curvilinear coordinates $(x_1,x_2)\\in\\omega$  range over a bounded open domain $\\omega$ (with Lipschitz boundary $\\partial\\omega$) of the $Ox_1x_2$ plane, see Figure \\ref{Fig1}.\nLet us denote the partial derivative with respect to $x_\\alpha$ by $\\partial_\\alpha f=\\frac{\\partial f}{\\partial x_\\alpha}\\,\\,$, for any function $f$. We designate by $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2 \\}$ the (covariant) base vectors in the tangent plane of $S^0$ and by $\\boldsymbol{n}^0$ the unit normal to $S^0$ given by\n\\begin{equation}\\label{2}\n   \\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0 = \\dfrac{ \\partial\\boldsymbol{y}^0}{\\partial x_\\alpha}\\,,\\qquad \\boldsymbol{n}^0=  \\dfrac{ \\boldsymbol{a}_1\\times\\boldsymbol{a}_2}{ \\| \\boldsymbol{a}_1\\times\\boldsymbol{a}_2\\|}\\,\\,.\n\\end{equation}\nThe reciprocal (contravariant) basis $\\{\\boldsymbol{a}^1,\\boldsymbol{a}^2 \\}$ of the tangent plane is defined by\n$ \\boldsymbol{a}^\\alpha\\cdot\\boldsymbol{a}_\\beta =\\delta^\\alpha_\\beta$ (the Kronecker symbol). We also use the notations\n$$\\boldsymbol{a}_3=\\boldsymbol{a}^3=\\boldsymbol{n}^0, \\quad a_{\\alpha\\beta}=\\boldsymbol{a}_\\alpha\\cdot \\boldsymbol{a}_\\beta\\,,\\quad a^{\\alpha\\beta}=\\boldsymbol{a}^\\alpha\\cdot\\boldsymbol{a}^\\beta,\\quad a=\\sqrt{\\det(a_{\\alpha\\beta})_{2\\times 2}}>0.$$\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{Fig1}\n\\put(-398,43){\\small{$O$}} \\put(-411,20){\\small{$\\boldsymbol{e}_1$}} \\put(-378,45){\\small{$\\boldsymbol{e}_2$}}\n\\put(-432,2){$x_1$} \\put(-316,31){$x_2$}\n    \\put(-415,64){\\small{$\\boldsymbol{e}_3$}} \\put(-372,14){$\\omega$}\n    \\put(-357,88){$\\boldsymbol{y}^0(x_1,x_2)$} \\put(-228,82){$\\boldsymbol{R}(x_1,x_2)$}\n    \\put(-195,60){$\\boldsymbol{y}(x_1,x_2)$} \\put(-410,105){$\\boldsymbol{Q}^{0}(x_1,x_2)$}\n    \\put(-293,130){${S}^0$} \\put(-264.5,145){\\small{$\\boldsymbol{d}^0_1$}} \\put(-264,173){\\small{$\\boldsymbol{d}^0_2$}} \\put(-311,191){\\small{$\\boldsymbol{d}^0_3$}}\n    \\put(-67.5,173.5){\\small{$\\boldsymbol{y} $}} \\put(-33,167){$\\boldsymbol{d}_1$} \\put(-39.5,195){\\small{$\\boldsymbol{d}_2$}} \\put(-81,202){\\small{$\\boldsymbol{d}_3$}}\n\\put(-220,204){$\\boldsymbol{\\chi}( \\boldsymbol{y}^0)$}\n\\put(-220,225){$\\boldsymbol{Q }^e( \\boldsymbol{y}^0)$} \\put(-307,150){\\small{$\\boldsymbol{y}^0$}}\n \\put(-72,120){$ {S}$}\n\\caption{The base surface $S^0$ of the shell in the initial configuration, the base surface $S$ in the deformed configuration, and the fictitious planar reference configuration $\\omega$. The orthonormal triads of vectors $\\{\\boldsymbol{e}_i\\}\\,$, $\\{\\boldsymbol{d}_i^0 \\}$ and $\\{\\boldsymbol{d}_i\\}$ are related through the relations $ \\boldsymbol{d}_i=\\boldsymbol{Q }^e\\boldsymbol{d}_i^0=\\boldsymbol{R}\\boldsymbol{e}_i \\,$ and $\\boldsymbol{d}_i^0=\\boldsymbol{Q}^{0}\\boldsymbol{e}_i\\,$, where $\\boldsymbol{Q}^e  $ is the elastic rotation field, $\\boldsymbol{Q}^{0}$ is the  initial rotation, and $\\boldsymbol{R}$ is the total rotation field.}\n\\label{Fig1}      \n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{Fig2}\n     \\put(-354,14){$\\omega\\times (\\!-\\frac{h}{2}\\,,\\frac{h}{2}) $}\n    \\put(-271,115){$ \\boldsymbol{\\Theta}\\big( \\omega\\times(\\!-\\frac{h}{2}\\,,\\frac{h}{2})\\big) $}\n \\put(-94,105){$ \\boldsymbol{\\varphi}\\big( \\omega\\times(\\!-\\frac{h}{2}\\,,\\frac{h}{2})\\big) $}\n \\put(-294,63){$\\boldsymbol{F}^0\\!=\\!\\boldsymbol{P} $}\n    \\put(-172,16){$ \\bar{\\boldsymbol{F}}$}\n \\put(-150,125){$ \\boldsymbol{F}^e$}\n\\caption{\\small{Multiplicative decomposition of the total deformation gradient $\\,\\bar{F}\\,=\\,F^eF^0\\,$ into the elastic shell deformation gradient $F^e$ and the initial deformation gradient $F^0=P$. Interpretation in terms of reconstructed three-dimensional quantities. The elastic response is governed by $F^e$. The curved initial configuration corresponds to the intermediate stress free configuration in multiplicative plasticity.}}\n\\label{Fig2}      \n\\end{center}\n\\end{figure}\n\n\nFor the deformed configuration of the shell, let $\\boldsymbol{y}(x_1,x_2)$ denote the position vector (relative to $O$) and $\\{ \\boldsymbol{d}_i(x_1,x_2)\\}$ the orthonormal triad of directors attached to the point with initial curvilinear coordinates $(x_1,x_2)$. The deformed configuration is completely characterized by the functions\n\\begin{equation}\\label{3}\n    \\boldsymbol{y}=\\boldsymbol{\\chi}(\\boldsymbol{y}^0),\\qquad \\boldsymbol{Q}^e= \\boldsymbol{d}_i\\otimes \\boldsymbol{d}_i^0\\in SO(3),\n\\end{equation}\nwhere $\\boldsymbol{\\chi}:S^0\\rightarrow\\mathbb{R}^3$ represents the deformation of the base surface and the proper orthogonal tensor field $\\boldsymbol{Q}^e$ is the (effective)  elastic rotation. The displacement vector is defined as usual by $\\boldsymbol{u}= \\boldsymbol{y}-\\boldsymbol{y}^0\\,$.\n\nWe mention that the role of the triads of directors $\\{\\boldsymbol{d}_i^0 \\}$ and $\\{\\boldsymbol{d}_i\\}$ is to determine the structure tensor $\\boldsymbol{Q}^{0}$ and the rotation tensor $\\boldsymbol{Q}^{e}$ of the shell, respectively. Thus, the directors do not describe here any microstructure of the material. According to the derivation procedure of the 6-parameter shell model, the kinematical fields $\\boldsymbol{y}$ and $\\boldsymbol{Q}^e$ are uniquely defined as the work--conjugate averages of 3D deformation distribution over the shell thickness, whose virtual values enter the virtual work principle of the shell (see \\cite{Pietraszkiewicz-book04,Pietraszkiewicz-book02}).\n\nIn view of \\eqref{1} and \\eqref{3}, the deformed configuration can alternatively be characterized by the functions\n$$\\boldsymbol{y}=\\boldsymbol{y}(x_1,x_2)= \\boldsymbol{\\chi}\\big( \\boldsymbol{y}^0(x_1,x_2)\\big),\\qquad\n\\boldsymbol{R}(x_1,x_2)=\\boldsymbol{Q}^e\\boldsymbol{Q}^{0}= \\boldsymbol{d}_i(x_1,x_2)\\otimes \\boldsymbol{e}_i\\in SO(3),$$\nwhere the vector $\\boldsymbol{y}$ and the orthogonal tensor $\\boldsymbol{R}$ are fields defined over $\\omega$.  The  orthogonal tensor field   $\\boldsymbol{Q}^e$ represents the \\emph{elastic rotation} tensor between the reference and deformed configurations \\cite{Pimenta04,Pimenta09}. The tensor $\\boldsymbol{Q}^{0}$ is the     \\emph{initial rotation}  field, while $\\boldsymbol{R}= \\boldsymbol{Q}^e\\boldsymbol{Q}^{0} $ describes the\n\\emph{total rotation} from the fictitious planar reference configuration $\\omega$ (endowed with the triad $\\{\\boldsymbol{e}_i\\}$) to the deformed configuration $S$. We mention that the tensors $\\boldsymbol{Q}^{0}$   and $\\boldsymbol{R}$ are also called the \\emph{structure tensors} of the reference and deformed configurations, respectively \\cite{Pietraszkiewicz-book04,Eremeyev06}.\nThe following relations hold\n\\begin{equation}\\label{4}\n    \\boldsymbol{Q}^e=\\boldsymbol{R}\\boldsymbol{Q}^{0,T}\\,,\\qquad \\boldsymbol{d}_i^0=\\boldsymbol{Q}^{0} \\boldsymbol{e}_i\\,,\\qquad \\boldsymbol{d}_i=\\boldsymbol{Q}^e\\boldsymbol{d}_i^0= \\boldsymbol{R}  \\boldsymbol{e}_i\\,.\n\\end{equation}\nUsually, the initial directors $\\,\\boldsymbol{d}_i^0\\,$ are chosen such that $\\,\\boldsymbol{d}_3^0 = \\boldsymbol{n}^0$ and $ \\boldsymbol{d}_\\alpha^0$ belong to the tangent plane of $S^0$ (see Remark \\ref{rem6}). This assumption is not necessary in general and we do not use it in the proof of our existence result.\n\nLet $\\boldsymbol{F}=\\text{Grad}_s\\boldsymbol{y}=\\partial_\\alpha \\boldsymbol{y}\\otimes\\boldsymbol{a}^\\alpha$ denote the (total) shell deformation gradient tensor. The strong form of the equations of equilibrium for 6-parameter shells can be written in the form \\cite{Eremeyev06}\n\\begin{equation}\\label{5}\n    \\mathrm{Div}_s\\, \\boldsymbol{N}+\\boldsymbol{f}=\\boldsymbol{0},\\qquad \\mathrm{Div}_s\\, \\boldsymbol{M} + \\mathrm{axl}(\\boldsymbol{N}\\boldsymbol{F}^T-\\boldsymbol{F}\\boldsymbol{N}^T)\n    +\\boldsymbol{c}=\\boldsymbol{0},\n\\end{equation}\nwhere $\\boldsymbol{N}$ and $\\boldsymbol{M}$ are the internal surface stress resultant and stress couple tensors of the 1$^{st}$ Piola--Kirchhoff type, while $\\boldsymbol{f}$ and $\\boldsymbol{c}$ are the external surface resultant force and couple vectors applied to points of $S$, but measured per unit area of $S^0\\,$. The operators Grad$_s$ and  Div$_s$ are the surface gradient and surface divergence, respectively, defined intrinsically in \\cite{Gurtin-Murdoch-75,Murdoch-90}. The superscript $(\\cdot)^T$ denotes the transpose and axl$(\\,\\cdot)$ represents the axial vector of a skew--symmetric tensor.\n\nLet $\\boldsymbol{\\nu}$ be the external unit normal vector to the boundary curve $\\partial S^0$ lying in the tangent plane. We consider boundary conditions of the type \\cite{Pietraszkiewicz04,Pietraszkiewicz11}\n\\begin{equation}\\label{6}\n\\begin{array}{l}\n\\boldsymbol{N}\\boldsymbol{\\nu}=\\boldsymbol{n}^*,\\qquad \\boldsymbol{M}\\boldsymbol{\\nu}=\\boldsymbol{m}^*\\qquad\\mathrm{along}\\,\\,\\,\\partial S^0_f\\,,\\\\\n    \\quad\\boldsymbol{y}=\\boldsymbol{y}^* ,\\qquad\\quad\\,\\, \\boldsymbol{R}=\\boldsymbol{R}^* \\qquad\\mathrm{along}\\,\\,\\,\\partial S^0_d\\,,\n    \\end{array}\n\\end{equation}\nwhere $\\partial S^0=\\partial S^0_f\\cup \\partial S^0_d $ is a disjoint partition  of $S^0\\,$ ($\\partial S^0_f\\cap \\partial S^0_d=\\emptyset$) with length($\\partial S^0_d )>0$. Here, $\\boldsymbol{n}^*$ and $\\boldsymbol{m}^*$ are the external boundary resultant force and couple vectors respectively, applied along the deformed boundary $\\partial S$, but measured per unit length of $\\partial S^0_f\\subset\\partial S^0\\,$. We denote by $\\partial \\omega_f$ and $\\partial \\omega_d$ the subsets of the boundary curve $\\partial \\omega$ which correspond to $\\partial S^0_f$ and $\\partial S^0_d$ respectively, through the mapping $\\boldsymbol{y}^0\\,$.\n\nThe weak form associated to these local balance equations for shells has been presented in \\cite{Libai98,Pietraszkiewicz-book04,Pietraszkiewicz04}.\n\n\n\n\n\n\n\\section{Elastic shell strain and curvature measures}\\label{sect3}\n\n\n\nAccording to \\cite{Eremeyev06,Pietraszkiewicz-book04}, the elastic shell strain tensor $\\boldsymbol{E}^e$ in the material representation is given by\n\\begin{equation}\\label{7}\n    \\boldsymbol{E}^e=\\boldsymbol{Q}^{e,T}\\text{Grad}_s\\,\\boldsymbol{y}- \\text{Grad}_s\\,\\boldsymbol{y}^0=\\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}- \\partial_\\alpha \\boldsymbol{y}^0 \\big) \\otimes \\boldsymbol{a}^\\alpha\\,,\n\\end{equation}\nsince $\\text{Grad}_s\\boldsymbol{y}=\\partial_\\alpha \\boldsymbol{y} \\otimes\\boldsymbol{a}^\\alpha\\,$.\nIt is useful to write  the strain tensor $\\boldsymbol{E}^e$ in the alternative form\n\\begin{equation*}\n\\begin{array}{l}\n\\boldsymbol{E}^e=\\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y} - \\boldsymbol{a}_\\alpha\\big) \\otimes \\boldsymbol{a}^\\alpha=  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{a}^\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{a}^3\\big)- \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{a}^i\\big) \\\\\n\\quad\\,\\,\\, =  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big) - {1\\!\\!\\!\\:1 } _3\n\\end{array}\n\\end{equation*}\nor equivalently, since $\\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big)= \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1}$,\n\\begin{equation}\\label{13}\n\\begin{array}{l}\n    \\boldsymbol{E}^e  =  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1} - {1\\!\\!\\!\\:1 } _3 \\,\\,\\,=\\,\\, \\, \\overline{\\boldsymbol{U}}^{\\,e} - {1\\!\\!\\!\\:1 } _3\\,\\,,\\vspace{4pt}\\\\\n    \\quad\\text{with}\\qquad \\overline{\\boldsymbol{U}}^{\\,e}= \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1},\n    \\end{array}\n\\end{equation}\nwhere $ {1\\!\\!\\!\\:1 } _3=\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_i$ is the identity tensor and  $\\overline{\\boldsymbol{U}}^{\\,e}$ represents the non-symmetric \\emph{elastic shell stretch tensor}, which can be seen as the 2D analog of the 3D non-symmetric Biot--type stretch tensor \\cite{Neff_Biot07} or the first Cosserat deformation tensor \\cite[page 123, eq. (43)]{Cosserat09neu} for the shell.\nLet us denote by  $\\boldsymbol{P}$  the tensor defined by\n\\begin{equation}\\label{14}\n    \\boldsymbol{P}  =  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i = \\partial_\\alpha \\boldsymbol{y}^0\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\,.\n\\end{equation}\nThen, from \\eqref{13} and \\eqref{14} we get\n\\begin{equation}\\label{14,1}\n\\begin{array}{l}\n    \\boldsymbol{E}^e =\\, \\overline{\\boldsymbol{U}}^{\\,e} - {1\\!\\!\\!\\:1 } _3\\, = \\,\\boldsymbol{Q}^{e,T} \\big( \\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^{e}\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\boldsymbol{P}^{-1}- {1\\!\\!\\!