Content-Type: multipart/mixed; boundary="-------------0011171011652" This is a multi-part message in MIME format. ---------------0011171011652 Content-Type: text/plain; name="00-458.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-458.keywords" Periodic Coulomb model, fonctionnal density, Thomas-Fermi-von Weizsaker ---------------0011171011652 Content-Type: application/x-tex; name="b.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="b.tex" \documentclass[11pt]{article} \usepackage{times,amssymb,amsfonts,amsmath} %\usepackage{showkeys} %\usepackage{comments} % verbatim, s'utilise avec \verb"..." %\usepackage{french} \textheight 20.5cm \textwidth 15cm \oddsidemargin 0.5cm \newcommand{\be}{\begin{eqnarray}} \newcommand{\ee}{\end{eqnarray}} \newcommand{\beno}{\begin{eqnarray*}} \newcommand{\eeno}{\end{eqnarray*}} \newcommand{\carre}{$\rule{6pt}{6pt}$} \newcommand{\fer}[1]{(\ref{#1})} \newcommand{\xv}{\mathbf{x}} \newcommand{\yv}{\mathbf{y}} \newcommand{\bh}{\mathbf{h}} \newcommand{\bm}{\mathbf{m}} \newcommand{\cI}{\mathcal I} \newcommand{\cJ}{\mathcal J} \newcommand{\cL}{\mathcal L} \newcommand{\cN}{\mathcal N} \newcommand{\cX}{\mathcal X} \newcommand{\cY}{\mathcal Y} \newcommand{\cB}{\mathcal B} \newcommand{\tD}{\widetilde D} \newcommand{\ma}{\alpha} \newcommand{\mb}{\beta} \newcommand{\md}{\delta} \newcommand{\mg}{\gamma} \newcommand{\mL}{\Lambda} \newcommand{\mO}{\Omega} \newcommand{\mt}{\theta} \newcommand{\mT}{\Theta} \newcommand{\eps}{\epsilon} \newcommand{\epsr}{\epsilon_\rho} \newcommand{\C}{{\mathbb C}} \newcommand{\R}{\mbox{$\mathbb R$}} \newcommand{\N}{\mbox{$\mathbb N$}} \newcommand{\T}{{\mathbb T}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\nlp}[2]{\|#1\|_{_{#2}}} \newcommand{\conv}{\rightarrow} \newcommand{\cE}{\mathcal{E}} \newcommand{\Ekin}{\cE_{kin}} \newcommand{\Ekinloc}{\cE_{kin}^{loc}} \newcommand{\Eee}{\cE_{ee}} %% %% sec 3,4,5 %% \newcommand{\naps}[2]{|#1|_{_{#2,\ma}}} % norme alpha pseudo : |1|_{2,alpha} \newcommand{\na}[2]{\|#1\|_{_{#2,\ma}}} % norme alpha : ||1||_{2,alpha} \newcommand{\noaps}[1]{|#1|_{_{0,\ma}}} % norme 0,alpha pseudo %% %% sec 4,5 %% \newcommand{\nepsr}{\noaps{\epsr}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} \newcommand{\Log}{\mbox{Log}} \newcommand{\Vol}{\mathrm{Vol}} \newcommand{\disp}{\displaystyle} %% %% NEWTHEOREM %% \newtheorem{defi}{Definition}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}{Theorem} \newtheorem{remark}{Remark}[section] \newtheorem{cor}{Corollary} \newtheorem{corol}{Corollary}[section] \newtheorem{prop}[lemma]{Proposition} %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% \begin{document} \title{Local density approximations for the energy of a periodic Coulomb model } %% %% AUTHORS : %% \author{ %------------------ Olivier BOKANOWSKI% \footnote{ LSMA (Univ. Paris 7) and LAN (Univ. Paris 6); C.P. 7012, Universit\'e Paris 7, 2 Place Jussieu, F--75251 Paris Cedex 05, France}, %------------------- Beno\^\i t GREBERT% \footnote{D\'epartement de Math\'ematiques, UMR 6629, Universit\'e de Nantes. 2, rue de la Houssini\`ere, 44072 NANTES Cedex 03, France.}, %------------------- Norbert J.\ MAUSER% \footnote{ Inst.\ f.\ Mathematik, Univ Wien, Strudlhofg.\ 4, A--1090 Wien, Austria.} %and %Courant Institute, 251 Mercer Street, NY-10012-1185 NY, USA} % % %------------------ } \date{02.11.2000} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{small} {\bf Abstract:} We deal with local density approximations for the kinetic and exchange energy term, $\cE_{kin}(\rho )$ and $\cE_{ex}(\rho )$, of a periodic Coulomb model. We study asymptotic approximations of the energy when the number of particles goes to infinity and for densities close to the constant averaged density. For the kinetic energy, we recover the usual combination of the von-Weizs\"acker term and the Thomas-Fermi term. Furthermore, we justify the inclusion of the Dirac term for the exchange energy and the Slater term for the local exchange potential. \end{small} \vspace{0.5 cm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Introduction} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The aim of this work is to provide a justification of the Thomas-Fermi-von Weizs\"acker and Thomas-Fermi-von Weizs\"acker-Dirac models \cite{L1} in a crystal. We use the method of deformations (local scaling transformations) \cite{PSK}, \cite{BG1}, \cite{BG3} of wave functions of constant electron density. % in the "high density limit" (\cite{LS1}, \cite{GS}, \cite{BM1}), % i.e. the limit $N \rightarrow \infty$ at constant volume of a (periodic) box % (cf. Appendix for results in the thermodynamic limit). To this end we consider a cubic crystal with $N$ electrons and $P$ nuclei in the elementary cell $\Omega =[-\frac{L}{2},\frac{L}{2}]^3$. % Note that we state all results for the 3-dimensional case only. Following \cite{CBL1}, \cite{CBL2} we regard the periodic Hamiltonian: \be \label{Hperdef} H_{per} := - \sum_{i=1}^{N}\Delta_i + \sum_{i=1}^{N}V_{ext}(x_i) + \sum_{i0,\, \int_{\Omega}\rho =N ,\, \sqrt \rho \in H^1_{loc}(\R^3) \} $$ where $C^{0,\ma}(\R^3/L\Z^3)$ denotes the space of H\"olderian functions of degree $\ma$ on the torus $\R^3/L\Z^3$ or equivalently the space of periodic function in the H\"older space $C^{0,\ma}(\R^3)$. \noindent We introduce the semi norm $\naps{\cdot}{0}$ defined by \be \label{eq:naps} \naps{\eps}{0} :=\sup_{ x\neq y \in \R^3} \frac{|\eps(y)-\eps(x)|}{|y-x|^\ma} \ . \ee \medskip \noindent Our results are based on the crucial assumption that $\rho$ is close to the averaged constant electron density $\rho_0$. Precisely, we shall assume that $L^\ma\nepsr$ is small, with the definitions % \beno \rho_0 := \frac{N}{|\mO|} = \frac{N}{L^3} \eeno and \beno \epsr (x) := \frac{\rho(x)-\rho_0}{\rho_0}\ . \eeno Extending \cite{BM1}, \cite{BGM1} we demonstrate how the heuristic idea of the ``free electron approximation'' can be used in a mathematically rigorous way by using the method of deformations (local scaling transformations) of plane waves \cite{PSK}, \cite{KL}, \cite{BG2}, \cite{BG3}. \noindent In this article we essentially prove that (see section 2 for precise statements): \begin{itemize} {\it \item The exact kinetic energy, as a functional of the density $\rho$, is equal to the Thomas-Fermi-von Weizs\"acker functional up to small remainder terms depending on~$L^\ma \eps_{\rho}$ and~$N$. \item The global energy, as a functional of the density $\rho$, is majorized by the Thomas-Fermi-von Weizs\"acker-Dirac functional up to small remainder terms depending on $L^\ma \eps_{\rho}$ and $N$.