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+{"text":"\\section{Introduction and preliminaries}\n\n\\subsection{Homomorphisms and derivations}\n\nThe study of additive mappings from a ring into another ring which\npreserve squares was initiated by G.~Ancochea in \\cite{Anc42} in\nconnection with problems arising in projective geometry. Later,\nthese results were strengthened by (among others) Kaplansky\n\\cite{Kap} and Jacobson--Rickart \\cite{JR}.\n\n Let $R, R'$ be rings, the mapping $\\varphi:R\\rightarrow R'$ is called a \\emph{homomorphism} if\n\\[\n \\varphi(a+b)=\\varphi(a)+\\varphi(b)\n\\qquad \\left(a, b\\in R\\right)\n\\]\nand\n\\[\n \\varphi(ab)=\\varphi(a)\\varphi(b)\n\\qquad \\left(a, b\\in R\\right).\n\\]\nFurthermore, the function $\\varphi:R\\to R'$ is an\n\\emph{anti-homomorphism} if\n\\[\n \\varphi(a+b)=\\varphi(a)+\\varphi(b)\n\\qquad \\left(a, b\\in R\\right)\n\\]\nand\n\\[\n \\varphi(ab)=\\varphi(b)\\varphi(a)\n\\qquad \\left(a, b\\in R\\right).\n\\]\n\nHenceforth, $\\mathbb{N}$ will denote the set of the positive\nintegers. Let $n\\in\\mathbb{N}, n\\geq 2$ be fixed. \nThe function $\\varphi:R\\rightarrow R'$ is called an $n$-Jordan\nhomomorphism if\n\\[\n\\varphi(a+b)=\\varphi(a)+\\varphi(b) \\qquad \\left(a, b\\in R\\right)\n\\]\nand\n\\[\n \\varphi(a^{n})=\\varphi(a)^{n}\n\\qquad \\left(a\\in R\\right).\n\\]\nFinally, we remark that in case $n=2$ we speak about homomorphisms\nand Jordan homomorphisms, respectively. It was G.~Ancochea who\nfirstly dealt with the connection of Jordan homomorphisms and\nhomomorphisms, see \\cite{Anc42}. These results were\ngeneralized and extended in several ways, see for instance\n\\cite{JR}, \\cite{Kap}, \\cite{Zel68}. \n\n\nLet $n\\in\\mathbb{N}$, we say that a ring $R$ is \\emph{of\ncharacteristic larger than $n$} if $n!x=0$ implies that $x=0$. The\nring $R$ is termed to be a \\emph{prime ring} if\n$ a, b\\in R$ and $aRb=\\left\\{0\\right\\}$ imply that either $a=0$ or $b=0$. \nIn \\cite{Her} I.N.~Herstein proved that \nif $\\varphi$ is a Jordan homomorphism of a ring $R$\n\\emph{onto} a prime ring $R'$ of characteristic different from $2$\nand $3$, then either $\\varphi$ is a homomorphism or an\nanti-homomorphism.\n\n\nFurthermore, in the above-mentioned paper \\cite{Her}, \nnot only Jordan homomorphisms but also $n$-Jordan\nmappings were considered and he proved the following statement. \n\\begin{thm}\nLet $\\varphi$ be an $n$-Jordan\nhomomorphism from a ring $R$ \\textbf{onto} a prime ring $R'$ and assume that $R'$ has \ncharacteristic larger than $n$, suppose further that $R$ has a unit\nelement. Then $\\varphi=\\varepsilon\\tau$ where $\\tau$ is either a\nhomomorphism or an anti-homomorphism and $\\varepsilon$  is an\n$(n-1)$st root of unity lying in the center of $R'$.\n\\end{thm}\n\n\nBesides homomorphisms, derivations also play a key role in the theory of rings and fields. Concerning this notion, we will follow \\cite[Chapter 14]{Kuc}. \n\nLet $Q$ be a ring and let $P$ be a subring of $Q$.\nA function $d\\colon P\\rightarrow Q$ is called a \\emph{derivation}\\index{derivation} if it is additive,\ni.e. \n\\[\nd(x+y)=d(x)+d(y)\n\\quad\n\\left(x, y\\in P\\right)\n\\]\nand also satisfies the so-called \\emph{Leibniz rule}\\index{Leibniz rule}, i.e.  equation\n\\[\nd(xy)=d(x)y+xd(y)\n\\quad\n\\left(x, y\\in P\\right). \n\\]\n\nFundamental examples for derivations are the following ones. \n\n\nLet $\\mathbb{F}$ be a field, and let in the above definition $P=Q=\\mathbb{F}[x]$\nbe the ring of polynomials with coefficients from $\\mathbb{F}$. For a polynomial\n$p\\in\\mathbb{F}[x]$, $p(x)=\\sum_{k=0}^{n}a_{k}x^{k}$, define the function\n$f\\colon \\mathbb{F}[x]\\rightarrow\\mathbb{F}[x]$ as\n\\[\nf(p)=p',\n\\]\nwhere $p'(x)=\\sum_{k=1}^{n}ka_{k}x^{k-1}$ is the derivative of the polynomial $p$.\nThen the function $f$ clearly fulfills\n\\[\nf(p+q)=f(p)+f(q)\n\\]\nand\n\\[\nf(pq)=pf(q)+qf(p)\n\\]\nfor all $p, q\\in\\mathbb{F}[x]$. Hence $f$ is a derivation.\n\n\n\n Let $(\\mathbb{F}, +, \\cdot)$ be a field, and suppose that we are given a derivation \n $f\\colon \\mathbb{F}\\to \\mathbb{F}$. We define the mapping $f_{0}\\colon \\mathbb{F}[x]\\to \\mathbb{F}[x]$ in the following way. \nIf $p\\in \\mathbb{F}[x]$ has the form \n\\[\n p(x)=\\sum_{k=0}^{n}a_{k}x^{k}, \n\\]\nthen let \n\\[\n f_{0}(p)= p^{f}(x)=\\sum_{k=0}^{n}f(a_{k})x^{k}. \n\\]\nThen $f_{0}\\colon \\mathbb{F}[x]\\to \\mathbb{F}[x]$ is a derivation. \n\n\nIt is well-known that in case of additive functions, Hamel bases play an important role. \nAs \\cite[Theorem 14.2.1]{Kuc} shows in case of derivations, algebraic bases are fundamental. \n\n\\begin{thm}\\label{T14.2.1}\nLet $(\\mathbb{K}, +,\\cdot)$ be a field of characteristic zero, let $(\\mathbb{F}, +,\\cdot)$\nbe a subfield of $(\\mathbb{K}, +,\\cdot)$, let $S$ be an algebraic base of $\\mathbb{K}$ over $\\mathbb{F}$,\nif it exists, and let $S=\\emptyset$ otherwise.\nLet $f\\colon \\mathbb{F}\\to \\mathbb{K}$ be a derivation.\nThen, for every function $u\\colon S\\to \\mathbb{K}$,\nthere exists a unique derivation $g\\colon \\mathbb{K}\\to \\mathbb{K}$\nsuch that $g \\vert_{\\mathbb{F}}=f$ and $g \\vert_{S}=u$.\n\\end{thm}\n\n\n\n\n\n\n\\subsection{The symmetrization method}\n\nWhile proving our results, the so-called Polarization formula for multi-additive functions and the symmetrization method will play a key role. In this subsection the most important notations and statements are summarized. Here we follow the monograph \\cite{Sze91}. \n\n\n\\begin{dfn}\n Let $G, S$ be commutative semigroups, $n\\in \\mathbb{N}$ and let $A\\colon G^{n}\\to S$ be a function.\n We say that $A$ is \\emph{$n$-additive} if it is a homomorphism of $G$ into $S$ in each variable.\n If $n=1$ or $n=2$ then the function $A$ is simply termed to be \\emph{additive}\n or \\emph{bi-additive}, respectively.\n\\end{dfn}\n\nThe \\emph{diagonalization} or \\emph{trace} of an $n$-additive\nfunction $A\\colon G^{n}\\to S$ is defined as\n \\[\n  A^{\\ast}(x)=A\\left(x, \\ldots, x\\right)\n  \\qquad\n  \\left(x\\in G\\right).\n \\]\nAs a direct consequence of the definition each $n$-additive function\n$A\\colon G^{n}\\to S$ satisfies\n\\[\n A(x_{1}, \\ldots, x_{i-1}, kx_{i}, x_{i+1}, \\ldots, x_n)\n =\n kA(x_{1}, \\ldots, x_{i-1}, x_{i}, x_{i+1}, \\ldots, x_{n})\n \\qquad \n \\left(x_{1}, \\ldots, x_{n}\\in G\\right)\n\\]\nfor all $i=1, \\ldots, n$, where $k\\in \\mathbb{N}$ is arbitrary. The\nsame identity holds for any $k\\in \\mathbb{Z}$ provided that $G$ and\n$S$ are groups, and for $k\\in \\mathbb{Q}$, provided that $G$ and $S$\nare linear spaces over the rationals. For the diagonalization of $A$\nwe have\n\\[\n A^{\\ast}(kx)=k^{n}A^{\\ast}(x)\n \\qquad\n \\left(x\\in G\\right).\n\\]\n\nThe above notion can also be extended for the case $n=0$ by letting \n$G^{0}=G$ and by calling $0$-additive any constant function from $G$ to $S$. \n\nOne of the most important theoretical results concerning\nmultiadditive functions is the so-called \\emph{Polarization\nformula}, that briefly expresses that every $n$-additive symmetric\nfunction is \\emph{uniquely} determined by its diagonalization under\nsome conditions on the domain as well as on the range. Suppose that\n$G$ is a commutative semigroup and $S$ is a commutative group. The\naction of the {\\emph{difference operator}} $\\Delta$ on a function\n$f\\colon G\\to S$ is defined by the formula\n\\[\\Delta_y f(x)=f(x+y)-f(x)\n\\qquad\n\\left(x, y\\in G\\right). \\]\nNote that the addition in the argument of the function is the\noperation of the semigroup $G$ and the subtraction means the inverse\nof the operation of the group $S$.\n\n\\begin{thm}[Polarization formula]\\label{Thm_polarization}\n Suppose that $G$ is a commutative semigroup, $S$ is a commutative group, $n\\in \\mathbb{N}$.\n If $A\\colon G^{n}\\to S$ is a symmetric, $n$-additive function, then for all\n $x, y_{1}, \\ldots, y_{m}\\in G$ we have\n \\[\n  \\Delta_{y_{1}, \\ldots, y_{m}}A^{\\ast}(x)=\n  \\left\\{\n  \\begin{array}{rcl}\n   0 & \\text{ if} & m>n \\\\\n   n!A(y_{1}, \\ldots, y_{m}) & \\text{ if}& m=n.\n  \\end{array}\n  \\right.\n \\]\n\n\\end{thm}\n\n\\begin{cor}\n Suppose that $G$ is a commutative semigroup, $S$ is a commutative group, $n\\in \\mathbb{N}$.\n If $A\\colon G^{n}\\to S$ is a symmetric, $n$-additive function, then for all $x, y\\in G$\n \\[\n  \\Delta^{n}_{y}A^{\\ast}(x)=n!A^{\\ast}(y).\n\\]\n\\end{cor}\n\n\\begin{lem}\n\\label{mainfact}\n  Let $n\\in \\mathbb{N}$ and suppose that the multiplication by $n!$ is surjective in the commutative semigroup $G$ or injective in the commutative group $S$. Then for any symmetric, $n$-additive function $A\\colon G^{n}\\to S$, $A^{\\ast}\\equiv 0$ implies that\n  $A$ is identically zero, as well.\n\\end{lem}\n\n\n\\begin{dfn}\n Let $G$ and $S$ be commutative semigroups, a function $p\\colon G\\to S$ is called a \\emph{generalized polynomial} from $G$ to $S$ if it has a representation as the sum of diagonalizations of symmetric multi-additive functions from $G$ to $S$. In other words, a function $p\\colon G\\to S$ is a \n generalized polynomial if and only if, it has a representation \n \\[\n  p= \\sum_{k=0}^{n}A^{\\ast}_{k}, \n \\]\nwhere $n$ is a nonnegative integer and $A_{k}\\colon G^{k}\\to S$ is a symmetric, $k$-additive function for each \n$k=0, 1, \\ldots, n$. In this case we also say that $p$ is a generalized polynomial \\emph{of degree at most $n$}. \n\nLet $n$ be a nonnegative integer, functions $p_{n}\\colon G\\to S$ of the form \n\\[\n p_{n}= A_{n}^{\\ast}, \n\\]\nwhere $A_{n}\\colon G^{n}\\to S$ are the so-called \\emph{generalized monomials of degree $n$}. \n\\end{dfn}\n\n\\begin{rem}\n Obviously, generalized monomials \nof degree $0$ are constant functions and generalized monomials of degree $1$ are additive functions. \n\n Furthermore, generalized monomials of degree $2$ will be termed \\emph{quadratic functions}. \n\\end{rem}\n\n\n\n\n\\subsection{Polynomial functions}\n\n\nAs Laczkovich \\cite{Lac04} enlightens, on groups there are several polynomial notions. One of them is that we introduced in subsection 1.2, that is the notion of generalized polynomials. \nAs we will see in the forthcoming sections, not only this notion, but also that of \\emph{(normal) polynomials} will be important. The definitions and results recalled here can be found in \\cite{Sze91}. \n\nThroughout this subsection $G$ is assumed to be  a commutative group (written additively).\n\n\\begin{dfn}\n{\\it Polynomials} are elements of the algebra generated by additive\nfunctions over $G$. Namely, if $n$ is a positive integer,\n$P\\colon\\mathbb{C}^{n}\\to \\mathbb{C}$ is a (classical) complex\npolynomial in\n $n$ variables and $a_{k}\\colon G\\to \\mathbb{C}\\; (k=1, \\ldots, n)$ are additive functions, then the function\n \\[\n  x\\longmapsto P(a_{1}(x), \\ldots, a_{n}(x))\n \\]\nis a polynomial and, also conversely, every polynomial can be\nrepresented in such a form.\n\\end{dfn}\n\n\\begin{rem}\n For the sake of easier distinction, at some places polynomials will be called normal polynomials. \n\\end{rem}\n\n\n\n\n\\begin{rem}\n We recall that the elements of $\\mathbb{N}^{n}$ for any positive integer $n$ are called\n ($n$-dimensional) \\emph{multi-indices}.\n Addition, multiplication and inequalities between multi-indices of the same dimension are defined component-wise.\n Further, we define $x^{\\alpha}$ for any $n$-dimensional multi-index $\\alpha$ and for any\n $x=(x_{1}, \\ldots, x_{n})$ in $\\mathbb{C}^{n}$ by\n \\[\n  x^{\\alpha}=\\prod_{i=1}^{n}x_{i}^{\\alpha_{i}}\n \\]\nwhere we always adopt the convention $0^{0}=1$. We also use the\nnotation $\\left|\\alpha\\right|= \\alpha_{1}+\\cdots+\\alpha_{n}$. With\nthese notations any polynomial of degree at most $N$ on the\ncommutative semigroup $G$ has the form\n\\[\n p(x)= \\sum_{\\left|\\alpha\\right|\\leq N}c_{\\alpha}a(x)^{\\alpha}\n \\qquad\n \\left(x\\in G\\right),\n\\]\nwhere $c_{\\alpha}\\in \\mathbb{C}$ and $a=(a_1, \\dots, a_n) \\colon\nG\\to \\mathbb{C}^{n}$ is an additive function. Furthermore, the\n\\emph{homogeneous term of degree $k$} of $p$ is\n\\[\n \\sum_{\\left|\\alpha\\right|=k}c_{\\alpha}a(x)^{\\alpha} .\n\\]\n\\end{rem}\n\n\n\\begin{lem}[Lemma 2.7 of \\cite{Sze91}]\\label{L_lin_dep}\n Let $G$ be a commutative group,\n $n$ be a positive integer and let\n \\[\n  a=\\left(a_{1}, \\ldots, a_{n}\\right),\n \\]\nwhere $a_{1}, \\ldots, a_{n}$ are linearly independent complex valued\nadditive functions defined on $G$. Then the monomials\n$\\left\\{a^{\\alpha}\\right\\}$ for different multi-indices are linearly\nindependent.\n\\end{lem}\n\n\\begin{dfn}\nA function $m\\colon G\\to \\mathbb{C}$ is called an \\emph{exponential}\nfunction if it satisfies\n\\[\n m(x+y)=m(x)m(y)\n \\qquad\n \\left(x,y\\in G\\right).\n\\]\nFurthermore, on an  \\emph{exponential polynomial} we mean a linear\ncombination of functions of the form $p \\cdot m$, where $p$ is a\npolynomial and $m$ is an exponential function.\n\\end{dfn}\n\n\n\n\\begin{dfn}\n Let $G$ be an Abelian group and $V\\subseteq \\mathbb{C}^G$ a set of functions. We say that $V$ is {\\it translation invariant} if for every $f\\in V$ the function $\\tau_{g}f\\in V$ also holds for all $g\\in G$, where\n \\[\n  \\tau_{g}f(h)= f(h+g)\n  \\qquad\n  \\left(h\\in G\\right).\n \\]\n \\end{dfn}\n\n \n In view of Theorem 10.1 of Sz\\'ekelyhidi \\cite{Sze91}, any finite dimensional translation invariant linear \n space of complex valued functions on a commutative group consists of exponential polynomials. \n This implies that if $G$ is a commutative group, then any function \n $f\\colon G\\to \\mathbb{C}$, satisfying the functional equation \n \\[\n  f(x+y)= \\sum_{i=1}^{n}g_{i}(x)h_{i}(y) \n  \\qquad \n  \\left(x, y\\in G\\right)\n \\]\nfor some positive integer $n$ and functions $g_{i}, h_{i}\\colon G\\to \\mathbb{C}$ ($i=1, \\ldots, n$), \nis an exponential polynomial of degree at most $n$. \n \n\nThis enlightens the connection between generalized polynomials and polynomials. It is easy to see that \neach polynomial, that is, any function of the form \n\\[\n  x\\longmapsto P(a_{1}(x), \\ldots, a_{n}(x)), \n \\]\nwhere $n$ is a positive integer,\n$P\\colon\\mathbb{C}^{n}\\to \\mathbb{C}$ is a (classical) complex\npolynomial in\n $n$ variables and $a_{k}\\colon G\\to \\mathbb{C}\\; (k=1, \\ldots, n)$ are additive functions, is a generalized polynomial. The converse however is in general not true. A complex-valued generalized polynomial $p$ defined on a commutative group $G$ is a polynomial \\emph{if and only if} its variety (the linear space spanned by its translates) is of \\emph{finite} dimension. \nTo make the situation more clear, here we also recall Theorem 13.4 from Sz\\'ekelyhidi \\cite{Sze14}. \n \n \\begin{thm}\\label{thm_torsion}\n  The torsion free rank of a commutative group is finite \\emph{if and only if} every generalized polynomial on the group is a polynomial. \n \\end{thm}\n\n \n\\section{Results}\n\nIn this section $\\mathbb{F}$ is assumed to be a field with $\\mathrm{char}(\\mathbb{F})=0$.  \nLet further $n$ be a positive integer and $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{C}[x]$ be polynomials. Our aim is to prove characterization theorems for generalized polynomials $f\\colon \\mathbb{F}\\to \\mathbb{C}$ of degree at most $n$ that also fulfill equation \n\\[\n f(P(x))= Q(f(x))\n\\]\nfor each $x\\in \\mathbb{F}$. \n\n\nBefore presenting the results of this paper, we note that related problems have already been considered by Z.~Boros and E.~Garda--M\\'{a}ty\\'{a}s in \\cite{BorGar18, BorGar20, Gar19} and also by M.~Amou in \\cite{Amo20}. In these papers the authors consider monomial functions $f, g\\colon\\mathbb{R}\\to \\mathbb{R}$ of degree $n$, where $n\\in \\mathbb{N}$, $n\\geq 2$, which satisfy the conditional equation \n\\[\n x^{n}f(y)=y^{n}g(x)\n\\]\nfor all points $(x, y)$ on a specified planar curve. \n\nRoughly speaking, the following lemmata tell us that the problem investigated in this paper is meaningful in the sense that for any positive integer $n$, there \\emph{do exist} generalized polynomials of degree at most $n$ that satisfy the above identity. \n\n\n\\begin{lem}\\label{lemma3}\n Let $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$, $n$ be a positive integer and \n $\\varphi_{1}, \\ldots, \\varphi_{n}\\colon \\mathbb{F}\\to \\mathbb{C}$ be homomorphisms. Define the function $f$ on \n $\\mathbb{F}$ by \n \\[\n  f(x)= \\varphi_{1}(x)\\cdots \\varphi_{n}(x)\n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nThen the following statements hold true. \n\\begin{enumerate}[(i)]\n \\item The function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a generalized polynomial of degree $n$. \n \\item The function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a polynomial of degree $n$.\n \\item For any positive integer $k$, we have \n \\[\n  f(x^{k})=f(x)^{k} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\] \n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n Firstly, observe that the function $f$ defined on $\\mathbb{F}$ by \n \\[\n  f(x)= \\varphi_{1}(x)\\cdots \\varphi_{n}(x)\n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right)\n \\]\nis the trace of the symmetric $n$-additive mapping $F$ defined by \n\\[\n F(x_{1}, \\ldots, x_{n}) = \\frac{1}{n!}\\sum_{\\sigma\\in \\mathscr{S}_{n}}\\varphi_{1}(x_{\\sigma(1)})\\cdots \\varphi_{n}(x_{\\sigma(n)}) \n \\qquad \n \\left(x \\in \\mathbb{F}\\right). \n\\]\nThus $f$ is a generalized monomial of degree $n$. \nSecondly, in view of the definition of the function $f$ we immediately get that it is also a monomial of degree $n$. Indeed, let \n\\[\n P(x_{1}, \\ldots, x_{n})= x_{1}\\cdots x_{n} \n \\qquad \n \\left(x\\in \\mathbb{C}\\right)\n\\]\nand then $f$ can be written as $f= P\\circ \\varphi$, where $\\varphi\\colon \\mathbb{F}\\to \\mathbb{C}^{n}$ is \n\\[\n \\varphi(x)= \\left(\\varphi_{1}(x), \\ldots, \\varphi_{n}(x)\\right) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nThirdly, recall that in case $\\varphi$ is a homomorphism between $\\mathbb{F}$ and $\\mathbb{C}$, then \nwe also have \n\\[\n\\varphi(x^{k}) = \\varphi(x)^{k}\n\\]\nfor each $x\\in \\mathbb{K}$. Therefore, \n\\[\n f(x^{k}) = \\varphi_{1}(x^{k})\\cdots \\varphi_{n}(x^{k}) = \\varphi_{1}(x)^{k}\\cdots \\varphi_{n}(x)^{k} \n =\n \\left(\\varphi_{1}(x)\\cdots\\varphi_{n}(x)\\right)^{k}\n = f(x)^{k} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\end{proof}\n\n\n\\begin{lem}\\label{lemma4}\n Let $\\mathbb{F}\\subset \\mathbb{C}$ be a field\n and $d\\colon \\mathbb{F}\\to \\mathbb{C}$ be a derivation and consider the function \n $f\\colon \\mathbb{F}\\to \\mathbb{C}$ defined by \n \\[\n  f(x)= d(x^{n}) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nThen the following statements are satisfied. \n\\begin{enumerate}[(i)]\n \\item The function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a generalized polynomial of degree $n$. \n \\item The function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a polynomial of degree $n$.\n \\item For any positive integer $k\\geq 2$, we have \n \\[\n  f(x^{k})=kx^{(k-1)n}f(x)\n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\] \n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\n Let $\\mathbb{F}\\subset \\mathbb{C}$ be a field and  \n $d\\colon \\mathbb{F}\\to \\mathbb{C}$ be a derivation and let us consider the function \n $f\\colon \\mathbb{F}\\to \\mathbb{C}$ defined through \n \\[\n  f(x)= d(x^{n}) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nObserve that in this case the function $F_{n}\\colon \\mathbb{F}^{n}\\to \\mathbb{C}$ \ndefined by \n\\[\n F_{n}(x_{1}, \\ldots, x_{n})= d\\left(x_{1}\\cdots x_{n}\\right) \n \\qquad \n \\left(x_{1}, \\ldots, x_{n}\\in \\mathbb{F}\\right)\n\\]\nis a symmetric and $n$-additive function. Furthermore, its trace is $f$, showing that $f$ is a generalized monomial of degree $n$. \n\nAt the same time, since $d$ is a derivation, we also have that \n\\[\n d(x^{n})= nx^{n-1}d(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nyielding that \n\\[\n f(x)= nx^{n-1}d(x)= P(x, d(x)) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwith the two-variable complex polynomial\n\\[\n P(x, y)= nx^{n-1}y \n \\qquad \n \\left(x, y\\in \\mathbb{C}\\right). \n\\]\nTherefore, $f$ is a monomial of degree $n$. \n\nFinally, let $k\\geq 2$  be a positive integer. Then \n\\[\n f(x^{k})= d\\left((x^{k})^{n}\\right)= d(x^{kn})= kn x^{kn-1}d(x)\n =\n kx^{(k-1)n}\\cdot \\left(nx^{n-1}d(x)\\right)= kx^{(k-1)n}f(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\end{proof}\n\n\n\n\\begin{lem}\\label{lemma5}\n Let $n$ and $k$ be positive integers, $\\alpha\\in \\mathbb{N}^{n}$ be an n-dimensional multi-index, $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$ and \n $a_{1}, \\ldots, a_{n}\\colon \\mathbb{F} \\allowbreak\\to \\mathbb{C}$ be additive functions and \n $a=(a_{1}, \\ldots, a_{n})$. Then the mapping \n \\[\n  \\mathbb{F}\\ni x\\longmapsto a^{\\alpha}(x^{k})\n \\]\nis a generalized monomial of degree $|\\alpha|\\cdot k$. \n\\end{lem}\n\n\\begin{proof}\n Let $n$ and $k$ be positive integers, $\\alpha\\in \\mathbb{N}^{n}$ be an $n$-dimensional multi-index and \n $a_{1}, \\ldots, a_{n}\\colon \\mathbb{F} \\allowbreak\\to \\mathbb{C}$ be additive functions. \n If $\\alpha= (\\alpha_{1}, \\ldots, \\alpha_{n})$, then the function \n \\[\n \\mathbb{F}\\ni  x\\longmapsto a_{1}^{\\alpha_{1}}(x^{k})\\cdots a_{n}^{\\alpha_{n}}(x^{k}) \n \\]\nis the trace of the $|\\alpha|\\cdot k$-additive function $F$ defined by \n\\[\n F(x_{1, 1}, \\ldots, x_{\\alpha_{n}, k})= \n \\prod_{i=1}^{n}\\prod_{j=1}^{\\alpha_{i}}a_{i}(x_{j, 1}\\cdots x_{j, k}). \n\\]\n\\end{proof}\n\nLet $\\mathbb{F}$ be a field and denote $\\mathbb{F}^{\\times}$ the multiplicative group of the nonzero elements of $\\mathbb{F}$. \nObviously, every (normal) polynomial $p\\colon \\mathbb{F}^{\\times}\\to \\mathbb{C}$ is a generalized polynomial, too. Furthermore, in view of Theorem \\ref{thm_torsion} we know that these are the only generalized polynomials if the torsion free rank of $\\mathbb{F}^{\\times}$ is finite. For example if we take \n$\\mathbb{F}=\\mathbb{Q}$ (the field of the rationals), then this is the case. While for larger fields, the same does not hold in general. Despite Theorem \\ref{thm_torsion} gives an elegant criterion for the above problem, with the aid of this result it is not too easy to imagine how generalized polynomials look like. From one hand, this is the purpose of the remark below. \n\n\n\\begin{rem}\\label{rem3}\n Notice that the above lemma cannot be in general strengthened. To see this let \n $a\\colon \\mathbb{F}\\to \\mathbb{C}$ be a non-identically zero additive function. \n As the following proposition shows, the mapping \n \\[\n  \\mathbb{F} \\ni x\\longmapsto a(x^{2})\n \\]\n is a generalized monomial of degree two that is not necessarily a (normal) monomial. \n \\end{rem}\n \n \\begin{prop}\n  Let $\\mathbb{F}\\subset \\mathbb{C}$ be a field and $a\\colon \\mathbb{F}\\to\\mathbb{C}$ be a non-identically zero additive function. The mapping \n  \\[\n  \\mathbb{F} \\ni x\\longmapsto a(x^{2})\n  \\]\n  is a monomial of degree two if and only if \n  \\[\n   a(x)= \\varphi(d(x))+a(1)\\cdot \\varphi(x)\n   \\qquad \n   \\left(x\\in \\mathbb{F}\\right)\n  \\]\nor \n\\[\n a(x)= \\alpha \\varphi_{1}(x)+\\beta \\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwhere $\\alpha, \\beta$ are complex constants, $d\\colon \\mathbb{F}\\to \\mathbb{C}$ is a non-identically zero derivation and $\\varphi, \\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ are homorphisms such that $\\varphi_{1}$ and $\\varphi_{2}$ are linearly independent. \n \\end{prop}\n\n \n\\begin{proof} \n Let $\\mathbb{F}\\subset \\mathbb{C}$ be a field and $a\\colon \\mathbb{F}\\to\\mathbb{C}$ be a non-identically zero additive function.