\\:1 } _3\\,,\\vspace{4pt}\\\\\n    \\qquad\\,\\,\\,\\,\\overline{\\boldsymbol{U}}^{\\,e}= \\,\\boldsymbol{Q}^{e,T} \\big( \\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^{e}\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\boldsymbol{P}^{-1}.\n    \\end{array}\n\\end{equation}\nIn the sequel, it is useful to write the elastic shell strain tensors   in component form, relative to the fixed tensor basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$. Let $E^e=\\big(E^e_{ij}\\big)_{3\\times 3}$   be the matrix of components for the tensor $\\boldsymbol{E}^e=E^e_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\,$. In general, we decompose any second order tensor $\\boldsymbol{T}$ in the form $\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j$ and denote by  $T=\\big(T_{ij}\\big)_{3\\times 3}$ the matrix of components. Also, for any vector\n$\\boldsymbol{v}=v_{i }\\boldsymbol{e}_i$ we designate by $v=\\big(v_{i }\\big)_{3\\times 1}$ the column matrix of components.\n\\begin{remark}\\label{rem1}\nThe matrix of components $P=\\big(P_{ij}\\big)_{3\\times 3}$ for the tensor defined in \\eqref{14} can be specified in terms of its 3 column vectors as follows\n\\begin{equation}\\label{14bis}\n    P=\\Big(\\,\\partial_1 y^{0}\\,\\Big|\\, \\partial_2 y^{0}\\,\\Big|\\, n^0\\,\\Big)_{3\\times 3}=\\Big(\\,\\nabla y^{0}\\,\\Big|\\, n^0\\,\\Big)_{3\\times 3}= \\Big(\\,a_{1}\\,\\Big|\\, a_{2}\\,\\Big|\\, n^0\\, \\Big)_{3\\times 3}\\,\\,\\,.\n\\end{equation}\nWe mention that the tensor\n$\\boldsymbol{P}$ introduced in \\eqref{14} can be seen as a three-dimensional  (deformation) gradient\n\\begin{equation}\\label{14tris}\n\\begin{array}{c}\n\\boldsymbol{P}= \\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,x_3)_{\\big|x_3=0}\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\Theta}(x_1,x_2,x_3):=\\boldsymbol{y}^0(x_1,x_2)+x_3\\,\\boldsymbol{n}^0(x_1,x_2),\n\\end{array}\n\\end{equation}\nand it satisfies  $\\det\\boldsymbol{P} =  \\sqrt{\\det a_{\\alpha\\beta}}  =a>0$, so that the inverse $\\boldsymbol{P}^{-1}$ exists.\nThe mapping $\\,\\boldsymbol{\\Theta}:\\,\\omega\\times\\big(\\!-\\frac{h}{2}\\,,\\frac{h}{2}\\,\\big)\\rightarrow\\mathbb{R}^3$  has been introduced previously in \\cite{Ciarlet00,Ciarlet05,Neff_plate04_cmt} and employed for the geometrical description of 3D shells ($h$ denotes the thickness of the shell).\n\\end{remark}\nBy virtue of \\eqref{13} and \\eqref{14}, we obtain the following  matrix form for the strain tensor $\\boldsymbol{E}^e$\n\\begin{equation}\\label{15}\n    E^e= \\Big(\\,Q^{e,T}\\partial_1 y\\,\\Big|\\, Q^{e,T}\\partial_2 y\\,\\Big|\\, n^0\\,\\Big) \\,P^{-1} -  {1\\!\\!\\!\\:1 } _3\\,,\n\\end{equation}\nwhere $ {1\\!\\!\\!\\:1 } _3=\\big(\\delta_{ij}\\big)_{3\\times 3}$ is the unit matrix. The matrix $E^e$ can be written equivalently as\n\\begin{equation}\\label{15,1}\n\\begin{array}{l}\n    E^e \\,=\\, Q^{e,T}\\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} -  {1\\!\\!\\!\\:1 } _3\\,,\\qquad\\text{or}\\vspace{4pt}\\\\\n    E^e\\,=\\,\\, \\overline{U}^{\\,e}- {1\\!\\!\\!\\:1 } _3 \\,\\,= \\,\\, Q^{e,T}\\,F^e -  {1\\!\\!\\!\\:1 } _3 \\,\\, = \\,\\, Q^{e,T}\\bar{F}  \\,P^{-1}-   {1\\!\\!\\!\\:1 } _3\\,,\n    \\end{array}\n\\end{equation}\nwith\n\\begin{equation}\\label{15,2}\n\\begin{array}{l}\n    \\overline{U}^{\\,e}= Q^{e,T}\\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} = Q^{e,T}\\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1}\\,,\\vspace{4pt}\\\\\n    F^e:= \\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} = \\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\big(\\nabla \\Theta(x_1,x_2,0)\\big)^{-1}\\,,  \\vspace{4pt}\\\\\n    \\bar{F}\\,\\,:= \\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) = \\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) ,\\vspace{4pt}\\\\\n    F^0:=P=\\Big(\\,\\partial_1 y^{0}\\,\\Big|\\, \\partial_2 y^{0}\\,\\Big|\\, n^0\\,\\Big)= \\Big(\\,\\nabla y^0\\,\\Big|\\,   n^0\\,\\Big), \\vspace{4pt}\\\\\n    \\qquad\\qquad\\qquad\\,\\,\\,\\qquad\\bar{F}\\,\\,\\,=\\,\\,F^e\\,\\,F^0\\,.\n    \\end{array}\n\\end{equation}\nIn order to see a parallel with the classical  multiplicative decomposition  into elastic and plastic parts from finite elasto-plasticity \\cite{Neff_Cosserat_plasticity05,HutterSFB02}, we may interpret $F^e$ as an elastic shell mid-surface deformation gradient and $F^0=P$ as an initial deformation gradient. Both are gradients of suitably defined mappings, see Remark \\ref{rem2} and  Figure \\ref{Fig2}, in contrast to the case of elasto-plasticity. In our context, the elastic material response is defined in terms of the elastic part of the deformation, e.g. $\\,E^e\\,=\\, Q^{e,T}\\,F^e -  {1\\!\\!\\!\\:1 } _3 \\,$, cf. \\eqref{15,1}.\n\n\n\n\n\n\\begin{remark}\\label{rem2}\nAlthough the resultant shell model is truly a 2D theory, we may always consider artificially reconstructed three-dimensional quantities. In this sense, similar to the  context of Remark \\ref{rem1}, the tensor $\\,\\bar{\\boldsymbol{F}}=\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^e\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\,$, which has the components matrix  $\\,\\bar{F}\\,$ is a three-dimensional deformation gradient\n\\begin{equation}\\label{15,3}\n\\begin{array}{c}\n\\bar{\\boldsymbol{F}}= \\nabla \\, \\boldsymbol{\\varphi}(x_1,x_2,x_3)_{\\big|x_3=0}\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\varphi}(x_1,x_2,x_3):=\\boldsymbol{y}(x_1,x_2)+x_3\\,\\boldsymbol{Q}^e(x_1,x_2) \\boldsymbol{n}^0(x_1,x_2)\\\\\n\\qquad\\qquad\\qquad\\qquad\\,\\,\\, =   \\boldsymbol{y}(x_1,x_2)+x_3\\,\\boldsymbol{Q}^e(x_1,x_2)\\,      \\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,0)\\boldsymbol{e}_3.\n\\end{array}\n\\end{equation}\nHere, the mapping $\\,\\boldsymbol{\\varphi}:\\,\\omega\\times\\big(\\!-\\frac{h}{2}\\,,\\frac{h}{2}\\,\\big)\\rightarrow\\mathbb{R}^3$ is a 3D deformation of the body, in terms of the given 2D quantities $\\boldsymbol{y}(x_1,x_2)$ and $\\boldsymbol{Q}^e(x_1,x_2)$. Similarly,\n\\begin{equation*}\n\\begin{array}{c}\n\\boldsymbol{F}^e:= \\nabla \\, \\boldsymbol{\\varphi}^e\\Big(\\boldsymbol{\\Theta}(x_1,x_2,x_3)_{\\big|x_3=0}\\Big)\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\varphi}^e\\big(\\boldsymbol{\\Theta}(x_1,x_2,x_3)\\big):=\n \\boldsymbol{\\varphi}(x_1,x_2,x_3).\n\\end{array}\n\\end{equation*}\nHowever, we note that $\\bar{\\boldsymbol{F}}$ cannot be interpreted as the  3D deformation\ngradient of the real 3D shell, because in general the initial normals become\narbitrarily curved after deformation.\n\\end{remark}\n\n\nIn terms of the total rotation $\\boldsymbol{R}$ and the initial rotation $\\boldsymbol{Q}^{0}$, the elastic shell strain tensor is expressed by\n\\begin{equation}\\label{8}\n    \\boldsymbol{E}^e=\\boldsymbol{Q}^{0}\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} - \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0 \\big) \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nThen, we have\n$$\\boldsymbol{E}^e=\\boldsymbol{Q}^{0}\\big[\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} \\!\\!-\\! \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big)\\! \\otimes\\! \\boldsymbol{e}_\\alpha\\big] \\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big)= \\boldsymbol{Q}^{0}\\big[\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y}\\!-\\! \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big) \\!\\otimes\\! \\boldsymbol{e}_\\alpha\\big]\n \\boldsymbol{P}^{-1}$$\nwhich can be written in matrix form as follows\n\\begin{equation}\\label{16}\n    E^e= Q^0\\,H \\,P^{-1} \\quad\\text{with}\\quad H:=\\Big(\\,R^T\\partial_1 y- Q^{0,T}\\partial_1 y^{0}\\,\\Big|\\, R^T\\partial_2 y- Q^{0,T}\\partial_2 y^{0}\\,\\Big|\\,\\, 0\\,\\,\\Big)_{3\\times 3}\\, .\n\\end{equation}\\smallskip\n\nOn the other hand, the elastic shell curvature  tensor $\\boldsymbol{K}^e$ in the material description is defined by \\cite{Eremeyev06,Pietraszkiewicz-book04}\n\\begin{equation}\\label{9}\n     \\boldsymbol{K}^e=\\big[\\boldsymbol{Q}^{e,T}\\text{axl}(\\partial_\\alpha \\boldsymbol{R}  \\boldsymbol{R}^T)- \\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^{0}\\boldsymbol{Q}^{0,T})\\big] \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nIn order to write $\\boldsymbol{K}^e$ in a  form more convenient to us, we use relations of the type\n\\begin{equation}\\label{10}\n    \\tilde{\\boldsymbol{Q}}^{ T}\\text{axl}(\\partial_\\alpha\\tilde{ \\boldsymbol{Q}} \\, \\tilde{\\boldsymbol{Q}}^{ T})=  \\text{axl}(\\tilde{\\boldsymbol{Q}}^{ T} \\partial_\\alpha \\tilde{ \\boldsymbol{Q}}  ),\\qquad \\text{axl}(\\tilde{\\boldsymbol{Q}} \\boldsymbol{A}\\tilde{\\boldsymbol{Q}}^{ T}) = \\tilde{\\boldsymbol{Q}}  \\,\\text{axl}(\\boldsymbol{A}),\n\\end{equation}\nwhich hold true for any   orthogonal tensor $\\tilde{\\boldsymbol{Q}} \\in SO(3)$ and any skew--symmetric tensor $\\boldsymbol{A}\\in \\frak{so}(3)$ (see e.g., \\cite{Birsan-Neff-AnnRom12}). Using \\eqref{10} in \\eqref{9} we can write the elastic  curvature tensor $\\boldsymbol{K}^e$ in the equivalent forms\n\\begin{equation}\\label{11}\n    \\boldsymbol{K}^e= \\boldsymbol{Q}^{e,T}\\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^e  \\boldsymbol{Q}^{e,T}) \\otimes \\boldsymbol{a}^\\alpha= \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e ) \\otimes \\boldsymbol{a}^\\alpha ,\n\\end{equation}\nor,\n$$\\boldsymbol{K}^e= \\big[\\, \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e) \\otimes \\boldsymbol{e}_\\alpha\\,\\big]\n \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1}  = \\big[\\, \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e) \\otimes \\boldsymbol{e}_\\alpha\\,\\big]\n \\boldsymbol{P}^{-1}.\n$$\nThen, the matrix of components $K^e=\\big(K^e_{ij}\\big)_{3\\times 3}$ is given by\n\\begin{equation}\\label{17}\n    K^e= \\Big(\\,\\,\\text{axl}(Q^{e,T}\\partial_1 Q^e) \\,\\,\\Big|\\,\\, \\text{axl}(Q^{e,T}\\partial_2 Q^e) \\,\\,\\Big|\\, \\,0\\,\\,\\Big) \\,P^{-1}.\n\\end{equation}\n\nIf we express $\\boldsymbol{K}^e$ in terms of the total rotation $\\boldsymbol{R}$ and the initial rotation $\\boldsymbol{Q}^{0}$, we get\n\\begin{equation}\\label{12}\n    \\boldsymbol{K}^e= \\boldsymbol{Q}^{0} \\big[ \\text{axl}(\\boldsymbol{R}^T \\partial_\\alpha \\boldsymbol{R}  )- \\text{axl}(\\boldsymbol{Q}^{0,T}\\partial_\\alpha \\boldsymbol{Q}^{0})\\big] \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nThis relation can be written as\n\\begin{equation}\\label{12,1}\n\\begin{array}{c}\n    \\boldsymbol{K}^e=\\boldsymbol{K} - \\boldsymbol{K}^0,\\qquad \\text{with} \\qquad \\boldsymbol{K}:= \\boldsymbol{Q}^{0}   \\text{axl}(\\boldsymbol{R}^T \\partial_\\alpha \\boldsymbol{R}  ) \\otimes \\boldsymbol{a}^\\alpha\\,,\\vspace{4pt} \\\\\n      \\boldsymbol{K}^0:= \\boldsymbol{Q}^{0}  \\text{axl}(\\boldsymbol{Q}^{0,T}\\partial_\\alpha \\boldsymbol{Q}^{0})  \\otimes \\boldsymbol{a}^\\alpha\\,\\,=\\,\\,\\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^{0}\\,\\boldsymbol{Q}^{0,T}) \\otimes \\boldsymbol{a}^\\alpha\\,,\n    \\end{array}\n\\end{equation}\nwhere the tensor $\\boldsymbol{K}$ is the total  curvature tensor, while $\\boldsymbol{K}^0$ is the initial  curvature (or structure curvature tensor of $S^0$). In view of  \\eqref{12} and  \\eqref{12,1},\nthe matrix $K^e=\\big(K^e_{ij}\\big) $ is given by\n\\begin{equation}\\label{18}\n\\begin{array}{c}\n    K^e\\,\\,=\\,\\,Q^0\\,L\\,P^{-1}\\,\\,=\\,\\,K-K^0\\qquad\\text{with}  \\vspace{4pt}\\\\ L:= \\Big(\\,\\,\\text{axl}(R^T\\partial_1 R)-\\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\Big|\\,\\, \\text{axl}(R^T\\partial_2 R) -\\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\Big|\\, \\,0\\,\\,\\Big)_{3\\times 3}\\, ,\\vspace{4pt}\\\\\n    \\,\\, K= Q^0\\Big(\\,\\,\\text{axl}(R^T\\partial_1 R)  \\,\\,\\Big|\\,\\, \\text{axl}(R^T\\partial_2 R) \\,\\,\\Big|\\, \\,0\\,\\,\\Big)P^{-1}, \\vspace{4pt}\\\\\n    \\qquad K^0= Q^0\\Big(\\,\\, \\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\Big|\\,\\,  \\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\Big|\\, \\,0\\,\\,\\Big)P^{-1}.\n    \\end{array}\n\\end{equation}\nIn what follows, we shall use the expressions \\eqref{16} and \\eqref{18} of the elastic shell strain measures $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ written with tensor components in the basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$.\n\\begin{remark}\nAs expected, the case of zero strain and bending measures corresponds to a rigid body mode of the shell. Indeed, if $\\boldsymbol{E}^e=\\boldsymbol{0}$ and $\\boldsymbol{K}^e=\\boldsymbol{0}$, then from \\eqref{7} and \\eqref{11} we obtain\n$$\\partial_\\alpha\\boldsymbol{y}=\\boldsymbol{Q}^e\\,\\partial_\\alpha\\boldsymbol{y}^0\\qquad \\mathrm{and}\\qquad \\partial_\\alpha \\boldsymbol{Q}^e=\\boldsymbol{0}.$$\nHence, it follows that $\\boldsymbol{Q}^e$ is constant and\n$$\\boldsymbol{y}=\\boldsymbol{Q}^e \\boldsymbol{y}^0+ \\boldsymbol{c}\\qquad (\\boldsymbol{c}=\\mathrm{constant}),$$\nwhich means that the shell undergoes a rigid body motion with constant translation $\\,\\boldsymbol{c}\\,$ and constant rotation $\\,\\boldsymbol{Q}^e$.