} \end{itemize} \noindent The "high density limit" (\cite{LS1}, \cite{GS}, \cite{BM1}) is obtained by letting $N \rightarrow \infty$ for a given size of the periodicity cell (i.e $L=cst.$). In this limit and for our choice of the periodic model (with an equi-distribution of the external potential), the assumption $\eps_{\rho} \rightarrow 0$, is physically reasonable. Nevertheless this assumption is not yet mathematically proved in a general context (see \cite{LS1} for a proof in the Thomas-Fermi context).\\ The "thermodynamic limit" is obtained by letting $N \rightarrow \infty$ and $L \rightarrow\infty$ in such way that $\rho_0$ is constant (see e.g. \cite{LS1}, \cite{L1}, \cite{F1} for a discussion on different types of limits). In this case, there is no physical reason for the ground state density to be close to the constant density $\rho_0$. Thus the assumption $L^\ma\eps_{\rho} \rightarrow 0$ (i.e. $N^{\ma /3}\eps_{\rho} \rightarrow 0$) does not seem so relevant in the thermodynamic limit. \noindent That is why our results should be rather considered in the "high density limit" context than in the case where $L \conv \infty$. \medskip \noindent {\it Remark on the choice of the density space:} All our results are stated for density $\rho$ in the H\"older space $C^{0,\ma}(\R^3/L\Z^3)$ and thus depend on the choice of $\ma \in (0,1)$ ($\ma$ cannot be 0). However the choice of the H\"older space is only determined by Lemma~\ref{lem:def} (where we use Schauder estimates). This deformation Lemma can be established in Sobolev spaces (essentially using reference \cite{YE} instead of \cite{DM1}). Then all the results stated in section 2 can be established for density $\rho$ in the Sobolev space $H^2(\R^3/L\Z^3)$ (i.e. two derivatives in $L^2(\R^3/L\Z^3)$). In this case the basic assumption would be "$\nlp{\epsr}{H^2}$ small" instead of "$L^{\ma}\naps{\epsr}{0}$ small". This point of view seems, {\em a priori}, to be better when we consider the thermodynamic limit (since $L \conv\infty$). Nevertheless, the $\nlp{.}{H^2}$-norm is an integrated norm and thus when $L$ grows the domain of integration also grows and the condition "$\nlp{\epsr}{H^2}$ small" becomes more restrictive. \medskip \noindent The article is organized as follows: \noindent In section 2 we precisely state our results. \noindent In section 3 we describe the deformation method (as introduced in \cite{PSK}, \cite{BG1} and \cite{BG3}). We prove a deformation Lemma, based on a fundamental result of B. Dacorogna and J. Moser \cite{DM1}, which is used, in section 4, to estimate the kinetic energy functional with respect to its values at $\rho = \rho_0$. \noindent In section 4 we use precise estimates on the number of lattice points in a ball (given by number theory) to obtain a refined estimate of $\Ekin(\rho_0)$, the kinetic energy functional at $\rho = \rho_0$. Then we deduce Theorem~\ref{theo1} and Theorem~\ref{theo1'}. \noindent In section 5 we justify the so called $X_{\alpha}$ method which allows us to approximate the exchange energy at the Hartree Fock level and then to obtain an upper bound for the global energy functional. \medskip\noindent Part of the results have been announced in \cite{BGM1}, \cite{BGM2} and \cite{BGM3}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Our first result states that the exact kinetic energy, as a functional of the density, is equal to the Thomas-Fermi-von Weizs\"acker functional \cite{L1} up to small remainder terms depending on~$\eps_{\rho}$ and~$N$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% THEOREM 1 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{theo1} Let $0<\ma <1$. For densities $\rho \in D_{\ma}$, the functional $\mathcal{E}_{kin}(\rho )$ defined in \fer{calE} has the following behaviour in a neighborhood of $\rho =\rho_0$ and $N=+\infty$ : % \be \label{Tkinfirst} \mathcal{E}_{kin}(\rho ) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\mO \rho^{5/3}dx \big\{1 + O (L^{\ma}\naps{\epsr}{0}) + O(\frac{1}{N^{1/2}})\big\} \ee % where $C_F := \frac{3}{5} (6 \pi^2)^{2/3}$ is the Fermi constant (in our context)% \footnote{In general, $C_F := \frac{3}{5} (\frac{6 \pi^2}{s})^{2/3}$, where $s$ is the spin number. In our context, $s=1$.}.\\ Furthermore there exists $\eta(\ma) >0$ such that the error terms are uniform with respect to $(N, L, \rho)$ satisfying $L^{\ma}\naps{\epsr}{0} <\eta(\ma)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% fin THEOREM %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Note that this Theorem remains true if we make another choice of a periodic Hamiltonian (and thus of the periodic model). Actually Theorem~\ref{theo1} depends only on the choice of the space $\mL$ of wave functions. However, this result becomes physically relevant only if, for the chosen model, we are able to prove that the density of the ground state is close to the constant density. \begin{remark} In the "high density limit", estimate \fer{Tkinfirst} becomes $$ \mathcal{E}_{kin}(\rho ) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\mO \rho^{5/3}dx \big\{1 + O (\naps{\epsr}{0}) + O(\frac{1}{N^{1/2}})\big\} $$ while in the thermodynamic limit, estimate \fer{Tkinfirst} becomes $$ \mathcal{E}_{kin}(\rho ) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\mO \rho^{5/3}dx \big\{1 + O (N^{\ma /3}\naps{\epsr}{0}) + O(\frac{1}{N^{1/2}})\big\}. $$ \end{remark} \begin{remark} The $O(1/N^{1/2})$ term in \fer{Tkinfirst} can be slightly improved to $O((log N)^6/N^{5/9})$ (see Remark~\ref{rem:lem1}). The same remark holds for the estimates in Theorem~\ref{theo1'} and Theorem~\ref{theo3}. \end{remark} \noindent The estimate (\ref{Tkinfirst}) is also valid locally: denoting \be \label{eq:Tloc1} \Ekinloc(\Psi)(x) &:= & N \int_{\Omega^{N-1}} |\nabla_{x}\Psi(x,x_2,\dots ,x_N)|^2 dx_2\cdots dx_N \ee we have, for any wave function $\Psi\in\mL^c$ minimizing $\cE_{kin}(\rho)$, $$ \cE_{kin}(\rho) = \int_\mO \Ekinloc(\Psi)(x)\ dx $$ and \be \label{eq:Tloc2} \Ekinloc(\Psi)(x) & = & |\nabla\sqrt{\rho}(x)|^2 + C_F \rho^{5/3}(x) \big\{1 + O (L^{\ma}\naps{\epsr}{0}) + O(\frac{1}{N^{1/2}})\big\} \: . \ee \medskip \noindent Furthermore, if we consider only wave functions which are Slater determinants, i.e. $\Psi(x_1, ..., x_N) = {\scriptstyle \frac{1}{\sqrt{N!