\nAssume that the mapping appearing in the proposition is a monomial of degree two. Then there exist linearly independent additive functions \n$a_{1}, a_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ and complex constants $\\alpha_{i, j}$, $i, j=1, 2$ such that \n\\[\n a(x^{2})= \\alpha_{1, 1}a_{1}(x)^{2}+(\\alpha_{1, 2}+\\alpha_{2, 1})a_{1}(x)a_{2}(x)+\\alpha_{2, 2}a_{2}(x)^{2}\n\\]\nfor each $x\\in \\mathbb{F}$. Since both sides of the above identity are traces of symmetric bi-additive functions, we can use the Polarization Formula to get that \n\\[\n a(xy)= \\alpha_{1, 1}a_{1}(x)a_{1}(y)+\\frac{\\alpha_{1, 2}+\\alpha_{2, 1}}{2}\\left(a_{1}(x)a_{2}(y)+a_{1}(y)a_{2}(x)\\right)+\\alpha_{2, 2}a_{2}(x)a_{2}(y)\n\\]\nis fulfilled by any $x, y\\in \\mathbb{F}$. \nAfter some rearrangement, we have \n\\[\n a(xy)\n = a_{1}(x) \\cdot \\left(\\alpha_{1, 1}a_{1}(y)+\\frac{\\alpha_{1, 2}+\\alpha_{2, 1}}{2}a_{2}(y)+\n \\right)\n +a_{2}(x)\\cdot \\left(\\frac{\\alpha_{1, 2}+\\alpha_{2, 1}}{2} a_{1}(y)+\\alpha_{2, 2}a_{2}(y)\\right)\n\\]\nfor all $x, y\\in \\mathbb{F}$, which is a Levi-Civita equation on $\\mathbb{F}^{\\times}$ (on the multiplicative group of the non-zero elements of $\\mathbb{F}$). Applying the result of \\cite[pages 93--94]{Sze91}, we deduce that there exist complex constants $\\alpha, \\beta$, an additive function \n$l\\colon \\mathbb{F}^{\\times}\\to \\mathbb{C}$ and exponentials $m, m_{1}, m_{2}\\colon \\mathbb{F}^{\\times}\\to \\mathbb{C}$, $m_{1}$ and $m_{2}$ are linearly independent such that \n\\[\n a(x)= (\\alpha l(x)+\\beta) m(x) \n \\qquad \n \\left(x\\in \\mathbb{F}^{\\times}\\right)\n\\]\nor \n\\[\n a(x)= m_{1}(x)+m_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}^{\\times}\\right). \n\\]\nRecall that on the group $\\mathbb{F}^{\\times}$ multiplication as group operation is considered. Therefore these functions fulfill the following identities: \n\\[\n l(xy)= l(x)+l(y) \n \\qquad \n \\left(x, y\\in \\mathbb{F}^{\\times}\\right)\n\\]\nand \n\\[\n m(xy)= m(x)m(y) \n \\qquad \n \\left(x, y\\in \\mathbb{F}^{\\times}\\right)\n\\]\nas well as \n\\[\n m_{i}(xy)= m_{i}(x)m_{i}(y) \n \\qquad \n \\left(x, y\\in \\mathbb{F}^{\\times}, i=1, 2\\right). \n\\]\nAt the same time, the mapping $a$ was assumed to be additive, thus in the above representation we have \n\\[\n   a(x)=\\varphi(d(x))+ a(1)\\cdot\\varphi(x) \n   \\qquad \n   \\left(x\\in \\mathbb{F}\\right)\n  \\]\nor \n\\[\n a(x)= \\alpha \\varphi_{1}(x)+\\beta \\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwhere $\\alpha, \\beta$ are complex constants, $d\\colon \\mathbb{F}\\to \\mathbb{C}$ is a non-identically zero derivation and $\\varphi, \\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ are homomorphisms such that $\\varphi_{1}$ and $\\varphi_{2}$ are linearly independent. \n\nThe converse is obvious. \n\\end{proof} \n\n\n\\begin{rem}\n The proof of the previous proposition can be a starting point of further investigations. More precisely, if $n$ is a positive integer, $a\\colon \\mathbb{F}\\to \\mathbb{C}$ is a non-identically zero additive function and the mapping \n  \\[\n  \\mathbb{F} \\ni x\\longmapsto a(x^{2})\n  \\]\n  is a monomial of degree $n$, then there exist linearly independent additive functions \n  $a_{1}, \\ldots, a_{n}\\colon \\mathbb{F}\\to \\mathbb{C}$ and complex constants $\\alpha_{i, j}$, $i, j=1, \\ldots, n$ such that \n  \\[\n   a(x^{2})= \\sum_{i, j=1}^{n}\\alpha_{i, j}a_{i}(x)a_{j}(x) \n   \\qquad \n   \\left(x\\in \\mathbb{F}\\right). \n  \\]\nSince both sides of the above identity are traces of symmetric bi-additive functions, we can use the Polarization Formula to get that\n\\[\n a(xy)= \\sum_{i, j=1}^{n}\\frac{\\alpha_{i, j}+\\alpha_{j, i}}{2}\\left(a_{i}(x)a_{j}(y)+a_{j}(x)a_{i}(y)\\right)\n = \\sum_{i=1}^{n}\\widetilde{a_{i}}(x)\\widetilde{a_{j}}(y)\n\\]\nfor all $x, y\\in \\mathbb{F}^{\\times}$ which is a Levi-Civita equation on the group $\\mathbb{F}^{\\times}$. In other words, the mapping $a$ as a function restricted to the multiplicative group $\\mathbb{F}^{\\times}$, is a normal exponential polynomial of degree $n$. In view of the results of \\cite[page 43 and page 79]{Sze91}, \n\\[\n a(x)= \\sum_{j=1}^{k}P_{j}\\left(l_{j, 1}(x), l_{j, 2}(x), \\ldots, l_{j, n_{j}-1}(x)\\right)m_{j}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}^{\\times}\\right), \n\\]\nwhere $k, n_{1}, \\ldots, n_{k}$ are positive integers, $m_{1}, \\ldots, m_{k}$ are different, non-zero com\\-plex-valued exponentials on the group $\\mathbb{F}^{\\times}$, further \n$\\left\\{l_{j, 1}, \\ldots, l_{j, n_{j}-1} \\right\\}$ are sets of linearly independent, com\\-plex-valued additive functions defined on $\\mathbb{F}^{\\times}$ for $j=1, \\ldots, k$ and \n$P_{j}\\colon \\mathbb{C}^{n_{j}-1}\\to \\mathbb{C}$ are complex polynomials of degree at most $n_{j}-1$ and in $n_{j}-1$ variables for each $j=1, \\ldots, k$. \n\nWe conjecture that as a continuation, a result of Kiss--Laczkovich \\cite{KisLac18} might be useful. According to Theorem 1.1 of this paper, $a\\colon \\mathbb{F}\\to \\mathbb{C}$ is an additive function with $a(1)=0$ and \n$D\/j$, as a map from the group $\\mathbb{F}^{\\times}$ to $\\mathbb{C}$, is a generalized polynomial of degree at most $n$ if and only if $a$ is a derivation of order at most $n$. Here $j$ denotes the identity map from $\\mathbb{F}$ to $\\mathbb{C}$. \n\\end{rem}\n\n\\begin{ex}\n Let $d\\colon \\mathbb{C}\\to \\mathbb{C}$ be a non-identically zero derivation. Due to Theorem \\ref{T14.2.1} such a mapping does exist. Furthermore, due to Theorem 14.5.1 of \\cite{Kuc}, there exist non-trivial endomorphisms of $\\mathbb{C}$ (trivial endomorphisms are meant the identically zero and the identity mapping, resp.). Let $\\varphi\\colon \\mathbb{C}\\to \\mathbb{C}$ be a non-trivial endomorphism. Then the mapping $a$ defined on $\\mathbb{C}$ by \n \\[\n  a(x)= d(x)+\\varphi(x) \n  \\qquad \n  \\left(C\\right)\n \\]\nis clearly additive and we have \n\\[\n a(x^2)= d(x^{2})+\\varphi(x^{2})= 2xd(x)+\\varphi(x)^{2} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nshowing that the mapping \n\\[\n \\mathbb{C} \\ni x \\longmapsto a(x^{2})\n\\]\nis a generalized polynomial of degree exactly two and it is a (normal) polynomial of degree exactly \nthree. \n\nNote that the situation changes if the domain is the real field. Indeed, all homomorphisms \n$\\varphi\\colon \\mathbb{R}\\to \\mathbb{C}$ are trivial, i.e. they are either identically zero or they coincide with the identity map. Therefore if $d\\colon \\mathbb{R}\\to \\mathbb{C}$ is a non-identically zero derivation and we define \n\\[\n a(x)= d(x)+x \n \\qquad \n \\left(x\\in \\mathbb{R}\\right),\n\\]\nthen the mapping \\[\n \\mathbb{C} \\ni x \\longmapsto a(x^{2})\n\\]\nis a polynomial of degree exactly two. \n\\end{ex}\n\n\n\\begin{rem}\nFrom the above thoughts we obtain that if $a\\colon \\mathbb{F}\\to \\mathbb{C}$ is an additive function such that the mapping \n\\[\n \\mathbb{F} \\ni x \\longmapsto a(x^{2})\n\\]\nis a generalized monomial which is not a monomial, then for all positive integer $k\\geq 2$, the mapping \n\\[\n \\mathbb{F} \\ni x \\longmapsto a(x^{k})\n\\]\nis also a generalized monomial which is not a monomial. \n\nIndeed, observe that the above mapping is the trace of the symmetric and $k$-additive function $A_{k}$ defined on $\\mathbb{F}^{k}$ by \n\\[\n A_{k}(x_{1}, \\ldots, x_{k})= a(x_{1}\\cdots x_{k}) \n \\qquad \n \\left(x_{1}, \\ldots, x_{k}\\in \\mathbb{F}\\right). \n\\]\n\nIf there would exist a positive integer $k>2$ such that the mapping \n\\[\n \\mathbb{F} \\ni x \\longmapsto a(x^{k})\n\\]\nwould be a monomial, then all the translates of this mapping would also be generalized monomials, as well. \nBy the Polarization formula, we have  \n\\[\n \\Delta_{y}^{2}\\Delta^{k-2}_{1}a(x^{k})= k!A(y, y, 1, \\ldots, 1)= k!a(y^{2}) \n \\qquad \n \\left(x, y\\in \\mathbb{F}\\right). \n\\]\nFrom this we would deduce that the mapping \n\\[\n \\mathbb{F} \\ni y \\longmapsto a(y^{2})\n\\]\nis a monomial of degree two, which is a contradiction. \n\\end{rem}\n\n\n\\begin{lem}\\label{lemma_6}\n Let $k$ and $n$ be positive integers and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a generalized monomial of degree n, where $\\mathbb{F}$ is assumed to be a field with $\\mathrm{char}(\\mathbb{F})=0$. Then the mapping \n \\[\n  \\mathbb{F} \\ni x \\longmapsto f(x^{k})\n \\]\nis a generalized monomial of degree $n\\cdot k$. \n\\end{lem}\n\n\\begin{proof}\n Since $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a generalized monomial of degree $n$, there exists an \n $n$-additive function $F_{n}\\colon \\mathbb{F}^{n}\\to \\mathbb{C}$ such that its trace is the function $f$. In this case, the mapping \n $F_{nk}\\colon \\mathbb{F}^{nk}\\to \\mathbb{C}$ defined by \n \\begin{multline*}\n  F_{nk}(x_{1, 1}, \\ldots, x_{n, k}) =\n  F_{n}(x_{1, 1} \\cdots x_{1, k}, x_{2, 1} \\cdots x_{2, k}, \\ldots, x_{n, 1} \\cdots x_{n, k})\n  \\\\\n  \\left(x_{i, j}\\in \\mathbb{F}, i=1, \\ldots, n, j=1, \\ldots, k\\right)\n \\end{multline*}\nis an $(n\\cdot k)$-additive function whose trace is \n\\[\n F_{nk}(x, \\ldots, x)= F_{n}(x^{k}, \\ldots, x^{k})= f(x^{k})\n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\end{proof}\n\n\\begin{rem}\n The above lemma with a different proof can be found among others in \\cite{AicMoo21}. \n\\end{rem}\n\n\nNow we turn to deal with the main problem of this paper, which is the following. \nAssume $\\mathbb{F}$ to be a field. \nLet further $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{C}[x]$ be polynomials. Our aim is to prove characterization theorems for quadratic functions $f\\colon \\mathbb{F}\\to \\mathbb{C}$ that also fulfill equation \n\\[\n f(P(x))= Q(f(x))\n\\]\nfor each $x\\in \\mathbb{F}$. \n\n\nAs for the difficulty of the problem, it is an important condition that the function $f$ is supposed to be quadratic (that is, it is a \\emph{generalized} monomial of degree two). As the statement below shows, \nalthough the result is the same, the proof is much easier for normal monomials of degree two. \n\n\\begin{prop}\\label{Prop1}\n Let $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$ and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function that can be represented as \n \\[\n  f(x)= a_{1}(x)a_{2}(x) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right)\n \\]\n with the aid of the additive functions $a_{1}, a_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$. \nThen equation \n \\begin{equation}\\label{Eq1_simp}\n  f(x^{2})= f(x)^{2}\n \\end{equation}\nholds for each $x\\in \\mathbb{F}$ \\emph{if and only if} there exist complex-valued non-trivial homomorphisms  $\\varphi_{1}, \\varphi_{2}$ defined on $\\mathbb{F}$ such that \n\\[\n f(x)= f(1)\\cdot \\varphi_{1}(x)\\varphi_{2}(x)\n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nFurthermore, \n\\begin{enumerate}[(A)]\n \\item either $f(1)=0$ and then $f$ is identically zero;\n \\item or $f(1)=1$. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n Assume $\\mathbb{F}$ be a field and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function that can be represented as \n \\[\n  f(x)= a_{1}(x)a_{2}(x) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right)\n \\]\n with the aid of the additive functions $a_{1}, a_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$. \n Then equation \\eqref{Eq1_simp} in terms of the additive functions $a_{1}$ and $a_{2}$ is \n \\[\n  a_{1}(x^{2})a_{2}(x^{2})= a_{1}(x)^{2}a_{2}(x)^{2} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nSince both sides of this identity are traces of symmetric, $4$-additive functions, a simple application of the Polarization Formula leads to \n\\begin{multline}\\label{Eq1_simp_sym}\n a_{1}(xy)a_{2}(uv)+a_{1}(xu)a_{2}(yv)+a_{1}(xv)a_{2}(yu)\n \\\\\n +a_{1}(yu)a_{2}(xv)+a_{1}(yv)a_{2}(xu)+a_{1}(ux)a_{2}(xy)\n \\\\\n =\n a_{1}(x)a_{1}(y)a_{2}(u)a_{2}(v)+a_{1}(x)a_{1}(u)a_{2}(y)a_{2}(v)+a_{1}(x)a_{1}(v)a_{2}(y)a_{2}(u)\n \\\\\n +a_{1}(y)a_{1}(u)a_{2}(x)a_{2}(v)+a_{1}(y)a_{1}(v)a_{2}(x)a_{2}(u)+a_{1}(u)a_{1}(x)a_{2}(x)a_{2}(y)\n \\\\\n (x, y, u, v\\in \\mathbb{F}). \n\\end{multline}\nFrom this, with the substitution $x=y=u=v=1$, we immediately get that \n\\[\n a_{1}(1)\\,a_{2}(1)\\,\\left(a_{1}(1)\\,a_{2}(1)-1\\right)=0\n\\]\nand with the substitution $y=u=v=1$, \n\\[\n \\left(a_{1}(1)\\,a_{2}(1)-1\\right)\\,\\left(a_{1}(1)\\,a_{2}(x)+a_{2}(\n 1)\\,a_{1}(x)\\right)=0\n\\]\ncan be deduced for all $x\\in \\mathbb{F}$. \nAssume first that $a_{1}(1)\\,a_{2}(1)-1\\neq 0$, then $a_{1}(1)=0$ or $a_{2}(1)=0$. If $a_{1}(1)=0$, then \n\\begin{enumerate}[(A)]\n \\item either $a_{2}(1)\\neq 0$ and the above identity reduces to \n \\[\n  a_{2}(1)a_{1}(x)=0 \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nIn other words $a_{1}$ is the identically zero function. In this case $f$ is identically zero, too. \n\\item or $a_{2}(1)=0$ and from \\eqref{Eq1_simp_sym} we get that \n\\begin{multline*}\n a_{1}(1)\\,a_{2}(x\\,y)+a_{2}(1)\\,a_{1}(x\\,y)+\\left(\\left(2-2\\,a_{1}(\n 1)\\,a_{2}(1)\\right)\\,a_{1}(x)-a_{1}(1)^2\\,a_{2}(x)\\right)\\,a_{2}(y)\n \\\\\n +\n \\left(\\left(2-2\\,a_{1}(1)\\,a_{2}(1)\\right)\\,a_{2}(x)-a_{2}(1)^2\\,a_{\n 1}(x)\\right)\\,a_{1}(y)=0\n\\end{multline*}\nfor all $x, y\\in \\mathbb{F}$, that is, \n\\[\n a_{1}(x)\\,a_{2}(y)+a_{2}(x)\\,a_{1}(y)=0 \n \\qquad \n \\left(x, y\\in \\mathbb{F}\\right). \n\\]\nFrom this, \n\\[\n a_{1}(x)a_{2}(x)=0\n\\]\nfollows for all $x\\in \\mathbb{F}$, that is, $f$ is the identically zero function.\n\\end{enumerate}\nTherefore, from now on $a_{1}(1)\\,a_{2}(1)-1=0$ can be supposed. Note that without the loss of generality we can (and we do) assume that $a_{1}(1)=a_{2}(1)=1$. In this case equation \n\\eqref{Eq1_simp_sym} implies that \n\\[\n a_{2}(x\\,y)+a_{1}(x\\,y)=a_{2}(x)\\,a_{2}(y)+a_{1}(x)\\,a_{1}(y)\n\\]\nfor all $x, y\\in \\mathbb{F}$. From this we deduce that either \n\\begin{enumerate}[(A)]\n \\item $\\left\\{a_{1}, a_{2}\\right\\}$ is linearly dependent, that is \n \\[\n  a_{2}=ca_{1}(x) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right), \n \\]\nfrom which we get that $a_{1}\\equiv a_{2}$, since $a_{1}(1)=a_{2}(1)$. If so, then equation \\eqref{Eq1_simp} reduces to \n\\[\n a(x^{2})^{2}= \\kappa a(x)^{4} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nthat is $a$ is a homomorphism and \n\\[\n f(x)= f(1)a(x)^{2} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\item or $\\left\\{a_{1}, a_{2}\\right\\}$ is linearly independent. Then due to \\cite[pages 93-94]{Sze91}\n\\[\n a_{i}(x)= (\\alpha_{i}l(x)+\\beta_{i})m(x) \n \\qquad \n \\left(x\\in \\mathbb{F}^{\\times}, i=1, 2\\right)\n\\]\nor \n\\[\n a_{i}(x)= \\alpha_{i}m_{1}(x)+\\beta_{i}m_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}^{\\times}, i=1, 2\\right), \n\\]\nwhere $l\\colon \\mathbb{F}^{\\times}\\to \\mathbb{C}$ is a logarithmic function, \n$m, m_{1}, m_{2}\\colon \\mathbb{F}^{\\times}\\to \\mathbb{C}$ are multiplicative functions and \n$\\alpha_{1}, \\alpha_{2}$ and $\\beta_{1}, \\beta_{2}$ are complex constants. \n\nSubstituting the first form into equation \\eqref{Eq1_simp}, $\\alpha_{1}=0$ and $\\alpha_{2}=0$. That is, $a_{1}$ and $a_{2}$ are constant multiples of a (nonzero) homomorphism. This means that there exists linearly independent homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)= f(1)\\varphi_{1}(x)\\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nFinally, if we substitute the second form into \\eqref{Eq1_simp} then we get (among others) that \n\\[\n \\begin{array}{rcl}\n \\alpha_{1}\\alpha_{2}(1-\\alpha_{1}\\alpha_{2})&=&0\\\\\n  \\beta_{1}\\beta_{2}(1-\\beta_{1}\\beta_{2})&=&0\n \\end{array}\n\\]\nIf we would have $\\beta_{1}\\beta_{2}=1$, then the remaining equations, that is, \n\\[\n \\begin{array}{rcl}\n  \\alpha_{1}\\beta_{1}\\beta_{2}^{2}+\\alpha_{2}\\beta_{1}^{2}\\beta_{2}&=&0\\\\\n  \\alpha_{1}^{2}\\alpha_{2}\\beta_{2}+\\alpha_{1}\\beta_{2}^{2}\\beta_{1}&=&0\\\\\n  \\alpha_{1}\\beta_{2}+\\beta_{1}\\alpha_{2}-\\alpha_{1}^{2}\\beta_{2}^{2}-\\beta_{1}^{2}\\alpha_{2}-4\\alpha_{1}\\alpha_{2}\\beta_{1}\\beta_{2}&=&0\n \\end{array}\n\\]\nwould lead to a contradiction. The same concerns the case $\\alpha_{1}\\alpha_{2}=1$. \n\nThis means however that $\\alpha_{1}\\alpha_{2}=0$ and $\\beta_{1}\\beta_{2}=0$, that is, either $f$ is the \nidentically zero function, or there exists homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)=f(1)\\varphi_{1}(x)\\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\end{enumerate}\n\\end{proof}\n\n\n\\begin{rem}\n Roughly speaking the above statement says that among polynomials whose variety is at most two-dimensional, the solutions $f\\colon \\mathbb{F}\\to \\mathbb{C}$ of equation \\eqref{Eq1_simp} are of the form \n \\[\n  f(x)= f(1)\\varphi_{1}(x)\\varphi_{2}(x) \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right), \n \\]\nwith appropriate homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$. \n\nLet now $n$ be a fixed positive integer. \nIn the general case, that is, if $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a quadratic function whose variety is at most $n$-dimensional, then we have \n\\[\n f(x)=\\sum_{p, q=1}^{n}a_{p}(x)a_{q}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwith certain additive functions $a_{1}, \\ldots, a_{n}\\colon \\mathbb{F}\\to \\mathbb{C}$. \nIn this situation equation \\eqref{Eq1_simp} can be investigated with an analogous argument as in the proof of Proposition \\ref{Prop1}. \n\\end{rem}\n\n\nThe starting point of study of equation \\eqref{Eq1} among quadratic functions is the theorem below that can be found in \\cite{GseKisVin19}. Here we also recall its proof to enlighten the way to the general case. \n\n\\begin{thm}\\label{THM_main}\n Let $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$ and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function. \n Then equation \n \\begin{equation}\\label{Eq1}\n  f(x^{2})= f(x)^{2}\n \\end{equation}\nholds for each $x\\in \\mathbb{F}$ \\emph{if and only if} there exist non-trivial homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\allowbreak \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)= f(1)\\cdot \\varphi_{1}(x)\\varphi_{2}(x)\n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nFurthermore, \n\\begin{enumerate}[(A)]\n \\item either $f(1)=0$ and then $f$ is identically zero;\n \\item or $f(1)=1$. \n\\end{enumerate}\n\n\\end{thm}\n\n\\begin{proof}\n Since $f$ is a generalized monomial of degree $2$, there exists a\nsymmetric bi-additive function $F_{2}\\colon \\mathbb{F}^{2}\\allowbreak\n\\to \\mathbb{C}$ so that\n\\[\n F_{2}(x, x)= f(x)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\nDefine the symmetric $4$-additive mapping $F_{4}\\colon\n\\mathbb{F}^{4}\\to \\mathbb{C}$ through\n\\begin{multline*}\n F_{4}(x_{1}, x_{2}, x_{3}, x_{4})\n =\n F_{2}(x_{1}x_{2}, x_{3}x_{4})\n +  F_{2}(x_{1}x_{3}, x_{2}x_{4})\n + F_{2}(x_{1}x_{4}, x_{2}x_{3})\n \\\\\n -F_{2}(x_{1}, x_{2})F(x_{3}, x_{4})\n -F_{2}(x_{1}, x_{3})F(x_{2}, x_{4})\n -F_{2}(x_{1}, x_{4})F(x_{2}, x_{3})\n \\\\\n \\left(x_{1}, x_{2}, x_{3}, x_{4}\\in \\mathbb{F}\\right).\n\\end{multline*}\nSince\n\\[\n F_{4}(x, x, x, x)= 3 \\left(F_{2}(x^{2}, x^{2})-F_{2}(x, x)^{2}\\right)=3 \\left(f(x^{2})-f(x)^{2}\\right)=0\n \\qquad\n \\left(x\\in \\mathbb{F}\\right),\n\\]\nthe mapping $F_{4}$ has to be identically zero on\n$\\mathbb{F}^{4}$. Therefore, especially\n\\[\n0=F_{4}(1, 1, 1, 1)= 3F_{2}(1, 1)-3F_{2}(1, 1)^{2},\n\\]\nyielding that either $F_{2}(1, 1)=0$ or $F_{2}(1, 1)=1$. Moreover,\n\\[\n 0= F_{4}(x, 1, 1, 1) =\n 3F_{2}\\left(x , 1\\right)-3F_{2}\\left(1 , 1\\right)F_{2}\\left(x , 1\\right)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right),\n\\]\nfrom which either $F_{2}(1, 1)=1$ or $F_{2}(x, 1)=0$ follows for any $x\\in\n\\mathbb{F}$.\n\nUsing that\n\\[\n 0= F_{4}(x, x, 1, 1)\n =\n F_{2}(x^2 , 1)-F_{2}\\left(1 , 1\\right)\\,F_{2}\\left(x , x\\right)+2F_{2} \\left(x , x\\right)-2F^2_{2}\\left(x , 1\\right)\n \\qquad \n \\left(x\\in \\mathbb{F}\\right),\n\\]\nwe obtain that\n\\[\n \\left(F_{2}(1, 1)-2\\right)F_{2}\\left(x , x\\right)= F_{2}(x^2 , 1)-2F^2_{2}\\left(x , 1\\right)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\nNow, if $F_{2}(1, 1)=0$, then according to the above identities $F_{2}(x,\n1)=0 $ would follow for all $x\\in \\mathbb{F}$. Since $F_{4}(x,\nx, 1, 1)=0$ is also fulfilled by any $x\\in \\mathbb{F}$, this\nimmediately implies that\n\\[\n -2f(x)=-2F_{2}(x, x)= F_{2}(x^{2}, 1)-F_{2}(x, 1)^{2}=0\n \\qquad\n \\left(x\\in \\mathbb{F}\\right),\n\\]\ni.e., $f$ is identically zero.\n\nIn case $F_{2}(1, 1)\\neq 0$, then necessarily $F_{2}(1, 1)=1$ from which\n\\[\n -F_{2}(x, x)= F_{2}(x^{2}, 1)-2F_{2}(x, 1)^{2}\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\nDefine the non-identically zero additive function $a\\colon\n\\mathbb{F}\\to \\mathbb{C}$ by\n\\[\n a(x)=F_{2}(x, 1)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right)\n\\]\nto get that\n\\[\n f(x)=F_{2}(x, x)= -F_{2}(x^{2}, 1)+2F_{2}(x, 1)^{2}= 2a(x)^{2} -a(x^{2})\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\nSince $F_{4}(x, x, x, x)=0$ has to hold, the additive function\n$a\\colon \\mathbb{F}\\to \\mathbb{C}$ has to fulfill identity\n\\begin{equation}\\label{Eq3}\n -a(x^4)+a^2(x^2)+4a^2(x)a(x^2)-4a^4(x)=0\n \\qquad\n \\left(x\\in \\mathbb{F}\\right)\n\\end{equation}\ntoo.\n\nIn what follows, we will show that the additive function $a$ is of a\nrather special form.\n\nIndeed,\n\\[\n 0= F_{4}(x, y, z, 1)\n \\qquad\n \\left(x, y, z\\in \\mathbb{F}\\right)\n\\]\nmeans that $a$ has to fulfill equation\n\\[\n a(x)a(yz)+a(y)a(xz)+a(z)a(xy)= 2a(x)a(y)a(z)+a(xyz)\n \\qquad\n \\left(x, y, z\\in \\mathbb{F}\\right)\n\\]\nLet now $z^{\\ast}\\in \\mathbb{F}$ be arbitrarily fixed to have\n\\[\n a(x)a(yz^{\\ast})+a(y)a(xz^{\\ast})+a(z^{\\ast})a(xy)= 2a(x)a(y)a(z^{\\ast})+a(xyz^{\\ast})\n \\qquad\n \\left(x, y, z\\in \\mathbb{F}\\right).\n\\]\nDefine the additive function $A\\colon \\mathbb{F}\\to \\mathbb{C}$ by\n\\[\n A(x)=a(xz^{\\ast})-a(z^{\\ast})a(x)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right)\n\\]\nto receive that\n\\[\n A(xy)=a(x)A(y)+a(y)A(x)\n \\qquad\n \\left(x, y\\in \\mathbb{F}\\right),\n \\]\n which is a special convolution type functional equation.\nDue to Theorem 12.2 of \\cite{Sze91}, we get that\n\\begin{enumerate}[(a)]\n \\item the function $A$ is identically zero under any choice of $z^{\\ast}$, implying that $a$ has to be multiplicative.\n Note that $a$ is additive, too. Thus, for the quadratic mapping $f\\colon \\mathbb{F}\\to \\mathbb{C}$ there exists a\n homomorphism $\\varphi\\colon \\mathbb{F}\\to \\mathbb{C}$ such that\n \\[\n  f(x)=\\varphi(x)^{2}\n  \\qquad\n  \\left(x\\in \\mathbb{F}\\right).\n \\]\n\n\\item or there exists multiplicative functions $m_{1}, m_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$\nand a complex constant $\\alpha$ such that\n\\[\n a(x)=\\frac{m_{1}(x)+m_{2}(x)}{2}\n \\qquad\n \\left(x\\in \\mathbb{F}\\right)\n\\]\nand\n\\[\n A(x)=\\alpha \\left(m_{1}(x)-m_{2}(x)\\right)\n  \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\nDue to the additivity of $a$, in view of the definition of the\nmapping $A$, we get that $A$ is additive, too.\n\nThis however means that both the maps $m_{1}+m_{2}$ and\n$m_{1}-m_{2}$ are additive, from which the additivity of $m_{1}$ and\n$m_{2}$ follows, yielding that they are in fact homomorphisms.\n\nSince\n\\[\n F_{2}(x, x)= f(x)= 2a(x)^{2}-a(x^{2})\n \\qquad\n \\left(x\\in \\mathbb{F}\\right),\n\\]\nwe obtain for the quadratic function $f\\colon \\mathbb{F}\\to\n\\mathbb{C}$ that there exist homomorphisms $\\varphi_{1},\n\\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that\n\\[\n f(x)=\\varphi_{1}(x)\\varphi_{2}(x)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\n\\end{enumerate}\nSumming up, we received the following: identity\n\\[\n f(x^{2})= f(x)^{2}\n \\qquad\n \\left(x\\in \\mathbb{F}\\right)\n\\]\nholds for the quadratic function $f\\colon \\mathbb{F}\\to \\mathbb{C}$\nif and only if there exist homomorphisms $\\varphi_{1},\n\\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that\n\\[\n f(x)=f(1)\\cdot\\varphi_{1}(x)\\varphi_{2}(x)\n \\qquad\n \\left(x\\in \\mathbb{F}\\right).\n\\]\n\\end{proof}\n\n\n\\begin{cor}\n Let $n\\geq 2$ be a positive integer, $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$ and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function. \n Then equation \n \\begin{equation}\\label{Eq2}\n  f(x^{n})= f(x)^{n}\n \\end{equation}\nholds for each $x\\in \\mathbb{F}$ \\emph{if and only if} there exist  homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)= f(1)\\cdot \\varphi_{1}(x)\\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nfurthermore either $f(1)=0$ and then $f$ is identically zero, or $f(1)$ is an $(n-1)$\\textsuperscript{st} root of unity. \n\\end{cor}\n\n\\begin{proof}\n In case $n=2$, then in view of the previous theorem, there is nothing to prove. \n Thus $n>2$ can be supposed subsequently. \n \nLet $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$, $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ be homomorphisms and \n$\\lambda\\in \\mathbb{C}$ be such that either $\\lambda$ is zero, or it is an $(n-1)$\\textsuperscript{st} root of unity. \nDefine the function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ by \n\\[\nf(x)= \\lambda \\varphi_{1}(x)\\varphi_{2}(x) \n\\qquad \n\\left(x\\in \\mathbb{F}\\right). \n\\]\nSince every homomorphism is additive, Lemma \\ref{lemma3} immediately yields that $f$ is a quadratic function and we also have \n\\[\n f(x^{n})= \\lambda\\varphi_{1}(x^{n})\\varphi_{2}(x^{n})= \\lambda \\varphi_{1}(x)^{n}\\varphi_{2}(x)^{n}\n = \\lambda^{n}\\varphi_{1}(x)^{n}\\varphi_{2}(x)^{n}=  \\left(\\lambda \\varphi_{1}(x)\\varphi_{2}(x)\\right)^{n}\n =f(x)^{n}\n\\]\nfor each $x\\in \\mathbb{F}$, since $\\lambda= \\lambda^{n}$. \n  \nConversely, let $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function such that we additionally have that \n  \\[\n   f(x^{n})= f(x)^{n}\n  \\]\nfor all $x\\in \\mathbb{F}$. Since $f$ is a quadratic function, there exists a symmetric, bi-additive function $F_{2}\\colon \\mathbb{F}^{2}\\to \\mathbb{C}$ such that \n\\[\n f(x)= F_{2}(x, x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nEquation \\eqref{Eq2} with $x=1$ immediately yields that \n\\[\n f(1)=f(1)^{n}, \n\\]\nthat is, $F(1, 1)= f(1)$ is either zero, or it is an $(n-1)$\\textsuperscript{st} root of unity. \n\nFurthermore, equation \\eqref{Eq2} in terms of the function $F_{2}$ is \n\\[\n F_{2}(x^{n}, x^{n})= F_{2}(x, x)^{n} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nObserve that both the sides of the above identity are traces of symmetric and $2n$-additive functions, namely we have \n\\[\n \\frac{1}{(2n)!