\n\\end{remark}\n\\begin{remark}\nIn the case when the base surface $S^0$ of the initial configuration of the shell is planar we may assume that $S^0$ coincides with $\\,\\omega$. In this situation we have $\\,\\boldsymbol{a}_i=\\boldsymbol{e}_i\\,$, $\\boldsymbol{P}=  {1\\!\\!\\!\\:1 } _3\\,$, and\nthe above strain and curvature measures coincide with those defined for the Cosserat model of planar--shells introduced in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\\begin{remark}\nIn view of \\eqref{11} or \\eqref{17}, the elastic shell  curvature tensor $\\,\\boldsymbol{K}^e$ is an analog of the second Cosserat deformation tensor in the 3D theory, see the original Cosserats book \\cite[page 123, eq. (44)]{Cosserat09neu}.\n\\end{remark}\n\n\n\n\n\n\n\\section{Variational formulation for elastic shells}\\label{sect4}\n\n\n\n\nLet us denote the strain energy density of the elastic shell by $W=W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$. According to the hyperelasticity assumption, the    internal surface stress resultant $\\boldsymbol{N}$ and stress couple tensor $\\boldsymbol{M}$ are expressed by the constitutive equations in the form\n\\begin{equation}\\label{19}\n    \\boldsymbol{N}=\\boldsymbol{Q}^e\\,\\dfrac{\\partial\\, W}{\\partial \\boldsymbol{E}^e}\\,\\,,\\qquad \\boldsymbol{M}=\\boldsymbol{Q}^e\\,\\dfrac{\\partial\\, W}{\\partial \\boldsymbol{K}^e}\\,\\,.\n\\end{equation}\nIn this paper we assume that the strain energy density $W$ is a quadratic function of its arguments $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$. Thus, the considered model  is physically linear and geometrically non-linear. The explicit form of the strain energy function $W$ is presented in \\cite{Libai98,Eremeyev06} for isotropic, hemitropic or orthotropic elastic shells.\nIn general, the coefficients of the strain energy function $W$ depend on the structure curvature tensor $\\,\\boldsymbol{K}^0$, see \\cite{Eremeyev06}.\nIn \\cite{Chroscielewski11}, the case of composite (layered) shells is investigated and the expression of the energy density is established. These special cases will be discussed in Section \\ref{sect5}.\n\n\nConsider the usual Lebesgue spaces $\\big(L^p(\\omega),\\|\\cdot\\|_{L^p(\\omega)}\\big)$, $p\\geq 1$, and Sobolev space $\\big(H^1(\\omega),\\|\\cdot\\|_{H^1(\\omega)}\\big)$. We denote by $\\boldsymbol{L}^p(\\omega,\\mathbb{R}^3)$ (respectively $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$) the space of all vector fields $\\boldsymbol{v}=v_i\\boldsymbol{e}_i$ such that $v_i\\in L^p(\\omega)$ (respectively $v_i\\in H^1(\\omega)$). Similarly, we denote the sets\n$    \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})=\\{\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j \\,|\\, T_{ij}\\in H^1(\\omega)\\}$ ,   $\\boldsymbol{H}^1(\\omega,SO(3))= \\{\\boldsymbol{T}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3}) \\,|\\,\\boldsymbol{T}\\in SO(3)\\}$ , $ \\boldsymbol{L}^p(\\omega,\\mathbb{R}^{3\\times 3})=\\{\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j \\,|\\, T_{ij}\\in L^p(\\omega)\\}$ , $ \\boldsymbol{L}^p(\\omega,SO(3))= \\{\\boldsymbol{T}\\in \\boldsymbol{L}^p(\\omega,\\mathbb{R}^{3\\times 3}) \\,|\\,\\boldsymbol{T}\\in SO(3)\\}$.\nThe norm of a tensor $\\boldsymbol{T}$ is defined by $\\|\\boldsymbol{T}\\|^2=\\text{tr}(\\boldsymbol{T} \\boldsymbol{T}^T)=T_{ij}T_{ij}\\,$.\n\n\nConcerning the boundary-value problem \\eqref{5}, \\eqref{6}, we assume the existence of a function $\\Lambda(\\boldsymbol{y},\\boldsymbol{R})$ representing the potential of external surface loads $\\boldsymbol{f}$, $\\boldsymbol{c}$, and boundary loads $\\boldsymbol{n}^*$, $\\boldsymbol{m}^*$ (cf. \\cite{Pietraszkiewicz04}).\n\nWe consider the following two--field minimization problem associated to the deformation of elastic shells: find the pair $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ in the admissible set $\\mathcal{A}$ which realizes the minimum of the functional\n\\begin{equation}\\label{20}\nI(\\boldsymbol{y},\\boldsymbol{R})=\\int_{S^0} W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,\\mathrm{d}S - \\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\qquad\\mathrm{for}\\qquad (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A},\n\\end{equation}\nwhere d$S$ is the area element of the surface $S^0\\,$. The admissible set $\\mathcal{A}$ is defined by\n\\begin{equation}\\label{21}\n    \\mathcal{A}=\\big\\{(\\boldsymbol{y},\\boldsymbol{R})\\in\\boldsymbol{H}^1(\\omega, \\mathbb{R}^3)\\times\\boldsymbol{H}^1(\\omega, SO(3))\\,\\,\\big|\\,\\,\\,  \\boldsymbol{y}_{\\big| \\partial S^0_d}=\\boldsymbol{y}^*, \\,\\,\\boldsymbol{R}_{\\big| \\partial S^0_d}=\\boldsymbol{R}^* \\big\\},\n\\end{equation}\nwhere the boundary conditions are to be understood in the sense of traces.\nThe tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ are expressed in terms of $(\\boldsymbol{y},\\boldsymbol{R})$ through the relations \\eqref{8} and \\eqref{12}. If we write $W=W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)=\\tilde{W}(\\nabla\\boldsymbol{y},\\boldsymbol{R},\\nabla\\boldsymbol{R})$, then referring the integral to the (fictitious reference) domain $\\omega$, the change of variable formula clearly  gives\n\\begin{equation}\\label{22}\n\\begin{array}{l}\n    \\displaystyle{\\int_{S^0}} W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,\\mathrm{d}S=\\displaystyle{\\int_\\omega } W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,a(x_1,x_2)\\, \\mathrm{d}x_1\\mathrm{d}x_2 \\\\\n    \\qquad\\,\\,\\, = \\displaystyle{\\int_\\omega } \\tilde{W}\\big(\\nabla\\boldsymbol{y}(x_1,x_2),\\boldsymbol{R}(x_1,x_2),\\nabla\\boldsymbol{R}(x_1,x_2)\\big)\\, \\det\\big(\\nabla \\boldsymbol{\\Theta} (x_1,x_2,0)\\big)\\, \\mathrm{d}x_1\\mathrm{d}x_2 ,\n    \\end{array}\n\\end{equation}\nwhere $a=\\sqrt{\\det(a_{\\alpha\\beta})}$ is the notation introduced previously. The variational principle for the total energy of elastic shells with respect to the functional \\eqref{20} has been presented in \\cite{Pietraszkiewicz04}, Sect.2. We decompose the loading potential $\\Lambda(\\boldsymbol{y},\\boldsymbol{R})$ additively as follows\n\\begin{equation}\\label{23}\n\\begin{array}{c}\n    \\Lambda(\\boldsymbol{y},\\boldsymbol{R})=\\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R}) + \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R}),\\\\\n    \\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})= \\displaystyle{\\int_{S^0}}\\! \\boldsymbol{f}\\!\\cdot\\! \\boldsymbol{u}\\, \\mathrm{d}S + \\Pi_{S^0}(\\boldsymbol{R}), \\quad\n    \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R})= \\displaystyle{\\int_{\\partial S^0_f}}\\! \\boldsymbol{n}^*\\!\\cdot\\! \\boldsymbol{u}\\, \\mathrm{d}l + \\Pi_{\\partial S^0_f}(\\boldsymbol{R}).\n    \\end{array}\n\\end{equation}\nwhere $\\boldsymbol{u}=\\boldsymbol{y}-\\boldsymbol{y}^0$ is the displacement vector and d$l$ is the element of length along the boundary curve $\\partial S^0_f\\,$. In \\eqref{23}, $\\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})$ is the potential of the external surface loads $\\boldsymbol{f}, \\boldsymbol{c}$, while $\\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{Q}^e)$ is the potential of the external boundary loads $\\boldsymbol{n}^*, \\boldsymbol{m}^*$. The expression of the load potential functions $\\,\\,\\Pi_{S^0}\\,,  \\,\\Pi_{\\partial S^0_f}:\\boldsymbol{L}^2( \\omega ,SO(3))\\rightarrow\\mathbb{R}$ are not given explicitly, but they are assumed to be continuous and bounded operators. Of course, the integrals over $S^0$ and $\\partial S^0_f$ appearing in \\eqref{23} can be transformed like in \\eqref{22} into integrals over $\\omega$ and $\\partial \\omega_f\\,$, respectively.\n\nWe mention that one can consider more general cases of external loads in the definition of the loading potential \\eqref{23}, such as for example tracking loads.\n\n\n\\subsection{Main result: Existence of minimizers}\n\n\n\nThis theorem states the existence of minimizers to the minimization problem \\eqref{20}--\\eqref{23}.\n\\begin{theorem}\\label{th1}\nAssume that the external loads satisfy the conditions\n\\begin{equation}\\label{24}\n    \\boldsymbol{f}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^3),\\qquad  \\boldsymbol{n}^*\\in \\boldsymbol{L}^2(\\partial\\omega_f,\\mathbb{R}^3),\n\\end{equation}\nand the boundary data satisfy the conditions\n\\begin{equation}\\label{25}\n    \\boldsymbol{y}^*\\in\\boldsymbol{H}^1(\\omega ,\\mathbb{R}^3),\\qquad \\boldsymbol{R}^*\\in\\boldsymbol{H}^1(\\omega, SO(3)).\n\\end{equation}\nAssume that the following conditions concerning the initial configuration are fulfilled: $\\,\\boldsymbol{y}^0:\\omega\\subset \\mathbb{R}^2\\rightarrow\\mathbb{R}^3$ is a continuous injective mapping and\n\\begin{equation}\\label{26}\n    \\begin{array}{c}\n   \\boldsymbol{y}^0\\in\\boldsymbol{H}^1(\\omega ,\\mathbb{R}^3),\\qquad \\boldsymbol{Q}^{0}\\in\\boldsymbol{H}^1(\\omega, SO(3)),\n   \\end{array}\n\\end{equation}\n\\begin{equation}\\label{26,1}\n    \\begin{array}{c}\n   \\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0\\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^3)\\quad \\big(\\,\\mathrm{i.e.}\\,\\,\\, \\nabla\\boldsymbol{y}^0\\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^{3\\times 2})\\big), \\vspace{6pt}\\\\\n    \\det\\big(a_{\\alpha\\beta}(x_1,x_2)\\big)\\geq a_0^2 >0\\,\\,,\n    \\end{array}\n\\end{equation}\nwhere $a_0$ is a constant. The strain energy density $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ is assumed to be a quadratic convex function of $(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ and $W$ is coercive, in the sense that  there exists a constant $C_0>0$ with\n\\begin{equation}\\label{26bis}\n    W( {\\boldsymbol{E}^e}, {\\boldsymbol{K}^e})\\,\\geq\\, C_0\\,\\big( \\,   \\|\\boldsymbol{E}^e\\|^2 +  \\|\\boldsymbol{K}^e\\|^2\\,\\big).\n\\end{equation}\nThen, the minimization problem \\eqref{20}--\\eqref{23} admits at least one minimizing solution pair\n$(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in  \\mathcal{A}$.\n\\end{theorem}\n\\begin{remark}\nThe hypotheses \\eqref{26,1} can be written equivalently in terms of the tensor $\\boldsymbol{P}=\\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,0)$ as\n\\begin{equation}\\label{26,2}\n    \\begin{array}{c}\n   \\boldsymbol{P} \\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^{3\\times 3}),\\qquad \\det\\boldsymbol{P} \\geq a_0 >0\\,\\,,\n    \\end{array}\n\\end{equation}\nin view of the relations \\eqref{14} and \\eqref{14bis}. Since $\\boldsymbol{y}^0$ represents the position vector of the reference base surface $S^0$ (which is bounded), the conditions \\eqref{26}$_1$ and \\eqref{26,1}$_1$ can be written together in the form $\\boldsymbol{y}^0 \\in\\boldsymbol{W}^{1,\\infty}(\\omega ,\\mathbb{R}^3)$.\n\\end{remark}\n\\begin{proof}\nWe employ the direct methods of the calculus of variations. We show first that there exists a constant $C>0$ such that\n\\begin{equation}\\label{27}\n    |\\,\\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\,|\\,\\leq\\, \\,\\,C\\,\\big(\\,\\|\\boldsymbol{y}\\|_{H^1(\\omega)}+1\\big),\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in\\mathcal{A}.\n\\end{equation}\nIndeed, since $\\boldsymbol{a}_\\alpha\\in\\boldsymbol{L}^\\infty(\\omega,\\mathbb{R}^3)$ it follows that $a=\\sqrt{\\det(a_{\\alpha\\beta})}\\in{L}^\\infty(\\omega)$.\nWe also have $\\|\\boldsymbol{R}\\|^2=\\text{tr}(\\boldsymbol{R}\\boldsymbol{R}^T)=3$, $\\,\\forall\\boldsymbol{R}\\in SO(3)$. Taking into account the hypotheses \\eqref{24} and the boundedness of $\\,\\,\\Pi_{S^0}\\,$ and $ \\,\\Pi_{\\partial S^0_f}\\,$, we deduce from \\eqref{23} that\n\\begin{equation*}\n \\begin{array}{l}\n     |\\,\\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\,|\\,\\leq\\, |\\, \\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})\\, | + |\\,\n    \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R}) \\,|\\,\\leq   \\,\n     C_1\\,\\|\\boldsymbol{y}-\\boldsymbol{y}^0\\|_{L^2(\\omega)} +C_2\\, \\|\\boldsymbol{y}-\\boldsymbol{y}^0\\|_{L^2(\\partial\\omega_f)} \\\\\n     \\qquad +\\,|\\, \\Pi_{S^0}(\\boldsymbol{R})\\,|+ | \\,\\Pi_{\\partial S^0_f}(\\boldsymbol{R})\\,| \\,\\leq \\, C_3 \\|\\boldsymbol{y}\\|_{L^2(\\omega)}+C_4\\|\\boldsymbol{y}\\|_{H^1(\\omega)} +C_5\\,,\n     \\end{array}\n\\end{equation*}\nfor some positive constants $C_k>0$. Then, the inequality \\eqref{27} holds.\n\nIn what follows, we employ the component form of the elastic  strain tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$, written as matrices $E^e$ and $K^e$ in \\eqref{16} and \\eqref{18}, respectively. Let us show next that there exists a positive constant $\\lambda_0>0$ such that\n\\begin{equation}\\label{28}\n    \\|\\,\\boldsymbol{E}^e\\,\\|=\\|\\,E^e\\,\\|\\,\\geq\\, \\lambda_0\\,\\|\\,H\\,\\|\\,,\\qquad  \\|\\,\\boldsymbol{K}^e\\,\\|=\\|\\,K^e\\,\\|\\,\\geq\\, \\lambda_0\\,\\|\\,L\\,\\|\\,,\n\\end{equation}\nwhere the matrices $H=\\big(H_{ij}\\big)_{3\\times 3}$ and $L=\\big(L_{ij}\\big)_{3\\times 3}$ are introduced in \\eqref{16} and \\eqref{18}. Indeed, since $E^e=Q^0HP^{-1}$ and $Q^0\\in SO(3)$ we have\n\\begin{equation}\\label{29}\n    \\|\\,E^e\\,\\|^2=\\|\\,Q^0HP^{-1}\\,\\|^2=\\|\\,HP^{-1}\\,\\|^2=\\text{tr}\\big[ HP^{-1} (HP^{-1})^T \\big]=\\text{tr}\\big[ H(P^{T}\\!P)^{-1} H^T \\big].\n\\end{equation}\nFrom \\eqref{14bis} we deduce that\n\\begin{equation}\\label{30}\n    P^{T}P =\\begin{bmatrix} a_{11} & a_{12} &  0 \\\\\n                                 a_{12} & a_{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\qquad\\text{and therefore}\\qquad\n    (P^{T}P)^{-1}=\\begin{bmatrix} a^{11} & a^{12} &  0 \\\\\n                                 a^{12} & a^{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\,\\,.\n\\end{equation}\nInserting \\eqref{30} into \\eqref{29} we obtain\n\\begin{equation}\\label{31}\n     \\|\\,E^e\\,\\|^2=a^{\\alpha\\beta}H_{i\\alpha}H_{i\\beta} = a^{\\alpha\\beta}H_{1\\alpha}H_{1\\beta}+ a^{\\alpha\\beta}H_{2\\alpha}H_{2\\beta}+ a^{\\alpha\\beta}H_{3\\alpha}H_{3\\beta}\\,,\\,\\,\\,\\text{with}\\,\\,\\, H=\\big(H_{ij}\\big)_{3\\times 3}\\,,\n\\end{equation}\nsince $H_{i3}=0$ according to \\eqref{16}.\nIn virtue of \\eqref{26,1}, it follows that the matrix $\\big(a_{\\alpha\\beta}\\big)_{2\\times 2}$ and its inverse matrix\n$\\big(a^{\\alpha\\beta}\\big)_{2\\times 2}=\\big(a_{\\alpha\\beta}\\big)^{-1}$  satisfy\n$$\\big(a_{\\alpha\\beta}\\big)\n\\in\\boldsymbol{L}^\\infty(\\omega, \\mathbb{R}^{2\\times 2})\\quad\\text{and}\\quad \\big(a^{\\alpha\\beta}\\big)\n\\in\\boldsymbol{L}^\\infty(\\omega, \\mathbb{R}^{2\\times 2}).$$\nThen, the smallest eigenvalue of the positive definite symmetric matrix $\\big(a^{\\alpha\\beta}(x_1,x_2)\\big)_{2\\times 2}$ is greater than a positive constant $\\lambda_0^2>0$ and consequently\n\\begin{equation}\\label{32}\n    a^{\\alpha\\beta}\\!(x_1,x_2)\\,\\,v_\\alpha\\, v_\\beta\\,\\geq\\, \\lambda_0^2\\,v_\\gamma\\, v_\\gamma\\,,\\qquad \\forall\\,(x_1,x_2)\\in\\omega,\\,\\,\\, \\forall\\,v_1,v_2\\in\\mathbb{R}.\n\\end{equation}\nUsing  inequality \\eqref{32} for each individual sum in the right-hand side of \\eqref{31} we deduce that $\\|E^e\\|^2\\geq \\lambda_0^2\\,H_{i\\alpha}H_{i\\alpha}=\\lambda_0^2 \\,\\|H\\|^2$, i.e. the inequality \\eqref{28}$_1$ is proved. The proof of the inequality \\eqref{28}$_2$ is identical. In view of \\eqref{28}$_1$ and \\eqref{16} we have\n\\begin{equation*}\n\\begin{array}{l}\n\\|\\,E^e\\,\\|^2\\geq \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2} \\|\\,R^T\\partial_\\alpha y\\!-\\! Q^{0,T}\\partial_\\alpha y^{0}\\,\\|^2   \\\\\n \\qquad\\qquad=\n \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2}\\big( \\|\\,R^T\\partial_\\alpha y\\|^2\\!-2\\langle\\,   R^T\\partial_\\alpha y\\, , Q^{0,T}\\partial_\\alpha y^{0}\\,\\rangle +\\|\\,Q^{0,T}\\partial_\\alpha y^{0} \\|^2\\big)\\\\\n \\qquad\\qquad = \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2}\\big( \\|\\,\\partial_\\alpha y\\|^2-2\\langle\\,   R^T\\partial_\\alpha y\\, , Q^{0,T}\\partial_\\alpha y^{0}\\,\\rangle +\\|\\,\\partial_\\alpha y^{0} \\|^2\\big)\n ,\n \\end{array}\n\\end{equation*}\nwhere $\\langle\\,S,T\\,\\rangle=\\text{tr}[ST^T]$ is the scalar product of  two matrices $S,T$. Integrating over $\\omega$ and using the Cauchy--Schwarz inequality we obtain\n$$\\|\\,E^e\\,\\|^2_{L^2(\\omega)} \\geq\n \\lambda_0^2\\sum_{\\alpha=1}^2\\big( \\|\\,\\partial_\\alpha y\\|^2_{L^2(\\omega)}-2\\|\\, \\partial_\\alpha y\\|_{L^2(\\omega)} \\|\\,\\partial_\\alpha y^{0}\\,\\|_{L^2(\\omega)} +\\|\\,\\partial_\\alpha y^{0 } \\|^2_{L^2(\\omega)}\\big),$$\nor\n\\begin{equation}\\label{33}\n    \\|\\,\\boldsymbol{E}^e\\,\\|^2_{L^2(\\omega)} \\geq\n \\lambda_0^2\\,\\big( \\|\\,\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\big) - \\bar{C}_1\\|\\, \\boldsymbol{y}\\,\\|_{H^1(\\omega)}+\\bar{C}_2\\,,\n\\end{equation}\nfor some positive constants $\\bar{C}_1>0$, $\\bar{C}_2>0$. Let us show that the functional $I(\\boldsymbol{y},\\boldsymbol{R})$ is bounded from below over the admissible set $\\mathcal{A}$. By virtue of \\eqref{22}, \\eqref{26,1}$_2$ and \\eqref{27} we can write\n$$I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0 \\!\\!\\int_\\omega \\|\\,\\boldsymbol{E}^e\\,\\|^2\\,a\\, \\mathrm{d}x_1\\mathrm{d}x_2 - \\Lambda(\\boldsymbol{y},\\boldsymbol{R}) \\geq C_0\\,a_0 \\|\\,\\boldsymbol{E}^e\\,\\|^2_{L^2(\\omega)}  - C\\,\\big(\\,\\|\\boldsymbol{y}\\|_{H^1(\\omega)}+1\\big)$$\nand using \\eqref{33} we deduce that there exist the constants $\\bar{C}_3>0$ and $\\bar{C}_4$ such that\n\\begin{equation}\\label{34}\n    I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0\\,a_0 \\lambda_0^2\\,\\big( \\|\\,\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\big) - \\bar{C}_3\\|\\, \\boldsymbol{y}\\,\\|_{H^1(\\omega)}-\\bar{C}_4\\,,\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A},\n\\end{equation}\nwith $\\,a_0\\,$ specified by \\eqref{26,1}.\nWe observe that the vector field $\\boldsymbol{y}-\\boldsymbol{y}^*\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ satisfies $\\boldsymbol{y}-\\boldsymbol{y}^*=\\boldsymbol{0}$ on $\\partial\\omega_d\\,$. Applying the Poincar\\'e--inequality we infer the existence of a constant $c_p>0$ such that\n\\begin{equation}\\label{35}\n     \\| \\partial_1 (\\boldsymbol{y}-\\boldsymbol{y}^*)\\|^2_{L^2(\\omega)} +\\| \\partial_2 (\\boldsymbol{y}-\\boldsymbol{y}^*) \\|^2_{L^2(\\omega)}  \\,\\geq\\, c_p\\,\\|\\, \\boldsymbol{y}-\\boldsymbol{y}^*\\,\\|_{H^1(\\omega)}^2\\,\\,.\n\\end{equation}\nUsing inequalities of the type $\\|\\partial_\\alpha \\boldsymbol{y}\\|_{L^2(\\omega)}^2\\geq \\big(\\|\\partial_\\alpha (\\boldsymbol{y}-\\boldsymbol{y}^*)\\|_{L^2(\\omega)}- \\| \\partial_\\alpha \\boldsymbol{y}^*\\|_{L^2(\\omega)}\\big)^2$ and \\eqref{35} we find that\n\\begin{equation*}\n\\begin{array}{l}\n\\|\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\geq  c_p\\,\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}^2 \\\\\n\\qquad\\quad\\,\\,\\,\\, -2\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}\\big( \\|\\partial_1 \\boldsymbol{y}^*\\|_{L^2(\\omega)} +\\|\\partial_2 \\boldsymbol{y}^* \\|_{L^2(\\omega)}\\big)  +\n\\big(\\|\\,\\partial_1 \\boldsymbol{y}^*\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y}^* \\|^2_{L^2(\\omega)}\\big).\n\\end{array}\n\\end{equation*}\nFrom the last inequality and \\eqref{34} follows that there exist some constants $\\bar{C}_5>0$ and $\\bar{C}_6$ with\n\\begin{equation}\\label{36}\n    I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0\\,a_0 \\lambda_0^2\\,c_p\\,\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}^2-\\bar{C}_5\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)} +\\bar{C}_6\\,,\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}.\n\\end{equation}\nSince the constant $C_0\\,a_0 \\lambda_0^2\\,c_p>0$, the function $I(\\boldsymbol{y},\\boldsymbol{R})$ is bounded from below over   $\\mathcal{A}$.  Hence, there exists an infimizing sequence $\\big\\{(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty \\subset\\mathcal{A}$ such that\n\\begin{equation}\\label{37}\n    \\lim_{n\\rightarrow \\infty} I(\\boldsymbol{y}_n,\\boldsymbol{R}_n) = \\,\\inf\\, \\big\\{I(\\boldsymbol{y},\\boldsymbol{R})\\, \\big|\\,  (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}\\big\\}.\n\\end{equation}\nIn view of the conditions \\eqref{25} we have $I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)<\\infty$. The infimizing sequence $\\big\\{(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty$ can be chosen such that\n\\begin{equation}\\label{38}\n    I(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\,\\leq \\,I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)\\,< \\infty\\,, \\qquad \\forall\\,n\\geq 1.\n\\end{equation}\nTaking into account \\eqref{36} and \\eqref{38} we see that the sequence $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$. Then, we can extract a subsequence of $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$  (not relabeled) which converges weakly in  $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ and moreover, according to Rellich's selection principle, it converges strongly in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$, i.e. there exists an element $\\hat{\\boldsymbol{y}}\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ such that\n\\begin{equation}\\label{39}\n    \\boldsymbol{y}_n  \\rightharpoonup \\hat{ \\boldsymbol{y}} \\quad\\mathrm{in}\\quad \\boldsymbol{H}^1(\\omega, \\mathbb{R}^3),\\qquad \\mathrm{and}\\qquad \\boldsymbol{y}_n \\rightarrow\\hat{ \\boldsymbol{y}} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^3).\n\\end{equation}\nFor any $n\\in\\mathbb{N}$, let us denote by $\\boldsymbol{E}^e_n$ and $\\boldsymbol{K}^e_n$ the strain measures corresponding to the fields $(\\boldsymbol{y}_n,\\boldsymbol{R}_n)$, defined by the relations \\eqref{8} and \\eqref{12}. We have $\\boldsymbol{E}^e_n, \\boldsymbol{K}^e_n\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ and let $E_n^e,K_n^e$ be the matrices of components in the basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$, given by \\eqref{16} and \\eqref{18} in terms $\\{y_n,R_n\\}$. From \\eqref{20}, \\eqref{26,1}$_2$ , \\eqref{26bis}, \\eqref{27} and \\eqref{38} we get\n$$ C_0a_0\\,\\|\\,\\boldsymbol{K}^e_n\\,\\|^2_{L^2(\\omega)} \\leq\n\\int_\\omega W(\\boldsymbol{E}^e_n,\\boldsymbol{K}^e_n)\\,a(x_1,x_2)\\, \\mathrm{d}x_1\\mathrm{d}x_2 \\leq I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)+ C\\,\\big(\\,\\|\\boldsymbol{y}_n\\|_{H^1(\\omega)}+1\\big).\n$$\nSince $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$, it follows from the last inequalities that $\\big\\{ \\boldsymbol{K}^e_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. In view of \\eqref{28}$_2\\,$, we deduce that $\\big\\{\\text{axl}(\\boldsymbol{R}_n^{T}\\partial_\\alpha \\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$, or equivalently $ \\big\\{ \\partial_\\alpha \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, for $\\alpha=1,2$. Since $\\boldsymbol{R}_n\\in SO(3)$ we have $\\|\\boldsymbol{R}_n\\|^2=3$ and thus we can infer that the sequence $ \\big\\{ \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$\nis bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})$. Hence, there exists a subsequence of $ \\big\\{ \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ (not relabeled) and an element $\\hat{\\boldsymbol{R}}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})$ with\n\\begin{equation}\\label{40}\n    \\boldsymbol{R}_n \\rightharpoonup       \\hat{\\boldsymbol{R}}    \\quad\\mathrm{in}\\quad \\boldsymbol{H}^1(\\omega, \\mathbb{R}^{3\\times3}) , \\qquad\\mathrm{and}\\qquad          \\boldsymbol{R}_n  \\rightarrow    \\hat{\\boldsymbol{R}} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}).\n\\end{equation}\nWe can show for the limit that $\\hat{\\boldsymbol{R}}\\in SO(3)$. Indeed, since $\\boldsymbol{R}_n\\in SO(3)$ we have\n$$\\|\\,\\boldsymbol{R}_n \\hat{\\boldsymbol{R}}{}^T - {1\\!\\!\\!\\:1 } _3\\|_{L^2(\\omega)}= \\|\\,\\boldsymbol{R}_n ( \\hat{\\boldsymbol{R}}{}^T -\\boldsymbol{R}_n^T)\\|_{L^2(\\omega)}=  \\|\\,  \\hat{\\boldsymbol{R}}  -\\boldsymbol{R}_{n }\\|_{L^2(\\omega)} \\longrightarrow 0,\n$$\ni.e. $\\boldsymbol{R}_n\\hat{\\boldsymbol{R}}{}^T\\rightarrow {1\\!\\!\\!\\:1 } _3$ in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. On the other hand, we can write\n$$\\|\\,\\boldsymbol{R}_n \\hat{\\boldsymbol{R}}{}^T -\\hat{\\boldsymbol{R}} \\hat{\\boldsymbol{R}}{}^T\\|_{L^1(\\omega)}= \\|\\, ( \\boldsymbol{R}_{n }  -\\hat{\\boldsymbol{R}})\\hat{\\boldsymbol{R}}{}^T\\|_{L^1(\\omega)} \\leq 3  \\|\\,   \\boldsymbol{R}_{n }  -\\hat{\\boldsymbol{R}} \\|_{L^2(\\omega)}\\,  \\|\\,  \\hat{\\boldsymbol{R}} \\|_{L^2(\\omega)} \\longrightarrow 0,\n$$\nwhich means that $\\boldsymbol{R}_n\\hat{\\boldsymbol{R}}{}^T\\rightarrow\\hat{\\boldsymbol{R}}\\hat{\\boldsymbol{R}}{}^T$ in $\\boldsymbol{L}^1(\\omega,\\mathbb{R}^{3\\times 3})$. Consequently, we find $\\hat{\\boldsymbol{R}}\\hat{\\boldsymbol{R}}{}^T= {1\\!\\!\\!\\:1 } _3$ so that $\\hat{\\boldsymbol{R}}\\in\\boldsymbol{H}^1(\\omega,SO(3))$.\n\nBy virtue of the relations  $(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\in \\mathcal{A}$ and \\eqref{39}, \\eqref{40},   we derive that $\\hat{\\boldsymbol{y}}=\\boldsymbol{y}^*$ on $\\partial S^0_d$ and $\\hat{\\boldsymbol{R}}=\\boldsymbol{R}^*$ on $\\partial S^0_d\\,$ in the sense of traces. Hence, we obtain that the limit pair satisfies $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in \\mathcal{A}$.