}}} \det(\phi_j(x_i))$, with $\int_\Omega \phi_i \bar \phi_j = \delta_{ij} $, we obtain with the same assumption as in Theorem~\ref{theo1} an upper bound of order 2 in~$\eps_{\rho}$: %---------------------------- \begin{theorem}\label{theo1'} %---------------------------- Let $0<\ma <1$. For densities $\rho \in D_{\ma}$, we have: % \be \label{Tkintwo} \cE_{kin}(\rho ) \leq \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\mO \rho^{5/3}dx \big\{1 + O (L^{2\ma}\naps{\epsr}{0}^2) + O(\frac{1}{N^{1/2}})\big\} \: \ee % where $C_F := \frac{3}{5} (6 \pi^2)^{2/3}$ is the Fermi constant (in our context).\\ Furthermore there exists $\eta(\ma) >0$ such that the error terms are uniform with respect to $(N, L, \rho)$ satisfying $L^{\ma}\naps{\epsr}{0} <\eta(\ma)$. \end{theorem} Again, the upper bound \fer{Tkintwo} is valid locally, cf. Remark~\ref{rem:4.2} for a precise statement. \medskip \noindent On the other hand, the estimate (\ref{Tkinfirst}) allows us to prove, in our specific context, a well known conjecture of March and Young (cf. \cite{MY} and also \cite{L1}). Namely we have the following Corollary: %% %% COROLLAIRE %% \begin{cor} Let $0<\ma <1$. There exists $C(\ma)>0$ and $\eta(\ma) >0$ such that for any $ \rho \in D_{\ma}$ and for any $ N>0$, $L\geq 1$ satisfying $L^{\ma}\naps{\epsr}{0} <\eta(\ma)$ we have $$ \cE_{kin}(\rho ) \leq \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C(\ma) \int_\mO \rho^{5/3}dx. $$ \end{cor} %% %% %% \begin{remark}\label{rem1} An application of these techniques in the case of a system of $N$ fermions in $\R^3$ faces the problem to find an equivalent of $\rho_0$ in the non periodic case. Note that in \cite{BG1} we have proved the estimate $$ \Ekin(\rho) \leq C N^{2/3} \int_{\R^3} |\nabla\sqrt{\rho}|^2 dx $$ by deforming $\R^3$ onto the unit cube. \end{remark} \begin{remark}\label{rem2} We recently learned about a similar approach in \cite{DR}, based, however, on more formal estimations. \end{remark} \medskip \noindent Our second result concerns the local exchange potential occurring in the $X{\alpha}$ method as an approximation of the exchange potential $V_{ex}$ for the Hartree-Fock model. We define the Hartree Fock energy as follows: \be \label{1.1} E^{HF}:= \inf \left\{ \langle H_{per} \Psi,\ \Psi \rangle\mid\ \Psi \in \mL^c,\ \Psi= [\psi_j],\ \langle \psi_i,\psi_j \rangle = \delta_{ij} \right\} \ee where for a given family of $N$ functions (orbitals) $\psi_1,\dots,\psi_N$ in $L^2(\mO)$, $[\psi_j]$ denotes the following $N$-particle wave function called Slater determinant % \be \label{slatdet-def} [\psi_j](x_1,\dots,x_N):= \frac{1}{\sqrt{N!}} det(\psi_j(x_i))_{1 \leq i,j \leq N}. \ee % \noindent Since the constraints $\langle \psi_i,\psi_j \rangle = \delta_{ij}$, $1\leq i,j\leq N$ implies that $ ||\Psi||_{L^2}=1 $, we have $E_0\leq E^{HF}$. \noindent In this Hartree-Fock context of such Slater determinants the electron density writes $\rho_\Psi(x):= \sum_{j=1}^N |\psi_j(x)|^2$. The inter-electron energy \fer{Eee} can be decomposed into two terms (see \cite{PY}): \be \label{Eee1} E_{ee}(\Psi) = J(\rho_{\Psi}) + E_{ex}([\psi_j]) \ee with \be \label{J} J(\rho) := \frac{1}{2} \int_{\Omega^2} \rho(x) \rho(x')G(x,x') dx dx', \ee \be \label{1.21} E_{ex}([\psi_j]) := - \frac{1}{2} \int_{\Omega\times\Omega} |D(x,y,[\psi_j])|^2 G(x-y)\,dx\,dy\ , \ee and where \be \label{dens-matrix} D(x,y,[\psi_j]) :=\sum_{j=1}^N \psi_j(x)\overline{\psi_j}(y) \ee denotes the density matrix. Note that we stress explicitly the dependence on the orbitals entering via a Slater determinant by the notation $D(x,y,[\psi_j])$. \noindent In \fer{Eee1}, $J$ corresponds to the "classical" electrostatic self-repulsion energy, and only depends on the density $\rho$. The second term, the so-called exchange energy $E_{ex}$, takes into account the Pauli principle (a purely quantum effect) but {\em a priori} does not depend only on the density. \medskip \noindent The existence of a minimum for (\ref{1.1}), which is a difficult problem when posed on $\R^3$ \cite{LS2}, \cite{PLL}, can be more easily proved here since $\Omega$ is compact. This minimization gives (after a unitary transformation on the orbitals) the so-called Hartree-Fock equations % \be \label{1.3} -\Delta \psi_i + V_{ext}(x) \psi_i + (\frac{1}{|x|}*\rho)\,\psi_i + (V_{ex}\psi_i)(x) = %\sum_{j=1}^N \epsilon_{ij} \psi_j \epsilon_{i} \psi_i \ee % %where $(\epsilon_{ij})$ is a constant hermitian matrix and where $(\epsilon_{i})$ are the eigenvalues and where $V_{ex}$ is a non-local operator, defined by: \be \label{eq:vex} (V_{ex}\psi_j)(x) := - \int_\Omega D(x,y,[\psi_j]) G(x-y)\psi_j(y) \,dy . \ee % % There is no exact local expression for the complicated {\em exchange potential} $V_{ex}$. However, $V_{ex}$ can be astonishingly well approximated by $ -C_S \rho ^{1/3}(x)$ (for some constant $C_S$) as proposed by Slater \cite{Sl} and widely used under the name "$X{\alpha}$ method". We refer to \cite{PY} for a review of such approximations. A first mathematical approach to this Slater approximation based on deformations of plane waves has been given in \cite{BM1}. Following \cite{Sl} we first approximate the exact exchange potential $V_{ex}$ by the {\em average exchange potential} $V_{av}$: % \be \label{Vavdef} V_{av}(x,[\psi_j]):= - \int_{\Omega} \frac{|D(x,y,[\psi_j])|^2}{\rho(x)} G(x-y)\,dy. \ee % This formula comes from the "Slater averaging" of the HF exchange potential $(V_{ex}\psi_i)(x)$ by the weighted densities of the $i$-th wave function : $ \sum_{j=1}^N (V_{ex}\psi_i)(x) \frac{|\psi_i|^2}{\rho(x)}$. %of $ \frac{(V_{ex}\psi_i)(x)}{\psi_i(x)}$) The advantage of $V_{av}$ is that it can be used as a "local" approximation of (\ref{1.3}) : $ (V_{ex}.\psi_k)(x) \sim V_{av}(x,[\psi_j]) \psi_k(x) $. Furthermore, we can recover the exact exchange energy from $V_{av}$, even if $V_{av}$ is an approximation, since we have from (\ref{1.21}), (\ref{Vavdef}) : % \be \label{1.5} E_{ex}([\psi_j]) = \frac{1}{2} \int_\Omega \rho(x) V_{av}(x,[\psi_j])\ dx \ . \ee % \noindent The following asymptotic results is proved in section 5: % %------------------------------- THEOREM 2 % \begin{theorem}[Averaged exchange potential estimate] \label{theo2} Let $0<\alpha <1$. For each $\rho \in D_{\alpha}$ sufficiently close to $\rho_0$, and for a particular choice \footnote{This choice is explicit, cf. Section 5.1.