Content-Type: multipart/mixed; boundary="-------------0109240312392" This is a multi-part message in MIME format. ---------------0109240312392 Content-Type: text/plain; name="01-334.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-334.keywords" 1991 Mathematics Subject Classification 35B 65, 35Q 40 and 81Q 05 ---------------0109240312392 Content-Type: application/x-tex; name="density.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="density.tex" \documentclass[12pt]{amsart} \NeedsTeXFormat{LaTeX2e}[1994/12/01] \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amstext} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lemma}[thm]{Lemma} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{remark}[thm]{Remark} \newtheorem*{statement}{Statement} \newtheorem*{acknowledgement}{Acknowledgement} \newcounter{nummer} \newenvironment{araliste}{ \begin{list} {\mbox{\textup{(\arabic{nummer})}}} {\usecounter{nummer}\setlength{\leftmargin}{1cm}\setlength{\labelwidth}{1cm}}} {\end{list}} \newenvironment{latinliste}{ \begin{list} {\mbox{\textup{(\roman{nummer})}}} {\usecounter{nummer}\setlength{\leftmargin}{1cm}\setlength{\labelwidth}{1cm}}} {\end{list}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\dist}{{\operatorname{dist}}} \newcommand{\N}{{\mathbb N}} \newcommand{\R}{{\mathbb R}} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{The electron density is smooth away from the nuclei} \author[S. Fournais, M. and T. Hoffmann-Ostenhof and T. \O. S\o rensen]{S\o ren Fournais, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof and Thomas \O stergaard S\o rensen} \address[S. Fournais, T. Hoffmann-Ostenhof, T. \O. S\o rensen]{The Erwin Schr\"odinger International Institute for Mathematical Physics \\ Boltzmanngasse 9 \\ A-1090 Vienna \\ Austria} \address[S. Fournais, present address]{Laboratoire de Math\'{e}matiques \\ Universit\'{e} Paris-Sud - B\^{a}t 425 \\ F-91405 Orsay Cedex \\ France} \address[M. Hoffmann-Ostenhof]{Institut f\"ur Mathematik \\ Strudlhofgasse 4 \\ Universit\"at Wien \\ A-1090 Vienna \\ Austria} \address[T. Hoffmann-Ostenhof]{Institut f\"ur Theoretische Chemie \\ W\"ahringer\-strasse 17 \\ Universit\"at Wien \\ A-1090 Vienna \\ Austria} \address[T. \O. S\o rensen, present address]{Department of Mathematical Sciences \\ Aalborg University \\ Fredrik Bajers Vej 7G \\ DK-9220 Aalborg East \\ Denmark} \email[S. Fournais]{fournais@imf.au.dk} \email[M. Hoffmann-Ostenhof]{mhoffman@esi.ac.at} \email[T. Hoffmann-Ostenhof]{thoffman@esi.ac.at} \email[T. \O. S\o rensen]{sorensen@math.auc.dk} \thanks{Work supported by the Carlsberg Foundation, European Union TMR grant FMRX-CT 96-0001, Ministerium f\"ur Wissenschaft und Verkehr der Republik \"Osterreich, and the Austrian Science Foundation, grant number P12864-MAT} %\footnote{1991 Mathematics Subject Classification 35B 65, 35Q 40 and 81Q 05} \date{\today} \begin{abstract} We prove that the electron densities of electronic eigenfunctions of atoms and molecules are smooth away from the nuclei. \end{abstract} \maketitle \section{Introduction and Statement of the Results.} We consider an $N$-electron molecule with $L$ fixed nuclei whose non-relativistic Hamiltonian is given by \begin{multline} \label{Hmol} H_{N,L}(\mathbf R,\mathbf Z)=\sum_{j=1}^N\left({}-\Delta_j-\sum_{l=1}^L \frac{Z_l}{|x_j-R_l|}\right)\\ +\sum_{1\le i 0$. % \begin{remark} Since $\psi$ is continuous, (\ref{eq:decay}) is only an assumption on the behaviour near infinity. For references on the exponential decay of eigenfunctions, see e.g. Simon~\cite{Si-semi}. \end{remark} % \begin{remark} We only assume the exponential decay for simplicity. Our proofs extend to the case where the eigenfunction \(\psi\) decays faster than polynomially. \end{remark} The main result of this paper is the following theorem: % \begin{thm} \label{thm:density_smooth} Let \(\psi\) be an eigenfunction of \(H\), satisfying (\ref{eq:decay}). Let \(\rho, \rho_{2}\), and \(\gamma_{1}\) be as defined in \eqref{rho}, \eqref{rho2}, and \eqref{gamma}, and define furthermore \(n(x)=\rho_{2}(x,x)\). Let \(D=\{(x,x)\in\R^{6}\}\subset\R^{6}\), and define \begin{align*} \Sigma=\big(\{R_{1},\ldots,R_{L}\}\times\R^{3}\big)\cup \big(\R^{3}\times\{R_{1},\ldots,R_{L}\}\big)\subset\R^{6}. \end{align*} Then \begin{align} \rho, n&\in C^{\infty}(\R^{3}\setminus \{R_{1},\ldots,R_{L}\}) \intertext{and} \rho_{2} &\in C^{\infty}(\R^{6}\setminus (\Sigma \cup D )), \gamma_{1}\in C^{\infty}(\R^{6}\setminus\Sigma). \end{align} Furthermore, all the derivatives of $\rho$ satisfy an exponential decay estimate near infinity: Let $\gamma\in\N^{3}$ be a multi-index. Then for all $\epsilon > 0$ there exists a constant $c = c(\gamma, \epsilon)$ such that \begin{equation} \label{eq:rho_exponential} |\partial_{x}^{\gamma} \rho(x)| \leq c e^{-(\lambda-\epsilon) |x|}\quad \text{for all}\quad|x| > \max \{ |R_1|,\ldots, |R_L| \} + 1. \end{equation} \end{thm} Similar results hold for $n$-electron densities and for $n$-electron density matrices. \begin{remark} \label{rem:simplify} For simplicity we will only prove Theorem~\ref{thm:density_smooth} for atoms (i.e. for $L=1$, $R_1=0$) and only indicate the necessary modifications for the molecular case. Moreover, we only explicitly treat the density $\rho$, the proofs in the cases of $n, \rho_2$, and $\gamma_1$ being essentially the same. Finally, it is enough to prove smoothness of each term $\int \psi_j^2\,d\hat x_j$ in the sum \eqref{rho}, and we therefore restrict ourselves to $$ \int \psi_1^2\,d\hat x_1 = \int \psi^2(x,x_2,\ldots,x_N)\,dx_2 \cdots dx_N. $$ \end{remark} \section{Additional regularity of eigenfunctions.} We will need to know something about the regularity properties of $\psi$ in order to conclude that $\rho$ is smooth. In order to study the regularity of $\psi$ it is convenient to work in spaces of functions which are H\"older continuous. Let us recall the definition of H\"older continuity: \begin{defn} Let \(\Omega\) be a domain in \(\R^{n}\), \(k\in\N\), and \(\alpha\in(0,1]\). We say that a function \(u\) belongs to \(C^{k,\alpha}(\Omega)\) whenever \(u\in C^{k}(\Omega)\), and for all \(\beta\in\N^{n}\) with \(|\beta|=k\), and all open balls $B(x_{0},r)$ with $\overline{B(x_{0},r)}\subset\Omega $, we have \begin{align*} \sup_{x,y\in B(x_{0},r),\,x\neq y} \!\!\!\!\!\!\!\!\! \frac{|\partial^{\beta}u(x)-\partial^{\beta}u(y)|}{|x-y|^{\alpha}} \leq C(x_{0},r). \end{align*} For any domain \(\Omega'\), with \(\overline{\Omega'}\subset\Omega\), we now define the following norms: \begin{align} \nonumber |u|_{C^{k,\alpha}(\Omega')}=\sum_{|\beta| \leq k}\|\partial^{\beta} u\|_{L^{\infty}(\Omega')} +\sum_{|\beta| = k}\sup_{x,y\in\Omega',\,x\neq y} \frac{|\partial^{\beta}u(x)-\partial^{\beta} u(y)|}{|x-y|^{\alpha}}. \end{align} \end{defn} We will need the following result on elliptic regularity in order to conclude that the solutions of elliptic second order equations with bounded coefficients are $C^{1,\alpha}$. The proposition is a reformulation of Corollary 8.12 in Gilbarg and Trudinger~\cite{GandT}, adapted for our purposes: \begin{prop} \label{prop:GT} Let \(\Omega\) be a bounded domain in \({\mathbb R}^{n}\) and suppose \(u\in W^{1,2}(\Omega)\) is a weak solution of \(\Delta u +\sum_{j=1}^{n}b_{j}D_{j}u+Wu=g\) in \(\Omega\), where \(b_{j},W,g\in L^{\infty}(\Omega)\). Then \(u\in C^{1,\alpha}(\Omega)\) for all \(\alpha\in(0,1)\) and for any domain \(\Omega'\), \(\overline{\Omega'}\subset\Omega\) we have \begin{align*} |u|_{C^{1,\alpha}(\Omega')}\leq C\big(\sup_{\Omega}|u|+\sup_{\Omega}|g|\big) \end{align*} for \(C=C(n,M,\dist(\Omega',\partial\Omega))\), with \begin{align} \max_{j=1,\ldots,n}\{1,\|b_{j}\|_{L^{\infty}(\Omega)}, \|W\|_{L^{\infty}(\Omega)},\|g\|_{L^{\infty}(\Omega)}\} \leq M. \nonumber \end{align} \end{prop} Our regularity result on $\psi$ is