\n\nLet us construct the elements $\\hat{\\boldsymbol{E}^e}, \\hat{\\boldsymbol{K}^e} \\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ defined in terms of the fields $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ by the relations \\eqref{8} and \\eqref{12}. Then, the matrices of components $\\hat{E}^e,\\hat{K}^e$ are expressed in terms of the components $(\\hat{y},\\hat{R})$ by \\eqref{16} and \\eqref{18}, i.e.\n\\begin{equation}\\label{41}\n    \\begin{array}{l}\n     \\hat{E}^e= Q^0\\big(\\,\\hat{R}{}^T\\partial_1 \\hat{y}- Q^{0,T}\\partial_1 y^{0}\\,\\big|\\, \\hat{R}{}^T\\partial_2 \\hat{y}- Q^{0,T}\\partial_2 y^{0}\\,\\big|\\,\\, 0\\,\\,\\big) \\,P^{-1} ,\\vspace{4pt}\\\\\n     \\hat{K}^e=Q^0\\big(\\,\\text{axl}(\\hat{R}{}^T\\partial_1 \\hat{R})\\!-\\!\\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\big|\\,\\, \\text{axl}(\\hat{R}{}^T\\partial_2 \\hat{R})\\! -\\!\\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\big|\\, \\,0\\,\\big)P^{-1} \\!.\n     \\end{array}\n\\end{equation}\n\nNext, we want to show that there exist some subsequences (not relabeled) of $\\{\\boldsymbol{E}^e_n\\}$ and $\\{\\boldsymbol{K}^e_n\\}$ such that\n\\begin{equation}\\label{42}\n    \\boldsymbol{E}^e_n  \\rightharpoonup  \\hat{\\boldsymbol{E}^e} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}),\\quad\\text{and} \\qquad  \\boldsymbol{K}^e_n \\rightharpoonup \\hat{\\boldsymbol{K}^e}\\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}).\n\\end{equation}\nAs shown above, the sequence $ \\big\\{ \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3 })$. It follows that $ \\big\\{ \\partial_\\alpha \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3 })$ and then the sequence $ \\big\\{ \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3 })$, since $ \\boldsymbol{R}_n\\in SO(3)$. Consequently, there exists a subsequence (not relabeled) and an element $\\boldsymbol{\\xi}_\\alpha\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$ such that\n\\begin{equation}\\label{43}\n    \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{y}_n \\,\\,\\rightharpoonup \\,\\, \\boldsymbol{\\xi}_\\alpha\\qquad\\text{in} \\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3}).\n\\end{equation}\nOn the other hand, let $\\boldsymbol{\\phi}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3})$ be an arbitrary test function. Then, using the properties of the scalar product  we deduce\n\\begin{equation*}\n    \\begin{array}{l}\n    \\displaystyle{\\int_\\omega}\\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n - \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n    \\qquad\\quad=\n    \\displaystyle{\\int_\\omega}  \\hat{\\boldsymbol{R}}{}^T \\big( \\partial_\\alpha \\boldsymbol{y}_n - \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n        +\\displaystyle{\\int_\\omega}  \\big( \\boldsymbol{R}_n^T- \\hat{\\boldsymbol{R}}{}^T\\big) \\partial_\\alpha \\boldsymbol{y}_n  \\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n     \\qquad\\quad   =\\displaystyle{\\int_\\omega}  \\big( \\partial_\\alpha \\boldsymbol{y}_n - \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\cdot\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi} \\, \\mathrm{d}x_1\\mathrm{d}x_2  +\n        \\displaystyle{\\int_\\omega} \\!\\!\\big\\langle \\boldsymbol{R}_{n}\\!\\!-\\! \\hat{\\boldsymbol{R}}\\,, \\partial_\\alpha \\boldsymbol{y}_n \\!\\otimes\\!\\boldsymbol{\\phi}\\rangle \\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n    \\qquad\\qquad\\,\\, \\leq\\|\\boldsymbol{R}_{n}\\!-\\! \\hat{\\boldsymbol{R}}\\|_{L^2(\\omega)}\\|\\partial_\\alpha \\boldsymbol{y}_n \\!\\otimes\\!\\boldsymbol{\\phi}\\|_{L^2(\\omega)} \\!+\\!\\! \\displaystyle{\\int_\\omega}  \\big( \\partial_\\alpha \\boldsymbol{y}_n \\!- \\!\\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\!\\cdot\\!\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi} \\, \\mathrm{d}x_1\\mathrm{d}x_2 \\,,\n\\end{array}\n\\end{equation*}\nsince the relations \\eqref{39}, \\eqref{40} and $\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$ hold, and $\\|\\partial_\\alpha \\boldsymbol{y}_n\\otimes\\boldsymbol{\\phi}\\|$ is bounded. Thus, we get\n\\begin{equation}\\label{44}\n    \\displaystyle{\\int_\\omega}\\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n \\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\longrightarrow\n    \\displaystyle{\\int_\\omega}\\big( \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2,\\quad\\forall\\, \\boldsymbol{\\phi}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^3).\n\\end{equation}\nBy comparison of \\eqref{43} and \\eqref{44} we find $\\boldsymbol{\\ell}_\\alpha= \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{y}}\\,$, which means that\n$\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n   \\rightharpoonup\n     \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}$ in $ \\boldsymbol{L}^2(\\omega,\\mathbb{R}^3)$ ,\nor equivalently\n\\begin{equation}\\label{45}\n    \\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n  - \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0 \\big)\\quad \\rightharpoonup\\quad\n    \\big( \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}} -  {\\boldsymbol{R}}{}^T_0 \\partial_\\alpha \\boldsymbol{y}^{0}\\big)\\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^3).\n\\end{equation}\nTaking into account \\eqref{16}, \\eqref{41}$_1$ and the hypotheses \\eqref{26}, \\eqref{26,1}, we obtain from \\eqref{45} that $E^e_n\\rightharpoonup\\hat{E}^e$ in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, i.e. the relation \\eqref{42}$_1$ holds.\n\nTo prove \\eqref{42}$_2$ we start from the fact that the sequence $ \\big\\{ \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3 })$, as we proved previously. Then, there exists a subsequence (not relabeled) and an element $\\boldsymbol{ \\zeta}_\\alpha\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ such that\n\\begin{equation}\\label{46}\n    \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{R}_n \\rightharpoonup \\boldsymbol{ \\zeta}_\\alpha\\qquad\\text{in} \\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3}).\n\\end{equation}\nOn the other hand, for any test function $\\boldsymbol{\\mathit{\\Phi}}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3\\times 3})$ we can write\n\\begin{equation*}\n\\begin{array}{l}\n    \\displaystyle{\\int_\\omega}\\big\\langle \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_{n} - \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{R}}  \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2=\n    \\displaystyle{\\int_\\omega}\\big\\langle   \\hat{\\boldsymbol{R}}{}^T \\big(\\partial_\\alpha  \\boldsymbol{R}_{n} - \\partial_\\alpha\\hat{\\boldsymbol{R}}\\big) \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\vspace{2pt}\\\\\n     \\qquad+\n    \\displaystyle{\\int_\\omega}\\big\\langle \\big( \\boldsymbol{R}_n^T- \\hat{\\boldsymbol{R}}{}^T\\big) \\partial_\\alpha \\boldsymbol{R}_{n}  \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\leq\n     \\displaystyle{\\int_\\omega}\n     \\big\\langle \\partial_\\alpha \\boldsymbol{R}_{n} - \\partial_\\alpha \\hat{\\boldsymbol{R}}  \\,,\\, \\hat{\\boldsymbol{R}}\\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2\n     \\vspace{2pt}\\\\\n     \\qquad+\n     \\|\\boldsymbol{R}_{n}- \\hat{\\boldsymbol{R}}\\|_{L^2(\\omega)}\\,\\| \\partial_\\alpha \\boldsymbol{R}_n \\boldsymbol{\\mathit{\\Phi}}^T \\, \\|_{L^2(\\omega)} \\longrightarrow 0,\n\\end{array}\n\\end{equation*}\nsince $\\hat{\\boldsymbol{R}}\\boldsymbol{\\mathit{\\Phi}}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, $\\|\\partial_\\alpha \\boldsymbol{R}_n \\boldsymbol{\\mathit{\\Phi}}^T\\|$ is bounded, and relations \\eqref{40} hold. Consequently, we have\n$$\\displaystyle{\\int_\\omega}\\big\\langle \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_{n}\\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\longrightarrow\n    \\displaystyle{\\int_\\omega}\\big\\langle  \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{R}}\\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2,\\quad\\forall \\,\\boldsymbol{\\mathit{\\Phi}}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3\\times3}), $$\nand by comparison with \\eqref{46} we deduce that $\\boldsymbol{ \\zeta}_\\alpha= \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}}\\,$, i.e. the convergence $ \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_n\\rightharpoonup \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}}$ holds in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. It follows that\n$$\\big[\\text{axl}(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_n) - \\text{axl}(\\boldsymbol{R}^{T}_0 \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]\n \\,\\,\\,\\rightharpoonup \\,\\,\\,\n\\big[\\text{axl}(\\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}})- \\text{axl}(\\boldsymbol{R}^{T}_0 \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]\n\\quad\\text{in}\\,\\, \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3}),$$\nand from \\eqref{18}, \\eqref{26}, \\eqref{26,1} and \\eqref{41}$_2$ we derive that the convergence \\eqref{42}$_2$ holds true. \\medskip\n\nIn the last step of the proof we use the convexity of the strain energy density $W$. In view of \\eqref{42}, we have\n\\begin{equation}\\label{47}\n     \\int_\\omega W(\\hat{\\boldsymbol{E}^e},\\hat{\\boldsymbol{K}^e})\\,a(x_1,x_2)\\,\\mathrm{d}x_1\\mathrm{d}x_2\\,\\leq \\, \\liminf_{n\\to\\infty}  \\int_\\omega W( {\\boldsymbol{E}^e_n}, {\\boldsymbol{K}^e_n})\\,a(x_1,x_2)\\,\\mathrm{d}x_1\\mathrm{d}x_2.\n\\end{equation}\nsince $W$ is convex in $(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$. Taking into account the hypotheses \\eqref{24}, the continuity of the load potential functions      $\\,\\,\\Pi_{S^0}\\,$, $ \\,\\Pi_{\\partial S^0_f}\\,$,    and the convergence relations \\eqref{39}$_2$ and \\eqref{40}$_2\\,$, we deduce\n\\begin{equation}\\label{48}\n  \\Lambda(\\hat{\\boldsymbol{y}}, \\hat{\\boldsymbol{R}})=  \\lim_{n\\to\\infty} \\Lambda(\\boldsymbol{y}_n, \\boldsymbol{R}_n).\n\\end{equation}\nFrom \\eqref{20}, \\eqref{22}, \\eqref{47} and \\eqref{48} we get\n\\begin{equation}\\label{49}\n    I(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\,\\leq\\,  \\liminf_{n\\to\\infty} \\, I(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\,.\n\\end{equation}\nFinally, the relations \\eqref{37} and \\eqref{49} show that\n$$ I(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\,=\\,\n\\,\\inf\\, \\big\\{I(\\boldsymbol{y},\\boldsymbol{R})\\, \\big|\\,  (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}\\big\\}.\n$$\nSince $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in\\mathcal{A}$, we conclude that $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ is a minimizing solution pair of our minimization problem. The proof is complete.\n\\hfill\\end{proof}\n\\begin{remark}\nThe solution fields satisfy the following regularity conditions\n$$\\hat{\\boldsymbol{y}}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3})\n,\\qquad\n\\hat{\\boldsymbol{R}}\\in\\boldsymbol{L}^\\infty(\\omega,SO(3))\n\\cap \\boldsymbol{H}^1(\\omega,SO(3)) .\n$$\nThus, the position vector $ \\,\\hat{\\boldsymbol{y}}\\,$ and the total rotation field $\\,\\hat{\\boldsymbol{R}}\\,$ may fail to be continuous, according to the limit case of Sobolev embedding.\n\\end{remark}\n\\begin{remark}\nWe observe that the boundary conditions imposed on the orthogonal tensor $\\boldsymbol{R}$ can be relaxed in the definition of the admissible set $\\mathcal{A}$. Thus, one can prove the existence of minimizers for the minimization problem \\eqref{20} over the following larger admissible set\n\\begin{equation*}\n    \\tilde{\\mathcal{A}} =\\big\\{(\\boldsymbol{y},\\boldsymbol{R})\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3) \\times\\boldsymbol{H}^1(\\omega, SO(3))\\,\\,\\, \\big|\\,\\, \\,\\, \\boldsymbol{y}_{\\big| \\partial\\omega_d}=\\boldsymbol{y}^*  \\big\\}.\n\\end{equation*}\nThis assertion can be proved in the same way as the Theorem \\ref{th1}. For a discussion of   possible alternative boundary conditions for the   field $\\boldsymbol{R}$ on $\\partial\\omega_d$ we refer to the works \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Applications of the theorem and discussions}\\label{sect5}\n\n\n\n\nIn this section we present some important special cases for the choice of the energy density $W$ where the Theorem \\ref{th1} can be successfully applied to show the existence of minimizers.\n\nLet us discuss first on the choice of the 3 initial directors $\\{\\boldsymbol{d}_i^0\\}$ in the reference configuration, i.e. the specification of the proper orthogonal tensor $\\boldsymbol{Q}^{0}=\\boldsymbol{d}_i^0 \\otimes \\boldsymbol{e}_i\\,$. One judicious choice for the tensor $\\boldsymbol{Q}^{0}$ is the following\n\\begin{equation}\\label{49,1}\n    \\boldsymbol{Q}^{0}=\\text{polar}(\\boldsymbol{P})=\\text{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big),\n\\end{equation}\nwhere $\\boldsymbol{P}=\\boldsymbol{a}_{i}\\otimes\\boldsymbol{e}_i=\\partial_\\alpha \\boldsymbol{y}^0\\otimes\\boldsymbol{e}_\\alpha +\\boldsymbol{n}^0\\otimes \\boldsymbol{e}_3$ has been introduced previously in \\eqref{14} and $\\text{polar}(\\boldsymbol{T})$ denotes the orthogonal tensor given by the polar decomposition of any tensor $\\boldsymbol{T}$.