}\\sum_{\\sigma\\in \\mathscr{S}_{2n}}F_{2}(x_{\\sigma(1)}\\cdots x_{\\sigma(n)}, x_{\\sigma(n+1)} \\cdots x_{\\sigma(2n)})\n =\n \\frac{1}{(2n!)} \\sum_{\\sigma \\in \\mathscr{S}_{2n}}F_{2}(x_{\\sigma(1)}, x_{\\sigma(2)})\\cdots F_{2}(x_{\\sigma(2n-1)}, x_{\\sigma(2n)})\n\\]\nfor all $x_{1}, \\ldots, x_{2n}\\in \\mathbb{F}$. \nThis identity with the substitution \n\\[\n x_{1}= x, \\; x_{2}=x \\quad x_{i}=1 \\quad \\text{for } i=3, \\ldots, 2n\n\\]\nyields that there are complex constants $\\alpha$ and $\\beta$ depending only on $F_{2}(1, 1)$ such that \n\\[\n f(x)=F_{2}(x, x)= \\alpha a(x^{2})+\\beta a(x)^{2} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwhere the additive function $a\\colon \\mathbb{F}\\to \\mathbb{C}$ is defined by \n\\[\n a(x)= F_{2}(x, 1) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nWriting this form back into equation \\eqref{Eq2}, we deduce \n\\[\n \\alpha a(x^{2n})+\\beta a(x^{n})^{2}= \\left(\\alpha a(x^{2})+\\beta a(x)^{2}\\right)^{n} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nAgain, both the sides of this identity are traces of symmetric and $2n$-additive functions, therefore \n\\begin{multline*}\n \\frac{1}{(2n)!} \\sum_{\\sigma\\in \\mathscr{S}_{2n}} \\left[\\alpha a(x_{\\sigma(1)} \\cdots x_{\\sigma_{(2n)}})+\\beta a(x_{\\sigma_{(1)}} \\cdots x_{\\sigma_{(n)}})a(x_{\\sigma(n+1)} \\cdots x_{\\sigma(n)})\\right]\n\\\\= \n\\frac{1}{(2n)!} \\sum_{\\sigma\\in \\mathscr{S}_{2n}} \\prod_{k=0}^{n-1}(\\alpha a(x_{\\sigma(2k+1)}x_{\\sigma(2k+2)})+\\beta a(x_{\\sigma(2k+1)})a(x_{\\sigma(2k+2)}))\n\\end{multline*}\nfor each $x_{1}, \\ldots, x_{2n}\\in \\mathbb{F}$. Let now $x, y, z\\in \\mathbb{F}$ be arbitrary, then this identity with the substitutions \n\\[\n x_{1}=x, \\, x_{2}=y,\\, x_{3}=z \\quad  \\text{and} \\quad x_{i}=0 \\quad \\text{for} \\quad 4\\leq i \\leq 2n\n\\]\nleads to \n\\[\n Aa(xyz)+Ba(xy)a(z)+Ca(xz)a(y)\n +Da(yz)a(x)+Ea(x)a(y)a(z)=0 \n \\qquad \n \\left(x, y, z\\in \\mathbb{F}\\right), \n\\]\nthat is a similar equation that appear in the proof of Theorem \\ref{THM_main}. With an analogous thread we get that $a$ can be written as a sum of two homomorphisms that finally implies for the function $f$ that there are homomorphisms $\\varphi_{1}, \\varphi_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)= f(1)\\varphi_{1}(x)\\varphi_{2}(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nwhere $f(1)$ is either zero, or it is an $(n-1)$\\textsuperscript{st} root of unity. \n\\end{proof}\n\n\n\\section{Open problems and further perspectives}\n\n\nAs it is written at the beginning of the second section, the main aim of this paper was to \nprove characterization theorems for generalized polynomials $f\\colon \\mathbb{F}\\to \\mathbb{C}$ of degree at most $n$ that also fulfill equation \n\\[\n f(P(x))= Q(f(x))\n\\]\nfor each $x\\in \\mathbb{F}$, \nwhere $n$ is a positive integer and $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{C}[x]$ are polynomials. \nThe results presented in connection with this problem can be considered as initial steps. Thus we close this paper with several open questions and we would like to give some perspectives, too. \n\n\\begin{rem}\n Clearly, it is enough to consider the case $\\deg (P)=\\deg(Q)$. Indeed, if \n $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a generalized monomial of degree $n$, the due to Lemma \\ref{lemma_6}, the mappings \n \\[\n  \\mathbb{F}\\ni x \\longmapsto f(P(x)) \n  \\quad \n  \\text{and}\n  \\quad \n  \\mathbb{F}\\ni x \\longmapsto Q(f(x)) \n \\]\nare generalized polynomials of degree $n\\cdot \\deg(P)$ and $n\\cdot \\deg(Q)$, respectively. Furthermore, \nthey coincide at each point $x\\in \\mathbb{F}$. However, this is only possible if $\\deg (P)=\\deg(Q)$. \n\\end{rem}\n\nIn the second section, we studied only quadratic functions, i.e., generalized monomials of degree two. Therefore, we formulate the following. \n\n\\begin{opp}\nLet  $n\\in \\mathbb{N}$, $n\\geq 2$ and $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{C}[x]$ be polynomials of degree at least two and $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a generalized monomial of degree $n$. Prove or disprove that if \n\\[\n f(P(x))= Q(f(x)) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nthen and only then there exist homomorphisms $\\varphi_{1}, \\ldots, \\varphi_{n}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n f(x)= f(1)\\cdot \\varphi_{1}(x)\\cdots \\varphi_{n}(x)\n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\n\\end{opp}\n\n\n\\begin{opp}\n It might be promising to consider firstly the case \n \\[\n  P(x)= Q(x)=x^{2} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right), \n \\]\n because as the results of the previous section show, hopefully, the case \n \\[\n  P(x)= Q(x)=x^{n} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right)\n \\]\n where $n>2$, leads back to the case $n=2$. \n\\end{opp}\n\n\\begin{opp}\n In the special case we considered the above general problem, it turned our that the solutions of the functional equations are always (normal) monomials. Additionally, we also showed that the proof is much easier if we know this, see Proposition \\ref{Prop1}. We conjecture that this is true in general, too. Thus, prove or disprove that if the generalized monomial $f\\colon \\mathbb{F}\\to \\mathbb{C}$ solves equation \n \\[\n f(P(x))= Q(f(x)) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nthen $f$ is a monomial. \n\\end{opp}\n\n\n\\begin{rem}\n The above problem is clearly meaningful for polynomials $P$ and $Q$ with degree one. At the same time, in this case we cannot expect representations similar to that ones that appeared in the statements proved in the second section. Here we consider only the case quadratic functions. Accordingly, assume that for the quadratic function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ equation \n \\[\n  f(ax+b)= Af(x)+B \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right), \n \\]\nwith some $a, b\\in \\mathbb{F}$ and $A, B\\in \\mathbb{C}$. Let us denote $F\\colon \\mathbb{F}^{2}\\to \\mathbb{C}$ the uniquely determined symmetric, bi-additive function whose trace is $f$. With $x=0$ we immediately get that $f(b)=F(b, b)=B$. \nFurthermore, \n\\[\n F(ax+b, ax+b)=AF(x, x)+F(b, b)\n \\qquad \n \\left(x, \\in \\mathbb{F}\\right), \n\\]\nthat is, \n\\[\n F(ax, ax)+2F(ax, b)+F(b, b)= AF(x, x)+F(b, b)\n \\qquad \n \\left(x, \\in \\mathbb{F}\\right), \n\\]\nor after some simplification \n\\[\n \\left(F(ax, ax)-AF(x, x)\\right)+2F(ax, b)= 0\n\\]\nfor all $x\\in \\mathbb{F}$. Since the left hand side of this equation is a generalized polynomial of degree two which has to be identically zero, all of its monomial terms should vanish. \nThis means from one hand that \n\\[\n F(ax, b)=0\n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nespecially, $f(b, b)=B=0$. On the other hand, we also have \n\\[\n F(ax, ax)=AF(x, x)\n \\qquad \n \\left(x\\in \\mathbb{F}\\right), \n\\]\nfrom this however \n\\[\n F(ax, ay)=AF(x, y)\n \\qquad \n \\left(x, y\\in \\mathbb{F}\\right), \n\\]\nfollows, that is, the symmetric, bi-additive function is \\emph{semi-homogeneous}. \nFrom \\cite[Theorem 3]{GseKisVin20} it is known that a non-identically zero, symmetric and bi-additive function \n$F$ fulfilling the above semi-homogeneity exists if and only if there are injective homomorphism \n$\\delta_{1}, \\delta_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n \\delta_{1}(a)\\delta_{2}(a)= A. \n\\]\nSumming up, if $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a non-identically zero quadratic function such that \n\\[\n  f(ax+b)= Af(x)+B \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right), \n \\]\nwith some $a, b\\in \\mathbb{F}$ and $A, B\\in \\mathbb{C}$, then \n\\begin{enumerate}[(i)]\n \\item $B=0$\n \\item for the uniquely determined symmetric and bi-additive function $F\\colon \\mathbb{F}^{2}\\to \\mathbb{C}$, identity\n \\[\n  F(ax, b)=0\n \\]\nis satisfied for all $x\\in \\mathbb{F}$. \n\\item there are injective homomorphism \n$\\delta_{1}, \\delta_{2}\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n \\delta_{1}(a)\\delta_{2}(a)= A. \n\\]\n\\end{enumerate}\n\n\nObserve that for instance with $b=0$, with arbitrary fixed $a\\in \\mathbb{Q}$ and with \n$A=a^{2}$ the above identity is fulfilled by \\emph{any} quadratic function $f\\colon \\mathbb{F}\\to \\mathbb{C}$ (this is obviously consistent with the fact that quadratic functions are $\\mathbb{Q}$-homogeneous of degree two). This shows that in this case we do not get in general any information for the form of the involved quadratic function $f$. With an analogous method, we obtain the same for higher order generalized monomials. \n\\end{rem}\n\n\n\n\\begin{opp}\n In this paper only equations with one unknown function were considered. At the same time, the investigated problem can clearly be extended: let \n $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{C}[x]$ be polynomials, $f, g\\colon \\mathbb{F}\\to \\mathbb{C}$ be generalized polynomials (of possibly \\emph{different order}) \n such that $\\deg(f)\\deg(P)=\\deg(g)\\deg(Q)$. Prove or disprove that if \n \\[\n  f(P(x))= Q(g(x)) \n \\]\nholds for all $x\\in \\mathbb{F}$, then $f$ and $g$ can be represented as products of homomorphisms. \n\\end{opp}\n\n\\begin{rem}\n A particularly interesting and presumably the simplest case of the following problem is when $\\deg(g)=1$, i.e., when $g\\colon \\mathbb{F}\\to \\mathbb{C}$ is an additive function. \n \\end{rem}\n \n In connection to this problem we prove the following special case, which is expected to be successfully applied in the general case as well. \n \n \\begin{prop}\n  Let $\\mathbb{F}$ be a field with $\\mathrm{char}(\\mathbb{F})=0$, $f\\colon \\mathbb{F}\\to \\mathbb{C}$ be a quadratic function and $a\\colon \\mathbb{F}\\to \\mathbb{C}$ be an additive function. Then equation \n  \\[\n  f(x^{2})= a(x)^{4} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right)\n \\]\n holds if and only if here exists a homomorphism $\\varphi\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n a(x)=a(1)\\varphi(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right)\n\\]\nand \n\\[\n f(x)= a(1)^{4}\\varphi(x)^{2}\n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\n\\end{prop}\n\n\\begin{proof} \n Assume that $f\\colon \\mathbb{F}\\to \\mathbb{C}$ is a quadratic, while $a\\colon \\mathbb{F}\\to\\mathbb{C}$ is an additive function such that \n \\[\n  f(x^{2})= a(x)^{4} \n  \\qquad \n  \\left(x\\in \\mathbb{F}\\right). \n \\]\nSince both the sides of this equation are traces of symmetric, $4$-additive functions, we obtain that \n\\[\n \\frac{1}{3}\\left[F(x_1 x_2, x_3 x_4)+F(x_1 x_3, x_2 x_4)+F(x_1 x_4, x_2 x_3)\\right]\n = a(x_1)a(x_2)a(x_3)a(x_4) \n \\qquad\n \\left(x_1, x_2, x_3, x_4\\in \\mathbb{F}\\right). \n\\]\nHere $F\\colon \\mathbb{F}\\times \\mathbb{F}\\to \\mathbb{C}$ is the uniquely determined symmetric, bi-additive function for which $F(x, x)= f(x)$ is satisfied for all $x\\in \\mathbb{F}$. \n\nThis identity implies especially that \n\\[\n f(1)=F(1, 1)=a(1)^{4} \n \\qquad \n \\text{and}\n \\qquad \n F(x, 1)=a(1)^{3}a(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nFurthermore, we also have \n\\[\n 2F(x, y)= 3a(1)^{2}a(x)a(y)-F(xy, 1) \n \\qquad \n \\left(x, y\\in \\mathbb{F}\\right). \n\\]\nThus, \n\\[\n 2F(x, y)=3a(1)^{2}a(x)a(y)-a(1)^{3}a(xy) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nIn other words, \n\\[\n f(x)= \\frac{3}{2}a(1)^{2}a(x)^{2}-\\frac{1}{2}a(1)^{3}a(x^{2})  \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nSubstituting this back into the original equation we get that the additive function $a\\colon \\mathbb{F}\\to \\mathbb{C}$ has to fulfill \n\\[\n a(x)^{4}= \\frac{3}{2}a(1)^{2}a(x^{2})^{2}-\\frac{1}{2}a(1)^{3}a(x^{4})  \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\]\nAgain, after symmetrization or after applying \\cite[Theorem 15]{GseKisVin19}, we derive that there exists a homomorphism \n$\\varphi\\colon \\mathbb{F}\\to \\mathbb{C}$ such that \n\\[\n a(x)=a(1)\\varphi(x) \n \\qquad \n \\left(x\\in \\mathbb{F}\\right)\n\\]\nand \n\\begin{multline*}\n f(x)= \\frac{3}{2}a(1)^{2}a(x)^{2}-\\frac{1}{2}a(1)^{3}a(x^{2})\n \\\\\n = \n \\frac{3}{2}a(1)^{2}a(1)^{2}\\varphi(x)^{2}-\\frac{1}{2}a(1)^{3}a(1)\\varphi(x^{2})\n =\n a(1)^{4}\\varphi(x)^{2} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right). \n\\end{multline*}\n\\end{proof}\n\n\\begin{rem}\nUsing the ideas of the results of the third section, the case \n\\[\n f(x^{n})= a(x)^{2n} \n \\qquad \n \\left(x\\in \\mathbb{F}\\right)\n\\]\nwhere $n$ is a fixed positive integer, can be reduced to the above studied case. \n\\end{rem}\n\n\n\\begin{rem}\n Observe that during the proof of Lemmas \\ref{lemma3}, \\ref{lemma5}, \\ref{lemma_6} the fact that we considered complex-valued mappings, was not used at all. We remark that these statements as well as their proofs are exactly the same for functions defined on a field $\\mathbb{F}$ and mapping to another field $\\mathbb{K}$. \n \n The situation is slightly different in case of Lemma \\ref{lemma4}, since in that case derivations are involved. Nevertheless, Lemma \\ref{lemma4} also holds true (with an unchanged proof) for mappings defined on $\\mathbb{F}$ and having values in $\\mathbb{K}$, where $\\mathbb{F}\\subset \\mathbb{K}$ are fields. \n\\end{rem}\n\n\n\\begin{rem}\n We would also like to clarify why we considered only complex-valued functions. Obviously, the investigated problems are meaningful in a much more general setting. The importance of this condition lies in our method. Namely, in each case we showed firstly that the involved mappings $f\\colon \\mathbb{F}\\to \\mathbb{C}$ are exponential polynomials of the multiplicative group $\\mathbb{F}^{\\times}$. This enabled us to describe the unknown function completely. This method however relies on the notion of exponential polynomials and the theory of this notion is well-developed only for complex-valued mappings. Maybe with a different approach the general case can also be handled. \n \n The assumption that the field $\\mathbb{F}$ has to have zero characteristic is caused by the limitations of the Polarization formula, since $n$-additive functions are uniquely determined by their diagonalizations only if the characteristic of the domain is large enough or zero. \n\\end{rem}\n\n\n\\begin{opp}\n Let $\\mathbb{F}$ and $\\mathbb{K}$ be fields, $n$ be a positive integer, $P\\in \\mathbb{F}[x]$ and $Q\\in \\mathbb{K}[x]$ be polynomials. Determine those generalized monomials $f, g\\colon \\mathbb{F}\\to \\mathbb{K}$ of degree at most $n$ that also fulfill equation \n\\[\n f(P(x))= Q(f(x))\n\\]\nfor each $x\\in \\mathbb{F}$. \n\nA particularly interesting case of this problem is when at least one of the fields $\\mathbb{F}$ and $\\mathbb{K}$ has finite characteristic. \n\\end{opp}\n\n\n\n\n\\begin{ackn}\n The author would like to thank Professor \\textsc{Jos\u00e9 Mar\u00eda Almira} (University of Murcia, Spain) and Professor \\textsc{Rezs\u0151 Lovas} (University of Debrecen, Hungary) for their generous help and valuable comments that improved the quality of the paper. \n \n Project No.~K134191 has been implemented with the support provided by the National Research, Development and Innovation Fund of Hungary, financed under the K\\_20 funding scheme. The research of the author has partially been carried out with the help of the Project 2019-2.1.11-T\u00c9T-2019-00049, which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the T\u00c9T funding scheme.\n\\end{ackn}\n\n\n\n\n \\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA deep connection between glass\ntransition in molecular glass formers,  structural arrest in\ncolloidal systems, and  jamming transition in granular media\n\\cite{coniglio,LN,OHern1,Danna,Mehta,OHern2} has often been stressed in the\npast few years. In spite of the fact that these systems are very\ndifferent one from each other, varying suitably the control\nparameters, a slowdown and a subsequent structural arrest in a\nsolid-like disordered state are found in each of them. In\n\\cite{LN,OHern2} a possible phase diagram for jamming is\nsuggested, which takes into account the fact that jamming is\nobtained either raising the volume fraction or lowering the\ntemperature or lowering the applied stress. Colloidal suspensions\nand molecular glass formers are both thermal systems,\nand it is commonly accepted that both colloidal glass transition\nand molecular glass transition are of the same type despite of the\nfact that  different control parameters may drive the transition.\nThe case of granular materials is instead very different: They are\nathermal systems, since the thermal fluctuations are significantly\nless than the gravitational energy and the system cannot explore\nthe phase space without any external driving. Nevertheless an\nexceedingly slowing down is observed when a granular material is\nshaken at low shaking amplitude, or flows under a low shear\nstress, with strong analogies with the slowing down observed in\nglass formers. Experimental and numerical studies\n\\cite{Danna,OHern2,NCH,Mehta} have confirmed this connection,\nhowever its precise nature is still unclear \\cite{OHern1,OHern2}.\n\nIn the present paper in order to study this connection we apply a\nstatistical mechanics approach to granular media. This approach,\nwhich has been extensively developed in previous works \\cite{e1, fnc}, is\nbased on an elaboration of the original ideas suggested by\nEdwards \\cite{Edwards}. The basic assumption is that for a granular\nsystem subject to an external drive (e. g. tapping), after having\nreached stationarity, time averages coincide with suitable\nensemble averages over the ``mechanically stable\" states. We have\nshown \\cite{fnc} that this assumption works for different lattice\nmodels namely that a generalized Gibbs distribution of the stable\nstates describes with good approximation the stationary state\nattained by the system under tapping dynamics. Here each tap\nconsists in raising the bath temperature to a finite value (called\ntap amplitude) and, after a lapse of time (called tap duration)\nquenching the bath temperature back to zero. By cyclically\nrepeating the process  the system explores the space of the\nmechanically stable states.\n\nWe thus consider one of the above lattice model for which the\nstatistical mechanics approach works. The model is made up of hard\nspheres under gravity. Then we apply standard statistical\nmechanics methods in order to investigate analytically the\nexistence and the nature of a possible jamming transition. More\nprecisely we consider the Bethe-Peierls approximation using the\ncavity method \\cite{MP,Biroli}: By changing the control parameter\na phase transition from a fluid to a crystal is found, and, when\ncrystallization is avoided, a glassy phase appears. The nature of\nthis glassy phase is analogous to that found in mean field models\nfor glass formers \\cite{Kurchan, Biroli, PC03}: In particular we\nobserve a dynamical transition, where an exponentially high number\nof metastable states appears, and at a lower temperature a\nthermodynamic discontinuous phase transition to a glassy state. A\nbrief account of these calculations was given in a previous\nLetter \\cite{epl}. We also studied \\cite{epl} the model in\n$3d$ by means of numerical simulations, and we found that the\nmodel under taps has a transition from a fluid to a crystal, in a\nvery good agreement with he mean field approximation. However the\nnumerical simulation was not suitable to study the glass\ntransition since the model showed a strong tendency towards\ncrystallization.\n\nFor this reason we study here a variant of the model \\cite{PC03}\nwhich has the virtue of avoiding crystallization. We find that the\nsystem under gravity evolved by Monte Carlo taps presents features\ncharacteristic of real granular media \\cite{Bideau, Clement}, and\nat low tap amplitudes a dynamical transition with properties\nrecalling those of usual glass formers. In particular we observe a\ndynamical non linear susceptibility with a maximum at increasing\ntime: This behavior, typical of glass formers, is usually\ninterpreted as the sign of dynamic heterogeneities in the system.\n\nIn conclusions the results confirm early\nspeculations about the deep connection between the jamming\ntransition in granular\nmedia and the glass transition in usual glass formers, giving moreover a\nprecise interpretation to its nature.\n\n\n\nIn Sect. \\ref{meanfield} the mean field phase diagram is discussed. The details\nof calculations are presented in App.s \\ref{app1} and \\ref{app2}. In particular\nin App. \\ref{app2} the self-consistency equations obtained using the cavity\nmethod are shown.  In Sect. \\ref{hardsphere} the $3d$ model is presented and the\nnumerical results are shown.\n\n\\section{Mean field solution in the Bethe-Peierls approximation}\n\\label{meanfield}\nThe model is a monodisperse hard sphere system (with diameter $\\sqrt{2} a_0$)\nunder gravity, constrained to move\non the sites of a cubic lattice of spacing $a_0=1$. The Hamiltonian is given by:\n\\begin{equation}\n{\\cal H} = {\\cal H}_{HC} + mg \\sum_i n_i  z_i\n\\label{H1}\n\\end{equation}\nwhere $z_i$ is the height of site $i$, $g=1$ is the gravity acceleration,\n$m=1$ the grain mass, $n_i\\in\\{0,1\\}$ is the occupancy variable\n(absence or presence of a grain on site $i$)\nand ${\\cal H}_{HC}(\\{n_i\\})$ is the hard core term\npreventing two nearest neighbor sites being simultaneously occupied.\n\nWe have shown in previous papers \\cite{fnc} that\nthe model, Eq.~(\\ref{H1}), evolving by means of a tap dynamics can be described\nin good approximation by\na generalized Gibbs distribution of the ``mechanically stable'' states (i.e.\nthe states where the system is found at rest).  In particular the weight of a\ngiven state, $\\{n_i\\}$, is:\n\\begin{equation}\n\\mbox{e}^{-\\beta H(\\{n_i\\})}{\\cdot} \\Pi(\\{n_i\\}),\n\\end{equation}\nwhere $T_{conf} = \\beta^{-1}$ is a thermodynamic parameter,\ncalled ``configurational temperature'', characterizing the distribution.\nThe operator $\\Pi(\\{n_i\\})$ selects mechanically stable states:\n$\\Pi(\\{n_i\\})=1$ if $\\{n_i\\}$ is ``stable'', or else $\\Pi(\\{n_i\\})=0$.\nThe system partition function is thus the following \\cite{fnc}:\\begin{equation}\n{\\cal Z} =\\sum_{\\{n_i\\}} \\mbox{e}^{-\\beta H(\\{n_i\\})}{\\cdot} \\Pi(\\{n_i\\})\n\\label{Z}\n\\end{equation}\nwhere the sum runs over all microstates, $\\{n_i\\}$.\n\nIn the present section we show the phase diagram of the model, Eq.~(\\ref{H1}),\nobtained using a mean field theory in the Bethe-Peierls approximation (see\n\\cite{MP, Biroli} and ref.s therein),\nbased on a random graph (plotted in Fig.~\\ref{Blattice})\nwhich keeps into account that the gravity breaks up the symmetry along the $z$\naxis. This lattice is made up by $H$ horizontal layers\n(i.e., $z\\in\\{1,...,H\\}$).\nEach layer is a random graph of connectivity, $k-1=3$. Each site in layer\n$z$ is also\nconnected to its homologous site in $z-1$ and $z+1$\n(the total connectivity is thus $k+1$).\nLocally the graph has a tree-like structure but there\nare loops whose length is of order $\\ln N$, insuring geometric frustration.\nIn the thermodynamic limit only very long loops are present.\nThe details of calculations are given in appendices~\\ref{app1} and~\\ref{app2}\n(see also Ref.s~\\cite{epl, prl} where this mean field theory was\nfirst introduced).\n\n\\begin{figure}[ht]\n\\centerline{\n\\psfig{figure=fig4.eps,width=6cm,height=4cm,angle=0}\n}\n\\vspace{-.3cm}\n\\caption{In the mean field approximation,\nthe grains are located on a Bethe lattice, sketched in the figure,\nwhere each horizontal layer is a random graph of given connectivity.\nHomologous sites on neighboring layers are also linked and the overall\nconnectivity, $c$, of the vertices is $c\\equiv k+1=5$.}\n\\label{Blattice}\n\\end{figure}\nWe solve the recurrence equations found in the Bethe-Peierls approximation in\nthree cases: 1) A fluid-like homogeneous phase; 2) a crystalline-like\nphase characterized by the breakdown of the horizontal translational\ninvariance; 3)  a glassy phase described by a $1$-step Replica Symmetry\nBreaking (1RSB).  The details of the calculations are shown in Appendices.\n\nThe results of the calculations are summarized in Fig.~\\ref{phi_T}, where the\nbulk density at equilibrium, $\\Phi\\equiv N_s\/(2\\langle z\\rangle-1)$\n\\cite{notadens}\n(where $\\langle z\\rangle$ is the average height)\nis plotted as a function of the configurational temperature, $T_{conf}$, for a\ngiven value of the number of grains per unit surface, $N_s$.\nWe found that at high $T_{conf}$\na homogeneous solution corresponding to the fluid-like phase is found.\nBy lowering $T_{conf}$ at $T_m$\na phase transition to a crystal phase (an anti-ferromagnetic solution with a\nbreakdown of the translation invariance) occurs.