} of orthonormals orbitals $(\psi_j)_{j=1,\dots,N}$ with $\rho=\sum_{j=1}^N |\psi_j|^2$, we have uniformly for $x\in \Omega$, % \be \label{Vav-result} V_{av}(x,[\psi_j]) = - C_S\ \rho(x)^{1/3} \bigg\{ 1 + O((L^\ma\nepsr)^\beta) + O(\frac{1}{N^{1/3}}) \bigg\} \ee % where $\beta=\min(2,\frac{1}{1-\ma})$ and $C_S=\frac{3}{2} (\frac{6}{\pi})^{1/3}$ is the "Slater constant" \cite{Sl} (in our context).% \footnote{In general, $C_S := \frac{3}{2} (\frac{6}{s\pi})^{1/3}$, where $s$ is the spin number (for $s=1$ or $s=2$); in our context, $s=1$.} \end{theorem} % %------------------------------- As a consequence of Theorem \ref{theo2} and of \fer{1.5}, we obtain immediately: \begin{cor}[Exchange energy estimate]\label{cor:2} Under the same assumptions as in Theorem~\ref{theo2}: $$ E_{ex}([\psi_j]) = - \frac{C_S}{2} \ \int_{\Omega} \rho ^{4/3}(x)\ dx \ \bigg\{1 + O ((L^\ma \nepsr)^\beta)+ O(\frac{1}{N^{1/3}})\bigg\} $$ \end{cor} \noindent \begin{remark} Since $1/(1-\ma)>1+\ma$, Theorem~\ref{theo2} and Corollary~\ref{cor:2} still hold with $\beta=1+\ma$. \end{remark} \noindent This is a refined version of a result in \cite{BM1}. We use the work of Friesecke \cite{F1} in order to deal with planes waves whose wave numbers are in the Fermi sphere (instead of the cube used for simplicity in \cite{MY}, \cite{BM1}). \noindent Note also that Corollary~\ref{cor:2} gives a justification of the Dirac approximation $\int_{\Omega} \rho ^{4/3} dx$ (see \cite{D1}) of the exchange energy. In an other context, justification of the Dirac term (and also the Thomas-Fermi term) have also been obtained, see V. Bach~\cite{B}, C. Fefferman and L.A. Seco~\cite{FS}, G.M. Graf and J.P. Solovej~\cite{GS}. In \cite{B} and \cite{FS} the authors consider a Coulomb model in $\R^3$ and the Thomas-Fermi density $\rho_{TF}$ plays the role of $rho_0$. \medskip \noindent Finally our method of deformation gives an approximation of the energy $\cE(\rho)$. Nevertheless, we have to restrict ourself to wave functions of Slater determinant type which are deformation of plane waves . This is why we only obtain an upper bound for the global energy. Combining the kinetic energy and exchange potential estimations, we obtain in section 5: %% %% THEOREM 3 : Global energy %% \begin{theorem}\label{theo3} Let $0<\alpha <1$. For densities $\rho \in D_{\alpha}$, the functional $\cE(\rho )$ admits the following upper bound in a neighborhood of $\rho =\rho_0$ and $N=+\infty$: % \be \label{eq:global} \cE(\rho) & \leq & \int_{\Omega} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\Omega \rho^{5/3}(x)dx \bigg\{ 1 + O (L^{2\ma}\nepsr^2) + O(\frac{1}{N^{1/2}})\bigg\} \\ & & +\int_{\Omega} V_{ext}(x) \rho (x) dx + J(\rho) - \frac{C_S}{2} \int_{\Omega} \rho ^{4/3}(x) dx \bigg\{ 1 + O ((L^{\ma}\nepsr)^\mb) + O(\frac{1}{N^{1/3}})\bigg\} \nonumber \ee where $ J(\rho) = \frac{1}{2} \int_{\Omega^2} \rho(x) \rho(y)G(x,y)\ dx dy$ is the {\em Coulomb energy}, $C_F :=\frac{3}{5} (6 \pi^2)^{2/3}$ is the Fermi constant and $C_S :=\frac{3}{2} (\frac{6}{\pi})^{1/3}$ is the "Slater" constant (in our context). \\ Furthermore there exists $\eta(\alpha) >0$ such that the error terms are uniforms with respect to $(L, N, \rho)$ satisfying $L^{\ma}\nepsr{\epsr} <\eta(\alpha)$ and $L \geq 1$. \end{theorem} %% %% fin THEOREM %% \begin{remark}\label{rem:glob} In the "high density limit" ($L$ fixed), estimate \fer{eq:global} becomes (cf. Section 5.2): \beno \cE(\rho) & \leq & \int_{\Omega} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\Omega \rho^{5/3}(x)dx \bigg\{ 1 + O (\nepsr^2) + O(\frac{1}{N^{1/2}})\bigg\} \\ & & +\int_{\Omega} V_{ext}(x) \rho (x) dx + J(\rho) - \frac{C_S}{2} \int_{\Omega} \rho ^{4/3}(x) dx \nonumber \eeno while in the thermodynamic limit ($\rho_0$ fixed), estimate \fer{eq:global} becomes \beno \cE(\rho) & \leq & \int_{\Omega} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\Omega \rho^{5/3}(x)dx +\int_{\Omega} V_{ext}(x) \rho (x) dx \\ & & + J(\rho) - \frac{C_S}{2} \int_{\Omega} \rho ^{4/3}(x) dx \bigg\{ 1 + O ((N^{\ma/3}\nepsr)^\mb) + O(\frac{1}{N^{1/3}})\bigg\} \nonumber \eeno \end{remark} \begin{remark}\label{rem3} The error terms in Theorem~\ref{theo3} can also be bounded as follows: \beno \int_\Omega \rho^{5/3} \bigg\{ O(L^{2\ma}\nepsr^2) + O(\frac{1}{N^{1/2}}) \bigg\} \leq cst.\ \frac{N^{5/3}}{L^2}\ \bigg\{ L^{2\ma}\nepsr^2 + \frac{1}{N^{1/2}} \bigg\} \eeno % When compared with our result announced in \cite{BGM1}, the leading error term in the TFvW approximation of the kinetic energy is improved (we obtain $O(\frac{1}{N^{1/2}})$ instead of $O(\frac{1}{N^{1/3}})$ in \cite{BGM1}). This is technically more complicated but essential for combining it with the Dirac term, $\int \rho^{4/3} \sim cst.\,\frac{N^{4/3}}{L}$ which is now asymptotically larger than the improved error term, $ O(\frac{N^{5/3}}{L^2} \frac{1}{N^{1/2}}) = O(\frac{N^{7/6}}{L^2})$ (independently of $L \geq 1$). \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Deformations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Our crucial assumption is that $\rho$ is close to the averaged constant electron density $\rho_0$. \noindent Indeed, we shall start from wave functions of constant density $\rho_0$, and then use a deformation of the space close to identity in order to obtain wave functions of density $\rho$ (close to $\rho_0$). \noindent This idea has already been used in Density Functional Theory, see \cite{PSK,KL} or \cite{BG3,BG1,BG2}, or \cite{DR}. \begin{defi} \label{Def123} We say that $f$ is a periodic deformation on the cube $\mO$ with sidelenght $L$ if $f$ is a $C^1$ diffeomorphism on $\R^3$ satisfying $f(x+L\,m) = f(x) + L\,m$ for any $m\in \Z^3$ and $x\in\R^3$. This means that $f$ is a $C^1$ diffeomorphism of the torus $\R^3/L\Z^3$. \\ Further we denote $J_f(x):=\det(D f(x))$ the Jacobian of $f$. \end{defi} \noindent We use the following H\"older semi norm on $C^{0,\ma}(\R^3)$ $$ \naps{a}{0}:= \sup\{ \frac{|a(x) -a(y)|}{|x-y|^{\ma}}\ \mid \ x\neq y \in \R^3 \} $$ \noindent Based on a fundamental result of B. Dacorogna and J. Moser~\cite{DM1} we prove %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Lemme deformations (Lemma 3.1) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma} \label{lem:def} Let $0<\ma <1$ and $L\geq 1$. There exist $\eta (\ma) >0$ and $K(\ma)>0$ such that, for any $\rho \in D_{\ma}$ satisfying $L^{\ma}\naps{\epsr}{0} <\eta(\ma)$, there exists a periodic deformation of the cube $[-L/2,L/2]^3$, $f$, solution of the Jacobian equation $J_f(x)=\frac{\rho(x)}{\rho_0}$.