the following: \begin{lemma} \label{lem:smoothness} Let $\psi$ be an $N$-electron atomic eigenfunction satisfying the decay estimate (\ref{eq:decay}). Let $P$ and $Q$ be a partition of $\{1,\ldots,N\}$: \begin{align*} \{1,\ldots,N\} = P \cup Q,\,\,\,\,\,\,\, P \cap Q = \emptyset,\,\,\,\,\,\,\, P \neq \emptyset. \end{align*} Define $x_P$ as \begin{align*} x_{P} = \frac{1}{\sqrt{|P|}}\,\sum_{j \in P} x_j \in {\mathbb R}^3, \end{align*} and let $T$ be any orthogonal transformation such that $T(x_1,\ldots,x_N) = (x_P,x')$ with $x'\in {\mathbb R}^{3N-3}$. Let $\epsilon > 0$ be given and define $U_P \subset {\mathbb R}^{3N}$ as the open set: \begin{align} \label{eq:def_U_P} U_P = \Big\{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N}\,\Big|&\, |x_j| > \epsilon \mbox{ for } j \in P, \nonumber \\ & |x_j-x_k| > \epsilon \mbox{ for } j \in P, k \in Q \Big\}. \end{align} Then \begin{align*} \partial_{x_P}^{\gamma} (\psi \circ T^*) \in C^{0,1}(T U_P)\quad\text{ for all } \gamma \in {\mathbb N}^{3}. \end{align*} Furthermore, the following a priori estimate holds: \begin{align*} \big| \nabla \partial_{x_P}^{\gamma} (\psi\circ T^*)(x_P,x') \big| + \big| \partial_{x_P}^{\gamma} &(\psi\circ T^*)(x_P,x') \big| \nonumber \\ &\leq C e^{-\lambda |(x_P,x')|} \,\,\,\,\,\,\,\, \text{ for all } (x_P,x') \in T U_P \end{align*} for some $C=C(\gamma)>0$. \end{lemma} \begin{remark} One could interpret $U_P$ as a (very large) neighbourhood of a singularity ${\mathbf x}^0$ of $V$ where the electrons with coordinates $x_j$ with $j \in Q$ sit on the nucleus and the electrons with coordinates $x_j$ with $j \in P$ sit on each other away from the nucleus, i.e. ${\mathbf x}^0 = (x_1^0,\cdots,x_N^0)$, with \begin{eqnarray*} x_j^0 = 0 && \text{ for all } j \in Q, \\ x_j^0 = y^0 \neq 0 && \text{ for all } j \in P. \end{eqnarray*} Notice that $x_P$ is (up to a scalar multiple) the centre of mass of the electrons $j$ with $j \in P$. Loosely speaking Lemma~\ref{lem:smoothness} then says that in the neighbourhood $U_P$, $\psi$ is smooth with respect to the centre of mass coordinate $x_P$. One can also consider ${\mathbf x}^0$ as a two-cluster singularity---one group of electrons on each other at the nucleus, another group of electrons on top of each other away from the nucleus. It is, of course, possible to have many clusters. Lemma~\ref{lem:smoothness} is a special case of a more general theorem: If we have any number of clusters which are separated from each other and from the nucleus, then the eigenfunction $\psi$ can be differentiated any number of times with respect to the centre of mass of each cluster. We only need the two-cluster version (see Lemma~\ref{lem:smoothness}) in the present paper. The more general result will be used in a detailed investigation of the regularity of the wavefunction near all kinds of singularities of the potential (see~\cite{second_regularity}). \end{remark} \begin{remark} In the case of molecules the definition of $U_P$ is slightly different: \begin{align*} U_{\text{MOL}} = \Big\{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N}\,\Big|\, &\min_{1 \leq l \leq L} |x_j- R_l| > \epsilon \mbox{ for } j \in P, \\ & |x_j-x_k| > \epsilon \mbox{ for } j \in P, k \in Q \Big\}. \end{align*} Apart from that, Lemma~\ref{lem:smoothness} remains unchanged. \end{remark} Before we prove Lemma~\ref{lem:smoothness}, let us fix some notation. We may assume without loss of generality that $P = \{1,\ldots,N_1\}$, with $N_1 \leq N$. Then any orthogonal transformation $T$ satisfying the assumptions in the statement of Lemma~\ref{lem:smoothness} can be written as: $$ T = \left(\begin{array}{c} \begin{array}{cccccc} \frac{1}{\sqrt{N_1}}& \cdots& \frac{1}{\sqrt{N_1}}& 0 & \cdots & 0 \end{array} \\ \, \\ \, \\ \tilde{T} \\ \, \\ \, \end{array} \right), $$ with the first row being understood as $3\times 3$ matrices---first $N_1$ repetitions of $\frac{1}{\sqrt{N_1}}I_3$ and then $N-N_1$ repetitions of the $3 \times 3$ $0$-matrix. The remaining part of the matrix, $\tilde{T} \in M_{3N-3,3N}({\mathbb R})$ is such that the complete matrix $T$ is orthogonal. We will denote the $(3N-3)\times3$-columns of $\tilde{T}$ by $t_j$, i.e. $$ \tilde{T} = \left( \begin{array}{ccc} t_1&\cdots&t_N \\ \end{array} \right) , $$ with $t_j \in M_{3N-3,3}({\mathbb R})$. Then we get: \begin{equation} \label{eq:T_stjerne} T^* = \left( \begin{array}{cccccc} \frac{1}{\sqrt{N_1}}&&& t_1^*& & \\ \vdots && &\vdots &&\\ \frac{1}{\sqrt{N_1}}&&& t_{N_1}^*& \\ 0 &&& t_{N_1+1}^*& \\ \vdots &&& \vdots& \\ 0 &&& t_{N}^* & \end{array} \right). \end{equation} \begin{proof} For the proof of Lemma~\ref{lem:smoothness} we first proceed as in M. and T. Hoffmann-Ostenhof and \O stergaard S\o rensen~\cite{firstWien}: We make the `Ansatz' \begin{align} \label{eq:ansatz} \psi = e^{F-F_1}\psi_1, \end{align} with \begin{align*} F= \sum_{j=1}^N{}-\frac{Z}{2} |x_j| + \sum_{1\leq j< k\leq N} \frac{1}{4}|x_j-x_k|, \end{align*} and \begin{align*} F_1 = \sum_{j=1}^N{}-\frac{Z}{2} \sqrt{|x_j|^2+1} + \sum_{1\leq j< k\leq N} \frac{1}{4}\sqrt{|x_j-x_k|^2+1}. \end{align*} Observing (see \eqref{potential} with $L=1$, $R_1 = 0$), \begin{equation} \label{eq:choice_of_F} \Delta F = V, \end{equation} we get the following equation for $\psi_1$ using \eqref{Schroe} and (\ref{eq:ansatz}): \begin{equation} \label{eq:eq_for_psi1} \Delta \psi_1 + 2\nabla(F-F_1)\cdot \nabla \psi_1 +\big(|\nabla(F-F_1)|^2-\Delta F_1+E\big)\psi_1 = 0. \end{equation} Furthermore, $F_1$ has the same behaviour as $F$ at infinity, so we get the following estimates: \begin{align} \label{eq:bounds_coef} \|\partial^{\beta}F_{1}\|_{L^\infty({\mathbb R}^{3N})},\, \| F - F_1 \|_{L^\infty({\mathbb R}^{3N})},&\,\|\nabla(F-F_1)\|_{L^\infty({\mathbb R}^{3N})} \nonumber \\ &\leq C(N,Z,\beta)\,, \,\beta\in\N^{3N}, |\beta|>0. \end{align} We will first investigate the necessary regularity properties of the prefactor $e^{F-F_1}\circ T^*$ in $TU_P$ with respect to $x_P$. Then we will differentiate the equation (\ref{eq:eq_for_psi1}) with respect to $x_P$ and analyse the regularity of $\partial_{x_P}^{\gamma}(\psi_1 \circ T^*)$ in $TU_P$. In the coordinates defined by $T$ we have \begin{align} \label{eq:transf1} x_j &= \frac{x_P}{\sqrt{N_1}} + t_j^* x' &\mbox{ for } j \leq N_1, \\ \label{eq:transf2} x_j &= t_j^* x' &\mbox{ for } j > N_1. \end{align} So \begin{align*} x_j-x_k = (t_j^* -t_k^*)x' &\mbox{ for } j,k \leq N_1 \mbox{ or } j,k > N_1. \end{align*} In particular these last expressions are independent of $x_P$. So when we differentiate $F \circ T^*$ with respect to $x_P$, the only non-vanishing terms come from derivatives of $|x_j|\circ T^*$ with $j \leq N_1$ and $|x_j -x_k| \circ T^*$ with $j \leq N_1$, $k > N_1$. The definition of $U_P$ implies that the function ${\mathbf x} \mapsto |x_j|$ is smooth with bounded derivatives on $U_P$ for $j \leq N_1$, and ${\mathbf x} \mapsto |x_j-x_k|$ is smooth with bounded derivatives on $U_P$ for $j \leq N_1$, $k > N_1$. Thus \begin{align} \label{eq:FandF1} \|\nabla \partial_{x_P}^{\gamma} (F-F_1) \circ T^*\|_{L^{\infty}(T U_P)} &+ \|\partial_{x_P}^{\gamma} (F-F_1) \circ T^*\|_{L^{\infty}(T U_P)} \nonumber\\ &\leq C(\gamma) \mbox{ for all } \gamma \in {\mathbb N}^3. \end{align} Furthermore, we get using (\ref{eq:bounds_coef}) that \begin{equation} \partial_{x_P}^{\gamma} \left(e^{F-F_1} \circ T^*\right) \in C^{0,1}(T U_P) \cap L^{\infty}(T U_P) \mbox{ for all } \gamma \in {\mathbb N}^3. \end{equation} % Hence, due to (\ref{eq:ansatz}) and (\ref{eq:FandF1}), it remains to study the regularity of $\psi_1\circ T^*$. In the rest of the proof we will use the following notation: We will write $\tilde{F},\tilde{F}_1, \tilde{\psi}_1$ instead of $F\circ T^*, F_1 \circ T^*, \psi_1\circ T^*$. In particular we have the following relation \begin{align*} \psi \circ T^* = e^{\tilde{F}-\tilde{F}_1} \tilde{\psi}_1. \end{align*} Notice that since the Laplacian is invariant under orthogonal transformations we have the following equation for $\psi \circ T^*$: $$ -\Delta (\psi \circ T^*) + (V \circ T^*) (\psi \circ T^*) = E (\psi \circ T^*). $$ Now, once again using the invariance of the Laplacian and (\ref{eq:choice_of_F}), $$ \Delta \tilde{F} = V \circ T^*, $$ so we get the following equation for $\tilde{\psi_1}$ (compare with (\ref{eq:eq_for_psi1})): \begin{align} \label{eq:eq_for_psi1_T} &L \tilde{\psi}_{1} =0,\nonumber\\ &L=\Delta+2\nabla(\tilde{F}-\tilde{F}_1)\cdot\nabla +\big(|\nabla(\tilde{F}-\tilde{F}_1)|^2-\Delta \tilde{F}_1+E\big). \end{align} The analysis of $\tilde{\psi}_1$ will be based on the elliptic regularity result from Proposition~\ref{prop:GT}. We shall proceed by induction and for this we will need the following open sets indexed by $l \in {\mathbb N}$: % \begin{align*} U_l = \Big\{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N}\,\Big|\,& |x_j| > \epsilon(1-2^{-(l+1)}) \mbox{ for } j \in P, \\ &|x_j-x_k| > \epsilon(1-2^{-(l+1)}) \mbox{ for } j \in P, k \in Q \Big\}. \end{align*} % It is clear that for $l_1 < l_2$ we have $$ U_P \subset U_{l_2} \subset U_{l_1} \subset U_0. $$ We will prove the following statement: \begin{statement} For all $\gamma \in {\mathbb N}^3$ we have \begin{araliste} \item \label{regularity} $\partial_{x_P}^{\gamma} \tilde{\psi}_1 \in C^{1,\alpha}(T U_{|\gamma|})$ for all $\alpha \in (0,1)$. \item \label{H2} $\partial_{x_P}^{\gamma} \tilde{\psi}_1 \in W^{2,2}_{\text{loc}}(T U_{|\gamma|})$. \item \label{decay} There exists $c=c(\gamma)>0$ such that \begin{align*} |\nabla \partial_{x_P}^{\gamma} \tilde{\psi}_1 (x_P,x')|+|\partial_{x_P}^{\gamma} \tilde{\psi}_1 (x_P,x')| \leq c e^{-\lambda |(x_P,x')|} \end{align*} for all $(x_P,x')\in T U_{|\gamma|}$. \end{araliste} \end{statement} Let us start by noticing that once this statement is established Lemma~\ref{lem:smoothness} is proved. The proof of the statement proceeds by induction with respect to $|\gamma|$. In order to prove the statement for $|\gamma| =0$, let us look at the equation (\ref{eq:eq_for_psi1_T}). We get from Proposition~\ref{prop:GT} that \begin{align*} \tilde{\psi}_{1}\in C^{1,\alpha}(T U_{0}), \end{align*} since the coefficients of the equation (\ref{eq:eq_for_psi1_T}) are bounded on $T U_{0}$, due to (\ref{eq:bounds_coef}). This proves (\ref{regularity}) for $|\gamma|=0$. We next prove that $\tilde{\psi}_{1} \in W^{2,2}_{\text{loc}}(T U_0)$. This is accomplished as follows: If we use that $\tilde{\psi}_{1}\in C^{1,\alpha}(T U_{0})$ and that the coefficients in the equation (\ref{eq:eq_for_psi1_T}) (derivatives of $\tilde{F}-\tilde{F}_1$) are bounded (again using (\ref{eq:bounds_coef})), then it is easily seen from (\ref{eq:eq_for_psi1_T}) that $$ \Delta \tilde{\psi}_1 \in L^2_{\text{loc}}(TU_0). $$ Therefrom, we get via standard elliptic regularity results (see for instance Folland~\cite[Theorem 6.33]{Folland}) that $\tilde{\psi}_1 \in W^{2,2}_{\text{loc}}(T U_0)$. Next, we verify the exponential decay estimate (\ref{decay}) for $|\gamma|=0$. We know from the assumption (\ref{eq:decay}) that $\tilde{\psi}_1$ decays exponentially, but we also need to prove it for $\nabla \tilde{\psi}_1$. This is done exactly as in the induction step below, using the exponential decay estimate (\ref{eq:decay}). In order not to repeat the argument, we refer the reader to the induction step below. Suppose now that we have proved (\ref{regularity})-(\ref{decay}) for all $\gamma$ with $|\gamma| \leq k$. Take a $\gamma$ with length $|\gamma| = k+1$. Differentiating the equation (\ref{eq:eq_for_psi1_T}) for $\tilde{\psi}_1$ we get the following equation for $\partial_{x_{P}}^{\gamma}\tilde{\psi}_1$ (in the sense of distributions): \begin{align} \label{eq:paralel_diffentiated} &L(\partial_{x_{P}}^{\gamma}\tilde{\psi}_{1})=f_{\gamma},\nonumber\\ &L=\Delta+2\nabla(\tilde{F}-\tilde{F}_1)\cdot\nabla +\big(|\nabla(\tilde{F}-\tilde{F}_1)|^2-\Delta \tilde{F}_1+E\big),\nonumber\\ &f_{\gamma}=-\sum_{\sigma+\mu = \gamma, |\mu| < |\gamma|} \Big( 2\left[ \partial_{x_P}^{\sigma}\big(\nabla (\tilde{F}-\tilde{F}_1)\big) \right] \cdot \nabla \partial_{x_P}^{\mu} \tilde{\psi}_1 \nonumber\\ & \quad\quad +\left[ \partial_{x_P}^{\sigma}\big(|\nabla(\tilde{F}-\tilde{F}_1)|^2 -\Delta \tilde{F}_1+E \big) \right] \partial_{x_P}^{\mu} \tilde{\psi}_1 \Big). \end{align} In $f_{\gamma}$ only partial derivatives of $\tilde{\psi}_1$ of length $\leq k$ occur. It is clear that in (\ref{eq:FandF1}) we may replace $TU_P$ by $TU_0$. Therefore we get, using the induction hypothesis for (\ref{regularity}), that $f_{\gamma} \in L^{\infty}(TU_{|\gamma|-1})$. A priori $\partial_{x_{P}}^{\gamma}\tilde{\psi}_1$ only satisfies the equation (\ref{eq:paralel_diffentiated}) in the distributional sense. In order to apply Proposition~\ref{prop:GT} we need that $\partial_{x_{P}}^{\gamma}\tilde{\psi}_1 \in W^{1,2}_{\text{loc}}(U_{|\gamma|-1})$. However, this follows from the induction hypotheses for (\ref{H2}), and the definition of Sobolev spaces. Now, due to (\ref{eq:FandF1}), the coefficients of $L$ are bounded, and therefore we get from Proposition~\ref{prop:GT} that $\partial_{x_P}^{\gamma}\tilde{\psi}_1 \in C^{1,\alpha}(TU_{|\gamma|-1})$ for all $\alpha \in (0,1)$, proving (\ref{regularity}). This together with (\ref{eq:FandF1}) implies that $f_{\gamma} \in L^2_{\text{loc}}(TU_{|\gamma|-1})$. Therefore, we get (invoking once again that $\partial_{x_P}^{\gamma}\tilde{\psi}_1 \in C^{1,\alpha}(TU_{|\gamma|-1})$ for all $\alpha \in (0,1)$ and (\ref{eq:FandF1})), $$ \Delta (\partial_{x_{P}}^{\gamma} \tilde{\psi}_1) \in L^2_{\text{loc}}(TU_{|\gamma|-1}). $$ As before, via a standard elliptic regularity theorem (e.g. Folland~\cite[Lemma 6.32]{Folland}), we see that $\partial_{x_{P}}^{\gamma} \tilde{\psi}_1 \in W^{2,2}_{\text{loc}}(TU_{|\gamma|-1})$, and therefore (\ref{H2}) is proved. Finally, we prove the exponential decay of $\partial_{x_{P}}^{\gamma} \tilde{\psi}_1$. % Let us write $\epsilon_{l} = \epsilon/2^{l+3}$. Then we have for all ${\mathbf x} \in U_l$ that $\overline{B({\mathbf x},2\epsilon_l)} \subset U_{l-1}$. Since the equation (\ref{eq:paralel_diffentiated}) is satisfied on $TU_{|\gamma|-1}$ it is in particular satisfied on $TB({\mathbf x},2\epsilon_{|\gamma|})$ for any ${\mathbf x} \in U_{|\gamma|}$. Applying Proposition~\ref{prop:GT} with $\Omega' = TB({\mathbf x},\epsilon_{|\gamma|})$, $\Omega = TB({\mathbf x}, 2\epsilon_{|\gamma|})$, we get that \begin{align*} | \partial_{x_P}^{\sigma} \tilde{\psi}_1 |_{C^{1,\alpha}(\Omega')} \leq C\left( \sup_{\Omega} |\partial_{x_P}^{\sigma} \tilde{\psi}_1|+ \sup_{\Omega}| f_{\sigma} | \right). \end{align*} Notice, using (\ref{eq:FandF1}), that the coefficients in the differential operator $L$ are uniformly bounded on all of $TU_0$. Furthermore, the derivatives of $\tilde{F}$ and $\tilde{F}_1$ in the expression for $f_{\gamma}$ are also uniformly bounded on $U_0$---once again using (\ref{eq:FandF1}). Therefore, using the induction hypothesis (\ref{decay}), we get for all $(x_P,x')\in TU_{|\gamma|}$: \begin{align*} |\nabla \partial_{x_P}^{\sigma} \tilde{\psi}_1(x_P,x')| & \leq | \partial_{x_P}^{\sigma} \tilde{\psi}_1 |_{C^{1,\alpha}(\Omega')} \\ &\leq C\left( \sup_{\Omega} |\partial_{x_P}^{\sigma} \tilde{\psi}_1|+ \sup_{\Omega}| f_{\sigma} | \right) \\ & \leq C \sup_{ (y_P,y')\in TB((x_P,x'),2\epsilon_{|\gamma|})} e^{-\lambda |(y_P,y')|} \\ & = C e^{-\lambda |(x_P,x')|}. \end{align*} In this last line the constant $C$ does not depend on the position $(x_P,x')$ of the ball, since the coefficients of the equation (\ref{eq:paralel_diffentiated}) are uniformly bounded on all $TU_0$. This proves (\ref{decay}) and thus finishes the induction. \end{proof} \begin{remark} In