\nThe fluid phase still exist below $T_m$ as a metastable phase\ncorresponding to a supercooled fluid when crystallization\nis avoided. Finally a 1RSB solution (found with the cavity\nmethod \\cite{MP}), characterized by the presence of a large number of\nlocal minima in the free energy \\cite{MP}, appears\nat $T_D$, and becomes stable at a lower point $T_K$, where\na thermodynamic transition from the supercooled fluid\nto a 1RSB glassy phase takes place.\nThe temperature $T_D$, which is interpreted in mean field\nas the location of a  dynamical transition where the relaxation time diverges,\nin real systems might instead correspond to a crossover in the dynamics\n(see \\cite{Kurchan, Biroli, Toninelli} and Ref.s therein).\n$\\Phi(T_{conf})$ has a shape very similar to that observed in the\n``reversible regime'' of tap experiments \\cite{Knight,Bideau}.\nThe location of the glass transition, $T_K$, corresponds\nto a cusp in the function $\\Phi(T_{conf})$. The dynamical crossover point $T_D$\nmight  correspond to the position of a characteristic shaking amplitude\n$\\Gamma^*$ found in experiments and simulations where the ``irreversible''\nand ``reversible'' regimes approximately meet.\n\n\\begin{figure}[ht]\n\\vspace{-0.5cm}\n\\begin{center}\n\\mbox{\\epsfysize=8cm\\epsfbox{compaction.eps}}\n\\end{center}\n\\caption{ The density, $\\Phi\\equiv N_s\/(2\\langle z \\rangle-1)$,\nfor $N_s=0.6$ as a function of $T_{conf}$. $\\Phi_{max}$ is the maximum density\nreached by the system in the crystal phase.}\n\\label{phi_T}\n\\end{figure}\n\nIn Fig.~\\ref{MFPD} the phase diagram obtained by varying $N_s$ is shown. The\ndashed vertical line in figure corresponds to the value of $N_s$ chosen in\nFig.~\\ref{phi_T}.\n\\begin{figure}[ht]\n\\centerline{\n\\psfig{figure=fig1.ps,width=6cm,angle=-90}}\n\\caption{The system mean field phase diagram is plotted in the plane\nof its two control parameters $(T_{conf},N_s)$.}\n\\label{MFPD}\n\\end{figure}\n\nThe model, Eq.~(\\ref{H1}), simulated in $3d$ by means of Monte Carlo tap\ndynamics \\cite{epl} presents a transition from a fluid  to a crystal\nas predicted by the mean field approximation, density profiles in good\nagreement with the mean field ones, and in the fluid phase a large increase of\nthe relaxation time as a function of the inverse tap amplitude.\nIn the following section we study a more complex model for hard spheres,\nwhere an internal degree of freedom allows to\navoid crystallization \\cite{PC03}.\n\n\\section{Hard spheres with an internal degree of freedom}\n\\label{hardsphere}\nThe Hamiltonian of the model is\n\\begin{equation}\n{\\cal H}  =\\sum_{\\langle ij \\rangle} n_i n_j \\phi_{ij}(\\sigma_i,\\sigma_j)\n+ mg \\sum_i n_i  z_i,\n\\label{H2}\n\\end{equation}\nwhere $z_i$ is the height of site $i$, $g=1$ is the gravity acceleration,\n$m=1$ the grain mass, $n_i\\in\\{0,1\\}$ is the occupancy variable (absence\nor presence of a grain on site $i$), $\\sigma_i\\in\\{1,\\ldots,q\\}$ represents the\ninternal degree of freedom  (which we call spin), and\n$\\phi_{ij}(\\sigma_i,\\sigma_j)$ is the interaction energy between spins.\nDifferent values of the spin correspond to different positions of the\nparticle inside the cell.  It is reasonable that a few number\nof internal states might be enough to catch the main features\nof real systems.\n\nAs in Ref.~\\cite{PC03} we study a  simple realization of the model\ndescribed by Eq.~(\\ref{H2}). Interpreting the spin as position of the particle\nin the cell, our choice can be easily visualized in $2d$, as shown in Fig.~\n\\ref{fig-model}. We\npartition the space in square cells, and subdivide each cell into four\ninternal positions (namely $q=4$). When a cell is occupied by a particle in any\ngiven position, a hard-core repulsion excludes the presence of\nparticles in some of the internal states of the neighboring\ncells (namely the interaction $\\phi_{ij}(\\sigma_i, \\sigma_j)$ is chosen zero\nif the\npositions $\\sigma_i$ and $\\sigma_j$ are ``compatible'', and infinite otherwise).\nThis choice can be interpreted as a coarse grained version of a\nhard sphere system in the continuum.  In $3d$ we subdivide the\nspace into cubic cells, and considers six internal positions\ninstead of four.\n\n\\begin{figure}[ht!]\n\\hspace{-1cm}\\centerline{\\mbox{\\epsfysize=6cm\\epsfbox{MODELLO.eps}}}\n\\vspace{-2.5cm}\n\\caption{The model in two dimensions: the space is partitioned in square cells,\nand each cell can be occupied by at most one particle\nin anyone of the four shown positions (little circles). A particle in\nany given position (large shaded circle) excludes the presence of\nparticles in any of the black colored positions.}\n\\label{fig-model}\n\\end{figure}\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{phi.eps}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{The bulk density, $\\Phi\\equiv N\/L^2(2\\langle z\\rangle-1)$,\nis plotted as function of $T_\\Gamma$ for $\\tau_0=10~MCsteps\/particle$. The empty\ncircles correspond to stationary states, and the black stars to out of\nstationarity ones. $\\Phi_{max}$ is the maximum density reached by the system in\nthe crystal phase, $\\Phi_{max}=6\/7$.}\n\\label{phi}\n\\end{figure}\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{prof.eps}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{The density profile, $\\sigma (z)$, as function of the height, $z$, for\n$T_\\Gamma=~0.20$ and $\\tau_0=10~MCsteps\/particle$.}\n\\label{prof}\n\\end{figure}\n\nIn the Monte Carlo simulations, $N=~433$ grains are confined in a $3d$ box\nof linear size $L=~12$ (i.e. $N_s=~3$), between hard walls in the vertical\ndirection and with periodic boundary\nconditions in the horizontal directions.  We perform a standard Metropolis\nalgorithm on the system.  The particles,  initially prepared in a random\nconfiguration, are subject to taps, each one followed by a\nrelaxation process. During a tap, for a time $\\tau_0$ (called tap\nduration), the temperature is set to the value $T_{\\Gamma}$ (called tap\namplitude), so that particles have a finite probability, $p_{up}\\sim\ne^{-mg\/T_{\\Gamma}}$, to move upwards. During the relaxation the\ntemperature is set to zero, so that particles can only reduce the energy,\nand therefore can move only downwards. The relaxation stops when the\nsystem has reached a blocked state, where no grain can move downwards.\nOur measurements are performed at this stage when\nthe shake is off and the system is at rest. The time, $t$,\nis the number of taps applied to the system.\n\nIn the following the tap duration is fixed, $\\tau_0=10 MCsteps\/particle$, and\ndifferent tap amplitudes, $T_\\Gamma$, are considered.\nIn Fig.~\\ref{phi} the bulk density, $\\Phi \\equiv N\/\nL^2(2\\langle z \\rangle-1)$, is plotted as a function of $T_\\Gamma$:\n$\\Phi(T_\\Gamma)$ has a shape resembling that found in the\n``reversible regime'' of tap experiments \\cite{Knight, Bideau}, and moreover\nvery similar to that\nobtained in the mean field calculations and shown in Fig.~\\ref{phi_T}.\nAt low shaking amplitudes (corresponding to high bulk densities) a strong\ngrowth of the equilibration time (i.e. the time necessary to reach stationarity)\nis observed, and for the lowest values here considered (the black stars in\nFig.~\\ref{phi}) the system remains out of stationarity. In this region the\ndensity profile, $\\sigma (z)\\equiv 1\/L^2\\sum_i n_i(z)$ (where the sum $\\sum_i$\nis done over the sites $i$ in the layer $z=~1, \\dots, L$ \\cite{notapart}),\nis almost constant until a given layer and sharply decays to zero (see Fig.\n~\\ref{prof}), as found in real granular media \\cite{Clement}.\nIn conclusions the system here studied presents a jamming transition at low\ntap amplitudes as found in real granular media.\n\nIn order to test the predictions of the mean field calculations,\nin the following we measure quantities usually important in the\nstudy of glass transition: The relaxation functions, the\nrelaxation time\nand the dynamical susceptibility, connected to the presence a\ndynamical correlation length.\n\n\nIn particular we calculate the two-time autocorrelation\nfunctions:\n\\begin{equation}\nC(t,t_w)=\\frac{1}{N}\\sum_i\\overline{ n_i(t)n_i(t_w) \\vec {\\sigma} _i(t){\\cdot}\n\\vec{\\sigma} _i(t_w)},\n\\end{equation}\nwhere $\\vec {\\sigma} _i$ are unit length vectors, pointing\nin one of the six coordinate directions, representing the position of the\nparticles inside the cell; the average $\\overline{(\\ldots)} $ is done over\n$16-32$ different\nrealizations of the model obtained varying the random number generator in\nthe simulations, and the errors are calculated as the fluctuations\nover this statistical ensemble.\nFor values of $t_w$ long enough, the system\nreaches a stationary state, where the time translation invariance is\nrecovered, i.e., $C(t,t_w)=C(t-t_w)$.\nIn this time region, by averaging $C(t',t_w)$\nover $t'$ and $t_w$ such that $t=t'-t_w$ is fixed, we calculate\nthe ``equilibrium'' autocorrelation functions\n\\begin{equation}\n\\langle q(t) \\rangle= \\langle C(t'-t_w) \\rangle,\n\\end{equation}\nand the dynamical non linear susceptibility\n\\begin{equation}\n\\chi(t)=\\langle q(t)^2 \\rangle - \\langle q(t) \\rangle^2.\n\\end{equation}\nAs shown in Fig.~\\ref{tti}, at low values of the tap amplitudes, $T_\\Gamma$,\ntwo-step decays appear, well fitted in the intermediate time region,\nby the $\\beta-$correlator predicted by the mode coupling theory for \nsupercooled liquids \\cite{mct, notaMCT} (the continuous\ncurve in Fig.~\\ref{tti}), and at long time by stretched exponentials (the dashed\ncurve in figure).\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{relax.eps}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{The ``equilibrium'' autocorrelation function, $\\langle q(t) \\rangle$,\nplotted as function\nof $t$, for tap amplitudes $T_\\Gamma=~0.60,~0.50,~0.425,\n~0.40,~0.385,~0.365,~ 0.36$  (from bottom to top). The continuous line in\nfigure is the $\\beta-$correlator of the mode coupling theory \nwith exponent parameters\n$a=~0.30$ and $b=~0.52$.  The dashed line is a stretched exponential $\\propto\nexp[-(t\/\\tau)^\\beta]$ with $\\beta=~0.70$.}\n\\label{tti}\n\\end{figure}\nThe relaxation time, $\\tau$, is defined as $\\langle q(\\tau)\n\\rangle \\sim 0.1$. \n\nIn Fig.~\\ref{tau_phi} the relaxation time, $\\tau$, is plotted as a function of \nthe density, $\\Phi$. As found in many glass forming liquids, $\\tau(\\Phi)$ is \nwell fitted by a Vogel-Fulcher for the entire range, even if we can identify a \nfirst region where $\\tau(\\Phi)$ is fitted with good approximation by a power \nlaw. The power law divergence can be interpreted as a mean field behavior, \nfollowed by a hopping regime.\nNote that the model, Eq.~(\\ref{H2}), studied in absence of gravity\nby means the usual Monte Carlo Metropolis \\cite{PC03}, exhibits a\ndivergence of the relaxation time as a power law, and no crossover\nto a hopping regime is observed. We suggest that in the present\ncase the tap dynamics favors the equilibration\nvia hopping precesses.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{tau_phi.eps}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{The relaxation time, $\\tau$, as function of the bulk\ndensity, $\\Phi$. The continuous line is a Vogel-Fulcher,\n$e^{A\/(\\Phi_c-\\Phi)}$,with $\\Phi_c=0.81\\pm 0.01$ and $A=0.49\\pm 0.10$. The \ndashed line is a power law, $(\\Phi_D-\\Phi)^{-\\gamma_1}$, with\n$\\Phi_D=0.76\\pm 0.01$ and $\\gamma_1=2.04\\pm 0.10$.}\n\\label{tau_phi}\n\\end{figure}\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{tau.eps}}\n\\end{center}\n\\vspace{-0.5cm} \\caption{The relaxation time, $\\tau$, as function\nof the tap amplitude inverse, $T_\\Gamma^{-1}$. The dashed line\nis a power law, $(T_\\Gamma-T_D)^{-\\gamma_2}$, with\n$T_D=0.40\\pm 0.01$ and $\\gamma_2=1.52\\pm 0.10$. The continuous line\nis an Arrhenius fit, $e^{A\/T_\\Gamma}$, with $A=17.4\\pm 0.5$ (the data in this \nregion are also well fitted by both a super-Arrhenius and\nVogel-Fulcher laws).}\n\\label{figure3}\n\\end{figure}\nIn Fig.~\\ref{figure3} the relaxation time, $\\tau$, is plotted as a function of \nthe tap amplitude, $T_\\Gamma$: A clear crossover\nfrom a power law to a different regime is again observed around a\ntap amplitude $T_D$, corresponding to the value of the density, $\\Phi(T_D)\n\\simeq \\Phi_D$, where a similar crossover has been found in Fig.~\\ref{tau_phi}.\n\nThe divergence of the relaxation time at vanishing tap amplitude\nis consistent with the experimental data of Philippe\nand Bideau \\cite{Bideau} and D'Anna {\\em et al.} \\cite{Danna}. \nTheir findings are in fact consistent with an Arrhenius\nbehavior as function of the experimental tap\namplitude intensity. However a direct comparison with our data is not\npossible since we do not know the relation between the\nexperimental tap amplitude and the tap amplitude in our simulations.\nA more direct comparison would be possible if the experimental data \nwere plotted as function of the bulk density, as we did in Fig.~\\ref{tau_phi}.\n\nThe dynamical non linear susceptibility, $\\chi(t)$, plotted in\nFig.~\\ref{chi} at different $T_\\Gamma$, exhibits a maximum at a\ntime, $t^*(T_\\Gamma)$. The presence of a maximum in the  dynamical non linear\nsusceptibility is typical of glassy systems \\cite{franz, glotzer}.\nIn particular the value of the maximum, $\\chi(t^*)$, diverges in\nthe $p$-spin model \\cite{franz} as the dynamical transition is\napproached from above, signaling the presence of a diverging\ndynamical correlation length.\nIn the present case the value of the maximum increases as $T_\\Gamma$\ndecreases (except at very low $T_\\Gamma$ where the maximum seems to\ndecrease \\cite{nota_max}). The growth of $\\chi(t^*)$ in our model \nsuggests the presence of a growing\ndynamical length also in granular media.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\epsfysize=7cm\\epsfbox{susc.eps}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{The dynamical non linear susceptibility, $\\chi(t)$, (normalized by\n$\\chi(t_0)$, the value at $t_0=1$) as a function of $t$, for tap amplitudes\n$T_\\Gamma=\n0.60,~0.50,~0.425, ~0.41, ~0.40,~0.385, 0.3825$  (from left to right).}\n\\label{chi}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn conclusions using standard methods of statistical mechanics we\nhave investigated  the jamming transition in a model for granular\nmedia. We have shown a deep connection  between the jamming\ntransition in granular media and the glass transition in usual\nglass formers. As in usual glass formers the mean field\ncalculations obtained using a statistical mechanics approach to\ngranular media predict a dynamical transition at a finite\ntemperature, $T_D$, and, at a  lower temperature, $T_K$, a\nthermodynamics discontinuous phase transition to a glass phase. In\nfinite dimensions 1) the dynamical transition becomes only a\ndynamical crossover as also found in usual glass formers\n\\cite{Kurchan, Biroli,Toninelli} (here the relaxation time,\n$\\tau$, as a function of both the density and the tap amplitude,  \npresents a crossover\nfrom a power law to a different regime); and 2) the thermodynamics\ntransition temperature, $T_K$, seems to go to zero (the relaxation\ntime, $\\tau$, seems to diverge only at $T_\\Gamma\\simeq 0$, even if a\nvery low value of the transition temperature is consistent with\nthe data).\n\n\\begin{acknowledgments}\nWe would like to thank M. Pica Ciamarra for many interesting discussions and \nsuggestions. Work supported by EU Network Number  MRTN-CT-2003-504712, \nMIUR-PRIN 2002, MIUR-FIRB 2002, CRdC-AMRA, INFM-PCI.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methodology}\\label{sec:methodology}\n\nIn the following, we will present a general formulation of the fragment molecular orbital (FMO) LC-TD-DFTB. As a starting point, we review the theory behind LC-TD-DFTB\\cite{humeniuk_long-range_2015, lutsker_implementation_2015}. This is followed by the formulation of the FMO-ansatz for the description of the electronic ground state. Subsequently, we present how the molecular orbitals of the total system can be calculated and introduce a theoretical approach for calculating electronically excited states. This is concluded with a description of the necessary Hamiltonian matrix elements and the computational procedure that is used in our implementation. \nThroughout the paper we use atomic units and the following notation convention: atoms are denoted as uppercase letters from A-H and molecular fragments from I-Z without indices. Uppercase letters with indices denote matrix elements and bold uppercase letters matrices. For molecular orbital (atomic orbital) indices lowercase (greek) letters are used. \nIt should be noted that in the following we use the terms fragment and monomer synonymously to refer to a molecular unit. Likewise, we use (molecular) cluster and aggregate synonymously to refer to a large number of molecular units that are not covalently linked. \n\n\\subsection{Long-range corrected DFTB (LC-DFTB)}\nAs shown previously, the formalism of LC-DFTB can be derived by a Taylor expansion of a long-range corrected full density functional (e.g. LC-PBE) around a reference density that is given by a sum of atomic densities, as described in detail elsewhere \\cite{humeniuk_long-range_2015, lutsker_implementation_2015}. In the following, we will only provide a brief summary of the working equations. After employing the usual tight-binding (TB) approximations\\cite{elstner_self-consistent-charge_1998} the total energy in SCC LC-DFTB is given by:\n\\begin{align}\nE=& \\sum_{\\mu \\nu} P_{\\mu \\nu} H_{\\mu \\nu}^{0}  \\tag{band structure energy}\\\\\n&+ \\frac{1}{2} \\sum_{\\mu, \\sigma, \\lambda, \\nu} \\Delta P_{\\mu \\sigma} \\Delta P_{\\lambda \\nu}(\\mu \\sigma \\mid \\lambda \\nu) \\tag{Coulomb energy} \\\\\n&-\\frac{1}{4} \\sum_{\\mu, \\sigma, \\lambda, \\nu} \\Delta P_{\\mu \\sigma}  \\Delta P_{\\lambda \\nu}(\\mu \\lambda \\mid \\sigma \\nu)_{\\operatorname{lr}}\\tag{exchange energy}\\\\\n& + \\sum_{A, B} V_{A B}^{\\mathrm{rep}}\\left(R_{A B}\\right)\\tag{repulsion energy} \n\\label{eq:dftb_energy}\n\\end{align}\nHere $P_{\\mu \\nu}$ denotes the electron density matrix elements and $H_{\\mu \\nu}^{0}$ the one-electron Hamiltonian matrix elements. The electron density difference matrix is defined as $\\Delta \\mathbf{P} = \\mathbf{P} - \\mathbf{P}^0$, where $\\mathbf{P}^0$ is the diagonal reference electron density matrix. The deviation of the Mulliken charge $q_A$ on Atom A from the charge of the neutral Atom $q^0_A$ is given by\n\\begin{align}\n    \\Delta q_A &= q_A - q^0_A \\\\\n    &= \\sum_{\\mu \\in A} \\sum_{\\nu} \\left[\\mathbf{P}_{\\mu\\nu} S_{\\mu\\nu} - \\mathbf{P}^0_{\\mu\\nu} S_{\\mu\\nu} \\right],\n\\end{align}\nwhere $S_{\\mu\\nu}$ represents a matrix element of the overlap matrix.\n\nApplying the tight-binding approximations to the two-electron integrals in the Coulomb and exchange part of the energy and neglecting all 3- and 4-center integrals gives the following expressions: \n\\begin{align*}\n(\\mu \\lambda \\mid \\sigma \\nu) &=\\iint \\phi_{\\mu}(r_1) \\phi_{\\lambda}(r_1) \\frac{1}{r_{12}} \\phi_{\\sigma}(r_2) \\phi_{\\nu}(r_2) \\mathrm{d} 1 \\mathrm{d} 2 \\\\\n& \\approx \\sum_{A, B} \\gamma_{A B} q_{A}^{\\mu \\lambda} q_{B}^{\\sigma \\nu}  \\numberthis{}{}\n\\label{eq:2e_approx_ao}\n\\end{align*}\nwith the transition charges on atom $A$ (in the atomic orbital basis) defined as\n\\begin{equation}\nq_{A}^{\\mu \\lambda}=\\frac{1}{2}(\\delta(\\mu \\in A)+\\delta(\\lambda \\in A)) S_{\\mu \\nu}\n\\label{eq:q_between_aos}\n\\end{equation}\nThe $\\bm{\\gamma}$-matrices for charge fluctuation interactions can be calculated assuming that the charge fluctuations can be represented by spherical Gaussian functions, leading to\n\\begin{equation}\n\\gamma_{A B}=\\frac{\\operatorname{erf}\\left(C_{A B} R\\right)}{R},\n\\label{eq:gamma}\n\\end{equation}\nwhere $R$ is the distance between the atomic centers $A$ and $B$ and $C_{A B}=(2\\left(\\sigma_A^2+\\sigma_B^2\\right))^{-\\frac{1}{2}}$ depends on the widths $\\sigma_A$ and $\\sigma_B$ of the charge clouds on the two atoms. The widths are determined by the atom-specific Hubbard parameters $U_A$ as $\\sigma_A= (\\sqrt{\\pi} U_A)^{\\frac{1}{2}}$.\nIn LC-DFTB the Coulomb potential is separated into a long-range and a short range part where the position of the smooth transition between the two regimes is controlled by the long-range radius $R_{\\mathrm{lr}}$\n\\begin{equation}\n\\frac{1}{r}=\\underbrace{\\frac{1-\\operatorname{erf}\\left(\\frac{r}{R_{\\mathrm{lr}}}\\right)}{r}}_{\\text {short range }}+\\underbrace{\\frac{\\operatorname{erf}\\left(\\frac{r}{R_{\\mathrm{lr}}}\\right)}{r}}_{\\text {long range }}.\n\\end{equation}\nThe short range term is already included by the local exchange-correlation functional employed in LC-DFTB. The electron integrals of the screened Coulomb potential for the long-range contribution can be calculated as \n\\begin{align*}\n(\\mu \\lambda \\mid \\sigma \\nu)_{\\mathrm{lr}} &=\\iint \\phi_{\\mu}(r_1) \\phi_{\\lambda}(r_1) \\frac{\\operatorname{erf}\\left(\\frac{r_{12}}{R_{\\mathrm{lr}}}\\right)}{r_{12}} \\phi_{\\sigma}(r_2) \\phi_{\\nu}(r_2) \\mathrm{d} 1 \\mathrm{d} 2 \\\\\n& \\approx \\sum_{A, B} \\gamma_{A B}^{\\operatorname{lr}} q_{A}^{\\mu \\lambda} q_{B}^{\\sigma \\nu}. \\numberthis{}{}\n\\label{eq:2e_approx_ao_lr}\n\\end{align*}\n The long-range $\\gamma$-matrix is defined similarly to the $\\gamma$-matrix in eq. \\ref{eq:gamma},\n\\begin{equation}\n\\gamma_{A B}^{\\operatorname{lr}}=\\frac{\\operatorname{erf}\\left(C_{A B}^{\\mathrm{lr}} R\\right)}{R},\n\\end{equation}\nwhere\n\\begin{equation}\nC_{A B}^{\\mathrm{lr}}=\\frac{1}{\\sqrt{2\\left(\\sigma_A^2+\\sigma_B^2+\\frac{1}{2} R_{\\mathrm{lr}}^2\\right)}}\n\\end{equation}\ndepends on the range-separation parameter $R_{\\mathrm{lr}}$.\nThe transformation of the two-electron repulsion integrals into molecular orbital (MO) basis leads to the following expressions:\n\\begin{equation}\n    (pq \\mid rs ) \\approx \\sum_{A} \\sum_{B} q_{A}^{pq} \\gamma_{AB} q_{B}^{rs}\n    \\label{eq:2e_approx_mo}\n\\end{equation}\nand \n\\begin{equation}\n    (pq \\mid rs )_{\\mathrm{lr}} \\approx \\sum_{A} \\sum_{B} q_{A}^{pq} \\gamma_{AB}^{\\mathrm{lr}} q_{B}^{rs}\n    \\label{eq:2e_approx_mo_lr}\n\\end{equation}\nThe atom-centered transition charges between MOs are defined as: \n\\begin{equation}\nq_{A}^{i j}=\\frac{1}{2} \\sum_{\\mu \\in A} \\sum_{\\nu}\\left(c_{\\mu}^{i} c_{\\nu}^{j}+c_{\\nu}^{i} c_{\\mu}^{j}\\right) S_{\\mu \\nu}\n\\label{eq:q_between_mos}\n\\end{equation}\nVariational minimization of the LC-DFTB energy with respect to the molecular orbitals leads to the corresponding Hamiltonian, that is defined as: \n\\begin{align*}\nH_{\\mu \\nu}^{\\mathrm{SCC}} &=H_{\\mu \\nu}^{0}+\\frac{1}{2} S_{\\mu \\nu} \\sum_{C}\\left(\\gamma_{A C}+\\gamma_{B C}\\right) \\Delta q_{C} \\\\\n&-\\frac{1}{8} \\sum \\Delta P_{\\alpha \\beta} S_{\\mu \\alpha} S_{\\beta \\nu}\\left(\\gamma_{\\mu \\beta}^{\\mathrm{lr}}+\\gamma_{\\mu \\nu}^{\\mathrm{lr}}+\\gamma_{\\alpha \\beta}^{\\mathrm{lr}}+\\gamma_{\\alpha \\nu}^{\\mathrm{lr}}\\right) \\numberthis{}{} \\label{eq:scc_hamiltonian}\n\\end{align*}\nThe ground state MOs and their energy are obtained by a self consistent procedure for solving the following general eigenvalue problem\n\\begin{equation}\n    \\mathbf{Hc} = \\epsilon\\mathbf{Sc}\n\\end{equation}\n\nBeing a semiempirical theory, (LC)-DFTB is heavily dependent on a parametrization for the electronic Hamiltonian, $H_{\\mu \\nu}^{0}$, the overlap matrix $S_{\\mu \\nu}$ and the repulsive potentials  $V_{A B}^{\\mathrm{rep}}$. The parametrization usually starts with the computation of pseudoorbitals, the tabulation of Slater-Koster files and ends with the fitting of repulsive potentials \\cite{elstner_density_2014,koskinen_density-functional_2009,spiegelman_density-functional_2020}. A benchmark of the used LC-DFTB parametrization for organic and biological molecules is given in Ref.\\cite{vuong_parametrization_2018}.\n\n\n\\subsection{Fragment molecular orbital LC-DFTB (FMO-LC-DFTB)}\nThe fragment molecular orbital method in combination with DFTB was developed by Nishimoto, Fedorov and Irle\\cite{nishimoto_density-functional_2014}. Recently, it was adopted to LC-DFTB by Niehaus and Irle\\cite{vuong_fragment_2019} and this theory forms the basis for our ground-state calculations. Therefore, we will only restate the most important equations in the following and refer to Ref. \\cite{nishimoto_density-functional_2014} and Ref. \\cite{vuong_fragment_2019} for a more detailed description. Only the case of molecular clusters --- molecules that are not covalently connected --- is considered in this work, and therefore, we do not introduce the hybrid orbital projection operator \\cite{nakano_fragment_2000}.\nThe total energy of the entire system in the self-consistent FMO-LC-DFTB is given by\n\\begin{equation}\nE=\\sum_{I}^{N} E_{I}+ \\frac{1}{2} \\sum_{I}^{N} \\sum_{J}^N \\left(E_{I J}-E_{I}-E_{J}\\right).