\\ Furthermore $f\in C^{1,\ma}(\R^3)$ and $f$ satisfies \be \label{estif} \noaps{D(f-Id)} \leq K(\ma)\ \nepsr, \ee \be \label{estif2} \nlp{D(f-Id)}{\infty} \leq (\sqrt{3}L)^{\ma}K(\ma)\ \nepsr. \ee \end{lemma} \begin{remark} At this point we need to assume that $\rho$ is H\"older continuous since we want to use Schauder estimates (see below). \end{remark} \begin{remark} Following \cite{DM1}, we can prove the existence (but not estimate \fer{estif}) of the periodic deformation in Lemma \ref{lem:def} without assuming $\noaps{\epsr}$ small. \end{remark} \noindent Before proving Lemma~\ref{lem:def} we explain how we use it in order to deform a wave function of density $\rho_0$ into a wave function of density $\rho \in D_\ma$. \noindent Let $\rho $ be a density in $D_{\ma}$, $f$ be the periodic deformation associated to $\rho$ given by Lemma~\ref{lem:def} and $\Psi$ be a ($N$-particle) wave function in $\mL^c$. Then $(J_f)^{1/2}$ and $\Psi(f(x_1),\dots,f(x_N))$ are both in $H^1 \cap C^0$ and we define the deformed wave function $T_f(\Psi)$ by \be \label{def:Tf} T_f\Psi(x_1,\dots,x_N) := \prod_{j=1}^{N}(J_f(x_j))^{1/2} \Psi(f(x_1),\dots,f(x_N)), \ee which is again in $\mL^c$. By a straightforward change of variables one gets $$ \rho_{_{T_f\Psi}}(x) = \frac{\rho (x)}{\rho_0} \rho_{\Psi}(f(x)). $$ As in \cite{BG1}, one easily deduces \begin{lemma} \label{Tf:isometry} For $\rho \in D_{\ma} $, the operator $T_f$ induces an isometry (for the $L_2$ norm) from \linebreak $\{ \Psi \in \mL^c \ \mid \ \rho_{\Psi}= \rho_0\}$ onto $\{ \Psi \in \mL^c \ \mid \ \rho_{\Psi}=\rho\}$. \end{lemma} \noindent {\em Proof of Lemma~\ref{lem:def}.} We follow closely the lines of the proof of \cite{DM1} Lemma 4. \noindent Let $\T_L$ be the torus $\T_L:=\R^3/L\Z^3$. We define $$ \cX:=\{ b\in C^{0,\ma}(\T_L, \R)\ | \ \int_\Omega b =0\} $$ and $$ \cY:=\{v\in C^{1,\ma}(\T_L, \R^3)\ | \ \int_\mO v =0 \}. $$ Note that a function in $\cX$ has to vanish somewhere in $\mO$ and therefore the $\naps{\cdot}{0}$ semi norm is a norm on $\cX$. In the same way, if $v \in \cY$ then $v$ and any of its partial derivatives have to vanish somewhere in $\mO$ (use the periodicity). Thus the semi norm $\naps{Dv}{0}$ is a norm on $\cY$ that we will denote $\nlp{v}{\cY}$: $$ \nlp{v}{\cY}:= \naps{Dv}{0}\ . $$ We note for the sequel that if $u\in C^{0,\ma}$ with $\int_{\Omega}u =0$, then \be\label{normes1} \nlp{u}{\infty} \leq (\sqrt{3}L)^{\ma}\naps{u}{0} \ee % and in particular if $v\in \cY$ \be\label{normes} \nlp{Dv}{\infty} \leq (\sqrt{3}L)^{\ma}\nlp{v}{\cY} \ee For $b\in \cX$, let $a\in C^{2,\ma}(\T)$ be the unique solution of the Laplace equation $$ \Delta a = b $$ satisfying $\int_\Omega a =0$. By Schauder estimates (see for instance \cite{Lady}), there exists a constant $C= C(\ma,L)$ such that $$ \naps{D^2 a}{0} \leq C\ (\naps{b}{0} + \nlp{b}{\infty}+ \nlp{a}{\infty})\ . $$ Considering the Fourier series representation of $a$ and $b$ ( $a(x) = \sum_{k\in \frac{2\pi}{L} \Z^3} \hat{a}(k)e^{ikx}$ and $b(x) = \sum_{k\in \frac{2\pi}{L} \Z^3} \hat{b}(k)e^{ikx}$ ) the relation $\Delta a = b $ with $\int_{\Omega}a =0$ writes $$ \hat{a}(k) =\frac{1}{k^2}\hat{b}(k)\ \mbox{ for } k\neq 0 \mbox{ and } \hat{a}(0) =0 \ . $$ Therefore, with $C'=C'(L) = \left( \sum_{k\in \frac{2\pi}{L} \Z^3} \frac{1}{k^4} \right)^{1/2}$ $$ \nlp{a}{\infty}\leq \sum_{k\in \frac{2\pi}{L} \Z^3}\frac{|\hat{b}(k)|}{k^2} \leq C' \left( \sum_{k\in \frac{2\pi}{L} \Z^3}|\hat{b}(k)|^2 \right)^{1/2} = C'L^{-3/2} \nlp{b}{L^2} \leq C' \nlp{b}{\infty} \ . $$ Thus using \fer{normes1} one concludes that there exists a constant $K= K(\ma,L)$ such that \be\label{eq:lady} \naps{D^2 a}{0} \leq K\ \naps{b}{0} \ . \ee Note that $K$ does in fact not depend on $L$ as can be verified by a scaling argument:\\ Let $\tilde{a}(x):= \left( \frac{1}{L}\right)^2 a\left(\frac{1}{L} x\right)$ and $\tilde{b}(x):=b\left(\frac{1}{L} x\right)$. Then $\Delta \tilde{a} = \tilde{b}$ on $\mO_1=[-1/2,1/2]^3$ and $\int_{\mO_1} \tilde{a}=0$. Thus $\naps{D^2 \tilde{a}}{0} \leq K(\ma,1)\ \naps{\tilde{b}}{0}$ and as $\naps{D^2 a}{0}=\left( \frac{1}{L}\right)^{\ma}\naps{D^2 \tilde{a}}{0}$ and $\naps{b}{0}= \left( \frac{1}{L}\right)^{\ma}\naps{\tilde{b}}{0}$, one obtains \fer{eq:lady} with $K(\ma) = K(\ma,1)$. \medskip \noindent Note that $v=\nabla a$ is in $\cY$ and satisfies $ \mathrm{div}\,v= b$. We can then define a bounded linear operator $\cL: \cX \rightarrow \cY$ which associates to every element $b$ in $\cX$ an element $v=\cL(b)$ in $\cY$ satisfying $$ \mathrm{div}\,v = b $$ and \be \label{estim:L} \nlp{\cL(b)}{\cY} = \naps{Dv}{0} \leq K(\ma)\,\naps{b}{0}\ . \ee In order to solve $ J_{f} =\rho/\rho_0 = 1 + \epsr$, we look for $f(x)$ in the form $$ f(x)= x + v(x)\ . $$ Hence the Jacobian equation on $f$ is equivalent to the problem of finding a vector field $v(x)$ such that \beno \mathrm{div}(v) + Q(Dv) = \epsr \eeno where for any $3\times 3$ matrix $A$, \be \label{Q} Q(A) := \det(Id + A)-1-\mbox{tr}(A). \ee Now we define \be \label{NN} \cN(v) := \epsr - Q(Dv) \ee and we remark that a solution of the Jacobian problem, $\mathrm{div}(v)= \cN(v)$, is obtained from the following fixed-point equation % \be \label{v} v = \cL\cN(v) . \ee % \medskip \noindent Note that denoting $v=(v_1,v_2,v_3)$, \be \label{Q2} Q(Dv)= \det (Dv) + \sum_{1\leq i 0$ we define $\cN (r)$ as the number of discrete points in a ball as follows \beno \cN (r) = \# \left[\Z^3 \cap B(0,r) \right] \eeno where $B(0,r)$ denotes the Euclidean ball with center $0$ and radius $r$. % The function $r \rightarrow \cN (r)$ is increasing with values in $\N$. Let $(N_j)_{j\in \N}$ be the increasing sequence of values of $\cN (r)$ and $r_j$ ($j\in \N$) be the minimal value of $r$ such that $\cN (r) = N_j. $ \noindent {}From \cite{Sk}, we learn the following two non-trivial estimates (cf. \cite{Hl} and \cite{He} for the first estimate): % \be \label{estim Nj} N_j = \frac{4}{3} \pi r_j^3 + O(r_j^{3/2}) \: \ee and % \be \label{estimdif Nj} N_{j+1} - N_j = O(r_j^{3/2}) \: . \ee (Note that $N_{j+1} - N_j$ is equal to the number of points of $\Z^3$ on the sphere $S(0, r_{j+1})$.) \noindent Let $N \in \N$ given. There exists $j\in \N$ such that $N_j \leq N < N_{j+1}$ and thus by \fer{estim Nj} and \fer{estimdif Nj}, \be \label{estim N} N_j= N + O(\sqrt{N}) \: . \ee Using \fer{estimdif Nj} and \fer{eq:T0def} we have, $$ T_0(N_{j+1}) - T_0(N_j)\leq r_{j+1}^2 \, O(r_j^{3/2})\: . $$ Thus, as $T_0(N_j)\leq T_0(N) \leq T_0(N_{j+1})$, we conclude \be \label{estim TN1} T_0(N) = T_0(N_j) + O(r_j^{7/2})\: . \ee \noindent It remains to calculate $T_0(N_j)$. We define \be \label{eq:Kj} K_{N_j} := B(0, r_j) \cap \Z^3 \ . \ee Denote $Q:= [-1/2, 1/2]^3$, $Q_k = Q + k$ ($k\in \Z^3$) and $D_j := \cup_{k \in K_{N_j}}\ Q_k.