the case of molecules we modify $F$ and $F_1$ as follows: \begin{align*} F= \sum_{l=1}^L \sum_{j=1}^N{}-\frac{Z_l}{2} |x_j - R_l| + \sum_{1\leq j< k\leq N} \frac{1}{4}|x_j-x_k|, \end{align*} and \begin{align*} F_1 = \sum_{l=1}^L \sum_{j=1}^N{}-\frac{Z_l}{2} \sqrt{|x_j-R_l|^2+1} + \sum_{1\leq j< k\leq N} \frac{1}{4}\sqrt{|x_j-x_k|^2+1}. \end{align*} The rest of the proof is analogous. \end{remark} \section{The proof of Theorem~\ref{thm:density_smooth}} As noted in Remark~\ref{rem:simplify} it is enough to prove smoothness of each individual term in~\eqref{rho}. We therefore redefine $\rho$ by $$ \rho(x) = \int \psi^2(x,x_2,\ldots,x_N)\,dx_2\cdots dx_N. $$ Lemma~\ref{lem:smoothness} will be essential in order to prove the smoothness of $\rho$. It suffices to prove that $\rho \in C^{\infty}({\mathbb R}^3 \setminus \overline{B(0,R)})$ for all $R >0$. Therefore, let us assume that $|x|>R>0$. \begin{remark} In the case of molecules we assume \begin{align*} \min_{1\leq l \leq L} |x - R_l| > R > 0, \end{align*} and prove that $\rho \in C^{\infty}({\mathbb R}^3 \setminus \left( \cup_{l=1}^L\overline{B(R_l,R)}\right))$. \end{remark} Let $\chi_1,\chi_2$ be a partition of unity in ${\mathbb R}_{+}$: $\chi_1+\chi_2 =1$, $\chi_1(x) = 1 $ on $[0,R/(4N)]$, $\supp \chi_1 \subset [0,R/(2N)]$ and $\chi_j \in C^{\infty}({\mathbb R_{+}})$ for $j=1,2$. We combine the $\chi_j$'s to make a partition of unity in ${\mathbb R}^{3N}$. Obviously: \begin{align*} 1 = \prod_{1\leq j 0$. \begin{remark} To motivate the determination of $P$ and hence the coordinate $x_P$ with respect to which we are allowed to differentiate, let us consider the following example: Let $N=3$ and $I=\{(1,2),(2,3)\}$ and $J = \{(1,3)\}$. Then we have on the support of $\phi_I$ that $|x_1-x_2| \leq \tfrac{R}{6}$ (due to the $\supp \chi_1$) and since $|x_1| > R,$ $|x_2| \geq \tfrac{5R}{6}$. Further, due to the $\supp \chi_2$, we have $|x_1-x_3| \geq \tfrac{R}{12}$. Suppose now, we would choose $P = \{ 1,2\}$ and $Q= \{3\}$, then, according to (\ref{eq:def_U_P}), $$ U_P = \{ (x_1,x_2,x_3) \big| |x_1|>\epsilon, |x_2|>\epsilon, |x_1-x_3|>\epsilon, |x_2-x_3|>\epsilon \}, $$ for some $\epsilon > 0$. But then $\supp \phi_I \not \subset U_P$, since $\supp \phi_I$ contains points with $x_2=x_3$. On the other hand one easily checks that the choice $P=\{1,2,3\}$ is the right one. This example shows that we cannot just choose $P$ to be $\{1\} \cup \{j \leq N \big| (1,j) \in I \}$. On the other hand $P$ cannot be too big: For $N=3$, $I=\{(1,2)\}$, $J=\{(1,3),(2,3)\}$ it is easily seen that with $P = \{1,2,3\}$, $\supp \phi_I$ contains points with $x_3=0$ and therefore $\supp \phi_I \not \subset U_P$. Physically speaking, we divide the $N$ electrons into $2$ clusters. The electrons $j$ with $j \in P$ define the `maximal cluster' of electrons containing the electron $1$. This will be done via an equivalence relation below. Note that the (three-dimensional) variable $x_{P} = \tfrac{1}{\sqrt{|P|}}\sum_{j \in P} x_j$ is (up to a scalar multiple) the centre (centre of mass) of the maximal cluster. \end{remark} Let $\sim$ denote the equivalence relation on $\{1,\ldots,N\}^2$ generated by $I$ and let $P$ denote the equivalence class of $1$. Explicitly this means that $j \sim k$ if either $j=k$ or there exists a sequence $j_1,\ldots,j_l$ with $j_{s}\in\{1,\ldots,N\}$ for $1\le s\le l$ and with $j_s \neq j_t$ for $s\neq t$, such that \begin{latinliste} \item \(\quad(j,j_1) \in I\) or \((j_1,j) \in I\), \item \(\quad(j_{s},j_{s+1}) \in I\) or \((j_{s+1},j_{s}) \in I\), for $ 1 \leq s \leq l-1$, \item \(\quad(j_l,k) \in I\) or \((k,j_{l}) \in I\). \end{latinliste} Clearly \begin{equation} \label{eq:sequence_length} l \leq N-2. \end{equation} Thus $P= \{ j \,|\, j \sim 1 \}$, $Q = \{1,\ldots, N\} \setminus P$. Notice that $P \neq \emptyset$. In order to be able to apply Lemma~\ref{lem:smoothness}, we have to show that $\supp \phi_I \subset U_P$ (with a suitable choice of $\epsilon$ in the definition of $U_P$). Let $j \in P$, then $j \sim 1$ and we can choose a sequence $j_1,\ldots,j_l$ according to the above. Taking into account $|x_1| > R$ we have $$ |x_j| \geq R - |x_1-x_j|. $$ Further, with $$ |x_1 - x_j| \leq |x_1-x_{j_1}| + \sum_{s=1}^{l-1} |x_{j_s}-x_{j_{s+1}}| + | x_{j_l}-x_j|, $$ the length scale of the cut-off's and (\ref{eq:sequence_length}) we obtain that \begin{align} \label{eq:support1} j \in P \Rightarrow \supp \phi_{I} \subset \{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N} \, \big| \, |x_j| > R/4\}. \end{align} \begin{remark} In the case of molecules we get \begin{align*} j \in P \Rightarrow \supp \phi_{I} \subset \{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N} \, \big| \, \min_{1\leq l \leq L}|x_j - R_l| > R/4\}. \end{align*} \end{remark} Furthermore, suppose $j \in P$, $k \in Q$, then it is clear that $(j,k) \in J$ or $(k,j) \in J$ (because if $(j,k) \in I$, then $1\sim j\sim k$ and therefore $k \in P$), and therefore: \begin{align} \label{eq:support2} j \in P, k\in Q \Rightarrow \supp \phi_{I} \subset \{ (x_1,\ldots,x_N) \in {\mathbb R}^{3N} \, \big| \,|x_j-x_k| > R/(4N)\}. \end{align} Using (\ref{eq:support1}) and (\ref{eq:support2}) we see that $\supp \phi_I \subset U_P$, with $\epsilon = \tfrac{R}{4N}$ in the definition (\ref{eq:def_U_P}) of $U_P$. Hence we get from Lemma~\ref{lem:smoothness} that $\psi\circ T^*$ is (infinitely often) differentiable with respect to the coordinate $x_P = \tfrac{1}{\sqrt{|P|}}\sum_{j\in P} x_j$ on the support of $\phi_{I}\circ T^*$. Denote $g_I= \psi^2 \phi_I$, and note that all partial derivatives of $\phi_I$ are bounded. We get from Lemma~\ref{lem:smoothness} that \begin{align} \label{eq:diff_and_decay} &\partial_{x_P}^{\gamma} (g_I \circ T^*) \in C^{0,1}({\mathbb R}^{3N}), \nonumber \\ &|\partial_{x_P}^{\gamma} (g_I \circ T^*) (x_P,x')| \leq c_{\gamma} e^{-\lambda |(x_P,x')|} \mbox{ for all } (x_P,x') \in {\mathbb R}^{3N}. \end{align} Now, we are ready to prove the smoothness of the electron density. We calculate (using the notation from (\ref{eq:T_stjerne}), and \eqref{eq:transf1} and \eqref{eq:transf2}): \begin{eqnarray} \label{eq:delta} \rho_I(x) &=&\int_{{\mathbb R}^{3N-3}} g_I(x,x_2,\ldots,x_N)\,dx_2\cdots\,dx_N \nonumber\\ &=& \int_{{\mathbb R}^{3N}} g_I(x_1,\ldots,x_N)\delta(x-x_1)\,dx_1dx_2\cdots\,dx_N \nonumber\\ &=& \int_{{\mathbb R}^{3N}} (g_I\circ T^*)(x_P,x') \delta(x-\tfrac{x_P}{\sqrt{N_1}}-t_1^*x') \,dx_P\,dx'\nonumber\\ &=& \int_{{\mathbb R}^{3N-3}} (g_I\circ T^*)(\sqrt{N_1}(x-t_1^*x'),x') \,dx' \end{eqnarray} Using (\ref{eq:diff_and_decay}) and Lebesgue integration theory, we obtain via the chain rule: \begin{align*} \partial_x^{\gamma} \rho_I(x) = & \partial_{x}^{\gamma} \Big(\int \big(g_I\circ T^*\big)\big(\sqrt{N_1}(x-t_1^*x'),x'\big) \,dx' \Big)\\ =& \big( \sqrt{N_1} \big)^{|\gamma|} \int \big(\partial_{x_P}^{\gamma}(g_I\circ T^*)\big) \big(\sqrt{N_1}(x-t_1^*x'),x'\big) \,dx'. \end{align*} This proves that $\rho$ is smooth away from the nucleus. The exponential decay of the derivatives of $\rho_I$ is a consequence of (\ref{eq:diff_and_decay}). This can be seen by a similar calculation as in (\ref{eq:delta}) but in reversed order: \begin{eqnarray*} |\partial_x^{\gamma} \rho_I(x)| &\leq & \big( \sqrt{N_1} \big)^{|\gamma|} \int \left| \big(\partial_{x_P}^{\gamma}(g_I\circ T^*)\big) \big(\sqrt{N_1}(x-t_1^*x'),x'\big) \right|\,dx' \\ &\leq & c \int e^{-\lambda |(\sqrt{N_1}(x-t_1^*x'),x')|} \,dx'\\ &=& c \int (e^{-\lambda |{\mathbf x}|} \circ T^*)(x_P,x') \delta\big(x-\tfrac{x_P}{\sqrt{N_1}}-t_1^*x'\big) \,dx_P\,dx'\\ &=&c \int e^{-\lambda |{\mathbf x}|} \delta(x-x_1) \,dx_1\cdots dx_N \\ &=& c\int e^{-\lambda |(x,x_2,\ldots,x_N)|} \,dx_2\cdots dx_N. \end{eqnarray*} Let us write $(x,x_2,\ldots,x_N) = (x,z)$. 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