\n\\label{eq:fmo_energy}\n\\end{equation}\nHere $E_I$ ($E_J$) are the energies of fragment $I$ ($J$) and $E_{IJ}$ is the energy of the pair $IJ$. The total number of fragments is denoted as $N$, while we will use $N_I$ for the number of atoms in fragment $I$. \nThe energies $E_X$ where $X$ is either $I$ or $IJ$ can be further separated into a part that depends only on the isolated molecule and another one that accounts for the environment of the molecule:\n\\begin{equation}\nE_{X}=E_{X}^{\\prime}+E_{X}^{\\mathrm{em}}\n\\label{eq:frag_energy}\n\\end{equation}\nThe former is the so-called internal energy, $E_{X}^{\\prime}$, which is equivalent to the total energy calculated in conventional LC-DFTB of entity $X$. The embedding energy, $E_{X}^{\\mathrm{em}}$, accounts for the Coulomb interaction between the entity $X$ and all other fragments: \n\\begin{equation}\nE_{X}^{\\mathrm{em}}=\\sum_{A \\in X} \\sum_{K \\neq X}^{N} \\sum_{C \\in K} \\gamma_{A C} \\Delta q_{A}^{X} \\Delta q_{C}^{K}\n\\label{eq:embedding_energy_frag}\n\\end{equation}\nInserting eq. \\ref{eq:frag_energy} and eq. \\ref{eq:embedding_energy} in eq. \\ref{eq:fmo_energy} allows to rewrite the total FMO energy as: \n\\begin{equation}\\label{eq:total_fmo_energy}\nE=\\sum_{I}^{N} E_{I}^{\\prime}+ \\frac{1}{2} \\sum_{I}^{N} \\sum_{J}^N \\left(E_{I J}^{\\prime}-E_{I}^{\\prime}-E_{J}^{\\prime}\\right)+ \\frac{1}{2} \\sum_{I}^{N} \\sum_{J}^{N} \\Delta E_{I J}^{\\mathrm{em}}\n\\end{equation}\nThe last term on the r.h.s. is the difference in embedding energy of the pair and corresponding embedded but noninteracting monomers. \n\\begin{equation}\\label{eq:embedding_energy}\n\\Delta E_{I J}^{\\mathrm{em}} = E_{I J}^{\\mathrm{em}} - E_{I}^{\\mathrm{em}} - E_{J}^{\\mathrm{em}} = \\sum_{A \\in I J} \\sum_{K \\neq I, J}^{N} \\sum_{C \\in K} \\gamma_{A C} \\Delta \\Delta q_{A}^{I J} \\Delta q_{C}^{K}\n\\end{equation}\nHere $\\Delta \\Delta q_{A}^{I J}$ denotes the difference between charges of pair $IJ$ and charges of fragments $I$ and $J$ for atom $A$:\n\\begin{align}\n\\Delta \\Delta q_{A}^{I J}   &= \\Delta q_{A}^{I J} - ( \\Delta q_{A}^{I} \\oplus \\Delta q_{A}^{J}) \\\\\n\\Delta q_{A}^{I} \\oplus \\Delta q_{A}^{J}  &= \\Delta q_{A}^{I}  \\text { for } A \\in I \\text { and } A \\notin J,\\\\\n                            &= \\Delta q_{A}^{J} \\text { for } A \\in J \\text { and } A \\notin I, \n\\end{align}\nWe employ the electrostatic dimer (ES-DIM) approximation \\cite{nakano_fragment_2002} for pairs, in which the fragments are so far separated, that their orbital overlap will be zero. The energy of the far-separated pairs is given by \n\\begin{equation}\\label{eq:esdim_energy}\nE_{I J}^{\\prime} = E_I^{\\prime}+E_J^{\\prime}+\\sum_{A \\in I} \\sum_{B \\in J} \\gamma_{A B} \\Delta q_A^I \\Delta q_B^J.\n\\end{equation}\nIt should be noted that although it is possible to separate the total energy into internal and embedding energies, the tight-binding Hamiltonian of an entity (\\textit{cf.} eq. \\ref{eq:scc_hamiltonian}) also depends on the electron density or, more specifically, the charges of all the other fragments\n\\begin{equation}\\label{eq:fragment_hamiltonian}\nH_{\\mu \\nu}^{X}=H_{\\mu \\nu}^{S C C, X}+V_{\\mu \\nu}^{X}\n\\end{equation}\nwhere, $V_{\\mu \\nu}^{X}$, is the electrostatic potential (ESP) that acts on fragment $X$ and is defined as: \n\\begin{equation}\n    V_{\\mu \\nu}^{X} = \\frac{1}{2} S_{\\mu \\nu}^X \\sum_{K \\notin I}^N \\sum_{C \\in K}^{N_K} \\left(\\gamma_{A C}+\\gamma_{B C}\\right) \\Delta q_{C} \\label{eq:electrostatic_potential}\n\\end{equation}\nThis makes it necessary to perform the SCC iterations for fragments self-consistently so that one iteration step per fragment is made to update all charges simultaneously. The obtained self-consistent charges are afterwards used for the calculation of the SCC iterations on pairs for which the ES-DIM approximation is not used. \\\\\n\nCalculating the electronic ground state in the context of FMO(-DFTB) allows one to make accurate calculations of the energy, but one does not obtain MOs of the entire system. Instead, one obtains only the MOs for each of the fragments and all fragment pairs that are not affected by the ES-DIM approximation. \nSince they are obtained by independent calculations such MOs are not orthogonal. However, it is possible, as shown by Tsuneyuki \\textit{et al.} \\cite{tsuneyuki_molecular_2009}, to construct orthogonal MOs for the whole system and to transform the Hamiltonian into such basis by using L\u00f6wdin's approach\\cite{lowdin_nonorthogonality_1950}\n\\begin{equation}\\label{LCMO_fock_4}\n    \\mathbf{H}^{\\prime}=\\mathbf{S}^{-1 \/ 2} \\mathbf{H}^{\\mathrm{LCMO}} \\mathbf{S}^{-1 \/ 2}.\n\\end{equation}\nSince the computation of the matrix inverse of the total overlap is computationally demanding and the overlap matrix in the basis of the fragments and pairs is almost diagonal, we approximate it in first order by\n\\begin{equation}\n    \\mathbf{S}^{-1 \/ 2} \\approx \\frac{3}{2} \\mathbf{1} - \\frac{1}{2} \\mathbf{S}\n    \\label{eq:invers_S_approx}\n\\end{equation}\nwhere the overlap matrix matrix elements are given by\n\\begin{equation}\n    S_{pq}^{IJ} = \\langle \\varphi_p^I | \\varphi_q^J \\rangle\n\\end{equation}\nand $\\varphi_p^I$ is the $p$-th MO of fragment $I$. \nThe Hamiltonian of the whole system, $\\mathbf{H}^{\\mathrm{LCMO}}$, can be constructed from the Hamiltonian matrices of the corresponding fragments and pairs as\n\\begin{equation}\\label{LCMO_fock_1}\n    \\mathbf{H}^{\\mathrm{LCMO}}=\\sum_{I}^N \\oplus \\mathbf{H}^{I}+ \\frac{1}{2} \\sum_{I}^N \\sum_{J \\neq I}^N \\oplus\\left(\\mathbf{H}^{I J}-\\mathbf{H}^{I} \\oplus \\mathbf{H}^{J}\\right).\n\\end{equation}\nThe $\\oplus$ sign indicates that each block in the total Hamiltonian matrix is filled with the Hamiltonian of the corresponding fragment ($\\mathbf{H}^{I}$) or fragment pair ($\\mathbf{H}^{IJ}$). \n\nIn contrast to the diagonal Hamiltonian matrix of the fragments \n\\begin{equation}\\label{LCMO_fock_2}\n  \\epsilon_{a}^X    =  \\delta_{a b} H^X_{a b},\n\\end{equation}\nwhere $\\epsilon_{a}$ is the $a$-th MO energy of fragment $X$, the Hamiltonian of pair $IJ$ accounts for the fact that the MOs between the fragments might not be strictly orthogonal\n\\begin{equation}\\label{LCMO_fock_3}\n    H^{IJ}_{ab} = \\sum_{r} \\epsilon_{r}^{IJ} \\langle \\varphi_a^I \\mid \\varphi_r^{IJ} \\rangle \\langle \\varphi_b^J \\mid \\varphi_r^{IJ} \\rangle \n\\end{equation}\nwhere the sum runs over all MOs of the pair $IJ$. Of course, it would now be possible to determine the eigenvectors of the Hamiltonian and thereby obtain the molecular orbitals of the entire system. In principle, once the MOs of the whole system are available, the excited states could be calculated using standard TD-DFTB procedure. While this seems straightforward, the disadvantage is that the calculation of the excited states are as costly as in a non-FMO approach and one would lose the scalability of the method. \n\n\\subsection{Excited states in the frame of FMO-LC-DFTB}\\label{sec:quasi_diabatic_states}\nAn alternative way to compute excited states in the framework of the FMO-Ansatz is motivated by the idea that an excited state wavefunction of the whole system can be expressed as a linear combination of basis states of the respective fragments and fragment pairs. Since we restrict ourselves to the interaction between two fragments in FMO, we also use only locally excited (LE) states on one fragment and charge-transfer (CT) states between two fragments as basis states. The electronic excited state wavefunction, $|\\Psi\\rangle$, of a large molecule assembly can then be expressed as \n\\begin{equation}\n\\left|\\Psi\\right\\rangle=\\sum_{I}^N \\sum_{m}^{N_{\\mathrm{LE}}} c_I^m \\left|\\mathrm{LE}_I^m \\right\\rangle+ \\sum_{I}^{N} \\sum_{J \\neq I}^{N} \\sum_{m}^{N_{\\mathrm{CT}}} c_{I\\rightarrow J}^m \\left| \\mathrm{CT}_{I \\to J}^{m}\\right\\rangle    \n\\label{eq:es_wf}\n\\end{equation}\nwhere the coefficients for fragment LE and CT configuration state functions (CSFs) are given by $c_I^m$ and $c_{I\\rightarrow J}^m$, respectively. The coefficients are obtained by solving the eigenvalue problem $\\mathbf{Hc} = \\mathbf{Ec}$. Notice that we assume that the basis states are orthogonal such that $\\mathbf{S} = \\mathbf{1}$. However, in the calculation of the Hamiltonian matrix elements the overlap matrix is explicitly included. This is in line with other semi-empirical methods that employ the same approximation. On the basis of FMO Hartree-Fock theory this approach was introduced by Fujita and Mochizuki\\cite{fujita_development_2018}. However, we will show that our definition of the CT states differs and how this impacts not only the energies but also the scaling of the method. \nThe ansatz in eq. \\ref{eq:es_wf} allows one to choose the energetically lowest basis states by setting the number of LE states per fragment ($N_{\\mathrm{LE}}$) and the number of CT states per pair ($N_{\\mathrm{CT}}$) to appropriate values. The number of eigenvalues for the corresponding eigenvalue problem then is given by $N \\cdot N_{\\mathrm{LE}} + (N^2 - N) \\cdot N_{\\mathrm{CT}}$. Furthermore, since these basis states are per definition locally excited or charge-transfer states the adiabatic states' eigenvectors (cf. \\ref{eq:es_wf}) contain the state character information. In principle, it would also be possible to extend this approach to doubly or even higher excited states.  \\\\\nThe LE states, $\\left|\\mathrm{LE}_{\\mathrm{I}}^m\\right\\rangle$, are calculated as singly excited states of a fragment. We restrict ourselves in this work to singlet  states, but it should be noted that this approach can be extended to triplet excited states in a straightforward way. The $m$-th excited state (S$_m$) on fragment $I$ defines the following LE basis state: \n\\begin{align}\n    \\left|\\mathrm{LE}_{\\mathrm{I}}^m\\right\\rangle &= \\sum_{i \\in I}\\sum_{a \\in I} T_{i a}^{m(I)}\\frac{1}{2}\\left(a_{\\mathrm{Ia}, \\alpha}^{\\dagger} a_{\\mathrm{Ii}, \\alpha}+a_{\\mathrm{Ia}, \\beta}^{\\dagger} a_{\\mathrm{Ii}, \\beta}\\right)|G\\rangle \\\\\n    &= \\sum_{i \\in I}\\sum_{a \\in I}  T_{i a}^{m(I)} | \\Phi_{I}^{i\\to a} \\rangle\n    \\label{eq:LE_states}\n\\end{align}\nHere, $a_{I a, \\alpha}^{\\dagger}$ and $a_{I a, \\beta}$ are the creation and annihilation operators of the $\\alpha$ and $\\beta$ spin electron in the $a$th orbital of fragment $I$, and $|G\\rangle$ is the ground-state wavefunction.\n$\\mathbf{T}^{m(I)}$ is the one-particle transition density matrix of the $m$-th excited state of fragment $I$ in the MO basis and $|\\Phi_{I}^{i\\to a} \\rangle$ is the configuration state function of the excitation from the occupied orbital $i$ to the virtual orbital $a$ on fragment $I$.\\\\\nSimilarly, the $m$-th intermolecular CT state from fragment $I$ (hole) to fragment $J$ (particle) can be defined as: \n\\begin{align}\n    \\left|\\mathrm{CT}_{I \\to J}^{m}\\right\\rangle &=\\sum_{i \\in I}\\sum_{a \\in J} T_{i a}^{m(IJ)} \\frac{1}{2}\\left(a_{\\mathrm{Ja}, \\alpha}^{\\dagger} a_{\\mathrm{Ii}, \\alpha}+a_{\\mathrm{Ja}, \\beta}^{\\dagger} a_{\\mathrm{Ii}, \\beta}\\right)|G\\rangle \\\\\n    &= \\sum_{i \\in I}\\sum_{a \\in J} T_{i a}^{m(IJ)} | \\Phi_{I \\to J}^{i\\to a} \\rangle\n    \\label{eq:CT_states}\n\\end{align}\nThe transition between the fragments $I$ and $J$ is restricted to the occupied orbitals of monomer $I$ and the virtual orbitals of monomer $J$. The definition of the CT state differs to the approach of Fujita and Mochizuki\\cite{fujita_development_2018} as they use the following  single configuration state function as a CT state\n\\begin{equation}\n    \\left|\\mathrm{CT}_{I \\to J}^{i \\to a}\\right\\rangle=\\frac{1}{2}\\left(a_{\\mathrm{Ja}, \\alpha}^{\\dagger} a_{\\mathrm{Ii}, \\alpha}+a_{\\mathrm{Ja}, \\beta}^{\\dagger} a_{\\mathrm{Ii}, \\beta}\\right)|G\\rangle =  | \\Phi_{I \\to J}^{i\\to a} \\rangle.\n    \\label{eq:CT_states_fujita}\n\\end{equation}\nBoth approaches have their merits, and we will briefly explain below why we define CT states as shown in equation \\ref{eq:CT_states}. The latter necessitates solving the CIS matrix for the lowest $N_{\\mathrm{CT}}$ charge-transfer states of each pair. Therefore, the construction of the basis states will no longer scale linearly with the number of fragments but quadratically. Nevertheless, this approach has the advantage that all electronic couplings between CT states on a pair are omitted since these are eigenstates of the pair. Furthermore, in both approaches, a restriction of the considered CT states or occupied and virtual orbitals is necessary. Using equation \\ref{eq:CT_states}, we can choose to use the $N_{\\mathrm{CT}}$ lowest energy CT states and ensure that all relevant orbitals for these states are considered, whereas with eq. \\ref{eq:CT_states_fujita}, the selection of states is made by considering $N_{\\mathrm{hole}}$ occupied and  $N_{\\mathrm{particle}}$ virtual MOs. A higher number of included orbitals quickly leads to very large eigenvalue problems as the number of considered CT states scales with $(N^2 - N) \\cdot N_{\\mathrm{hole}} \\cdot N_{\\mathrm{particle}}$. We will show in section \\ref{sec:accuracy_excited_states} how the accuracy of the excitation energies and the computational scaling depend on using either eq. \\ref{eq:CT_states} or \\ref{eq:CT_states_fujita} for the definition of charge-transfer states and that the usage of eq. \\ref{eq:CT_states} is generally preferable. \n\n\\subsection{Hamiltonian matrix elements}\\label{sec:DiabaticHamiltonian}\nThe definition of basis states in eq. \\ref{eq:LE_states} and \\ref{eq:CT_states} allows to express the matrix elements of the system Hamiltonian. We will first focus on the excitation energies (diagonal matrix elements) of LE and CT states. The energies are calculated by a LC-DFTB\\xspace calculation of the fragment (pair)\n in the framework of linear-response LC-TD-DFT, which was adapted to tight-binding DFT by Niehaus \\textit{et al.}\\cite{niehaus_tight-binding_2001} and has been extended to include the LC correction by some of us\\cite{humeniuk_long-range_2015}. In linear-response TD-DFT excited states are computed by solving the non-Hermitian eigenvalue problem \n\\begin{equation}\n    \\left(\\begin{array}{ll}\n\\mathbf{A} & \\mathbf{B} \\\\\n\\mathbf{B} & \\mathbf{A}\n\\end{array}\\right)\\left(\\begin{array}{l}\n\\mathbf{X} \\\\\n\\mathbf{Y}\n\\end{array}\\right)=\\bm{\\omega}\\left(\\begin{array}{cc}\n\\mathbf{1} & 0 \\\\\n0 & -\\mathbf{1}\n\\end{array}\\right)\\left(\\begin{array}{l}\n\\mathbf{X} \\\\\n\\mathbf{Y}.\n\\end{array}\\right)\\label{eq:Casida}\n\\end{equation}\nIn the framework of LC-TD-DFTB\\xspace the matrices $\\mathbf{A}$ and $\\mathbf{B}$ are defined as\\cite{humeniuk_long-range_2015}\n\\begin{align}\n    A_{i a, j b} &=\\delta_{i j} \\delta_{a b}\\left(\\epsilon_a-\\epsilon_i\\right)+2 \\sum_{A}\\sum_{B} q^{ia}_A \\gamma_{AB} q_B^{jb} - \\sum_{A} \\sum_{B} q^{ij}_A \\gamma_{AB}^{\\mathrm{lr}} q^{ab}_B \\\\\n    B_{i a, j b} &= 2 \\sum_{A}\\sum_{B} q^{ia}_A \\gamma_{AB} q_B^{jb} - \\sum_{A} \\sum_{B} q^{ib}_A \\gamma_{AB}^{\\mathrm{lr}} q^{aj}_B,\n\\end{align}\nwhere $\\epsilon_i$ is the energy of the $i$-th molecular orbital. The orbital energies $\\epsilon_i$ are represented by the diagonal elements of the orthogonalized total Hamiltonian $H^\\prime_{ii}$. Using the Tamm-Dancoff (TDA) approximation ($\\mathbf{B} = 0$), eq. (\\ref{eq:Casida}) is reduced to a simple Hermitian eigenvalue problem\n\\begin{equation}\n    \\mathbf{A} \\mathbf{X} = \\bm{\\omega} \\mathbf{X},\n\\end{equation}\nwhich can be solved iteratively for the lowest eigenvalues by employing the algorithm developed by Davidson\\cite{davidson_iterative_1975}. This is in principle equivalent to the configuration interaction singles (CIS) procedure. The energies of the CT states are calculated in analogy to the local excitations, where the excitations are restricted to the transition from the occupied orbitals of one monomer to the virtual orbitals of the other one.\n\nTo calculate the off-diagonal matrix elements of the Hamiltonian, the couplings between the basis states are required.\nIn accordance with Ref. \\cite{fujita_development_2018}, the Hamiltonian can be split into one electron\n\\begin{equation}\\label{One_electron_hamiltonian}\n    \\langle \\Phi_{I \\to J}^{i\\to a}|H_{1e}|\\Phi_{K \\to L}^{j \\to b} \\rangle = \\delta_{I K} \\delta_{i j} H_{a b}^{\\prime}-\\delta_{J L } \\delta_{a b} H_{i j}^{\\prime}\n\\end{equation}\nand two-electron contributions\n\\begin{equation}\\label{two_electron_hamiltonian}\n    \\langle \\Phi_{I \\to J}^{i\\to a} | H_{2 e} |\\Phi_{K \\to L}^{j \\to b} \\rangle =2\\left(i^{(I)} a^{(J)} \\mid j^{(K)} b^{(L)}\\right)-\\left(i^{(I)} j^{(K)} \\mid a^{(J)} b^{(L)}\\right),\n\\end{equation}\nwhich were derived using the usual Slater-Condon rules. The two-electron coupling matrix elements of the Hamiltonian require the calculation of the four-center electron repulsion integrals (\\textit{cf}. eq. \\ref{two_electron_hamiltonian}). In contrast to the work of Fujita and Mochizuki\\cite{fujita_development_2018}, the off-diagonal elements between the basis states that involve three or four fragments are not neglected.\n\n\n\\subsubsection{LE-LE matrix elements}\n The coupling between locally excited states is given only by the two-electron part, as the one electron coupling between locally excited states vanish due to the Slater-Condon rule,\n\\begin{align*}\n\\left\\langle \\mathrm{LE}_{I}^{m}\\left|H \\right| \\mathrm{LE}_{J }^{n}\\right\\rangle &=\\sum_{ia \\in I} \\sum_{jb \\in J} T_{i a}^{m(I)} T_{jb}^{n(J)}[2(i a \\mid j b) \\\\\n &-(ij \\mid ab)] \\label{le_le_coulomb} \\numberthis{}{}  \\\\\n\\approx& 2 \\sum_{A\\in I} \\sum_{B \\in J} q_{\\mathrm{tr}, A}^{m(I)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(J)}\\\\\n-& \\sum_{A\\in IJ} \\sum_{B \\in IJ} \n\\sum_{ia \\in I} \\sum_{jb \\in J} T_{i a}^{(\\mathrm{Im})} T_{jb}^{(J n)} q_{A}^{ij} \\gamma_{AB}^{\\mathrm{lr}} q_{B}^{ab}, \\numberthis{}{} \\label{le_le_coupling}\n\\end{align*}\nwhere\n\\begin{equation}\n    q_{\\mathrm{tr}, A}^{m(I)} = \\sum_{ia} q_{A}^{ia} T_{i a}^{m(I)} \n\\end{equation}\nis the transition charge of the $m$-th excited state of the fragment $I$ on atom $A$. $q_{A}^{ij}$ and $q_{B}^{ab}$ are the atom-centered transition charges between the occupied and virtual orbitals of the two monomer fragements $I$ and $J$ (\\textit{cf}. eq. (\\ref{eq:q_between_mos})). The first term of the LE-LE coupling (\\textit{cf}. eq. \\ref{le_le_coulomb}) characterizes the Coulomb interaction between the transition densities of the respective excitations. The second term represents the exchange interaction.\nIn the case of LE states on far separated fragments, fragment pairs for which the ES-DIM approximation is used, the exchange contribution can be neglected as the overlap between both states will be zero: \n\\begin{equation}\n    \\left\\langle \\mathrm{LE}_{I}^{m}\\left|H\\right| \\mathrm{LE}_{J }^{n}\\right\\rangle \\approx 2 \\sum_{A\\in I} \\sum_{B \\in J} q_{\\mathrm{tr}, A}^{m(I)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(J)}\n\\end{equation}\nTherefore, only the long-range Coulomb interaction is taken into account in this case. \n\n\\subsubsection{LE-CT matrix elements}\nThe Hamiltonian matrix elements between an LE state on fragment $I$ and a CT state which is formed by linear combination of holes on fragment $J$ and electrons on fragment $K$ contains a one-electron part\n\\begin{align*} \n    \\left\\langle\\mathrm{\\mathrm{LE}}_{I}^{m}\\left|H_{1 e}\\right| \\mathrm{CT}_{J \\to K}^{n}\\right\\rangle = & \\delta_{I J} \\sum_{ia \\in I}\\sum_{b \\in K} T_{ia}^{m(I)} T_{ib}^{n(IK)} H_{a b}^{\\prime} \\\\ &-\\delta_{I K}\n    \\sum_{ia \\in I}\\sum_{j \\in J} T_{i a}^{m(I)}T_{ja}^{n(JI)} H_{i j}^{\\prime}\n    \\label{eq:LE_CT_1e_coupling_2} \\numberthis{}{}\n\\end{align*}\nas well as a contribution from the two-electron interaction\n\\begin{align*}\n\\left\\langle \\mathrm{LE}_{\\mathrm{I}}^m\\left|H_{2 e}\\right| \\mathrm{CT}_{J \\to K}^{n}\\right\\rangle &= \\sum_{ia \\in I} \\sum_{j \\in J} \\sum_{b \\in K}  T_{i a}^{m(I)} T_{j b}^{n(JK)} [2(i a \\mid jb ) \\\\ &-( i j \\mid ab )] \\numberthis{}{} \\\\\n\\approx 2\\sum_{A \\in I}\\sum_{B \\in JK}& q_{\\mathrm{tr}, A}^{m(I)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(JK)} \\\\\n- \\sum_{A \\in IJ}\\sum_{B \\in IK\n} &\\sum_{ia \\in I}  \\sum_{j \\in J}\\sum_{b \\in K}T_{ia}^{m(I)} T_{jb}^{n(JK)} q_{A}^{ij} \\gamma_{AB}^{\\mathrm{lr}} q_{B}^{ab}. \\label{le_ct_coupling} \\numberthis{}{}\n\\end{align*}\nThe first and second term in eq. \\ref{le_ct_coupling} correspond to the Coulomb and exchange interaction between the transition densities of the LE and CT state. The one-electron contribution is only non-zero if the CT state shares one of its fragment with the LE state and if the other fragment is in spatial proximity, so that the ES-DIM approximation is not used, because otherwise the matrix element $H_{pq}^{\\prime}$ will be zero. If both fragments of the CT state are far apart (ES-DIM approx.) from the fragment of the LE state then the exchange term of eq. \\ref{le_ct_coupling} will be zero and is neglected. In this case the matrix element becomes\n\\begin{equation}\n    \\left\\langle \\mathrm{LE}_{\\mathrm{I}}^m\\left|H\\right| \\mathrm{CT}_{J \\to K}^{n}\\right\\rangle  \n\\approx 2\\sum_{A \\in I}\\sum_{B \\in JK} q_{\\mathrm{tr}, A}^{m(I)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(JK)}\n\\end{equation}\nand only the Coulomb interaction is calculated. \n\n\n\\subsubsection{CT-CT matrix elements}\nThe one-electron CT--CT coupling vanishes for $I \\neq K$ or $K \\neq L$ (\\textit{cf}. eq. \\ref{One_electron_hamiltonian}), and thus, is reduced to the diagonal contributions, which are included in the LC-TD-DFTB calculation of the CT state. \nThe two-electron off-diagonal coupling between two charge-transfer states is given by\n\\begin{align*}\n&\\left\\langle \\mathrm{CT}_{I \\to J}^{m(IJ)}\\left|H \\right| \\mathrm{CT}_{K \\to L}^{n (KL)}\\right\\rangle = \\\\ \n&\\sum_{i \\in I}\\sum_{a \\in J}\\sum_{j \\in K}\\sum_{b \\in L} T_{ia}^{m(IJ)} T_{jb}^{n(KL)}  [2(i a \\mid j b) -(i j \\mid a b)] \\numberthis{}{} \\\\\n&\\approx 2 \\sum_{A\\in IJ} \\sum_{B \\in KL} q_{\\mathrm{tr}, A}^{m(IJ)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(KL)}\\\\\n&- \\sum_{i \\in I}\\sum_{a \\in J}\\sum_{j \\in K}\\sum_{b \\in L} \\sum_{A \\in IK} \\sum_{B \\in JL} T_{ia}^{m(IJ)} T_{jb}^{n(KL)}q_{A}^{ij} \\gamma_{AB}^{\\mathrm{lr}} q_{B}^{ab} \\numberthis{}{} \\label{ct_ct_coupling}\n\\end{align*}\nAs shown for the LE-LE and LE-CT matrix elements, we also neglect the exchange part of the coupling for the CT-CT couplings in case if either fragment $I$ and $K$ or $J$ and $L$ are different fragments that are far apart. Thus in this case the matrix element simplifies to\n\\begin{equation}\n    \\left\\langle \\mathrm{CT}_{I \\to J}^{m(IJ)}\\left|H\\right| \\mathrm{CT}_{K \\to L}^{n (KL)}\\right\\rangle \n\\approx 2 \\sum_{A\\in IJ} \\sum_{B \\in KL} q_{\\mathrm{tr}, A}^{m(IJ)} \\gamma_{AB} q_{\\mathrm{tr}, B}^{n(KL)} \\label{eq:CT_CT_ESDIM}\n\\end{equation}\n\n\\section{Computational procedure}\\label{sec:computational_procedure}\n\\begin{figure}[tbh!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_1.pdf}\n    \\caption{Flowchart of the computational procedure of the entire FMO-LC-TDDFTB\\xspace calculation.}\n    \\label{flowchart}\n\\end{figure}\nHerein, the sequence of the steps necessary to carry out a FMO-LC-TDDFTB\\xspace calculation is briefly summarized. The flowchart displayed in Fig. \\ref{flowchart} shows the \norder of the different steps of a FMO-LC-TDDFTB\\xspace calculation and how the various parts of the calculation are connected. \n\nIn the first step, the input geometry is partitioned into the different monomer, pair and ES-DIM pair fragments. The ES-DIM approximation\\cite{nakano_fragment_2002} is applied if the closest distance between two fragments exceeds the threshold value of twice the sum of the van der Waals radii of the closest atoms. \n\nSubsequently, the coupled lc-DFTB SCC iterations of the monomer fragments are performed until the convergence of all monomer calculations is achieved. As the electrostatic potential between all monomer fragments is required to calculate the monomer Hamiltonian (\\textit{cf.} eq. \\ref{eq:fragment_hamiltonian}), the SCC iterations for all fragments are computed simultaneously in order to update the charges of all monomers in each step. The final self-consistent charges are stored, as they are necessary for the calculation of the pair, ES-DIM pair and embedding energies. The Anderson acceleration is used to speed up the convergence of the SCC routine \\cite{anderson_iterative_1965,zhang_globally_2020}. \n\nThen, using the electrostatic potential between all fragments (\\textit{cf}. eq. \\ref{eq:electrostatic_potential}) obtained from the previous step, the pair energies are calculated. The charges of the pair fragments are stored so that they can be used in the calculation of the embedding energy. The ES-DIM pair energy (\\textit{cf}. eq. \\ref{eq:esdim_energy}) and the embedding energy (\\textit{cf}. eq. \\ref{eq:embedding_energy}) are determined, using the charges of the monomer and pair fragments obtained from the previous SCC routines. The results of the fragment calculations are combined according to eq. (\\ref{eq:total_fmo_energy}), which yields the ground-state energy  of the FMO-LC-TDDFTB\\xspace approach.