\n\\begin{remark}\\label{rem6}\nIf the tensor $\\boldsymbol{Q}^{0}$ is given by \\eqref{49,1}, then the (initial) directors $\\boldsymbol{d}_\\alpha^0$ belong to the tangent plane at any point of $S^0$ and $\\boldsymbol{d}_3^0= \\boldsymbol{n}^0$. Indeed, let $\\boldsymbol{P}=\\boldsymbol{Q}^{0}\\boldsymbol{U}^0$ be the polar decomposition of $\\boldsymbol{P}$.\nUsing the matrices of components in the $\\{\\boldsymbol{e}_{i}\\otimes\\boldsymbol{e}_j\\}$ tensor basis, we write this relation as $P=Q^0U^0$, and from \\eqref{30} we derive consecutively\n\\begin{equation*}\n    U^{0,T} U^{0}=P^{T}P =\\begin{bmatrix} a_{11} & a_{12} &  0 \\\\\n                                 a_{12} & a_{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\\quad\n    U^0=\n    \\begin{bmatrix} u_{11}^0 & u_{12}^0 &  0 \\\\\n                                 u_{12}^0 & u_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\\quad\n    \\big(U^0\\big)^{-1}=\n    \\begin{bmatrix} \\bar{u}_{11}^0 & \\bar{u}_{12}^0 &  0 \\\\\n                                 \\bar{u}_{12}^0 & \\bar{u}_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\n\\end{equation*}\nwhere $u_{\\alpha\\beta}^0$ and $\\bar{u}_{\\alpha\\beta}^0$ are some given real functions of $(x_1,x_2)$. In view of \\eqref{14bis}, it follows\n\\begin{equation}\\label{49,2}\n   Q^0= P\\,\\big(U^0\\big)^{-1}= \\Big(\\,a_{1}\\,\\Big|\\, a_{2}\\,\\Big|\\, n^0\\, \\Big)_{3\\times 3}\n   \\begin{bmatrix} \\bar{u}_{11}^0 & \\bar{u}_{12}^0 &  0 \\\\\n                                 \\bar{u}_{12}^0 & \\bar{u}_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\qquad\\Rightarrow\\qquad Q^0e_3=n^0,\n\\end{equation}\nfrom which we can see that the third column of the matrix $\\,Q^0$ is equal to $\\,n^0$.\nOn the other hand, by the definition \\eqref{1}$_{2\\,}$, the initial rotation field $ \\boldsymbol{Q}^{0}$ is given by $ \\boldsymbol{Q}^{0}= \\boldsymbol{d}_i^{0}\\otimes  \\boldsymbol{e}_i\\,$ and the matrix $Q^0$ can be written in column form as\n\\begin{equation}\\label{49,3}\n   Q^0 = \\Big(\\,d_{1}^0\\,\\Big|\\, d_{2}^0\\,\\Big|\\, d_3^0\\, \\Big)_{3\\times 3}\\,\\,\\,.\n\\end{equation}\nIf we compare \\eqref{49,2} and  \\eqref{49,3} we find that $d_3^0=n^0$. Thus, we have\n$\\boldsymbol{d}_3^{0}= \\boldsymbol{n}^{0}$ and  $\\{\\boldsymbol{d}_1^0,\\boldsymbol{d}_2^0\\}$ is an orthonormal basis in the tangent plane, at any point of $S^0$.\n\nIf we choose the tensor $\\boldsymbol{Q}^{0}$ as in \\eqref{49,1}, then in order to satisfy \\eqref{26}$_2$ we need to consider an additional regularity assumption on the initial configuration, namely\n$$\\mathrm{polar}(\\boldsymbol{P})=\\mathrm{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big)\\in \\boldsymbol{H}^1(\\omega,SO(3)),$$\nwhich is equivalent to $\\mathrm{Curl}\\big[\\mathrm{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big)\\big]\\in \\boldsymbol{L}^2(\\omega,SO(3))\\,$, cf. \\cite{Neff_curl08}. A stronger sufficient condition is $\\boldsymbol{\\Theta}\\in \\boldsymbol{W}^{1,\\infty}(\\omega,\\mathbb{R}^3)\\cap  \\boldsymbol{H}^2(\\omega,\\mathbb{R}^3)$.\n\\end{remark}\n\nIt is possible to simplify the form of the equations in the case of an orthogonal parametrization of the initial surface $S^0\\,$. If we   assume that the curvilinear coordinates $(x_1,x_2)$ are   such that the basis $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal, then the initial surface $S^0$ is formally parametrized by orthogonal arc-length coordinates \\cite{Chroscielewski11} and we have\n\\begin{equation}\\label{50}\n    \\boldsymbol{a}_\\alpha=\\boldsymbol{a}^\\alpha,\\qquad a_{\\alpha\\beta}= a^{\\alpha\\beta}=\\delta_{\\alpha\\beta}\\,.\n\\end{equation}\n\\begin{remark}\nThe \\emph{Theorema Egregium} (Gauss) can be put into the following form: the Gaussian curvature $K$ can be found given the full knowledge of the first fundamental form of the surface and expressed via the first fundamental form and its partial derivatives of first and second order (the Brioschi formula). Therefore, the Gaussian curvature of an embedded smooth surface in $\\mathbb{R}^3$ is invariant under local isometries, i.e. if the parametrization $\\boldsymbol{y}^0:\\omega\\subset\\mathbb{R}^2\\rightarrow \\mathbb{R}^3$ of the surface from a flat reference configuration $\\omega$ is given such that  $\\big(\\nabla \\boldsymbol{y}^0\\big)^T \\nabla \\boldsymbol{y}^0= {1\\!\\!\\!\\:1 } _2$ (the basis $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal), then the curvature $K$ of the surface $\\boldsymbol{y}^0(\\omega)$ is necessarily zero. This is only the case for developable surfaces.\n\n\nFor general surfaces it is therefore impossible to determine, even locally, an orthonormal parametrization. However, in FEM approaches one may think in a discrete pointwise manner as in \\cite{Chroscielewski11}.\n\nFor example, let $S^0$ be a cylindrical surface (which is a developable surface) with generators parallel to $\\boldsymbol{e}_3\\,$. The position vector $\\boldsymbol{y}^0$ is given by\n$$\\boldsymbol{y}^0=\\boldsymbol{y}^0(\\theta,z)=r_0\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\nr_0\\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2+ z \\, \\boldsymbol{e}_3\\qquad (r_0>0\\,\\,\\mathrm{constant}).\n$$\nChoosing the curvilinear coordinates $x_1=\\theta, x_2=z$, we have\n$$\\boldsymbol{a}_1= \\partial_1 \\boldsymbol{y}^0=- \\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\n\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2,\\quad\n\\boldsymbol{a}_2=\\partial_2 \\boldsymbol{y}^0=\\boldsymbol{e}_3\n,\\quad\n\\boldsymbol{n}^0=\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\n\\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2\\,,$$\nso that $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal.\n\\end{remark}\\smallskip\n\nIn view of \\eqref{49,1}--\\eqref{50}, we obtain in this case that $\\boldsymbol{Q}^{0}=\\boldsymbol{P}$ (since $\\boldsymbol{U}^0= {1\\!\\!\\!\\:1 } _3$ and $\\text{polar}(\\boldsymbol{P})=  \\boldsymbol{P}\\in SO(3)$) and the directors $\\{\\boldsymbol{d}_i^0\\}$ in the reference configuration coincide with $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ in each point of $S^0\\,$, i.e.\n\\begin{equation}\\label{51}\n    \\boldsymbol{d}_\\alpha^0=\\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0\\,,\\qquad\n    \\boldsymbol{d}_3^0= \\boldsymbol{a}_3=\\boldsymbol{n}^0,\\qquad \\boldsymbol{a}_i= \\boldsymbol{Q}^{0}\\boldsymbol{e}_i\\,,\\qquad \\boldsymbol{d}_i=\\boldsymbol{R} \\boldsymbol{Q}^{0,T}\\boldsymbol{a}_i\\,.\n\\end{equation}\nThe expressions of the elastic strain measures $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ may be simplified  in this situation. By virtue of \\eqref{50} and \\eqref{51} we get\n\\begin{equation}\\label{52}\n\\begin{array}{l}\n    \\boldsymbol{Q}^{0} \\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} -  \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big) = \\boldsymbol{Q}^{0}  \\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} -  \\boldsymbol{a}_{ \\alpha} \\vspace{3pt}\\\\\n    \\qquad\\qquad\\quad\\qquad\\qquad\\qquad\n     = \\big( \\boldsymbol{a}_i\\cdot\\boldsymbol{Q}^{0}  \\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i} \\big) \\boldsymbol{a}_{ i}\n     = \\big( \\boldsymbol{d}_i\\cdot\\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i} \\big) \\boldsymbol{a}_{ i}\\,.\n     \\end{array}\n\\end{equation}\nWe can write\n$\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}=(\\boldsymbol{d}_{i}\\otimes\\boldsymbol{e}_i)^T (\\partial_\\alpha \\boldsymbol{d}_{j}\\otimes\\boldsymbol{e}_j)= (\\boldsymbol{d}_{i}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} )\n\\boldsymbol{e}_i\\otimes \\boldsymbol{e}_j\\,$,\nso that we find\n$$\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R})= \\dfrac{1}{2}\\,e_{ijk}(\\boldsymbol{d}_{k}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} )\\boldsymbol{e}_i\\,,\\qquad \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})= \\dfrac{1}{2}\\,e_{ijk}(\\boldsymbol{a}_{k}\\cdot \\partial_\\alpha \\boldsymbol{a}_{j} )\\boldsymbol{e}_i\\,,\n$$\nwhere $e_{ijk}$ is the permutation symbol. The last relations and $\\boldsymbol{Q}^{0,T} \\boldsymbol{a}_i =\\boldsymbol{e}_i$ yield\n\\begin{equation}\\label{53}\n\\begin{array}{c}\n    \\boldsymbol{Q}^{0} \\big[\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}) - \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]= \\big[\\big(\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}) - \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})\\big)\\!\\cdot\\! \\boldsymbol{e}_i\\big]\\boldsymbol{a}_i \\\\\n    =\n    \\dfrac{1}{2}\\,e_{ijk}\\big[ (\\boldsymbol{d}_{k}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} ) - (\\boldsymbol{a}_{k}\\cdot \\partial_\\alpha \\boldsymbol{a}_{j} )\\big]\\boldsymbol{a}_i\\,.\n    \\end{array}\n\\end{equation}\nUsing the relations \\eqref{8}, \\eqref{12}, \\eqref{52} and \\eqref{53} we decompose the strain tensor $\\boldsymbol{E}^e$ and the curvature tensor $\\boldsymbol{K}^e$ in the basis $\\{\\boldsymbol{a}_i\\otimes \\boldsymbol{a}_\\alpha\\}$ as follows\n\\begin{equation}\\label{54}\n\\begin{array}{l}\n    \\boldsymbol{E}^e=\\tilde{E}^e_{i\\alpha}\\boldsymbol{a}_i\\otimes\\boldsymbol{a}_\\alpha\\,,\\qquad\n    \\tilde{E}^e_{i\\alpha}= \\boldsymbol{d}_i  \\cdot   \\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i}\\,,\\qquad\n        \\boldsymbol{K}^e=\\tilde{K}^e_{i\\alpha}\\boldsymbol{a}_i\\otimes\\boldsymbol{a}_\\alpha\\,,\\\\\n        \\tilde{K}^e_{1\\alpha}= \\boldsymbol{d}_3\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{2} \\! -\\! \\boldsymbol{a}_3\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{2}, \\quad\n    \\tilde{K}^e_{2\\alpha}= \\boldsymbol{d}_1\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{3} \\!- \\! \\boldsymbol{a}_1\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{3}, \\quad\\!\n    \\tilde{K}^e_{3\\alpha}= \\boldsymbol{d}_2\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{1} \\!-\\! \\boldsymbol{a}_2 \\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{1} .\n\\end{array}\n\\end{equation}\nFor later reference, we introduce the notations\n\\begin{equation}\\label{54,1}\n\\boldsymbol{E}^e_{\\parallel}=\\boldsymbol{E}^e- (\\boldsymbol{n}^0\\otimes\\boldsymbol{n}^0) \\boldsymbol{E}^e,\\qquad\\boldsymbol{K}^e_{\\parallel}=\\boldsymbol{K}^e- (\\boldsymbol{n}^0\\otimes\\boldsymbol{n}^0) \\boldsymbol{K}^e\\,.\n\\end{equation}\nThen, from \\eqref{54} we get\n\\begin{equation}\\label{55}\n\\begin{array}{l}\n    \\boldsymbol{E}^e_{\\parallel}=\\tilde{E}^e_{\\alpha\\beta}\\boldsymbol{a}_\\alpha\\otimes\\boldsymbol{a}_\\beta\\,,\n    \\qquad\n     \\boldsymbol{n}^0\\boldsymbol{E}^e=\\boldsymbol{E}^{e,T}\\boldsymbol{n}^0 =\\tilde{E}^e_{3\\alpha}\\boldsymbol{a}_\\alpha\\,, \\\\\n     \\boldsymbol{K}^e_{\\parallel}=\\tilde{K}^e_{\\alpha\\beta}\\boldsymbol{a}_\\alpha\\otimes\\boldsymbol{a}_\\beta\\,,\n    \\qquad\n     \\boldsymbol{n}^0\\boldsymbol{K}^e=\\boldsymbol{K}^{e,T}\\boldsymbol{n}^0 =\\tilde{K}^e_{3\\alpha }\\boldsymbol{a}_\\alpha\\,.\n     \\end{array}\n\\end{equation}\nIf we denote the matrices by $\\tilde{E}^e=\\big(\\tilde{E}^e_{ij}\\big)_{3\\times 3}\\,$, $\\tilde{K}^e=\\big(\\tilde{K}^e_{ij}\\big)_{3\\times 3}\\,$, and also\n\\begin{equation}\\label{55bis}\n\\begin{array}{l}\n    \\tilde{E}^e_{\\parallel}=   \\begin{bmatrix} \\tilde{E}^e_{11} & \\tilde{E}^e_{12}  \\\\\n                                 \\tilde{E}^e_{21} & \\tilde{E}^e_{22}    \\end{bmatrix}=\n    \\begin{bmatrix} \\boldsymbol{d}_1  \\cdot    \\partial_1 \\boldsymbol{y} \\!-\\!1 \\,\\, & \\,\\,\n    \\boldsymbol{d}_1  \\cdot   \\partial_2 \\boldsymbol{y} \\\\\n    \\boldsymbol{d}_2  \\cdot     \\partial_1 \\boldsymbol{y} \\,\\, & \\,\\, \\boldsymbol{d}_2  \\cdot   \\partial_2 \\boldsymbol{y}\\!-\\!1\n    \\end{bmatrix}, \\vspace{4pt}\\\\\n    \\big(\\tilde{E}^{e,T}n^0\\big)= \\begin{bmatrix}       \\tilde{E}^e_{31} & \\tilde{E}^e_{32}    \\end{bmatrix}=\n     \\begin{bmatrix}    \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{y} \\,\\, & \\,\\,\n     \\boldsymbol{d}_3 \\! \\cdot   \\partial_2 \\boldsymbol{y}     \\end{bmatrix},\n    \\vspace{4pt}\\\\\n     \\tilde{K}^e_{\\parallel}= \\begin{bmatrix} \\tilde{K}^e_{11} & \\tilde{K}^e_{12}  \\\\\n                                 \\tilde{K}^e_{21} & \\tilde{K}^e_{22}    \\end{bmatrix} =\n    \\begin{bmatrix} \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{d}_2 \\!-\\! \\boldsymbol{a}_3 \\! \\cdot    \\partial_1 \\boldsymbol{a}_2 \\,\\, &\n    \\,\\, \\boldsymbol{d}_3 \\! \\cdot    \\partial_2 \\boldsymbol{d}_2 \\!-\\! \\boldsymbol{a}_3 \\! \\cdot    \\partial_2 \\boldsymbol{a}_2   \\\\\n    \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{d}_3 \\!