$ By a direct calculation we have, \be \label{estim} T_0(N_j) & = & \sum_{k \in K_{N_j}} |k|^2 =\sum_{k \in K_{N_j}} (\int_{Q_k} |u|^2 du - \frac{1}{4} ) \nonumber \\ &= &\int_{D_j} |u|^2 du - \frac{1}{4} N_j \: . \ee % On the other hand denoting by $B_r$ the ball in $\R^3$ of center $0$ and radius $r$, we have $$ B_{r_j -\sqrt{3}/2 } \subset D_j \subset B_{ r_j +\sqrt{3}/2 } \: . $$ Therefore \beno \hspace{-1cm}\bigg|\int_{D_j}|u|^2 du - \int_{B_{r_j}}|u|^2 du\bigg| & = & \bigg|\int_{D_j \backslash B_{r_j -\sqrt{3}/2}} |u|^2 du - \int_{B_{r_j} \backslash B_{r_j -\sqrt{3}/2} } |u|^2 du \bigg| \eeno \beno \hspace{2cm} & \leq & \max_\pm \bigg|(r_j \pm \sqrt{3}/2)^2 \Vol( D_j \backslash B_{r_j -\sqrt{3}/2}) - (r_j \mp \sqrt{3}/2)^2 \Vol(B_{r_j}\backslash B_{r_j-\sqrt{3}/2})\bigg| \\ & \leq & r_j^2 \left| \Vol( D_j \backslash B_{r_j -\sqrt{3}/2}) - \Vol(B_{r_j} \backslash B_{r_j -\sqrt{3}/2})\right| \\ & & \ \ + \ O(r_j)\ \left(\Vol(D_j \backslash B_{r_j -\sqrt{3}/2}) + \Vol (B_{r_j} \backslash B_{r_j -\sqrt{3}/2})\right) \\ & \leq & r_j^2 \left|\Vol(D_j) -\Vol (B_{r_j})\right| + O (r_j^3)\ . \eeno % As $\Vol(D_j)= N_j$, we conclude, using \fer{estim Nj}, that \be \label{estim2} \big|\int_{D_j}|u|^2 du - \int_{B_{r_j}}|u|^2 du\big| = O(r_j^{7/2})\: . \ee Furthermore, a simple calculation gives, \be \label{int-boule} \int_{B(o, r_j)}|u|^2 du = \frac{4 \pi}{5}r_j^5 \: . \ee \noindent Combining \fer{estim}, \fer{estim2} and \fer{int-boule}, we obtain, % \be \label{tnj} T_0(N_j) = \frac{4 \pi}{5}r_j^5 + O(r_j^{7/2})\: . \ee % Then using successively \fer{estim TN1}, \fer{tnj}, \fer{estim Nj} and \fer{estim N} we get, % \beno T_0(N) &= &T_0(N_j) + O(r_j^{7/2})\\ & = &\frac{4 \pi}{5}r_j^5 + O(r_j^{7/2})\\ & = &\frac{4 \pi}{5} (\frac{3 }{4\pi})^{5/3} N_j^{5/3} + O(N_j^{7/6})\\ & = &\frac{3 }{5} (\frac{3 }{4\pi})^{2/3} N^{5/3} + O(N^{7/6}) \: . \eeno \carre \begin{remark}\label{rem:lem1} Using \cite{V}, the error term $O(r^{3/2})$ in \fer{estim Nj} can be improved to $O(r^{4/3} ( \log r)^6)$. Also using \cite{Sk}, the error term $O(r^{3/2})$ in \fer{estimdif Nj} can be improved to $O(r^{1+\eta})$ (for any given $\eta>0$). These improvements lead to the following estimate for $T_0(N)$: $$ T_0(N) = \frac{1}{(2\pi)^2} C_F\ N^{5/3} \big( 1+ O\big( \frac{(\log N)^6}{N^{5/9}}\big) \big) \: . $$ \end{remark} %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of Theorem~\ref{theo1'}} %% %% \noindent To prove Theorem~\ref{theo1'} we would like to use Lemma~\ref{lem:Ekin2} and thus we first need to "symmetrize" $K$. \noindent Notice that by definition, for each $N_j$ the set $K_{N_j}$ is symmetric (cf. Definition~\ref{Ksym}). In particular, we can summarize formulas \fer{eq:t00}, \fer{estim N} and \fer{estim TN1} as follows: % \begin{lemma}\label{lem:2} For any $N\in \N^*$ there exists $n\in \N^*$ and a symmetric subset of $\Z^3$, $K_{n}$, of cardinal~$n$, such that $$ T(N) = (1+ O(\frac{1}{\sqrt{N}})) \sum_{k\in K_{n}} |k|^2 $$ and $$ n \leq N \leq n + O(\sqrt{N})\: . $$ \end{lemma} {\em Proof.} \ \ It suffices to use $n=N_j$ where $N_j$ is defined by as in the proof of Lemma \ref{lem:1} (i.e. such that $N_j\leq N < N_{j+1}$) and $K_n=K_{N_j}$ as defined in \fer{eq:Kj}. \carre \medskip \noindent This Lemma allows us to prove Theorem~\ref{theo1'}. \medskip \noindent {\em Proof of Theorem~\ref{theo1'}.}\ \ Let $\rho \in D_\ma$ be the density of an $N$-wave function with $N\in \N$ fixed.\\ Let $K\subset \Z^3$ be a minimizer for $T_0(N)$. Let $K_n \subset \Z^3 $ as in Lemma~\ref{lem:2}. In particular $K_n\subset K$ and $Card(K\backslash K_n) = O(\sqrt{N})$. Let $f$ be the periodic deformation constructed in Lemma~\ref{lem:def}. As $\Psi^f_K$ has density $\rho$, we have % \beno \Ekin (\rho) \leq E_{kin} (\Psi^f_K). \eeno % In the case $K = K_n$ (i.e. if $K$ is symmetric), we obtain, using Lemma~\ref{lem:Ekin2}, that \beno E_{kin}(\Psi^f_K) & = & \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + E_{kin}(\Psi_K)(1 + O(L^{2\ma}\nepsr^2)) \: . \eeno Then using Proposition~\ref{prop:Ekin0} we conclude (with $C_F := \frac{3}{5} (6 \pi^2)^{2/3}$), \beno E_{kin}(\Psi_K^f) & = & \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_{\mO} \rho_0^{5/3}\ (1 + O(\frac{1}{\sqrt{N}}) + O(L^{2\ma}\nepsr^2)) \label{eq:e1} \\ & = & \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_{\mO} \rho^{5/3}\ (1 + O(\frac{1}{\sqrt{N}}) + O(L^{2\ma}\nepsr^2)) \label{eq:e2} \eeno % where we used that $ \int_{\mO} \epsr = 0$ which implies \be\label{eq:rho-rho0} \int_{\mO} \rho^{5/3}=\int_{\mO} \rho_0^{5/3} \ (1 + O(L^{2\ma}\nepsr^2)) \: . \ee Hence inequality \fer{Tkintwo} follows. \medskip \noindent In the general case $K\neq K_n$, we cannot use Lemma~\ref{lem:Ekin2} but we still have, % \be\label{estim'} \Ekin (\rho) &\leq & E_{kin} (\Psi^f_K) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx \, +\, S(f,\Psi_K) \ee where following \fer{Tkinpsiest1}, \beno S(f,\Psi_K)= \int_\mO J_f(x)\ \sum_{k \in K} | Df(x)^T \frac{2 \pi}{L} k|^2\ dx\ . \eeno % We can decompose $S(f,\Psi_K)$ as follows: \be \label{S} S(f,\Psi_K)=S_1(f,\Psi_K) + S_2(f,\Psi_K) \ee with $$ S_1(f,\Psi_K) :=\frac{1}{|\mO|} \int_\mO J_f(x)\ \sum_{k \in K_n} | Df(x)^T \frac{2 \pi}{L} k|^2\ dx\ . $$ and $$ S_2(f,\Psi_K) :=\frac{1}{|\mO|} \int_\mO J_f(x)\ \sum_{k \in K\backslash K_n} | Df(x)^T \frac{2 \pi}{L} k|^2\ dx\ . $$ \noindent Notice that $S_1(f,\Psi_K)= S(f,\Psi_{K_n})$. Therefore, as $K_n$ is symmetric, using Lemma~\ref{lem:Ekin2}, $$ S_1(f,\Psi_K) = E_{kin}(\Psi_{K_n})(1 + O(L^{2\ma}\nepsr^2)). $$ Using $E_{kin}(\Psi_{K_n})=(\frac{2 \pi}{L})^2\ \sum_{K_n}|k|^2$ and Lemma~\ref{lem:2}, we have: $$ S_1(f,\Psi_K) = (\frac{2 \pi}{L})^2\ T(N) \left( 1 + O(\frac{1}{\sqrt{N}})+ O(L^{2\ma}\nepsr^2) \right) \ . $$ Using \fer{eq:TNdef0} and Proposition~\ref{prop:Ekin0}, we obtain \be \label{S1} S_1(f,\Psi_K)\ =\ C_F \int_{\mO} \rho^{5/3}\ \left( 1 + O(\frac{1}{\sqrt{N}})+ O(L^{2\ma}\nepsr^2) \right) \ee where we have used again~\fer{eq:rho-rho0}.