\n\nThereafter, the LCMO-Hamiltonian matrix is built according to eq. \\ref{LCMO_fock_4}--\\ref{LCMO_fock_3}, using the orbital energies, orbital coefficients and overlap matrices of the fragments. The $\\mathbf{H'}$ matrix is obtained by subsequent L\u00f6wdin orthogonalization. In order to construct the LE and CT basis states, LC-TD-DFTB\\xspace excited state calculations are performed for both the monomers, the pairs and the ES-DIM pairs, where the orbital energies of the respective fragments are replaced by the diagonal matrix elements of the $\\mathbf{H'}$ matrix. \n\nSubsequently, the off-diagonal matrix elements of the Hamiltonian are calculated according to the expressions for the couplings between the LE and CT states  given in eq. \\ref{le_le_coupling} -- \\ref{eq:CT_CT_ESDIM}. \nAs the calculation of the exchange terms in the eq. (\\ref{le_le_coupling}), (\\ref{le_ct_coupling}) and (\\ref{ct_ct_coupling}) includes all possible transitions between the orbitals of the monomer pairs, a screening of the one-particle transition density matrices $\\mathbf{T}$ was introduced, where only matrix elements over a certain threshold are considered. Thus, the number of transitions is limited to the actually contributing occupied and virtual orbitals of the fragments. \n\nAt the end of the FMO-LC-TDDFTB\\xspace calculation, the Davidson diagonalization is utilized to obtain the excited state energies and coefficients, which are then used to calculate the oscillator strengths.\n\nThe LC-TD-DFTB\\xspace and FMO-LC-TDDFTB\\xspace methods were implemented in our own software package DIALECT, which is available on Github\\cite{noauthor_dialect_2022}. \n\nAll calculations regarding the benchmarks and scaling tests were performed on a single core of a computing node with dual E5-2680 Xeon CPUs (2.4 GHz, 14 cores each), 188 Gb of DDR4 memory and a SATA hard drive. \n\nThe DFTB parameter set ob2-split\\cite{vuong_parametrization_2018} was employed in all calculations. In the case of the benchmark calculations, a long-range radius of 3.03~$a_0$ was used. In order to compensate the overestimation of the charge-transfer energies in the pentacene clusters, IP-EA tuning\\cite{stein_reliable_2009} was employed and an optimized long-range radius was used for these systems. This procedure will be described in greater detail in section \\ref{ch:pentacene_application}.\n\nThe Mercury\\cite{macrae_it_2020} program was employed to generate the various clusters of the anthracene, pentacene and perylen bisimide systems from their respective crystal structures\\cite{mason_crystallography_1964,holmes_nature_1999,lin_halochromic_2011}.\n\nIn case of the scans of potential energy curves of the pyrene dimer, the DFT-D3 dispersion correction\\cite{grimme_consistent_2010,grimme_effect_2011} was calculated using the simple-dftd3\\cite{ehlert_simple-dftd3_2021} program and the ob2-split dispersion parameters\\cite{vuong_parametrization_2018}.\n\\begin{figure*}[tb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_2.pdf}\n    \\caption{(a) Potential energy curves of the ground state and the first 6 excited states of the pyren dimer along the $\\pi$--$\\pi$ stacking coordinate $R_z$. (b) Potential energy curves of the ground state and the first 4 excited states along the parallel shift coordinate $R_x$ at a $\\pi$--$\\pi$ stacking distance of 3.1 \\AA{}. (c) The contributions of the locally excited and charge transfer states to the excited states shown in b) for different values of $R_x$. For each monomer 20 local excitations and for each pair 15 charge-transfer states were used.}\n    \\label{dimer_scan}\n\\end{figure*}\n\\section{Results}\\label{sec:results}\n\\subsection{Accuracy of FMO-LC-TDDFTB\\xspace excited states}\\label{sec:accuracy_excited_states}\nThe accuracy of the FMO-LC-TDDFTB\\xspace methodology is evaluated calculating the excitation energies of $\\pi$-stacked dimers\nof pyrene. The potential energy curves of the first 6 excited states were calculated  as a function of the $\\pi$--$\\pi$ stacking distance $R_z$ and the parallel shift coordinate $R_x$. As a reference we performed calculations with the conventional LC-TD-DFTB\\xspace approach. The main purpose of this comparison is to test the FMO approach and the validity of the employed approximations and not to benchmark the general performance. The latter has been thoroughly benchmarked in previous works.\\cite{vuong_parametrization_2018, bold_benchmark_2020, fihey_performances_2019} In the case of the FMO-LC-TDDFTB\\xspace calculation, 20 locally excited states for each monomer and 15 charge-transfer states for each pair were included. In Fig. \\ref{dimer_scan}a, the electronic state energies of the FMO-LC-TDDFTB\\xspace and LC-TD-DFTB\\xspace methods are shown for a scan of the $\\pi$--$\\pi$ stacking distance $R_z$. While there is a slight deviation from the LC-TD-DFTB\\xspace energies at distances from 2.75~\\AA{} to 3.0~\\AA{}, the deviation between the two methods decreases with increasing interfragment distances. From a distance of 3.5~\\AA{} onwards, the results of the FMO-LC-TDDFTB\\xspace calculation show an excellent agreement with the LC-TD-DFTB\\xspace energies. Below distances of 2.75~\\AA{}, which already belong to the repulsive part of the potential energy curves, the deviation from the reference LC-TD-DFTB\\xspace calculation results from the non-orthogonality of the MOs.\n\nFig. \\ref{dimer_scan}b shows the potential energy curves of the pyrene dimer for a scan of the parallel shift coordinate $R_x$ at a $\\pi$--$\\pi$ stacking distance of 3.1~\\AA{}. Here, the results of the FMO-LC-TDDFTB\\xspace calculation are in good agreement with the LC-TD-DFTB\\xspace reference. \n\nAs the excited state Hamiltonian is constructed using LEs and CTs as basis states, the analysis of the excited state composition becomes easily accessible. The contributions of the LE and CT states to the first 4 excited states for different parallel shift distances are shown in Fig. \\ref{dimer_scan}c. As the shift along $R_x$ decreases, the contribution of the charge-transfer states increases. While the S$_1$, S$_2$ and S$_3$ state show CT character of up to 30\\%, 20\\% and 12\\%, the $S_4$ onlyconsists of local excitations for all $R_x$ values.\n\nIn addition, the influence of the number of basis states on the accuracy of the excited states energies was investigated. To this end, the mean absolute errors (MAEs) of the first 6 excited states were calculated for different numbers of LE and CT states. Table \\ref{Tab:MAE_pyrene_scan} shows the MAEs of the molecular aggregate for $\\pi$--$\\pi$ stacking distances of 2.5, 2.75, 3.0, 3.5, 4.0 and 5.0 \\AA{}. The results confirm that the accuracy of the excited state energies of the dimer calculated at the FMO-LC-TDDFTB\\xspace level is satisfactory. At an intermolecular distance of 4.0 \\AA{} the errors are as small as 2.7--5.6~meV. At 3.5~\\AA{}, the MAEs are 4.8--9.9 meV, showing a an almost perfect agreement to the LC-TD-DFTB\\xspace energies.\n\n\\begin{table}[tbh]\n\\centering\n\\caption{Mean absolute errors (MAEs) of the first 6 excited states for the pyrene dimer at various interplanar distances $R_z$ and number of basis states.}\n\\label{Tab:MAE_pyrene_scan}\n\\setlength\\tabcolsep{6.5pt}\n\\begin{tabular}{@{}cc|rrrrrr@{}}\n\\toprule\n     &      & \\multicolumn{6}{c}{MAE \/ meV}        \\\\ \\midrule\nN$_{\\mathrm{LE}}$ & N$_{\\mathrm{CT}}$ & 2.5 \\AA{}& 2.75 \\AA{} & 3.0 \\AA{}  & 3.5 \\AA{}& 4.0 \\AA{}& 5.0 \\AA{}\\\\ \\midrule\n5    & 5    & 433.7 & 136.4& 34.5  & 9.9 & 5.6 & 2.5 \\\\\n10   & 5    & 406.8 & 118.2& 23.7 & 5.9 & 3.2 & 1.4 \\\\\n10   & 10   & 192.9 & 74.2 & 21.6  & 5.6 & 3.2 & 1.4 \\\\\n15   & 10   & 194.6 & 73.7 & 21.6  & 5.6 & 3.2 & 1.4 \\\\\n20   & 15   & 186.2 & 72.7& 21.9  & 5.6 & 3.2 & 1.4 \\\\\n30   & 20   & 178.4 & 71.0& 20.6  & 4.8 & 2.7 & 1.2 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\nAs expected, increasing the number of basis states improves the accuracy of the excited states energies. However, Table \\ref{Tab:MAE_pyrene_scan} shows that a further increase from 20 LE and 15 CT to 30 LE and 20 CT states yields only a marginal improvement of the MAEs as the additional states have a negligible electronic coupling to the first 6 excited states.\nAt all physically relevant intermolecular separations, the FMO-LC-TDDFTB\\xspace method almost perfectly reproduces the PES for the pyrene dimer. The deviations start to be visible at distances around 3~\\AA{}. Below 2.75~\\AA{}, which is already in the strongly repulsive part of the potential, the non-orthogonality of the MOs causes the deviation from the results of the LC-TD-DFTB\\xspace reference as the error in the couplings between the LE and CT states increases. Nevertheless, at distances at around 2.75~\\AA{} the MAEs are still relatively small; thus, the model should be sufficient for most physically realistic situations.\n\nTo evaluate the accuracy of our implementation of the charge-transfer states (\\textit{cf.} eq. \\ref{eq:CT_states}) in comparison to the approach of Fujita \\textit{et al.}\\cite{fujita_development_2018} (\\textit{cf.} eq. \\ref{eq:CT_states_fujita}), the mean absolute errors of the first 40 excited states from the LC-TD-DFTB\\xspace reference of a $\\pi$-stacked system of four pyrene monomers were calculated within both implementations. Fig. \\ref{ct_comparison} shows the results for stacking distances of 3.5 and 4.0 \\AA{}. For both methods, the number of LE states was set to 10 and the number of CT states was varied. As explained in section \\ref{sec:quasi_diabatic_states}, in our implementation, the dimension of the Hamiltonian grows linearly as we consider the lowest $N_{\\mathrm{CT}}$ states, and in the case of of Fujita's and Mochizuki's approach, the dimension of the Hamiltonian grows quadratically as all transitions between $N_{\\mathrm{hole}}$ occupied and $N_{\\mathrm{particle}}$ virtual orbitals are considered as separate CT states. As the amount of the charge-transfer states is increased for both implementations, the MAEs decrease in value. While both approaches achieve approximately the same accuracy, our method necessitates a much smaller size of the Hamiltonian. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_3.pdf}\n    \\caption{Comparison of the MAEs of both implementations of the charge-transfer states according to eq. (\\ref{eq:CT_states}) and (\\ref{eq:CT_states_fujita}) for different amounts of CT states at $\\pi$--$\\pi$ stacking distances of 3.5 and 4.0~\\AA{} for a system of four stacked pyrene molecules.}\n    \\label{ct_comparison}\n\\end{figure}\n\\begin{figure}[bth!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_4.pdf}\n    \\caption{Simulated absorption spectra of an anthracene cluster containing 48 fragments (1152 atoms). The transitions to the first 40 excited states of the system were convolved using a Lorentzian line shape with a broadening of 2~meV. Two LE states and one CT state were used for each monomer and pair, respectively.}\n    \\label{spectrum_tda_vs_fmo}\n\\end{figure}\n\nIn order to investigate the accuracy of the FMO-LC-TDDFTB\\xspace method regarding the energies and oscillator strengths of larger systems, we compared the absorption spectra of an anthracene cluster composed of 48 fragments (1152 atoms) including the first 40 excited states for both theoretical methods. The basis was constructed using two LE states for each monomer and one CT state for each pair. The individual vertical transitions were convolved using a Lorentzian line-shape function with a width of 2~meV. The comparison between the calculated spectra is depicted in Fig. \\ref{spectrum_tda_vs_fmo} and shows a slight blue shift of the FMO-LC-TDDFTB\\xspace energies by about 20--30 meV. However, since the general error in TD-DFT and, in particular, in LC-TD-DFTB\\xspace excitation energies is on the order of 0.5~eV\\cite{vuong_parametrization_2018, fihey_performances_2019}, the additional error due to the FMO approach is hardly significant. Aside from slightly underestimating the intensity of the first peak, our approach could sufficiently reproduce the anthracene cluster's LC-TD-DFTB\\xspace absorption spectrum.\n\n\\subsection{Computational scaling of FMO-LC-TDDFTB\\xspace}\nTo investigate the computational efficiency of the developed method for the calculation of excited states, we compare the wall times of the method against conventional LC-TD-DFTB\\xspace for anthracene clusters containing up to 360 fragments (8640 atoms). In the case of the LC-TD-DFTB\\xspace approach, the number of fragments was limited to 48 (1152 atoms) due to the rapid increase in computational demand. For each cluster, the first 40 excited singlet states were calculated, and as in the previous anthracene calculation (\\textit{cf.} Fig. \\ref{spectrum_tda_vs_fmo}) two LE states per monomer and one CT state per pair were used for FMO-LC-TDDFTB\\xspace computations.\nThe wall times of the FMO-LC-TDDFTB\\xspace and LC-TD-DFTB\\xspace calculations are compared in Fig. \\ref{wall_time_fmo_vs_dftb}. In the case of the anthracene cluster containing 48 fragments (1152 atoms), a speedup factor of approx. $1.5\\times 10^4$ was achieved. Whereas the LC-TD-DFTB\\xspace calculation, using a full active space, took almost 11 days, the fragment approach finished in 60~seconds. \n\n\\begin{figure}[tbh!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_5.pdf}\n    \\caption{Wall time comparison for an excited state calculation of anthracene clusters of different size between FMO-LC-TDDFTB\\xspace and LC-TD-DFTB\\xspace. The difference between using all molecular orbitals for the active and a reduced active space of only considering 20~\\% of the MOs is also shown.}\n    \\label{wall_time_fmo_vs_dftb}\n\\end{figure}\n\nFurthermore, we analyzed the timings of the LC-TD-DFTB\\xspace method using only 20~\\% of the active orbital space. As expected, this reduction in size of the $\\mathbf{A}$ matrix in the TDA-DFTB approach significantly accelerates the computation of excited states for the anthracene cluster consisting of 48 fragments which took 6293~seconds. However, the FMO-LC-TDDFTB\\xspace approach is still around 100 times faster for this system and the speedup grows to factor of more than 200 for the next bigger anthracene cluster containing 70 fragments (1680~atoms). \nIt should be noted that a restriction of the active spaces only lowers the prefactor of the method and not scaling. In addition, the restriction of the active orbitals leads to a loss of accuracy in (LC)-TD-DFTB as shown in Fig. \\ref{spectrum_tda_vs_truncated_tda}. While the energies and oscillator strength of FMO-LC-TDDFTB\\xspace with only two LE states and one CT state has shown a good agreement with LC-TD-DFTB\\xspace, the reduction of the active orbital space to 20~\\% leads to significantly larger errors of around 150~meV.\n\n\\begin{figure}[bth!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_6.pdf}\n    \\caption{Simulated absorption spectrum (LC-TD-DFTB\\xspace: black, LC-TD-DFTB\\xspace with 20~\\% active space: blue) of an anthracene cluster containing 48 fragments (1152 atoms). The transition to the first 40 excited states of the system were convolved using a Lorentzian line shape with a broadening of 2~meV.}\n    \\label{spectrum_tda_vs_truncated_tda}\n\\end{figure}\n\n\\begin{figure*}[bth!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_7.pdf}\n    \\caption{Timings of the FMO-LC-TDDFTB\\xspace excited state calculations for various a) anthracene and b) perylene bisimide clusters. The runtime scaling in relation to the system size is shown for different parts of the calculation on the right hand side of each figure. }\n    \\label{log_scaling}\n\\end{figure*}\n\nIn order to gain an insight into the time requirements of the different steps involved in a FMO-LC-TDDFTB\\xspace calculation, the total wall times have been split into separate parts. Fig. \\ref{log_scaling}a shows the timings of five steps: \n\\begin{enumerate}\n    \\item SCC: ground-state calculation of the entire systems\n    \\item $\\bm{H^{\\prime}}$ matrix: construction and L\u00f6wdin orthogonalization of the Hamiltonian matrix according to eq. \\ref{LCMO_fock_4} --\\ref{LCMO_fock_3}\n    \\item Basis states: calculation of LE and CT basis states of individual fragments and pairs according to section \\ref{sec:DiabaticHamiltonian}\n    \\item Couplings: computation of one- and two-electron coupling matrix elements between all LE and CT states according to eq. \\ref{le_le_coupling} -- \\ref{eq:CT_CT_ESDIM}\n    \\item Davidson: diagonalization of the Hamiltonian using the iterative Davidson procedure. \n\\end{enumerate}\n\nIn addition, the asymptotic scaling of the respective parts of the calculation was evaluated. With the increase in the size of the anthracene clusters, the Hamiltonian size grows up to a dimension of $129960^2$ for the system of 360 fragments (8640~atoms). Thus, the most time demanding steps are the calculation of the electronic couplings between the basis states and the following diagonalization for the lowest eigenvalues, which scale with factors of $\\mathcal{O}(N^{3.6})$ and $\\mathcal{O}(N^{5.1})$, respectively. The SCC routine implemented in our code shows the same nearly linear scaling of $\\mathcal{O}(N^{1.1})$, which has been reported previously\\cite{nishimoto_density-functional_2014}.\nAs the construction of the Fock matrix is limited by the scaling of the matrix multiplications during the L\u00f6wdin orthogonalization, it scales with a factor of $\\mathcal{O}(N^{2.9})$. Compared to the other parts of the calculation, the scaling of the construction of the basis states is negligible with a factor of $\\mathcal{O}(N^{1.9})$. \nThe combination of all steps of the procedure results in an effective scaling of $\\mathcal{O}(N^{3.4})$ for the total run time.\n\nA molecular system involving larger fragments was investigated to verify the effective scaling of the FMO-LC-TDDFTB\\xspace method. To this end, different clusters of substituted perylene bisimides (PBIs), where each monomer contains 142 atoms, were created, and the first 40 singlet excited states were calculated. Herefore, a basis of 2 locally excited states for each monomer and one charge-transfer state for each pair was employed. As shown in Fig. \\ref{log_scaling}b, the asymptotic scaling of the FMO approach remained approximately the same, if the Davidson diagonalization and the electronic couplings are excluded. Since the PBI clusters consist of distinctly less monomers than the anthracene clusters, the Hamiltonian is also significantly smaller. Subsequently, the fit of the scaling, which is strongly determined by the timings of the bigger clusters, results in smaller factors for the PBI systems, and thus, the total run time only scales with a factor of $\\mathcal{O}(N^{2.2})$. \nAs the system size grows above 100 fragments (approx. 15000 atoms), the calculation of the Hamiltonian becomes the most time consuming factor of the whole calculation.\nSubsequently, a further increase in the number of fragments would also result in a higher effective scaling factor as the Hamiltonian grows and the calculation of the couplings becomes the dominant aspect. \n\n\n\\subsection{Application to Pentacene} \\label{ch:pentacene_application}\nAfter demonstrating that FMO-LC-TDDFTB\\xspace can simulate excited states and absorption spectra with significantly reduced effort and, in return, at nearly the same accuracy as LC-TD-DFTB\\xspace, we wish to show the scope of possible applications by performing calculations on large pentacene clusters which serve as a model for bulk pentacene. This system has recently gained a lot of experimental and theoretical attention since it serves as a prototype for organic materials whose properties are determined by an interplay between local (Frenkel) and CT excitations.\\cite{zimmerman_mechanism_2011,zirzlmeier_singlet_2015,beljonne_charge-transfer_2013}\nThe FMO-LC-TDDFTB\\xspace method was used to simulate the absorption spectrum of pentacene clusters of different sizes, which were generated from crystal structure data. As a first step, an adjustment of the long-range radius was required since the energies of the pentacene aggregates' charge-transfer states are overestimated with the default radius of 3.03 $a_0$. Subsequently, ionization potential and electron affinity (IP-EA) tuning\\cite{stein_reliable_2009} were employed to estimate a reasonable value for the long-range radius by minimizing the following expression\n\\begin{equation}\n    (\\mathrm{IP}-\\epsilon_{\\mathrm{HOMO}})^2+(\\epsilon_{\\mathrm{LUMO}}-\\mathrm{EA})^2.\n\\end{equation}\nA scan of the radius yielded a minimum value of 6.875~$a_0$ as the optimal long-range radius which was used for all subsequent pentacene simulations. The ionization potential of 6.59~eV and the electron affinity of 1.35~eV were taken from experimental data of pentacene \\cite{coropceanu_hole-_2002,crocker_electron_1993}. A value of 6.67~eV and 1.77~eV  was obtained using the optimally tuned radius for $\\epsilon_{\\mathrm{HOMO}}$ and $\\epsilon_{\\mathrm{LUMO}}$, respectively.\n\\begin{figure}[bth!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_8.pdf}\n    \\caption{Comparison between the experimental spectrum of a pentacene film \\cite{hammer_influence_2021} and the oscillator strengths of the FMO-LC-TDDFTB\\xspace calculation of pentacene clusters of various sizes. The energies of the clusters are red-shifted by 0.7 eV.}\n    \\label{pentacene_spectrum_vs_exp}\n\\end{figure}\n\nFor each of the pentacene clusters, which contain up to 319 fragments (11484 atoms), a calculation of the singlet excited state energies was performed. The basis states were constructed using three LE states for each monomer and one CT state for each pair. Depending on the system size, up to 3000 excited states have been calculated. The comparison between the experimental spectrum of a pentacene film\\cite{hammer_influence_2021} and our results is shown in Fig, \\ref{pentacene_spectrum_vs_exp}. The calculated oscillator strengths of the different pentacene clusters have been convolved with a Lorentzian line shape function of 10 meV width. In addition, the spectra were red-shifted by 0.7 eV to compensate for the error of the absorption energy. In the literature similar shifts are used even in the case of TD-DFT.\\cite{green_excitonic_2021, hoche_excimer_2021, rohr_exciton_2018} \n\nOverall, the calculated oscillator strengths for the first two bands of the absorption spectrum are in good agreement with the experimental spectrum. As the size of the pentacene clusters increases, the intensity of the first band grows in strength, while the oscillator strength of the second band shows a slight decrease in magnitude. Comparing the third and fourth absorption bands of the spectra, the deviation in the energy and intensity are apparent. While the intensity of the third peak is overestimated by a factor of two, the magnitude of the forth band is severely underestimated by FMO-LC-TDDFTB\\xspace. However, these discrepancies may partially stem from the fact that the influence of the vibrational modes is fully neglected in the present theory, which play a significant role in the case of the pentacene clusters.\\cite{benkyi_calculation_2019,craciunescu_accurate_2022, qian_herzbergteller_2020}\n\nStudying the excited states of large molecular systems leads to the question of how many monomers in a cluster contribute to an excited state. To this end, the participation numbers of the natural transition orbitals (NTOs)\\cite{luzanov_application_1976,luzanov_interpretation_1980} of the excited states of the pentacene clusters were calculated according to the expression\n\\begin{equation}\n    \\mathrm{PR}_{\\mathrm{NTO}}=\\frac{\\left(\\sum_i \\lambda_i\\right)^2}{\\sum_i \\lambda_i^2},\n\\end{equation}\nwhere $\\lambda_i$ are singular values which were obtained by a singular value decomposition (SVD) of the transition density matrix between the electronic ground state and a singlet excited state \\cite{plasser_new_2014}. Due to the fact that the pentacene monomers in the crystal structure all share the same internal coordinates, the locally excited states of the various monomers are nearly degenerate, which results in unrealistically high participation numbers for a perfectly arranged structure. In order to model the structural disorder that is present in a real system, we optimized the monomer geometry in the ground state using LC-DFTB\\xspace and sampled coordinates of the monomers from the ground-state canonical harmonic Wigner distribution.\\cite{bonacic-koutecky_theoretical_2005}\nSubsequently, the NTO participation numbers were calculated for the singlet excited states. In order to interpret the data, which showed strongly fluctuating participation numbers of the various states, the simple moving average was calculated by convolving the data with a rectangular window function. To guarantee a consistent approach, the width of the window was chosen to be 20\\% of the complete number of data points of the respective system. The resulting participation numbers of the different pentacene clusters are shown in Fig.