-\\! \\boldsymbol{a}_1 \\! \\cdot    \\partial_1 \\boldsymbol{a}_3 \\,\\,   & \\,\\, \\boldsymbol{d}_1 \\! \\cdot    \\partial_2 \\boldsymbol{d}_3  \\!-\\! \\boldsymbol{a}_1 \\! \\cdot    \\partial_2 \\boldsymbol{a}_3   \\end{bmatrix},\n                                \\vspace{4pt} \\\\\n    \\big(\\tilde{K}^{e,T}n^0\\big)= \\begin{bmatrix}       \\tilde{K}^e_{31} & \\tilde{K}^e_{32}    \\end{bmatrix}=\n    \\begin{bmatrix}   \\boldsymbol{d}_2  \\cdot    \\partial_1 \\boldsymbol{d}_1 \\!-\\! \\boldsymbol{a}_2  \\cdot    \\partial_1 \\boldsymbol{a}_1  \\,\\, & \\,\\, \\boldsymbol{d}_2  \\cdot    \\partial_2 \\boldsymbol{d}_1 \\!-\\! \\boldsymbol{a}_2  \\cdot    \\partial_2 \\boldsymbol{a}_1  \\end{bmatrix},\n     \\end{array}\n\\end{equation}\n then the relations \\eqref{54} and \\eqref{55} can be written in matrix form\n\\begin{equation}\\label{55,1}\n\\begin{array}{l}\n    \\tilde{E}^e=\\begin{bmatrix} \\tilde{E}^e_{11} & \\tilde{E}^e_{12} &  0 \\\\\n                                 \\tilde{E}^e_{21} & \\tilde{E}^e_{22} &  0 \\\\\n                                 \\tilde{E}^e_{31} & \\tilde{E}^e_{32} & 0\n    \\end{bmatrix}\\! = \\!\\! \\begin{bmatrix} \\big(\\,\\tilde{E}^e_{\\parallel}\\,\\big)_{2\\times 2} & 0_{2\\times 1}  \\vspace{10pt} \\\\\n                                 \\big(\\tilde{E}^{e,T}\\!n^0\\big)_{1\\times 2} & 0\n    \\end{bmatrix}_{3\\times 3}\\!\\! \\!\\!=\\!\\! \\begin{bmatrix} \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{y} \\!-\\!1 & \\boldsymbol{d}_1 \\! \\cdot   \\partial_2 \\boldsymbol{y} &  \\!\\!0 \\\\\n                                \\boldsymbol{d}_2 \\! \\cdot     \\partial_1 \\boldsymbol{y}  & \\boldsymbol{d}_2 \\! \\cdot   \\partial_2 \\boldsymbol{y}\\!-\\!1 & \\!\\! 0 \\\\\n                                 \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{y}  & \\boldsymbol{d}_3 \\! \\cdot   \\partial_2 \\boldsymbol{y} &  \\!\\!0\n    \\end{bmatrix}\\!,\\vspace{4pt}\\\\\n    \\tilde{K}^e=\\begin{bmatrix} \\tilde{K}^e_{11} & \\tilde{K}^e_{12} &  0 \\\\\n                                 \\tilde{K}^e_{21} & \\tilde{K}^e_{22} &  0 \\\\\n                                 \\tilde{K}^e_{31} & \\tilde{K}^e_{32} & 0\n    \\end{bmatrix}  =  \\begin{bmatrix} \\big(\\,\\tilde{K}^e_{\\parallel}\\,\\big)_{2\\times 2} & 0_{2\\times 1}  \\vspace{10pt} \\\\\n                                 \\big(\\tilde{K}^{e,T}\\!n^0\\big)_{1\\times 2} & 0\n    \\end{bmatrix}_{3\\times 3} \\vspace{4pt}\\\\\n    \\qquad\\qquad = \\tilde{K}- \\tilde{K}^0 = \\begin{bmatrix} \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{d}_2  & \\boldsymbol{d}_3 \\! \\cdot    \\partial_2 \\boldsymbol{d}_2  &  0 \\\\\n                                \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{d}_3   & \\boldsymbol{d}_1 \\! \\cdot    \\partial_2 \\boldsymbol{d}_3  &  0 \\\\\n                                \\boldsymbol{d}_2 \\! \\cdot    \\partial_1 \\boldsymbol{d}_1   & \\boldsymbol{d}_2 \\! \\cdot    \\partial_2 \\boldsymbol{d}_1  & 0\n    \\end{bmatrix} -\n    \\begin{bmatrix} \\boldsymbol{a}_3 \\! \\cdot    \\partial_1 \\boldsymbol{a}_2  & \\boldsymbol{a}_3 \\! \\cdot    \\partial_2 \\boldsymbol{a}_2  &  0 \\\\\n                                \\boldsymbol{a}_1 \\! \\cdot    \\partial_1 \\boldsymbol{a}_3   & \\boldsymbol{a}_1 \\! \\cdot    \\partial_2 \\boldsymbol{a}_3  &  0 \\\\\n                                \\boldsymbol{a}_2 \\! \\cdot    \\partial_1 \\boldsymbol{a}_1   & \\boldsymbol{a}_2 \\! \\cdot    \\partial_2 \\boldsymbol{a}_1  & 0\n    \\end{bmatrix}\n         \\end{array}\n\\end{equation}\nThese expressions are completely similar to the strain measures for planar--shells introduced in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\nLet us discuss next some important classes of elastic shells.\n\n\n\\subsection{Isotropic shells}\n\n\nIn the resultant 6-parameter theory of shells, the strain energy density for isotropic shells has been presented in various forms. The simplest expression of $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ has been proposed in the papers \\cite{Pietraszkiewicz-book04,Pietraszkiewicz10} in the form\n\\begin{equation}\\label{56}\n   \\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= \\,\\,\\,C\\big[\\,\\nu \\,(\\mathrm{tr} \\boldsymbol{E}^e_{\\parallel})^2 +(1-\\nu)\\, \\mathrm{tr}(\\boldsymbol{E}^{e,T}_{\\parallel}  \\boldsymbol{E}^e_{\\parallel} )\\big]  + \\alpha_{s\\,}C(1-\\nu) \\, \\boldsymbol{n}^0 \\boldsymbol{E}^e \\boldsymbol{E}^{e,T}  \\boldsymbol{n}^0 \\\\\n    \\qquad\\qquad\\qquad\\,\\, +\\,D \\big[\\,\\nu\\,(\\mathrm{tr} \\boldsymbol{K}^e_{\\parallel})^2 + (1-\\nu)\\, \\mathrm{tr}(\\boldsymbol{K}^{e,T}_{\\parallel}  \\boldsymbol{K}^e_{\\parallel} )\\big]  + \\alpha_{t\\,}D(1-\\nu) \\,    \\boldsymbol{n}^0 \\boldsymbol{K}^e \\boldsymbol{K}^{e,T}  \\boldsymbol{n}^0,\n\\end{array}\n\\end{equation}\nwhere $C=\\frac{E\\,h}{1-\\nu^2}\\,$ is the stretching (in-plane) stiffness of the shell, $D=\\frac{E\\,h^3}{12(1-\\nu^2)}\\,$ is the bending stiffness, $h$ is the thickness of the shell, and $\\alpha_s\\,$, $\\alpha_t$ are two shear correction factors. Also,    $E $ and $ \\nu$ denote the Young modulus and  Poisson ratio of the isotropic and homogeneous material. By the numerical treatment of non-linear shell problems, the values of the shear correction factors have been set to  $\\alpha_s=5\/6$, $\\alpha_t=7\/10$ in \\cite{Pietraszkiewicz10}. The value $\\alpha_s=5\/6$ is a classical suggestion, which has been previously deduced analytically by Reissner in the case of plates \\cite{Reissner45,Naghdi72}.\nAlso, the value $\\,\\alpha_t=7\/10\\,$ was proposed earlier in \\cite[see p.78]{Pietraszkiewicz-book79} and has been suggested in the work \\cite{Pietraszkiewicz79}.\nHowever, the discussion concerning the possible values of  shear correction factors for shells is long and controversial in the literature \\cite{Naghdi72,Naghdi-Rubin95}.\n\nWith the help of the matrices \\eqref{55bis}, we can express the strain energy density \\eqref{56}   in the alternative form\n\\begin{equation}\\label{57}\n\\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= \\\\\n    =C(1\\!-\\!\\nu)\\big(\\|\\text{dev}_2\\,\\text{sym}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\!+ \\|\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\big)\\!+\n    C \\dfrac{1\\!+\\!\\nu}{2}\\big(\\text{tr}\\, \\tilde{E}^e_{\\parallel}\\big)^2\n         + \\alpha_sC(1\\!-\\!\\nu)\\,\\|\\tilde{E}^{e,T}n^0  \\|^2 \\vspace{2pt}\\\\\n    +D(1\\!-\\!\\nu)\\big(\\|\\text{dev}_2\\,\\text{sym}\\, \\tilde{K}^e_{\\parallel}\\,\\|^2\\!+ \\|\\text{skew}\\, \\tilde{K}^e_{\\parallel}\\,\\|^2\\big)\\!\n         + D \\dfrac{1\\!+\\!\\nu}{2}\\big(\\text{tr}\\, \\tilde{K}^e_{\\parallel}\\big)^2 + \\alpha_tD(1\\!-\\!\\nu)\\,\\|\\tilde{K}^{e,T}n^0   \\|^2\\!,\n\\end{array}\n\\end{equation}\nwhere $\\,\\text{sym}\\, X= \\frac{1}{2}\\big(X+X^T\\big)$ is the symmetric part, $\\,\\text{skew}\\, X= \\frac{1}{2}\\big(X-X^T\\big)$ is the skew-symmetric part,\nand $\\,\\,\\text{dev}_2\\, X= X-\\frac{1}{2}\\big(\\text{tr}\\,X\\big) {1\\!\\!\\!\\:1 } _2\\,\\,$ is the deviatoric part of any $2\\times 2$ matrix  $X $. The coefficients in \\eqref{57} are expressed in terms of the Lam\\'e constants of the material $\\lambda$ and $\\mu$  by the relations\n\\begin{equation*\n    C \\dfrac{1\\!+\\!\\nu}{2}\\,=h\\,\\dfrac{\\mu(2\\mu\\!+\\!3\\lambda)}{2\\mu+\\lambda}\\,,\\quad\\! C(1\\!-\\!\\nu)=2\\mu h,\\quad\\!\n    D \\dfrac{1\\!+\\!\\nu}{2}\\,=\\dfrac{h^3}{12}\\,\\,\\dfrac{\\mu(2\\mu\\!+\\!3\\lambda)}{2\\mu+\\lambda}\\,,\\quad\\!\n    D(1\\!-\\!\\nu)=\\dfrac{\\mu h^3}{6}\\,.\n\\end{equation*}\nThen, we obtain  that the given quadratic form \\eqref{57} is positive definite if and only if the coefficients $E$ and $\\nu$ satisfy the inequalities\n\\begin{equation}\\label{57,1}\n    E>0,\\qquad -1<\\nu<\\dfrac{1}{2}\\,\\,.\n\\end{equation}\nIn terms of  the  Lam\\'e moduli of the material,   the  inequalities \\eqref{57,1} are equivalent to\n$$\\mu>0,\\qquad 2\\mu+3\\lambda>0.$$\nThese conditions are guaranteed by the positive definiteness of the 3D quadratic elastic strain energy for isotropic materials.\nThus, we find that the strain energy $W$ is convex and satisfies the coercivity condition \\eqref{26bis}, so that the hypotheses of Theorem \\ref{th1} are fulfilled. Applying Theorem \\ref{th1} we obtain (under suitable assumptions on the given load and boundary data, and the reference configuration $( \\boldsymbol{y}^0,\\boldsymbol{Q}^{0})$) the existence of minimizers for isotropic shells with strain energy density in the form \\eqref{56}.\\smallskip\n\nIn \\cite{Eremeyev06}, Eremeyev and Pietraszkiewicz have proposed a more general form of the strain energy density, namely\n\\begin{equation}\\label{58}\n    \\begin{array}{l}\n   2 W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)=  \\alpha_1\\big(\\mathrm{tr}  \\boldsymbol{E}^e_{\\parallel}\\big)^2 +\\alpha_2  \\mathrm{tr} \\big(\\boldsymbol{E}^e_{\\parallel}\\big)^2    + \\alpha_3 \\mathrm{tr}\\big(\\boldsymbol{E}^{e,T}_{\\parallel}  \\boldsymbol{E}^e_{\\parallel} \\big)  + \\alpha_4     \\boldsymbol{n}^0 \\boldsymbol{E}^e \\boldsymbol{E}^{e,T}  \\boldsymbol{n}^0 \\\\\n    \\qquad\\qquad\\qquad + \\beta_1\\big(\\mathrm{tr}  \\boldsymbol{K}^e_{\\parallel}\\big)^2 +\\beta_2  \\mathrm{tr} \\big(\\boldsymbol{K}^e_{\\parallel}\\big)^2    + \\beta_3 \\mathrm{tr}\\big(\\boldsymbol{K}^{e,T}_{\\parallel} \\boldsymbol{K}^e_{\\parallel} \\big)  + \\beta_4     \\boldsymbol{n}^0 \\boldsymbol{K}^e \\boldsymbol{K}^{e,T}  \\boldsymbol{n}^0.\n\\end{array}\n\\end{equation}\nThe eight coefficients $\\alpha_k\\,$, $\\beta_k$ ($k=1,2,3,4$) can depend in general on the structure curvature tensor  $ \\boldsymbol{K}^0=\\text{axl}\\big(\\partial_\\alpha \\boldsymbol{Q}^{0}\\boldsymbol{Q}^{0,T}\\big) \\otimes \\boldsymbol{a}^\\alpha$ of the reference configuration. For the sake of simplicity, we assume in our discussion that the coefficients $\\alpha_k$ and $\\beta_k$ are constant.\nWe can decompose the strain energy density  \\eqref{58}  in the  in-plane part  $W_{\\text{plane}}(\\boldsymbol{E}^e)$ and the curvature part $W_{\\text{curv}}(\\boldsymbol{K}^e)$ and write their expressions using the matrices of components \\eqref{55bis} in  form\n\\begin{equation}\\label{59}\n    \\begin{array}{c}\n    W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= W_{\\text{plane}}(\\boldsymbol{E}^e\\big)+ W_{\\text{curv}}(\\boldsymbol{K}^e\\big)\\,,\n\\end{array}\n\\end{equation}\n\\begin{equation*}\n    \\begin{array}{c}\n    2W_{\\text{plane}}(\\boldsymbol{E}^e)= (\\alpha_2\\!+\\!\\alpha_3) \\| \\text{sym}\\, \\tilde{E}^e_{\\parallel}\\|^2\\!+ (\\alpha_3\\!-\\!\\alpha_2)\\|\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\|^2\\!+ \\alpha_1\\big(\\mathrm{tr}  \\tilde{E}^e_{\\parallel}\\big)^2 +  \\alpha_4\\|\\tilde{E}^{e,T}n^0  \\|^2, \\vspace{4pt}\\\\\n    2 W_{\\text{curv}}(\\boldsymbol{K}^e)= (\\beta_2\\!+\\!\\beta_3) \\| \\text{sym}\\, \\tilde{K}^e_{\\parallel}\\|^2\\!+ (\\beta_3\\!-\\!\\beta_2)\\|\\text{skew}\\, \\tilde{K}^e_{\\parallel}\\|^2\\!+ \\beta_1\\big(\\mathrm{tr}  \\tilde{K}^e_{\\parallel}\\big)^2\n    + \\beta_4\\|\\tilde{K}^{e,T}n^0  \\|^2.\n    \\end{array}\n\\end{equation*}\nThe in-plane part of the energy density \\eqref{59} can be written equivalently as\n\\begin{equation}\\label{59,1}\n \\begin{array}{l}\n    2W_{\\text{plane}}(\\boldsymbol{E}^e)= \\underbrace{(\\alpha_2+\\alpha_3)\\, \\|\\,\\text{dev}_2 \\,\\text{sym}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2 }_{\\text{in-plane shear--stretch energy}}  \\,+\\, \\underbrace{(\\alpha_3-\\alpha_2)\\,\\|\\,\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\,}_{\\text{in--plane drill rotation energy}}\\vspace{4pt}\\\\\n    \\qquad\\qquad\\quad\\,\\,\\, +\\,\\underbrace{\\Big(\\alpha_1+\\dfrac{\\alpha_2 + \\alpha_3}{2}\\Big)\\big(\\mathrm{tr} \\, \\tilde{E}^e_{\\parallel}\\big)^2}_{\\text{in-plane elongational stretch energy}}  \\,+\\,\\underbrace{\\, \\alpha_4\\,\\|\\,\\tilde{E}^{e,T}n^0 \\, \\|^2}_{\\text{transverse shear energy}}.\n    \\end{array}\n\\end{equation}\nThe above forms of the strain energy $W$ are expressed in terms of the components of the tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ in the basis $\\{\\boldsymbol{a}_i\\otimes \\boldsymbol{a}_\\alpha\\}\\,$, i.e. in terms of the elements of the matrices \\eqref{55bis}.\nDenoting with $\\,\\,\\mu_c^{\\text{drill}}\\,$ the coefficient $\\,(\\alpha_3-\\alpha_2)\\,$  in   \\eqref{59,1}, we remark that the   term\n\\begin{equation}\\label{59,2}\n    \\mu_c^{\\text{drill}}\\,\\|\\,\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\,,\\qquad \\text{with}\\qquad \\mu_c^{\\text{drill}}\\,:=\\,\\alpha_3-\\alpha_2\\,,\n\\end{equation}\ndescribes the quadratic in-plane drill rotation energy of the shell. We call the coefficient $\\,\\,\\mu_c^{\\text{drill}}\\,$  the \\emph{linear in-plane rotational couple modulus}, in analogy to the Cosserat couple modulus in the three-dimensional Cosserat theory \\cite{Neff_zamm06}.\n\\begin{remark}\nThe planar isotropic Cosserat shells have been investigated also in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}, using a model derived directly from the 3D equations of Cosserat elasticity. We mention that the expressions \\eqref{59}, \\eqref{59,1} of the strain energy density are essentially the same as the strain energy of the Cosserat model for planar shells \\cite{Neff_plate04_cmt}. By comparing these two approaches (6-parameter resultant shells and Cosserat model) we deduce the following identification of the constitutive coefficients $\\alpha_1\\,,...,\\alpha_4$\n\\begin{equation}\\label{59,3}\n    \\alpha_1=h\\,\\dfrac{2\\mu\\lambda}{2\\mu+\\lambda}\\,,\\quad \\alpha_2=h(\\mu-\\mu_c),\\quad \\alpha_3=h(\\mu+\\mu_c),\\quad \\alpha_4=\\kappa\\, h (\\mu+\\mu_c),\n\\end{equation}\nwhere $\\,\\mu_c\\,$ is the Cosserat couple modulus of the 3D continuum, and $\\,\\kappa\\,$ is a formal shear correction factor. From \\eqref{59,2}, \\eqref{59,3} we observe that\n\\begin{equation}\\label{59,4}\n    \\mu_c^{\\mathrm{drill}}\\,=\\,\\alpha_3-\\alpha_2\\,=\\,2h\\,\\mu_c\\,,\n\\end{equation}\nwhich means that the in-plane rotational couple modulus $\\,\\mu_c^{\\mathrm{drill}}\\,$ of the Cosserat shell model is determined by the Cosserat couple modulus $\\,\\mu_c\\,$ of the 3D Cosserat material.\n\nThe relations \\eqref{59,3} are similar to the corresponding relations in the linear theory of micropolar plates, see \\cite[Eqs.(45)]{Altenbach-Erem09}. From a mathematical viewpoint, the difference between the two sets of relations consists in the notations used and the value of the shear correction factor.\n\\end{remark}\n\n\nLooking at \\eqref{59} and \\eqref{59,1} we observe that  the quadratic form $ W(\\boldsymbol{E}^e,\\boldsymbol{K}^e) $ is positive definite  if and only if the coefficients verify the conditions\n\\begin{equation}\\label{60}\n    \\begin{array}{l}\n    2\\alpha_1+\\alpha_2+\\alpha_3>0,\\quad \\alpha_2+\\alpha_3>0,\\quad \\alpha_3-\\alpha_2>0, \\quad\\alpha_4>0,\\\\\n    2\\beta_1+\\beta_2+\\beta_3>0,\\quad \\beta_2+\\beta_3>0,\\quad \\beta_3-\\beta_2>0,\\quad  \\beta_4>0,\n\\end{array}\n\\end{equation}\nProvided that the conditions \\eqref{60} are satisfied, the strain energy function $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ is convex and coercive in the sense of \\eqref{26bis}. By virtue of Theorem \\ref{th1}, in this case the minimization problem associated to the deformation of isotropic elastic shells admits at least one solution.\n\n\\begin{remark}\nThe same conditions \\eqref{60} have been imposed in \\cite{EremeyevLebedev11} to establish existence results in the \\emph{linearized} theory of micropolar (6-parameter) shells.\n\\end{remark}\n\\begin{remark}\nThe case $\\,\\mu_c^{\\mathrm{drill}}=0\\,$ (i.e., $\\alpha_3-\\alpha_2=0$) is not uniformly positive definite. However, with a slight change of the resultant shell model, one can prove the existence of minimizers using similar methods as in \\cite{Neff_plate07_m3as}. A linearization of such a model  leads exactly to the Reissner kinematics with 5 degrees of freedom \\cite{Neff_plate07_m3as}, where the in-plane drill rotation is absent. The physical meaning of the in-plane rotational stiffness $\\,\\mu_c^{\\mathrm{drill}}=\\alpha_3-\\alpha_2\\,$ in the resultant shell model is not entirely clear to us.\n\nSince only two independent rotations are required to orient a unit director field, a distinctive feature of classical plate and shell theories is a rotation field defined in terms of only \\emph{two} independent degrees of freedom. Rotations about the director itself -- the so-called drill rotation -- are irrelevant and for that matter undefined in classical shell theory.\n\\end{remark}\n\n\n\n\\subsection{Orthotropic shells}\n\n\n\nThe constitutive equations for orthotropic shells have been presented in \\cite{Eremeyev06} within the 6-parameter resultant shell  theory. The expression of the strain energy density in terms of the tensor components defined in \\eqref{54} is given by\n\\begin{equation}\\label{61}\n   2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= C_{\\alpha\\beta\\gamma\\delta}^E\\, \\tilde{E}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta} +  D_{\\alpha\\beta}^E\\, \\tilde{E}^e_{3\\alpha } \\tilde{E}^e_{3\\beta} +\n      C_{\\alpha\\beta\\gamma\\delta}^K\\,   \\tilde{K}^e_{\\alpha\\beta} \\tilde{K}^e_{\\gamma\\delta}+      D_{\\alpha\\beta}^K\\, \\tilde{K}^e_{3\\alpha } \\tilde{K}^e_{3\\beta}\n\\end{equation}\nwhere $C_{\\alpha\\beta\\gamma\\delta}^E\\,$, $C_{\\alpha\\beta\\gamma\\delta}^K\\,$, $D_{\\alpha\\beta}^E\\,$ and $D_{\\alpha\\beta}^K\\,$ are material constants which satisfy the following symmetry relations\n$$ C_{\\alpha\\beta\\gamma\\delta}^E=  C_{\\gamma\\delta\\alpha\\beta}^E\\,,\\quad D_{\\alpha\\beta}^E= D_{\\beta\\alpha}^E\\,,\\quad\n      C_{\\alpha\\beta\\gamma\\delta}^K=  C_{\\gamma\\delta\\alpha\\beta}^K\\,,\\quad D_{\\alpha\\beta}^K= D_{\\beta\\alpha}^K\\,.$$\nWe observe that the quadratic function \\eqref{61} is coercive if and only if the following symmetric matrices are positive definite\n\\begin{equation}\\label{62}\n    \\begin{bmatrix} C_{1111}^E & C_{1122}^E  &  C_{1112}^E  &   C_{1121}^E   \\\\\n                    C_{1122}^E & C_{2222}^E  &   C_{2212}^E  &   C_{2221}^E   \\\\\n                    C_{1112}^E  & C_{2212}^E   &  C_{1212}^E &   C_{1221}^E  \\\\\n                    C_{1121}^E  &  C_{2221}^E   &  C_{1221}^E   &  C_{2121}^E   \\end{bmatrix},\n    \\begin{bmatrix} C_{1111}^K & C_{1122}^K  &  C_{1112}^K  &   C_{1121}^K   \\\\\n                    C_{1122}^K & C_{2222}^K  &   C_{2212}^K  &   C_{2221}^K   \\\\\n                    C_{1112}^K  & C_{2212}^K   &  C_{1212}^K &   C_{1221}^K  \\\\\n                    C_{1121}^K  &  C_{2221}^K   &  C_{1221}^K   &  C_{2121}^K   \\end{bmatrix},\n    \\begin{bmatrix} D_{11 }^E & D_{12}^E   \\\\\n                    D_{12}^E & D_{22}^E    \\end{bmatrix},\n    \\begin{bmatrix} D_{11 }^K & D_{12}^K   \\\\\n                    D_{12}^K & D_{22}^K    \\end{bmatrix}\\!.\n\\end{equation}\nIn the situation when the matrices \\eqref{62} are positive definite, then the strain energy $W$ given by \\eqref{61} satisfies the hypotheses of Theorem \\ref{th1}. Then, we can use our theoretical results to derive the existence of minimizers for orthotropic shells.\n\n\n\n\\subsection{Composite layered shells}\n\n\n\nLet us analyze the case of composite shells made of a finite number of individually homogeneous layers. According to \\cite{Chroscielewski11}, the strain energy density of such type of shells can be written by means of the tensor components \\eqref{54} in the form\n\\begin{equation}\\label{63}\n\\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= A_{\\alpha\\beta\\gamma\\delta} \\, \\tilde{E}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta} +  D_{\\alpha\\beta\\gamma\\delta} \\, \\tilde{K}^e_{\\alpha \\beta} \\tilde{K}^e_{\\gamma\\delta} +\n      B_{\\alpha\\beta\\gamma\\delta} ( \\tilde{E}^e_{\\alpha\\beta} \\tilde{K}^e_{\\gamma\\delta} + \\tilde{K}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta})\\vspace{4pt}\\\\\n      \\qquad\\qquad\\qquad\\quad\n      +      S_{\\alpha\\beta} \\, \\tilde{E}^e_{3\\alpha } \\tilde{E}^e_{3\\beta}\n        +      G_{\\alpha\\beta} \\,        \\tilde{K}^e_{3\\alpha } \\tilde{K}^e_{3\\beta}\\,,\n        \\end{array}\n\\end{equation}\nwhere $A_{\\alpha\\beta\\gamma\\delta}\\,$, $B_{\\alpha\\beta\\gamma\\delta}\\,$, $D_{\\alpha\\beta\\gamma\\delta}\\,$, $S_{\\alpha\\beta}\\,$ and $G_{\\alpha\\beta}\\,$ are the constitutive coefficients of composite elastic shells, which have been determined in \\cite{Chroscielewski11} in terms of the material\/geometrical parameters of the layers. They satisfy the symmetry conditions\n$$A_{\\alpha\\beta\\gamma\\delta}=  A_{\\gamma\\delta\\alpha\\beta}\\,,\\quad D_{\\alpha\\beta\\gamma\\delta} = D_{\\gamma\\delta\\alpha\\beta}\\,,\\quad\n      S_{\\alpha\\beta}=  S_{\\beta\\alpha}\\,,\\quad G_{\\alpha\\beta} = G_{\\beta\\alpha}\\,.$$\nIn the constitutive relation \\eqref{63} one can observe a multiplicative coupling of the strain tensor $\\boldsymbol{E}^e$ with the curvature tensor $\\boldsymbol{K}^e$ for composite shells. Let us denote by $A$, $D$ and $B$ the $4\\times 4$ matrices of material constants\n\\begin{equation*}\n\\begin{array}{c}\n    A=\\begin{bmatrix} A_{1111}  & A_{1122}   &  A_{1112}   &   A_{1121}    \\\\\n                    A_{1122}  & A_{2222}   &   A_{2212}   &   A_{2221}    \\\\\n                    A_{1112}   & A_{2212}    &  A_{1212}  &   A_{1221}   \\\\\n                    A_{1121}   &  A_{2221}    &  A_{1221}    &  A_{2121}    \\end{bmatrix},\\qquad\n    D=\\begin{bmatrix} D_{1111}  & D_{1122}   &  D_{1112}   &   D_{1121}    \\\\\n                    D_{1122}  & D_{2222}   &   D_{2212}   &   D_{2221}    \\\\\n                    D_{1112}   & D_{2212}    &  D_{1212}  &   D_{1221}   \\\\\n                    D_{1121}   &  D_{2221}    &  D_{1221}    &  D_{2121}    \\end{bmatrix},\\\\\n    B=\\begin{bmatrix} B_{1111}  & B_{1122}   &  B_{1112}   &   B_{1121}    \\\\\n                    B_{2211}  & B_{2222}   &   B_{2212}   &   B_{2221}    \\\\\n                    B_{1211}   & B_{1222}    &  B_{1212}  &   B_{1221}   \\\\\n                    B_{2111}   &  B_{2122}    &  B_{2112}    &  B_{2121}    \\end{bmatrix}.\n\\end{array}\n\\end{equation*}\nOne can show that the necessary and sufficient condition for the coercivity of the strain energy function \\eqref{63} is that the following matrices are positive definite\n\\begin{equation*}\n    \\mathbb{C}=\\begin{bmatrix} A_{4\\times 4 }  & B_{4\\times 4 }   \\\\\n                    B_{4\\times 4 } & D_{4\\times 4 }    \\end{bmatrix}_{8\\times 8}\\,,\\quad\n                    \\mathbb{S}= \\begin{bmatrix} S_{11 }  & S_{12}    \\\\\n                    S_{12}  & S_{22}     \\end{bmatrix}_{2\\times 2}\\,,\\quad\n                    \\mathbb{G}= \\begin{bmatrix} G_{11 }  & G_{12}    \\\\\n                    G_{12}  & G_{22}     \\end{bmatrix}_{2\\times 2}\\,.\n\\end{equation*}\nWith these notations, one may write the strain energy density \\eqref{63} in the matrix form\n\\begin{equation*}\n\\begin{array}{l}\n   2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= V\\,\\mathbb{C}\\,V^T+ \\big(\\,\\tilde{E}^e_{31}\\,,\\,\\tilde{E}^e_{32}\\big)\\, \\mathbb{S}\\, \\big(\\,\\tilde{E}^e_{31}\\,,\\,\\tilde{E}^e_{32}\\big)^T + \\big(\\,\\tilde{K}^e_{31}\\,,\\,\\tilde{K}^e_{32}\\big)\\, \\mathbb{G}\\, \\big(\\,\\tilde{K}^e_{31}\\,,\\,\\tilde{K}^e_{32}\\big)^T, \\vspace{4pt}\\\\\n   \\qquad\\quad\\text{with}\\qquad V= \\big(\\,\\tilde{E}^e_{11}\\,,\\,\\tilde{E}^e_{22}\\,,\\, \\tilde{E}^e_{12}\\,,\\,\\tilde{E}^e_{21}\\,,\\,\\tilde{K}^e_{11}\\,,\\,\\tilde{K}^e_{22}\\,,\\, \\tilde{K}^e_{12}\\,,\\,\\tilde{K}^e_{21}\\big)_{1\\times 8}\\,\\,.\n\\end{array}\n\\end{equation*}\nIn conclusion, if the matrices $\\mathbb{C}$, $\\mathbb{S}$ and $\\mathbb{G}$ are positive definite, then we can apply  Theorem \\ref{th1} for the strain energy density given by \\eqref{63} and prove the existence of minimizers for composite layered shells.\n\n\n\\begin{remark}\nThe results and conclusions presented above are obviously valid also in the case of \\emph{plates}, i.e. when the reference base surface $S^0$ is \\emph{planar}. However, many of the formulas  for general shells can be significantly simplified in the case of plates, since   the 3 orthonormal bases\n$\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}\\,$, $\\{\\boldsymbol{d}^0_1,\\boldsymbol{d}^0_2,\\boldsymbol{d}^0_3\\}$ and $\\{\\boldsymbol{e}_1,\\boldsymbol{e}_2,\\boldsymbol{e}_3\\}$ can be considered identical.\n\nThe corresponding existence results for 6-parameter geometrically non-linear plates (planar shells) has been presented in \\cite{Birsan-Neff-Plates} for isotropic or anisotropic materials, and in \\cite{Birsan-Neff-AnnRom12} for composite planar--shells. We mention that, in the case of isotropic plates, the existence theorem can be obtained from the more general results concerning Cosserat planar--shells presented by the second author in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\n\nIn a forthcoming contribution we will extend our existence results to the 6-parameter resultant shell model with physically non-linear behavior and show the invertibility of the reconstructed deformation gradient $\\bar{F}$.\n\n\n\\bigskip\\bigskip\n\\small{\\textbf{Acknowledgements.}\nThe first author (M.B.) is supported by the german state  grant: ``Programm des Bundes und der L\\\"ander f\\\"ur bessere Studienbedingungen und mehr Qualit\\\"at in der Lehre''. We thank our many friends who have made substantial comments on a preliminary version of the paper.\n\n\n\n\n\n\\bibliographystyle{plain}\n{\\footnotesize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}