~\\ref{participation_numbers}. \n\n\\begin{figure}[tbh!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{pictures\/figure_9.pdf}\n    \\caption{Average participation numbers of the excited states of the Wigner-sampled pentacene clusters.}\n    \\label{participation_numbers}\n\\end{figure}\n\nAs expected, as the size of the pentacene clusters increases, the number of monomers contributing to the excited states grows. However, a convergence of the participation numbers would also be expected after a specific system size is reached. \nIn the case of the pentacene aggregates, the increase in the average participation number seems to be slowing down gradually.\n\\begin{figure}[tb!]\n    \\centering\n    \\includegraphics[scale=0.074]{pictures\/figure_10_a.png}\n    \\includegraphics[scale=0.074]{pictures\/figure_10_b.png}\n    \\caption{Particle and hole density (hole: blue, particle: red) of an excited state of the Wigner sampled pentacene cluster.\n    The state represents a high participation number of the system. The isovalue of the plot is 0.0001.\n    }\n    \\label{particle_hole_density}\n\\end{figure}\n\nIn order to visualize electron and hole delocalization the particle and hole density of a chosen pentacene system, containing 99 fragments (3564 atoms), was computed. The densities, which are shown in Fig. \\ref{particle_hole_density}, were selected due to their relative large participation number of 22.7. Both local excitations and charge-transfer states are discernible.\n\nThus, we have proven the applicability of our method to large molecular systems. The FMO-LC-TDDFTB\\xspace method is sufficient to estimate the excited state properties of molecular aggregates and can be used to analyze the composition of the excited states.\n\n\\section{Introduction}\nIn materials science, solid-state physics, biochemistry and many other branches of molecular sciences, the study of excited states is essential for understanding photochemical and photophysical processes as well as the function of molecular materials. Simulations of the excited state properties necessitate an accurate description of the electronic structure of molecular systems, which was first accomplished with ab-initio quantum chemical methods\\cite{friesner_new_1991,gonzalez_progress_2012}.\n\nHowever, investigating excited states of extended systems in the framework of the ab-initio approach requires very long calculation times due to the high computational scaling with the system size. With the development of the time-dependent density functional theory (TD-DFT), a reasonably fast method of calculating the excited state properties of relatively large molecular systems (over 100 atoms) became available \\cite{casida_time-dependent_1995,casida_progress_2012,laurent_td-dft_2013}. \nNew developments of approximative methods for the prediction of the nature of the excited states, including linear-scaling TD-DFT\\cite{wu_linear-scaling_2011}, subsystem density functional theory\\cite{jacob_subsystem_2014} and multilayer QM\/MM approaches (ONIOM)\\cite{dapprich_new_1999,vreven_combining_2006}, have further increased the applicability of TD-DFT to large polyatomic systems. However, despite the multitude of recent developments, the calculation of the excited states of molecular systems consisting of thousand atoms or more is still not possible even with TD-DFT.\n\nThe semiempirical quantum mechanical (SQM) methods, which approximate the wavefunction or density functional based methods, provide a further alternative for investigating the excited states of molecular systems. In the various SQM theories, a minimal basis set is used and the electronic integrals are approximated by the partial or complete neglect of the differential overlap between the atomic basis functions, which leads to a sizable increase in the computational efficiency  \\cite{thiel_semiempirical_2014,christensen_semiempirical_2016}. Whereas these approaches have been studied extensively in the 1970s--1990s\\cite{dewar_ground_1977,dewar_development_1985,stewart_optimization_1989,kolb_beyond_1993,thiel_extension_1996,weber_orthogonalization_2000}, the development of the self-consistent charge (SCC) density functional tight-binding (DFTB) method\\cite{porezag_construction_1995,seifert_calculations_1996,elstner_self-consistent-charge_1998} has renewed the interest in semiempirical approaches in the last 20 years. Niehaus \\textit{et al.}\\cite{niehaus_tight-binding_2001} adapted the time-dependent density functional theory to the tight-binding formalism, enabling the calculation of excited state properties of large molecular assemblies of hundreds of atoms. The introduction of long-range corrections to the DFTB framework facilitated the investigation of excited state properties of molecular systems involving charge-transfer states \\cite{niehaus_importance_2005,humeniuk_long-range_2015,lutsker_implementation_2015}. The DFTB and TD-DFTB methods have been implemented in various software packages including DFTB+\\cite{hourahine_dftb_2020}, DFTBaby\\cite{humeniuk_dftbaby_2017}, ADF\\cite{te_velde_chemistry_2001}, CP2K\\cite{hutter_cp2k_2014}, and hotbit\\cite{koskinen_density-functional_2009}, which can be employed to predict the excited state properties of polyatomic systems of over 1000 atoms. \n\nRecently, Grimme developed the simplified Tamm-Dancoff approximation\\cite{grimme_simplified_2013} (sTDA) for the calculation of excited-state spectra of large molecules, which is based on a ground state DFT calculation, followed by the semiempirical treatment of the integrals involved in the linear-response TD-DFT calculation, similar to the TD-DFTB approach. Furthermore, this method was extended to the tight-binding formalism (sTDA-xTB)\\cite{grimme_ultra-fast_2016} by the parametrization of the ground-state Hamiltonian to reduce the computational requirements of the previous models. Further developments of this approach include the GFN-xTB methods by Grimme and coworkers, which can be employed to predict the ground state properties and structures of a multitude of molecular systems \\cite{grimme_robust_2017,bannwarth_gfn2-xtbaccurate_2019,bannwarth_extended_2021}. Additional notable semiempirical methods, which can be used to predict the excited state properties of large molecular systems are MNDO\\cite{dewar_ground_1977}, AM1\\cite{dewar_development_1985}, PMx\\cite{stewart_optimization_1989} and OMx\\cite{weber_orthogonalization_2000}. \n\nThe computational efficiency and applicability of all above methods can be further enhanced by using fragmentation-based approaches allowing the simulation of large molecular systems of over a million atoms. Prominent examples include the fragment molecular orbital\\cite{kitaura_fragment_1999,nakano_fragment_2000,nakano_fragment_2002,fedorov_multilayer_2005,mochizuki_configuration_2005,tsuneyuki_molecular_2009,brorsen_fully_2011,fujino_fragment_2015,tanaka_electron-correlated_2014} (FMO) method and the divide-and-conquer\\cite{yoshikawa_novel_2013,nishizawa_three_2016,nakai_development_2017,yoshikawa_gpuaccelerated_2019,nishimura_quantum_2021} (DC) scheme. In these fragmentation approaches, a partitioning scheme divides the system into multiple fragments. The properties of the complete system are then obtained by combining the results of the different fragments. In the case of the FMO method, the energy of the fragments -- the monomers and fragment pairs -- is calculated iteratively while considering the electrostatic embedding potential (ESP) resulting from the interaction of all fragments. The number of fragment pairs is determined by the closest atomic interfragment distance. The FMO and the DC approach have previously been employed to investigate complex molecular systems, such as proteins\\cite{sawada_role_2010}, polymers\\cite{nakata_derivatives_2014} or optoelectronic materials\\cite{uratani_simulating_2020}. Furthermore, Nishimoto, Fedorov and Irle combined the FMO method with the density functional tight-binding approach (FMO-DFTB) to enable the study of even larger systems \\cite{nishimoto_density-functional_2014,nishimoto_fmo-dftb_2021}. The FMO-DFTB method has been extended to include analytical ground-state gradients\\cite{nishimoto_large-scale_2015}, the polarizable continuum model\\cite{nishimoto_fragment_2016}, periodic boundary conditions\\cite{nishimoto_fragment_2021}, the long-range correction\\cite{vuong_fragment_2019} and many other theoretical concepts\\cite{nishimoto_third-order_2015,nishimoto_three-body_2017,nishimoto_adaptive_2018,fedorov_partition_2020}. It has been succesfully employed to study the properties of large molecular systems, such as the charge transport in covalent organic frameworks \\cite{kitoh-nishioka_multiscale_2017,kitoh-nishioka_linear_2021}.\n\nWhile these fragmentation-based methods provide excellent tools to calculate the ground state properties of molecular aggregates, the study of excited states is limited to only local excitations within the fragments\\cite{komoto_development_2019,komoto_large-scale_2020,uratani_fast_2020,uratani_non-adiabatic_2020,uratani_trajectory_2021}. Recently, several approaches have been developed to investigate the excited states properties of molecular clusters using an excitonic Hamiltonian, consisting of local excitations (LEs) and charge-transfer states (CTs)\\cite{green_excitonic_2021,li_ab_2017,wen_fragmentation-based_2017}. Fujita and Mochizuki combined the multilayer FMO\\cite{fedorov_multilayer_2005} and the transition density fragment interaction\\cite{fujimoto_transition-density-fragment_2012,fujimoto_theoretical_2013} method and introduced an excitonic Hamiltonian to calculate nonlocal excitations in molecular clusters. A basis consisting of fragment configuration state functions (CSFs), which define the locally excited and charge-transfer states, is used to construct the Hamiltonian\\cite{fujita_development_2018}. \n\nIn this work, we combine the FMO-LC-DFTB method with the construction of an excitonic Hamiltonian to calculate the excited states of large molecular systems consisting of hundreds of molecules. As a basis for the description of the excited states of large molecular assemblies, we use locally excited and charge-transfer states, which are calculated for monomers and pair dimers employing the LC-TD-DFTB\\xspace approach. While the diagonal elements of the Hamiltonian are represented by the energies of the basis states, the couplings between the basis states, which represent the off-diagonal matrix elements of the Hamiltonian, are calculated by utilizing the tight-binding formalism for the two-electron four-center integrals. The excited state energies of the full system are given by the diagonalization of the excitonic Hamiltonian. \nThe accuracy and the efficiency of the proposed theory has been tested by the comparison with the full LC-TD-DFTB\\xspace method for molecular dimers and large assemblies of molecules. The effective computational scaling was determined by the calculation of large clusters of anthracene and perylene bisimide aggregates. In addition, the applicability of our approach was confirmed by the calculation of the excited state properties of pentacene crystal models. The FMO-LC-TDDFTB\\xspace method was implemented in our own software package \\mbox{DIALECT}\\cite{noauthor_dialect_2022}, which is publicly available on Github and was written in the Rust programming language. \n\nAlthough Rust is still a very young language --- with the first stable version (1.0) released in 2015 ---  its potential for scientific computing applications becomes increasingly apparent\\cite{perkel_why_2020}. The Rust programming language does not only have a strong focus on the code and memory safety, but also shows performance comparable to the C, C++ and Fortran languages. Additionally, Rust offers in our opinion a friendlier syntax and provides a strongly growing repository of development tools. Therefore, we expect Rust to gain high popularity also in the field of quantum chemistry in the coming years.\n\nThe present work is structured as follows: In Sec. \\ref{sec:methodology}, the methodological framework of our approach is outlined. Subsequently, in Sec. \\ref{sec:computational_procedure}, the computational procedure and the different steps of a FMO-LC-TDDFTB\\xspace calculation are illustrated. The application of the proposed theory to pentacene and the results of the benchmark calculations regarding the accuracy and the effective scaling of the FMO-LC-TDDFTB\\xspace method are presented in Sec. \\ref{sec:results}. Finally, the conclusions and outlook are given in Sec. \\ref{sec:conclusions}.\n\\section{Conclusions and outlook}\\label{sec:conclusions}\nIn this work, we have developed a new method to calculate the excited states in large molecular assemblies, consisting of hundreds of molecules. \nTo this end, we based our approach on the fragment orbital density-functional tight-binding method\\cite{nishimoto_density-functional_2014,vuong_fragment_2019} (FMO-DFTB) and employed an excitonic Hamiltonian constructed from locally excited and charge-transfer configuration state functions.\nOur method has been implemented in the software package DIALECT, publicly available on Github\\cite{noauthor_dialect_2022}.\n\nWe have proven that the accuracy of the proposed theory is sufficient to reproduce the excited state LC-TD-DFTB\\xspace potential energy curves of $\\pi$--stacked molecular pyrene dimers. The mean absolute errors are below 80 meV at all physically relevant intermolecular separations for the pyrene systems, showing that for distances of 2.75 \\AA \\text{ } and above, our method produces satisfactory results. The applicability of the present theory was confirmed by the calculation of the excited state energies and oscillator strengths of a large anthracene cluster. The spectrum of the FMO-LC-TDDFTB\\xspace approach was in good agreement with the LC-TD-DFTB\\xspace reference. \n\nThe comparison of the wall times of the LC-TD-DFTB\\xspace and FMO-LC-TDDFTB\\xspace calculation showed excellent speedup factors for different anthracene cluster sizes of up to $1.5\\times 10^4$. In addition, the computational effective scaling of the proposed theory was evaluated for molecular aggregates involving differently sized monomers. The results showed that in the case of small monomers, the scaling is highly dependent on the calculation of the matrix elements of the Hamiltonian. However, if larger monomers like the substituted perylene bisimide are to be considered, the calculation of the matrix elements of the Hamiltonian only becomes the dominant factor of the effective scaling after the systems grows to a size of over 15000 atoms. The effective scaling of the anthracene and PBI systems was determined as $\\mathcal{O}(N^{3.4})$ and $\\mathcal{O}(N^{2.2})$, respectively.\n\nAt last, the application of our method to pentacene crystals confirmed its validity to large molecular systems. The calculated spectra of the pentacene clusters showed reasonable agreement to the experimental reference spectrum. We were able to analyze the number of monomers that contribute to the excited states of the system by calculating the NTO participation numbers, which were verified by the particle and hole density of a chosen pentacene cluster.\n\nIn the future, we plan to extend our approach to quantum-classical dynamics simulations of large molecular systems, like organic semiconductors or optoelectronic materials. To this end, we will implement the analytical gradients of the excited states to enable the investigation of exciton dynamics and charge transport simulations. Concerning the efficiency, graph theory methods will be used to accelerate the diagonalization of the Hamiltonian by partitioning the matrices involved in the Davidson algorithm into separate blocks, and thus reduce the dimensionality of the problem. In addition, we intend to apply the parametrization of the DFTB framework to a wider range of elements to facilitate the investigation of different molecular systems.\n\\section*{Conflicts of interest}\n\\vspace*{-2ex}\nThere are no conflicts to declare.\n\\section*{Acknowledgments}\n\\vspace*{-2ex}\nWe gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft via the grants MI1236\/6-1 and MI1236\/7-1.\n\\section*{Data availability}\n\\vspace*{-2ex}\nThe data that support the findings of this study are available on Github under \\url{https:\/\/github.com\/mitric-lab\/Data_for_FMO-LC-TDDFTB}.\n\\section*{References}\n\\vspace*{-2ex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOne of the important subjects in elementary particle physics is to understand the basic properties of hadronic matter described by Quantum Chromodynamics (QCD).\nThe phase diagram and equation of state of QCD have been investigated both theoretically and experimentally.\nA number of studies indicate that QCD at finite temperature and finite density has various phases such as the quark-gluon plasma phase with effective chiral symmetry and the color superconductivity phase.\n\nInvestigating QCD vacuum need non-perturbative method due to strong interactions of QCD.\nVarious non-perturbative methods are employed in order to study the phase structure of QCD.\nLattice gauge simulation is very powerful because it is the first principle calculation respecting the gauge symmetry.\nHowever it is difficult to maintain the chiral symmetry on the lattice.\nThe even more difficult problem which is known as the sign problem occurs at finite chemical potential because the Dirac operator becomes complex. \n\nOther non-perturbative methods, such as the mean-field approximation (MFA), the Schwinger-Dyson equations (SDE) or the large-$N$ expansion, have been applied to QCD.\nThese methods do not have the difficulty of maintaining the chiral symmetry nor the sign problem.\nHowever, their approximations break the gauge symmetry and its systematic improvement is not easy.\n\nIn this paper, we study the dynamical chiral symmetry breaking (D$\\chi$SB) by using the functional renormalization group (FRG), another non-perturbative method~\\cite{Aoki:2000wm,Pawlowski:2005xe,Gies:2006wv,Wegner:1972ih,Polchinski:1983gv,Wetterich:1992yh}.\nThe main idea of the FRG is solving the effective action $\\Gamma_\\Lambda$ generated by the integration of fluctuations with higher momentum modes $\\Lambda < |p| <\\Lambda _0$ where $\\Lambda$ is the infrared (IR) cut-off scale and the bare action is defined at the initial scale $\\Lambda_0$.\nThe evolution of the effective action with decreasing the IR cut-off scale is described by the functional differential equation, so-called the Wetterich equation~\\cite{Wetterich:1992yh,Morris:1993qb},\n\\begin{align}\n\\partial_t \\Gamma_\\Lambda =\\frac{1}{2} {\\rm STr} \n\\left\\{ \\left[ \\frac{\\overrightarrow \\delta}{\\delta \\Phi} \\Gamma_\\Lambda \\frac{\\overleftarrow \\delta}{\\delta \\Phi} +R_\\Lambda \\right] ^{-1}\\cdot (\\partial _tR_\\Lambda) \\right\\}.\n\\end{align}\nThis equation itself is exact. However, we cannot solve it exactly.\nTherefore we solve the effective action in the truncated theory space.\nThe improvement of approximation can be done by enlarging the theory space, which are easier and more systematic than the case of the MFA or the SDE.\n\nThe D$\\chi$SB has been investigated by using low energy effective models such as the Nambu--Jona-Lasinio (NJL) model~\\cite{Nambu:1961tp,Nambu:1961fr,Hatsuda:1994pi,Kodama:1999if,Shimizube:2003bv,Braun:2011pp,Braun:2011fw,Braun:2012zq} which is described by the four-Fermi interactions. \nRe-bosonization methods~\\cite{Aoki:1999dw,Gies:2001nw,Gies:2002hq} are often applied in order to obtain the physical quantities in macro scale from the interactions in micro scale via the FRG.\nIn such analyses, however, the theory space is widely expanded and the analysis becomes highly complicated.\nRecently, a new method called the ``weak solution'' is proposed in order to evaluate the macro physical values without re-bosonization.\nSo far, the weak solution method can be applied to a specific type of the approximated FRG equation corresponding to the Fermionic large-$N$ leading approximation.\n\nIn this paper we analyze the NJL model at finite temperature and finite density without re-bosonization method nor weak solution method.\nOur theory space and quantum corrections included can not be treated by the weak solution method up to the present.\nWe compare the behaviors of the RG flows of the four-Fermi coupling constant between the large-$N$ leading approximation and that beyond it.\nWe actually evaluate the RG flows of the inverse four-Fermi coupling constant although its physical and mathematical ground is not completely clear.\nIn this manner we define macro physics beyond the MFA in the NJL model.\nHowever our method does not allows us to evaluate a critical endpoint in the phase diagram. \nOur work aims to motivate future studies of phase diagrams including the first order phase transition.\n\nThis paper is organized as follows: In section \\ref{section2} we briefly explain the NJL model in view of FRG and our analysis method.\nThe RG equations of the four Fermi coupling constant are numerically analyzed at finite temperature and density in section \\ref{section3}.\nSummary and Discussion are given in section \\ref{section5}.\n\\section{Nambu--Jona-Lasinio model in view of FRG}\\label{section2}\nIn this paper, we study the NJL model as a low energy effective model of QCD. \nThe NJL Lagrangian is given by\n\\begin{align} \n{\\mathcal L}_{\\rm NJL}={\\bar \\psi}i{\\Slash \\partial}\\psi \n\t\t\t\t\t\t+\\frac{G}{2N}\\{ ({\\bar \\psi}\\psi)^2 + ({\\bar \\psi}i\\gamma_5\\psi)^2\\} ,\n\\end{align}\nwhere $G$ is the four-Fermi coupling constant and $N$ is the number of degrees of freedom for the Dirac Fermion ${\\bar \\psi}$, $\\psi$.\nThe four-Fermi interactions are generated by the QCD gauge interactions.\nThis Lagrangian is invariant under the chiral U(1) transformation:~$\\psi \\to e^{i\\gamma^5\\theta}\\psi$.\nIn present paper, we analyze the ${\\rm U}(1)_L \\times {\\rm U}(1)_R$ symmetric system to study basic structures and behaivors of the four-Fermi interaction at finite temperature and density.\nThe four-Fermi coupling constant in the effective action corresponds to the chiral fluctuation: $G \\sim \\langle ({\\bar \\psi}\\psi)^2\\rangle $.\nTherefore, the divergence of the four-Fermi coupling constant in the course of RG flow means the signal of the D$\\chi$SB as the second order phase transition. \n\nThe RG flow equation of the four-Fermi coupling constant for the $4d$ sharp cut-off scheme in the large-$N$ leading approximation is given by\n\\begin{align}\n\\partial _t g=-2g+2g^2\n\\end{align}\nwhere $t$ is the dimensionless RG scale defined by $t=\\log (\\Lambda_0\/\\Lambda)$, and $g=G\\Lambda^2\/4\\pi^2$ is the dimensionless four-Fermi coupling constant~\\cite{Aoki:1999dv}. \nThis RG equation has an ultraviolet fixed point $g^\\ast =1$. \nWhen we solve this equation with an initial value $g_0>1$, the RG flow diverges at the following finite scale,\n\\begin{align}\nt_{\\rm c}=\\frac{1}{2}\\log \\left( \\frac{g_0}{g_0-1} \\right) .\n\\end{align} \nWe cannot solve the RG flow after $t_{\\rm c}$ because of this divergence. \nRecently, the ``weak solution'' method~\\cite{Aoki:2014ola} is introduced in order to define the flows after this divergence of $g$. \nIn this section, we propose another analysis method to evaluate the RG flow of $g$. \\\\\n\\indent We define the RG flow equation of the inverse four-Fermi coupling constant, ${\\tilde g}=1\/g$, and we have the RG flow equation of ${\\tilde g}$ as follows,\n\\begin{align}\n\\label{rgeqgs}\n\\partial _t {\\tilde g}=2{\\tilde g}-2.\n\\end{align}\nThe solution of this equation is \n\\begin{align}\n{\\tilde g}(t)=({\\tilde g}_0-1)e^{2t}+1.\n\\end{align} \nThe flow with an initial value ${\\tilde g}_0<1$ reaches zero at $t_{\\rm c}$, ${\\tilde g}(t_{\\rm c})=0$, which corresponds to the divergence of $g$.\nIn order to interpret the RG flow with negative ${\\tilde g}$ values, the Euclidean NJL action is bosonized,\n\\begin{align}\nS_{\\rm NJL}=\\int d^4x \\left[\n{\\bar \\psi}({\\Slash \\partial} +\\gamma_0\\mu)\\psi -\\frac{h^2}{G}(\\sigma ^2+\\pi^2)-h{\\bar \\psi}(\\sigma+i\\gamma _5\\pi)\\psi\n\\right],\n\\end{align}\nwhere $h$ is the Yukawa coupling constant and, $\\sigma$ and $\\pi$ are auxiliary fields.\nThe inverse four-Fermi coupling constant corresponds to the mass of the mesonic potential:\n\\begin{align}\\label{mandg}\nm^2=\\frac{h^2}{G}\n\\end{align}\nTherefore the negative RG flow of ${\\tilde g}$ might indicate that the curvature of the mesonic potential at the origin becomes negative and the potential takes a double-well shape, thus, D$\\chi$SB develops.\nSuch a crude treatment has turned out to be right in case when the weak solution can be defined~\\cite{Aoki:2014ola}.  \nThis simple analysis method follows the RG flow after D$\\chi$SB and reach the IR limit $\\Lambda \\to 0$. \nIn the next section, this method is applied to the NJL model at finite temperature and finite density. \n\n\\section{The RG flow at finite temperature and density}\\label{section3}\n\\subsection{RG flow equations of the four-Fermi coupling constant}\n\nWe analyze the Euclidean effective action in the local potential approximation~\\cite{Hasenfratz:1985dm},\n\\begin{align}\\label{eaa}\n\\Gamma_\\Lambda[\\psi,{\\bar \\psi}]=\n\\int _0^\\beta d \\tau\n\\int d^3x\n\t\t\t\t\t\t\t\\left[{\\bar \\psi}({\\Slash \\partial}+\\mu)\\psi \n\t\t\t\t\t\t-\\frac{G}{2N}\\{ ({\\bar \\psi}\\psi)^2 + ({\\bar \\psi}i\\gamma_5\\psi)^2\\} \\right],\n\\end{align}\nwhere $\\beta$ is the inverse temperature $1\/T$ and $\\mu$ is the chemical potential.\nThe evolution of $\\Gamma_\\Lambda$ is described by the Wetterich equation. In this case it reads,\n\\begin{align}\n\\partial _t \\Gamma_\\Lambda[\\psi,{\\bar \\psi}]\n=-{\\rm Tr}\\left[ \\frac{\\partial _tR_\\Lambda}{\\Gamma^{(1,1)}_\\Lambda+R_\\Lambda} \\right],\n\\end{align}\nwhere $\\partial _t=-\\Lambda \\frac{\\partial}{\\partial \\Lambda}$ and $R_\\Lambda$ is the cut-off profile function for the momentum of Fermion defined in Appendix~\\ref{thresholdapp}.\nWe introduce the simple notation,\n\\begin{align}\n\\Gamma_\\Lambda^{(1,1)}=\\frac{\\overrightarrow \\delta}{\\delta {\\Psi}^{\\rm T}(-p)}\\Gamma_\\Lambda \\frac{\\overleftarrow \\delta}{\\delta \\Psi(p)},\n\\end{align}\nwith $\\Psi^{\\rm T}(-p):=(\\psi^{\\rm T}(-p),{\\bar \\psi}(p))$ and\n$$\n\\Psi(p):=\n\\begin{pmatrix}\n\\psi(p)\\\\\n{\\bar \\psi}^{\\rm T}(-p)\n\\end{pmatrix}.\n$$\nThe ``Tr'' denotes sum over momenta and internal indices.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=150mm]{4fermi.eps}\n\\end{center}\n\\caption{The quantum corrections to the four-Fermi interactions. The diagram surrounded by the dashed line is the large-$N$ leading term. The arrow denote the direction of particle number flow.}\n\\label{fourFermi}\n\\end{figure}\n\nLet us briefly explain how to introduce the RG flow equation of the four-Fermi coupling constant.\nThe modified inverse propagator $\\Gamma^{(1,1)}_\\Lambda+R_\\Lambda$ can be divided into the field-independent part and the field-dependent part,\n\\begin{align}\n\\Gamma^{(1,1)}_\\Lambda+R_\\Lambda ={\\mathcal S}_\\Lambda+{\\mathcal V}_\\Lambda[\\Psi],\n\\end{align}\nwith\n\\begin{align}\n{\\mathcal S}_\\Lambda :=\n\\begin{pmatrix}\n0\t&\t-i{\\Slash p}^{-}{}^ {\\rm T}\\\\\n-i{\\Slash p}^{+}\t&\t0\n\\end{pmatrix},\n\\end{align}\nand\n\\begin{align}\n{\\mathcal V}_\\Lambda \n&:=\\frac{\\overrightarrow \\delta}{\\delta {\\Psi}^{\\rm T}}\n\\left[\n-\\frac{G}{2N}\\{ ({\\bar \\psi}\\psi)^2 + ({\\bar \\psi}i\\gamma_5\\psi)^2\\} \n\\right]\n\\frac{\\overleftarrow \\delta}{\\delta \\Psi} \\nonumber \\\\\n&=\\frac{G}{N}\n\\begin{pmatrix}\n{\\mathcal C}\t&\t {\\mathcal A}^{\\rm T} + {\\mathcal B}^{\\rm T} \\\\\n{\\mathcal A} + {\\mathcal B}\t&\t{\\mathcal D}\n\\end{pmatrix},\n\\end{align}\nwhere we employ the following notations\n\\begin{align}\n{\\Slash p}^{+}& = {\\Slash p}+i\\mu\\gamma_0\t+{\\bvec {\\Slash p}}r_\\Lambda({\\bvec p})\n=\\left( p_0 +i\\mu ,{\\bvec p}(1+r_\\Lambda({\\bvec p})) \\right)_\\mu \\gamma_\\mu, \\\\\n{\\Slash p}^{-}& = {\\Slash p}-i\\mu \\gamma_0 + {\\bvec {\\Slash p}}r_\\Lambda({\\bvec p})\n=\\left( p_0 -i\\mu ,{\\bvec p}(1+r_\\Lambda({\\bvec p})) \\right)_\\mu \\gamma_\\mu,  \\nonumber \\\\\n{\\mathcal A}&= - \\{ \\delta_{ij} ({\\bar \\psi}\\psi) + i\\gamma^5_{ij}({\\bar \\psi}i\\gamma_5\\psi)\\} ,\\\\\n{\\mathcal B}&= -\\{ \\psi_i  {\\bar \\psi}_j + (i\\gamma_5\\psi)_i ({\\bar \\psi}i\\gamma^5)_j \\} , \\\\\n{\\mathcal C}&=  \\{ \\bar \\psi_i  {\\bar \\psi}_j +(\\bar \\psi i\\gamma_5)_i ({\\bar \\psi}i\\gamma^5)_j \\} ,\\\\\n{\\mathcal D}&=  \\{ \\psi_i  \\psi _j + (i\\gamma_5 \\psi )_i ( i\\gamma^5 \\psi)_j \\}.\n\\end{align}\nIn order to obtain the RG flow equation, we power expand the Wetterich equation with respect to ${\\mathcal V}_\\Lambda[\\Psi]$ as follows,\n\\begin{align}\n\\nonumber\n\\partial_t\\Gamma_\\Lambda\n&=-{\\rm Tr}[{\\tilde \\partial}_t~{\\rm ln}({\\mathcal S}_\\Lambda+{\\mathcal V}_\\Lambda[\\Psi])]\\\\\n\\label{modwet}\n&=-{\\rm Tr}[{\\tilde \\partial}_t~{\\rm ln}({\\mathcal S}_\\Lambda)]\n+{\\rm Tr}[{\\tilde \\partial}_t({\\mathcal S}_\\Lambda^{-1}{\\mathcal V}_\\Lambda[\\Psi])]\n-\\frac{1}{2}{\\rm Tr}[{\\tilde \\partial}_t({\\mathcal S}_\\Lambda^{-1}{\\mathcal V}_\\Lambda[\\Psi])^2]+\\cdots ,\n\\end{align}\nwhere ${\\tilde \\partial}_t$ acts only on the $\\Lambda$ dependence of function $R_\\Lambda$.\nWe evaluate the third term as follow:\n\\begin{align}\n-\\frac{1}{2}{\\rm Tr}[{\\tilde \\partial}_t({\\mathcal S}_\\Lambda^{-1}{\\mathcal V}_\\Lambda[\\Psi])^2]\n=\\frac{1}{2}\\int \\frac{d^4p}{(2\\pi)^4} {\\tilde \\partial}_t\\left[\\frac{1}{(p^+)^2} + \\frac{1}{(p^-)^2} \\right] \\frac{G^2}{N^2}\n\\left({\\rm tr}[\\gamma_\\mu {\\mathcal A}\\gamma_\\mu {\\mathcal A}]\\right) + \\cdots,\n\\end{align}\nwhere ${\\rm tr}[\\gamma_\\mu {\\mathcal A}\\gamma_\\mu {\\mathcal A}] = ({\\rm tr}[{\\bvec 1}_{\\rm spinor}]\\times  {\\rm tr}[{\\bvec 1}_{\\rm flavor}]) {\\mathcal O}_{\\rm 4Fermi}  =4N\\{ ({\\bar \\psi}\\psi)^2 + ({\\bar \\psi}i\\gamma_5\\psi)^2\\} $.\nThis term yields factor $N$, thus it is the large-$N$ leading term.\nOther terms including the operators ${\\mathcal B}$, ${\\mathcal C}$ and ${\\mathcal D}$ do not yield it.\nFor example, \n\\begin{align}\\label{example4}\n{\\rm tr}[\\gamma_\\mu\n {\\mathcal D}\\gamma_\\mu^{\\rm T} {\\mathcal C}]\n&={\\rm tr}[ (\\gamma_\\mu)_{ij} \n\\{ \\psi_j \\psi _k + (i\\gamma_5 \\psi )_j ( i\\gamma^5 \\psi)_k \\} \n(\\gamma_\\mu ^{\\rm T})_{kl}\n\\{ \\bar \\psi_l  {\\bar \\psi}_m +(\\bar \\psi i\\gamma_5)_l ({\\bar \\psi}i\\gamma^5)_m \\}\n] \\nonumber\\\\\n&=2\\{ ({\\bar \\psi}\\gamma_\\mu \\psi)^2 - ({\\bar \\psi}\\gamma_5\\gamma_\\mu\\psi)^2 \\}.\n\\end{align}\nUsing the Fierz transformation, we obtain the opertator $({\\bar \\psi}\\psi)^2 + ({\\bar \\psi}i\\gamma_5\\psi)^2$.\nAlthough the nonleading terms yields not only the scalar-type operator but also other four-Fermi operators such as a vector-type operator $({\\bar \\psi}\\gamma_\\mu \\psi)^2 + ({\\bar \\psi}\\gamma_5\\gamma_\\mu\\psi)^2$, these terms are dropped in our truncated theory space. \nThe possible ways of contraction between the operators are exhibited by the Feynman diagrams as shown in Fig.~\\ref{fourFermi}.\nIn particular , the first diagram in the dashed box is the large-$N$ leading term.\nThe large-$N$ leading approximation neglects other diagrams than the first one.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{NJLladders.eps}\n\\includegraphics[width=80mm]{NJLnonladders.eps}\n\\end{center}\n\\caption{The threshold functions $I_0$ (left) and $I_1$ (right) on ${\\tilde T}-{\\tilde \\mu}$ plane. }\n\\label{threshold}\n\\end{figure}\n\n\\indent We obtain the RG flow equation of the dimensionless four-Fermi coupling constant, \n\\begin{align}\\label{njlfinitetmuln}\n\\partial _t g&=-2g+\\frac{1}{3}\\{ 4I_0({\\tilde T},{\\tilde \\mu})-\\frac{1}{N}I_1({\\tilde T},{\\tilde \\mu})\\} g^2,\n\\end{align}\nwhere the rescaled temperature ${\\tilde T}=T\/\\Lambda$ and rescaled chemical potential ${\\tilde \\mu}=\\mu\/\\Lambda$, obeys the following equations respectively, \n\\begin{align}\n\\partial _t {\\tilde T}&={\\tilde T},\\\\\n\\partial _t {\\tilde \\mu}&={\\tilde \\mu}.\n\\end{align}\nThe ``form factor'' $I_0$ and $I_1$ are the threshold functions of ${\\tilde T}$ and ${\\tilde \\mu}$, which are given by the shell mode momentum integration of the large-$N$ leading term and the third diagram in Fig.~\\ref{fourFermi}, respectively, and they are defined in Appendix~\\ref{thresholdapp}.\nThe second and fourth diagrams in Fig.~\\ref{fourFermi} do not contribute to the RG flow equation of $g$ in the truncated effective action~(\\ref{eaa}).\nWe adopted the $3d$ optimized cut-off function~\\cite{Litim:2001up,Litim:2006ag} as the regulator function $R_\\Lambda$ which makes it easy to integrate in momentum space with the Matsubara summation~(see Appendix.~\\ref{thresholdapp}).\nThe large-$N$ leading approximated equation can be obtained by taking the limit $N\\to \\infty$, which makes $I_1$ term vanish. \n\nLet us now discuss the RG flows at finite temperature and density in a qualitative manner. \nFirst, we discuss the fixed point structure of the RG equation.\nThe non-trivial fixed point is given by\n\\begin{align}\ng^\\ast=\\frac{6}{4I_0({\\tilde T},{\\tilde \\mu})-\\displaystyle \\frac{1}{N}I_1({\\tilde T},{\\tilde \\mu})},\n\\end{align}\nhence, the fixed point moves with the RG evolutions of dimensionless temperature and density.\nWhen the threshold functions at finite temperature and density are smaller than these with the vanishing temperature and density, we obtain $g^\\ast_{T\\neq 0, \\mu\\neq 0}>g^\\ast_{T=0,\\mu=0}$.\nThus, the chiral symmetry tends to be restored by thermal and finite density effects.\n\nSecond, we briefly discuss the effects of the large-$N$ nonleading term at vanishing temperature and density. By taking the limit $T\\to 0$ and $\\mu\\to 0$, we obtain\n\\begin{align}\n\\partial_t g=-2g+\\frac{1}{3}\\left( 4-\\frac{1}{N}\\right) g^2,\n\\end{align}\nwhere the factor $4$ in the leading term is caused by the trace of the spinor space. \nNote that the difference of coefficients, $\\frac{4}{3}g^2$ here compared with $2g^2$ in Eq.~(\\ref{rgeqgs}), comes out of the difference of cut-off schemes.\nThe nonleading term has negative sign, and it suppresses D$\\chi$SB. \n\nFinally, we discuss finite temperature and density effects.\nWe show the threshold functions on ${\\tilde T}-{\\tilde \\mu}$ plane in Fig.~\\ref{threshold}.\nThe solutions of the RG equations of ${\\tilde T}$ and ${\\tilde \\mu}$ become simply the exponentially-growing solutions with increasing $t$, thus, the ratio ${\\tilde \\mu}\/{\\tilde T}$ is a scale independent constant, $\\mu\/T$.\nTherefore the RG flows of ${\\tilde T}$ and ${\\tilde \\mu}$ move on the straight line with the slope $\\mu\/T$ on ${\\tilde T}-{\\tilde \\mu}$ plane.\nNote that the threshold functions $I_0$ and $I_1$ with the vanishing temperature have the singular point at the Fermi surface $\\Lambda=\\mu$:\n\\begin{align}\nI_0(0,{\\tilde \\mu})&=1-\\theta({\\tilde \\mu}-1)-\\delta ({\\tilde \\mu}-1),\\\\\nI_1(0,{\\tilde \\mu})&=\\frac{1}{2(1+{\\tilde \\mu})^2}+\\frac{1}{(1-{\\tilde \\mu})^2}\\left( \\frac{1}{2}-\\theta({\\tilde \\mu}-1)\\right) +\\frac{1}{1-{\\tilde \\mu}}\\delta ({\\tilde \\mu}-1),\n\\end{align}\nwhere $\\theta({\\tilde \\mu}-1)$ is the step function.\nThe function $I_0$ at finite dimensionless chemical potential and the vanishing temperature is constant and positive value.\nIn particular, this function vanishes in the region $\\Lambda \\le \\mu$.\nOn the other hand, the function $I_1$ depends on ${\\tilde \\mu}$ in arbitrary scale. \nFurthermore, this function becomes quite large at the Fermi surface, therefore, the large-$N$ nonleading term is stronger than the leading term.\nThis means that the RG flow of the four Fermi coupling constant with the large-$N$ nonleading term tends to go to positive infinity at low temperature and high density.\nIn the next section, we numerically evaluate the RG flow equations.\n\n\\subsection{Numerical analysis}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=160mm]{flow.eps}\n\\end{center}\n\\caption{The RG flows in case of the large-$N$ leading at ${\\tilde \\mu}=0.4$ (left) and the large-$N$ nonleading with $N=1$ at ${\\tilde \\mu}=0.22$ (right). }\n\\label{flows}\n\\end{figure}\nWe solve the RG flow equations of the inverse four-Fermi coupling constant ${\\tilde g}$. \nThe numerical results of the RG flows on ${\\tilde g}-{\\tilde \\mu}$ plane are shown in Fig.~\\ref{flows}. \nThe left-hand side panel and the right-hand side panel are the large-$N$ leading approximation case and the nonleading extended case, respectively.\nHere, we used the initial value: ${\\tilde g}=0.3$ where the RG flow at the vanishing temperature and density goes to the negative region in the IR limit. \nFirst, let us discuss the large-$N$ leading case.\nThe RG flows of $\\tilde g$ at higher temperature cases ($T=0.2 \\Lambda_0$ and $T=0.15\\Lambda_0$) go to positive infinity. \nThe flow at $T=0.1\\Lambda_0$ goes to negative region crossing the origin ${\\tilde g}=0$, and comes back to positive region. \nEventually, this flow goes to positive infinity at the IR scale.\nThe RG flows at lower temperature cases ($T=0.01\\Lambda_0$ and $T=0.001\\Lambda_0$) go to negative region at the IR limit.\n\nThese flows can be evaluated only by the RG equation for the inverse four-Fermi coupling constant.\nIf we solve the RG equation for the four-Fermi coupling constant Eq.~(\\ref{njlfinitetmuln}), these flows stop in the middle of the RG scale.\nSolving the RG equations of the inverse four-Fermi coupling constant is useful to discuss the flows after D$\\chi$SB, although it is not strictly authorized for large-$N$ nonleading case.\nIt should be noted that to discuss RG flows towards positive infinity at IR scale do not always imply the restoration of the chiral symmetry.\nThe RG flow of the inverse four-Fermi coupling constant corresponds to the curvature at origin of the effective potential with respect to the expectation value $\\sigma \\sim \\langle {\\bar \\psi}\\psi \\rangle$.\nIn case of the first order phase transition, the curvature at origin may remain positive, even after the D$\\chi$SB. \nWe cannot distinguish between the restoration of chiral symmetry and the first order D$\\chi$SB.\nIn other words, we see the effective potential only at the neighborhood of origin.\nThe chiral phase transition including the first order phase transition should be investigated by searching the global minima of the effective potential by using the weak solution method or the re-bosonization method. \nAlthough our method can be applied only to the second order phase transition, our results may be used as references for advanced analysis of the chiral phase transition.\n\nSecond, let us discuss the nonleading extended case. \nThe RG flows at low temperature~($T=0.01\\Lambda_0$ and $T=0.001\\Lambda_0$) go to positive infinity at the IR limit.\nThese behaviors are quite different from the large-$N$ leading case.\nThe large-$N$ nonleading term becomes larger than the leading term because the threshold function $I_1$ has the singularity at $\\mu=\\Lambda$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=130mm]{phase.eps}\n\\end{center}\n\\caption{The chiral phase structure given by the large-$N$ leading case (blue and green line) and nonleading case (red and pink line). We used ${\\tilde g}_0=0.3$ as the initial value and $N=1$ for the nonleading case. \nThe phase boundaries assume the second order phase transition.}\n\\label{phase}\n\\end{figure}\nThe chiral phase diagram on ${\\mu\/\\Lambda_0}-T\/\\Lambda_0$ plane is shown in Fig.~\\ref{phase}.\nWe show the phase boundaries evaluated by the blowup solution of Eq.~\\eqref{njlfinitetmuln} and the RG equation of the inverse four-Fermi coupling constant.\nHere we call the solution of the RG equation for the inverse four-Fermi coupling constant simply the inverse solution.\n\nFirst, we discuss the difference of phase boundaries between the blowup solution case and the inverse solution case.\nSince the blowup solution stops the RG flow in the midst of scale, we cannot evaluate it after the chiral symmetry breaking.\nThen we notice that the phase boundary with the inverse solution is correct.\nIn high temeperature and low density region, both boundaries become same.\nBy contrast, in low temperature and high density region, the phase boundaris evaluated by the blowup solutions become larger than the inverse solutions.\nThat is, it is especially important to evaluate the RG equation of the inverse four-Fermi coupling constant in this region.\n\nNext, we discuss the difference of phase boundaries with the inverse solution between the large-$N$ leading case and nonleading case.\nThe D$\\chi$SB region becomes smaller in case of the large-$N$ nonleading. \nWe find drastic difference of phase boundaries at low temperature and high density region since the large-$N$ nonleading effect strongly suppresses the D$\\chi$SB as we previously mentioned.\nThe chiral susceptibility, the curvature at the origin of mesonic potential $m^2\\sim 1\/\\langle ({\\bar \\psi}\\psi)^2 \\rangle$, is sensitive to the nonleading effects at low temperature and high density.\n\nNote that the phase boundaries shown in Fig.~\\ref{phase} denote second order phase transition. \nAs mentioned below Eq.~\\eqref{mandg}, the inverse four-Fermi coupling constant corresponds to the curvature of the effective potential at the origin and its RG equation describes the change of the curvature.\nIn the present work, the phase boundary is determined by the sign of the inverse four-Fermi coupling constant in IR limit.\nThe broken phase potential with double well form has the negative curvature of the effective potential at the origin.\nThe symmetric phase has a single well form and the positive curvature of the effective potential at the origin.\nThus, the change from the negative curvature to the positive curvature mean the second order phase transition at least near the origin.\nOne of important statements in our study is that these pictures can be described by the RG flow of the inverse four-Fermi coupling constant.\n\nOn the other hand, when the first order phase transition occurs, there might be no global minimum at the origin of the effective potential even in case that the curvature of the effective potential at the origin changes to be positive.\nTherefore, it does not always mean we evaluate the global minimum of the effective potential in the present work.\nThat is, there is possibility of occurring the first order phase transition at larger density region than the second order boundary evaluated in present paper.\n\n\n\n\n\\section{summary and discussion}\\label{section5}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{yukawa.eps}\n\\end{center}\n\\caption{The quantum corrections to the four-Fermi interactions generated by the Yukawa interaction. The crossed ladder diagram (right) at the zero temperature and finite density has the singularity at the Fermi surface.}\n\\label{yukawa}\n\\end{figure}\nIn this work, we have studied the RG flows of the four-Fermi coupling constant in NJL model at finite temperature and density by using the FRG. \nWe solved the RG equations of the inverse four-Fermi coupling constant.\nThis manner allows to integrate the equation up to the IR scale without the difficulties of the divergence.\nWe have investigated the impact of the large-$N$ nonleading term on the RG flow equation. \nThe large-$N$ nonleading term has the singularity at the Fermi surface scale $\\Lambda=\\mu$ with the vanishing temperature, i.e. the quantum fluctuation due to the large-$N$ nonleading term becomes quite large, which makes the system more symmetric, as seen in Fig.~\\ref{phase} \n\nIn this analysis, the second order phase transition is investigated. \nIn order to investigate the first order phase transition with the critical end point in the NJL model, the global minima of the effective potential with respect to the chiral order parameter must be studied by using more technical methods such as the re-bosonization method or the weak solution method.\nCurrently, since the weak solution method can be applied to only the Fermionic large-$N$ approximated case,  we need to employ the re-bosonization method.\nFor example, a truncated effective action of the NJL model with bosonization is given by\n\\begin{align}\\nonumber\n\\Gamma _{\\Lambda}[\\psi, {\\bar \\psi}, \\phi]=\\int^{1\/T}_0d\\tau \\int d^3x \n& \\left\\{ Z_{\\psi,\\Lambda}{\\bar \\psi}({\\Slash \\partial} + \\mu \\gamma _0)\\psi \n+{\\bar h_{\\Lambda}}{\\bar \\psi}(\\sigma +i{\\vec \\tau}\\cdot {\\vec \\pi}\\gamma _5)\\psi \n\\right. \\\\\n\\label{bosoef}\n&+\\left. \\frac{G_\\Lambda}{2}[({\\bar \\psi}\\psi)^2+({\\bar \\psi}i\\tau^i\\gamma_5 \\psi)^2]\n+\\frac{Z_{\\phi,\\Lambda}}{2}(\\partial _{\\mu}\\phi^i)^2\n+U_{\\Lambda}(\\phi^2)\\right\\},\n\\end{align}\nwhere $\\phi^i=(\\sigma,{\\bvec \\pi})$.\nThe four-Fermi interaction is generated by the Yukawa interaction in this action through the diagrams shown in Fig.~\\ref{yukawa}.\nThe contribution from the crossed ladder diagram at finite density has the singularity at the Fermi surface.\nThe impact of this singularity to the chiral phase diagram on $T-\\mu$ plane should be investigated.\nThe paper~\\cite{Braun:2014ata,Mitter:2014wpa} which study QCD at vanishing temperature and density by using re-bosonization method indicates that the Yukawa coupling constant always gets the same value in the IR scale, i.e. the Yukawa coupling constant at the IR scale does not depend on the initial values. However, the impact of the singularity to the phase boundary is non-trivial.\nThe analysis of the effective action (\\ref{bosoef}) will be presented elsewhere.\n\nOn the other hand, the weak solution method is also powerful to analyze the dynamical chiral symmetry breaking~\\cite{Aoki:2014ola}.\nThis method is just developing to apply to many systems.\nTherefore, the present work becomes one of valuable benchmarks.\nWe are going to clearly discuss the relation between our results obtained in this paper and the phase diagram given by the weak solution method in future works.\n\n\n\\section*{Acknowledgements}\nWe thank Kenji Fukushima, Akira Onishi, Jan. M. Pawlowski, Fabian Rannecke, Daisuke Sato and Motoi Tachibana for discussions and valuable comments.\nM. Y. thanks the Yukawa Institute for Theoretical physics, Kyoto University.\nDiscussions during the YITP workshop YITP-T-13-05 on ``New Frontiers in QCD'' were useful to complete this work. \nM. Y. is supported by a Grant-in-Aid for JSPS Fellows (No.~25-5332). \nK-I. A. is supported by JSPS Grant-in-Aid for Challenging Exploratory Research (No.~25610103).\n\n\\begin{appendix}\n\\section{Threshold functions}\\label{thresholdapp}\nIn this appendix, we show the threshold functions which are obtained by the shell mode integration.\nWe first introduce the cut-off profile function $R_\\Lambda(p)$.\nIn case of finite temperature and density, we employ the $3d$ optimized cut-off function~\\cite{Litim:2001up,Litim:2006ag},\n\\begin{align}\\nonumber\nR_\\Lambda({\\bvec p})\n\t&=i{\\Slash {\\bvec p}}\\left( \\frac{\\Lambda}{|{\\bvec p}|} -1\\right) \\theta (1-|{\\bvec p}|\/\\Lambda)\\\\\n\t&=:i{\\Slash {\\bvec p}}~r_\\Lambda({\\bvec p}),\n\\end{align}\nwhere $\\theta (1-|{\\bvec p}|\/\\Lambda)$ is the Heaviside step function,\n\\begin{align}\n\\theta (1-|{\\bvec p}|\/\\Lambda)\n=\n\\begin{cases}\n1\t&\t(|{\\bvec p}|<\\Lambda),\\\\\n0\t&\t(|{\\bvec p}|>\\Lambda).\n\\end{cases}\n\\end{align}\n\nWe define the following threshold functions,\n\\begin{align}\n{\\hat I}_0(T,\\mu;\\Lambda) \n&=T\\sum ^{\\infty}_{n=-\\infty}\\int \\frac{d^3p}{(2\\pi)^3}{\\tilde \\partial}_t\n\\frac{1}{[(p_0+i\\mu)^2+{\\bvec p}^2(1+r_\\Lambda({\\bvec p}))^2]},\\\\\n{\\hat I}_1(T,\\mu;\\Lambda) \n&=T\\sum ^{\\infty}_{n=-\\infty}\\int \\frac{d^3p}{(2\\pi)^3}{\\tilde \\partial}_t\n\\frac{(p_0+i\\mu)(p_0-i\\mu)+{{\\bvec p}}^2(1+r_\\Lambda({\\bvec p}))^2}{[(p_0+i\\mu)^2+{{\\bvec p}}^2(1+r_\\Lambda({\\bvec p}))^2][(p_0-i\\mu)^2+{{\\bvec p}}^2(1+r_\\Lambda({\\bvec p}))^2]},\n\\end{align}\nwhere ${\\tilde \\partial}_t=\\partial _tr_\\Lambda\\cdot \\partial _{r} $ and $p_0=(2n+1)\\pi T$.\nWe can analytically calculate the above integration and summation and we obtain\n\\begin{align}\n\\nonumber\n{\\hat I}_0(T,\\mu;\\Lambda) \n&=\\frac{\\Lambda^2}{3} \n\\left. \\left[ \\left( \\frac{1}{2}-n_+\\right) + \\left( \\frac{1}{2}-n_-\\right)\t+\\frac{\\partial}{\\partial \\omega}[n_++n_-]\\right]\\right| _{\\omega\\to 1}\\\\\n&=\\frac{\\Lambda^2}{3} I_0({\\tilde T},{\\tilde \\mu}),\\\\\n\\nonumber\n{\\hat I}_1(T,\\mu;\\Lambda) \n&=\\frac{\\Lambda^2}{3}\n\\left. \\left[ \\frac{1}{(1+{\\tilde \\mu})^2} \\left( \\frac{1}{2}-n_+\\right) + \\frac{1}{(1-{\\tilde \\mu})^2}\\left( \\frac{1}{2}-n_-\\right)\t+\\frac{1}{1+{\\tilde \\mu}}\\frac{\\partial}{\\partial \\omega}n_+   +\\frac{1}{1-{\\tilde \\mu}}\\frac{\\partial}{\\partial \\omega}n_-\\right]\\right| _{\\omega\\to 1}\\\\\n&=\\frac{\\Lambda^2}{3} I_1({\\tilde T},{\\tilde \\mu}).\n\\end{align}\nHere we have used the formula,\n\\begin{align}\n\\sum^{\\infty}_{n=-\\infty}\\frac{1}{(2n+1)^2\\pi^2+\\beta^2E_\\pm^2}\n=\\frac{1}{\\beta E_\\pm}\\left( \\frac{1}{2}-n_\\pm\\right),\n\\end{align}\nwhere $n_\\pm$ is the Fermi-Dirac distribution function:\n\\begin{align}\nn_\\pm=\\frac{1}{\\exp(\\beta E_\\pm)+1}\n\\end{align}\nwith $E_\\pm=\\Lambda\\pm \\mu$ and $\\beta =1\/T$.\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgments}\n We thank Shailesh Chandrasekharan, Robert Ott, Arnab Sen, and Uwe-Jens Wiese for useful discussions. This work was supported by the Simons Collaboration on UltraQuantum Matter, which is a grant from the Simons Foundation (651440, P.Z.). D.B.~acknowledges support by the German Research Foundation (DFG), Grant ID BA 5847\/2-1. This work is part of and supported by the Interdisciplinary Center Q@TN \u2014 Quantum Science and Technologies at Trento, the DFG Collaborative Research Centre SFB 1225 (ISOQUANT), the Provincia Autonoma di Trento, and the ERC Starting Grant StrEnQTh (Project- ID 804305).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}