\\ On the other hand, there exists a constant $C$ independent of $N$ such that, $$ |S_2(f,\Psi_K)|\ \leq \ C\sum_{k \in K\backslash K_n} |k|^2 . $$ Using the fact that $\# K\backslash K_n = O(\sqrt{N})$ and $K_n = \Z^3 \cap B(0,r)$ with $r = O(N^{1/3})$ we get, % \be\label{S2} S_2(f,\Psi_K) = O(N^{7/6}) = O(\frac{1}{N^{1/2}}) \int_{\mO} \rho^{5/3} \ . \ee % Finally, combining \fer{estim'}, \fer{S}, \fer{S1} and \fer{S2} we obtain \be \label{kin:pfk} E_{kin} (\Psi^f_K) & = & \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_{\mO} \rho(x)^{5/3}\,dx\ \left(1 + O(\frac{1}{\sqrt{N}})+ O(L^{2\ma}\nepsr^2)\right) \ee which in particular, gives \fer{Tkintwo}. \carre \begin{remark}\label{rem:4.2} As in Remark~\ref{rem:4.1}, using $ \rho^{5/3} = \rho_0^{5/3}\left(1 + \frac{5}{3}\epsr + O (\nepsr^2) \right) $ instead of \fer{eq:rho-rho0}, we can prove the following local estimate (where $K$ and $f$ are defined as above): \beno \Ekinloc(\Psi_K^f)(x) & = & \sum_{j=1}^N |\nabla \psi_j^f(x)|^2 \\ & = & |\nabla\sqrt{\rho}(x)|^2 + C_F \rho^{5/3}(x) \big\{1 + O (L^{2\ma}\nepsr^2) + O(\frac{1}{N^{1/2}})\big\} \: . \eeno \end{remark} % % sec 5. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Justification of the TFvWD model and of the $X_{\alpha}$ method} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we prove Theorem~\ref{theo2} and Theorem~\ref{theo3}. \subsection{Proof of Theorem~\ref{theo2}} %--------------------------------------- Let $K=K_N$ be a set of $N$ wave vectors $k_j\in\Z^3$ %with $\# K=N$ minimizing $T_0(N)$ (see \fer{eq:T0def}) and $\Psi_K :=[\psi_j]$ be the associated Slater determinant (see \fer{eq:plane-waves}). Let $\rho$ be in $D_\ma$ with $L^\ma \nepsr< \eta_\ma$ and $f$ be the deformation defined in Lemma~\ref{lem:def}. Let $\Psi_K^f=[\psi_j^f]$ be the Slater determinant associated to the deformed plane waves (cf \fer{psijdef} - \fer{eq:dpw}).\\ We now prove that % \be \label{estim:Vav} V_{av}(x,[\psi_j^f]) = - C_S\ \rho(x)^{1/3} \bigg\{ 1 + O((L^\ma\naps{\epsr}{0})^{\beta}) + O(\frac{1}{N^{1/3}}) \bigg\}. \ee Recall that (cf. \fer{Vavdef}) $$ V_{av}(x_0,[\psi_j^f]) = - \int_{\Omega} \frac{|D(x_0,x,[\psi_j^f])|^2}{\rho(x_0)} G(x_0-x)\,dx $$ where in view of \fer{eq:dpw} and \fer{dens-matrix}, $$ D(x,y,[\psi_j^f])=J_f(x)^{1/2}\ J_f(y)^{1/2}\ D_K(f(x)-f(y)) $$ with the notation $$ D_{K}(h) := \frac{1}{|\mO|} \sum_{k\in K} e^{i\frac{2\pi}{L}\, k\cdot h} \ . $$ Thus by the change of variables $y=f(x)$ one obtains with $y_0=f(x_0)$: $$ V_{av}(x_0,[\psi_j^f])= - \rho_0^{-1} \int_{\mO} |D_K(y_0 - y)|^2 G(f^{-1}(y_0)- f^{-1}(y))\,dy $$ where we have used that $f$ satisfies the Jacobian equation $J_f(x)=\rho(x)/\rho_0$. Finally denoting $y=y_0+h$, and using the periodicity of $G$ and $D_K$, one has $$ V_{av}(x_0,[\psi_j^f])= - \rho_0^{-1} \int_{\mO} |D_K(h)|^2 A_f(x_0;h)\ dh $$ where $$ A_f(x_0;h):= G(f^{-1}(y_0)- f^{-1}(y_0+h)). $$ \noindent In order to prove \fer{estim:Vav}, it remains to find an asymptotic for $A_f(h)$ (this is done in Lemma \ref{lem:A}) and to find an approximation for $D_K$ (this is done in Lemma~\ref{lem:D}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Asymptotic for $A(h)$. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent Let $X$ denote the vector field in $\cY$ such that (cf. proof of Lemma~\ref{lem:def}) $$ f(x)=x+X(x). $$ \begin{lemma} \label{lem:A} %----------- Uniformly with respect to $h\in \mO$ and $x\in \mO$ we have: \beno A_f(x;h) = \frac{1}{|h|} \left\{ 1 + \frac{\< h ,\,DX(x)h \>}{|h|^2} + O( (L^\ma\nepsr)^{\beta} ) + O(\frac{|h|}{L}) \right\} . \eeno where $\beta=\min(2,\frac{1}{ 1-\ma})$. \end{lemma} \noindent {\em Proof of Lemma~\ref{lem:A}.}\ {}From \cite{LS1} we learn that $G(x)-\frac{1}{|x|}$ is Lipschitz on $\mO$ and thus $G(x)=\frac{1}{|x|} +O(\frac{1}{L})$ uniformly on $\mO$ (the $O(\frac{1}{L})$ factor can be obtained by a scaling argument, as in the proof of Lemma~\ref{lem:def}). Therefore we have, with $g:=f^{-1}$ and $y=f(x)$ \be \label{5.1} A_f(x;h) = |g(y+h)-g(y)|^{-1} + O(\frac{1}{L}). \ee Recall that by Lemma~\ref{lem:def} and \fer{normes} \be \label{eq:step1} \nlp{DX}{\infty}\leq (\sqrt{3}L)^\ma \noaps{DX} = O(L^\ma \nepsr)\ . \ee Let $Y$ be a vector field such that $g(y)=y+Y(y)$. We differentiate $g(f(x))=x$ and obtain, with $y=f(x)$, $$ DY(y) + DX(x) + DY(y) DX(x) =0 . $$ This leads to the following estimate, for the $L^\infty$-norm, \be \label{eq:step2} Dg(y) = I - DX(x) + O(L^{2\ma}\nepsr^2). \ee \indent We claim that, uniformly in $h,y\in\Omega$, \be \label{eq:step3} g(y+h) - g(y) = (I - DX(x))\,h + |h| \left\{O(L^{2\ma}\nepsr^2) + O(\nepsr\,|h|^\ma)\right\} . \ee Indeed, let $y_t=y+th$, and let $x_t$ be such that $y_t=f(x_t)$. We have $|x_t-x|\leq \nlp{Dg}{\infty} |y_t-y| \leq C |h|$, and thus $|DX(x_t)-DX(x)|\leq \noaps{DX} |x_t-x|^\ma \leq O(\nepsr\ |h|^\ma)$. In particular, $DX(x_t)=DX(x)+O(\nepsr\ |h|^\ma)$. Then, using~\fer{eq:step2}, we have $g(y+h)-g(y)=\int_0^1 Dg(y_t).h dt=(I-DX(x)).h + O(L^{2\ma}\nepsr^2 |h|) + O(\nepsr |h|^{1+\ma})$, which proves \fer{eq:step3}. % L^\ma \noindent Using~\fer{eq:step3} and~\fer{eq:step1} we obtain $$ g(y+h)-g(y) = h - DX(x).h + |h|.\left\{ O(L^{2\ma}\nepsr^{2}) + O(\nepsr\,|h|^\ma) \right\} . $$ Since $DX(x)=O(L^\ma\nepsr)$, we have (with $|\cdot|$ denoting the Euclidean norm in $\R^3$) $$ |g(y+h)-g(y)|^2 = |h|^2\ \left\{ 1 - 2\frac{\< h,\ DX(x).h\>}{\< h,\,h\>} + O(L^{2\ma}\nepsr^{2}) + O(\nepsr\,|h|^\ma) \right\} . $$ Then from \fer{5.1} we conclude $$ A_f(x;h) = \frac{1}{|h|} \left\{ 1 + \frac{\< h,\ DX(x).h\>}{\< h,\,h\>} + O(L^{2\ma}\nepsr^{2}) + O(\nepsr\,|h|^\ma) \right\} + O(\frac{1}{L}). $$ % By Young's inequality $ \disp \nepsr\,|h|^\ma = (L^\ma \nepsr) (\frac{|h|}{L})^\ma \leq (1-\ma)\ (L^\ma \nepsr)^{1/(1-\ma)} + \ma\ \frac{|h|}{L} $ and thus the estimate of Lemma~\ref{lem:A} is proved. \carre \medskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Approximation of $D_K$. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent It remains to estimate $D_K(h)$ for which we do not have a simple formula. Recall the following approximation which can be found in Friesecke \cite{F1}: %------------------- % Friesecke \cite{F1} %------------------- there exists a constant $c_0>0$ such that, for any $ r>0$ and for any $ x \in \R^3$ with $\nlp{x}{\infty}\leq \pi$, \be \label{Friesecke} \left|\sum_{k\in\Z^3\cap B_r} e^{i\,k.x} - \int_{B_r} e^{i\,k.x}\,dk \right| \leq c_0\,(1+ r^{3/2}) . \ee To make use of \fer{Friesecke}, we define $R_N>0$ such that the volume of the Euclidean ball of center $0$ and radius $R_N$ equals $N$ : \be\label{eq:nrn} N = \frac{4}{3}\pi R_N^3 \ \ee and we define a continuous analogue of $D_K$, for $R>0$, \beno \label{eq:DR} \tD_{R}(h) & :=& \frac{1}{|\mO|} \int_{B_{R}} e^{i\,\frac{2\pi}{L} k\cdot h} d^3k \\ & = & 4\pi(\frac{R}{L})^3\ \frac{\sin(t)-t\cos(t)}{t^3}, \qquad t=\frac{2\pi}{L}\ R|h| \ . \eeno \begin{lemma}\label{lem:D} Let $K=K_N$ and $R_N$ as above. When $N\conv \infty$, we have \be\label{FL2} \int_\mO \frac{|D_{K_N}(h)|^2}{\rho_0} \frac{dh}{|h|} & = & \int_\mO \frac{ |\tD_{R_N}(h)|^2 }{\rho_0}\frac{dh}{|h|} + O\big(\frac{1}{L} \big) \ee \end{lemma} {\em Proof of Lemma~\ref{lem:D} in the symmetric case.}\\ For the moment, we assume that $K=K_N$ is {\em symmetric}, (i.e. a {\em closed-shell} situation (cf e.g. \cite{F1}). We define the "Fermi radius" $k_F$ by: \be \label{eq:kf} k_F=k_F(K):= \max\{|k|,\ k\in K\} \ee Since $K$ is symmetric, we have $K= \Z^3 \cap B_{k_F}$. \\ Note that, with the notations of section 4, there exists $j\geq 1$ such that $N=N_j$ and $k_F=r_j$. In particular \fer{estim Nj} leads to \be \label{eq:RN} k_F = R_N\ (1 + O(\frac{1}{R_N^{3/2}}) ) \ . \ee By \fer{Friesecke} (with $x=\frac{2 \pi}{L} h$) we deduce that uniformly with respect to $h \in\mO=[-\frac{L}{2},\frac{L}{2}]^3$ and for $N$ large \be |D_{K_N}(h)-\tD_{k_F}(h)| \leq C\ L^{-3}{k_F}^{3/2} \label{eq:dokf} \ee where $C$ is a constant. \noindent Note also that \beno | \tD_{k_F}(h) -\tD_{R_N}(h) | & \leq & L^{-3}\Vol\left(B_{R_N} \triangledown B_{k_F}\right) \\ & \leq & \frac{4}{3} L^{-3} \max(k_F,R_N)^2 | k_F - R_N| \eeno (where $B_{R_N} \triangledown B_{k_F}$ denotes the symmetric difference $B_{R_N}\backslash B_{k_F} \cup B_{k_F} \backslash B_{R_N}$). Using \fer{eq:RN}, we deduce $|\tD_{R_N}(h) - \tD_{k_F}(h)|\leq C\ L^{-3} R_N^{3/2}$, and together with~\fer{eq:dokf} we conclude \be\label{eq:D0estim} |D_{K_N}(h)-\tD_{R_N}(h)| \leq C\ L^{-3} {R_N}^{3/2} \ . \ee Therefore, $|D_{K_N}(h)|^2 = |\tD_{R_N}(h)|^2 + O(L^{-3} R_N^{3/2}) |\tD_{R_N}(h)| + O(L^{-6} R_N^3)$, and we have \be \int_\mO \frac{\left| |D_{K_N}(h)|^2 - |\tD_{R_N}(h)|^2 \right|}{\rho_0} \frac{dh}{|h|} & \leq & O(L^{-3}R_N^{3/2}) I_{N,L} + O(L^{-6}R_N^3) J_{N,L} \label{eq:5.4} \ee where $I_{N,L}:=\int_\mO \frac{|\tD_{R_N}(h)|}{\rho_0} \frac{dh}{|h|} $ and $J_{N,L}:=\int_\mO \frac{1}{\rho_0} \frac{dh}{|h|} $. We easily obtain $I_{N,L}=O(L^2 R_N^{-2} log(R_N))$ and $J_{N,L}=O(L^5 R_N^{-3})$ (using the analytical expression of $\tD_{R_N}$ in \fer{eq:DR}, and a change of variables $t=\frac{2\pi}{L}R_N\,h$). Thus the right side of inequality~\fer{eq:5.4} is $O(L^{-1})$, as desired. \carre \medskip\noindent {\em Proof of Lemma~\ref{lem:D} in the general case.}\\ Recall that $K_N$ is a minimizer for $T_0(N)$ (note that now $K_N$ is not unique in general). \noindent We consider as in the proof of Lemma~\ref{lem:1} an index $j\in\N$ such that $N_j\leq N < N_{j+1}$ where the integers $N_j$ and $N_{j+1}$ correspond to symmetric sets $K_{N_j}$ and $K_{N_{j+1}}$ (that are minimizers for $T_0(N_j)$ and $T_0(N_{j+1})$ respectively). \noindent As in the symmetric case, we define $R_{N_k}$ by $N_k = \frac{4}{3} \pi R_{N_k}^3$ for $k=j$ and $k=j+1$. By \fer{estim Nj} and \fer{estimdif Nj} we deduce $N_{j+1}-N_j=O(\sqrt{N_j})$, $R_{N_j}\stackrel{N\conv\infty}{\sim} R_N$, and \be\label{eq:njrn} N_{j+1}-N_j= O\left(R_{N}^{3/2}\right). \ee We thus have \beno D_{K_N}(h) - D_{K_{N_j}}(h) &=& \frac{1}{|\mO|} \sum_{k\in K_N\backslash K_{N_j}} e^{i \frac{2\pi}{L}k.h}\\ &=& L^{-3} O\left(\# (K_N\backslash K_{N_j})\right)\\ &=& L^{-3} O(N-N_j) \\ &=& O(L^{-3} R_{N}^{3/2}) \eeno (for the fourth equality we use \fer{eq:njrn} and $N-N_j \leq N_{j+1} - N_j$). Using the fact that in the symmetric case we have, as in \fer{eq:D0estim}, the estimate $D_{K_{N_j}}(h)=\tD_{R_{N_j}}(h)+O(L^{-3} R_{N_j}^{3/2})$, we deduce $$ D_{K_N}(h)=\tD_{R_{N_j}}(h)+O(L^{-3} R_{N}^{3/2}). $$ Similarly, we can prove that $$ \tD_{R_{N_j}}(h) = \tD_{R_N}(h)+O(L^{-3} R_{N}^{3/2}) $$ (using $\Vol(B_{R_{N}}\backslash B_{R_{N_j}}) =N-N_j= O(R_{N}^{3/2})$), and we obtain $$ D_{K_N}(h)=\tD_{R_{N}}(h)+O(L^{-3} R_{N}^{3/2}). $$ Proceeding as in the proof of Lemma~\ref{lem:D} in the symmetric case, we obtain finally the estimate \fer{FL2} in all cases. \carre \medskip We can now end the proof of Theorem~\ref{theo2}. \medskip \noindent {\em Proof of Theorem~\ref{theo2}.}\\ Using successively Lemma~\ref{lem:A} and then Lemma~\ref{lem:D} we obtain the estimate \be V_{av}(x_0,[\psi_j^f]) & = & - \int_\mO \frac{ |D_K(h)|^2 }{\rho_0}\frac{1}{|h|} \left\{ 1 + \frac{\< h ,\,DX(x_0)h \>}{|h|^2} + O((L^\ma\nepsr)^{\beta}) + O(\frac{|h|}{L}) \right\} dh \nonumber \\ \hspace*{-1cm} & = & - \int_\mO \frac{ |\tD_{R_N}(h)|^2 }{\rho_0}\frac{1}{|h|} \left\{ 1 + \frac{\< h ,\,DX(x_0)h \>}{|h|^2} + O((L^\ma \nepsr)^{\beta}) + O(\frac{|h|}{L}) \right\} dh \nonumber \\ & & + O\big(\frac{1}{L} \big)\ . \label{eq:vav1} \ee Let us show that the zero-order term, \beno V_{av,0}:= - \int_\mO \frac{ |\tD_{R_N}(h)|^2 }{\rho_0}\frac{1}{|h|} dh \eeno and the first order term, \beno V_{av,1}:= - \int_\mO \frac{ |\tD_{R_N}(h)|^2 }{\rho_0}\frac{1}{|h|} \frac{\< h ,\,DX(x_0)h \>}{|h|^2} dh \eeno satisfy the following asymptotics: \be V_{av,0} &=& -C_S\,\rho_0^{1/3} + O(\frac{1}{R_N^2}), \label{eq:av-0} \\ V_{av,1} &=& -C_S\,\rho_0^{1/3} \left(\frac{1}{3}\epsr(x_0) + O(L ^{2\ma}\nepsr^2)\right) + o(\frac{1}{R_N^2}) \label{eq:av-1} \ee where $C_S$ is the Slater constant (cf \fer{Vav-result}).\\ % To prove (\ref{eq:av-0}) we replace $\tD_{R_N}(h)$ by its analytical expression (\ref{eq:DR}). Using a change of variables $t=\frac{2\pi}{L} R_N h$ and the identity $\rho_0=N/L^3 =\frac{4\pi }{3}(R_N/L)^3$, we obtain \beno V_{av,0} &= & - \int_{\frac{2\pi}{L}R_N\mO} \frac{\left( 4\pi (R_N/L)^3\ q(t) \right)^2}{ \frac{4\pi }{3}(R_N/L)^3} \frac{1}{(\frac{2 \pi}{L} R_N)^2} \frac{d^3 t}{|t|} \\ &= & -\frac{3}{\pi} \frac{R_N}{L}\ \int_{\frac{2\pi}{L}R_N\mO} \frac{q(t)^2}{|t|} d^3 t, \eeno where we have denoted $\disp q(t):=\frac{\sin(|t|)-|t|\cos(|t|)}{|t|^3}$ and $\frac{2\pi}{L}R_N\mO = [-\pi R_N,\pi R_N]^3$. A direct calculation gives \be \label{eq:pyintegral} \int_{\R^3} \frac{q(t)^2}{|t|} d^3 t = \pi \ee (see for instance Parr and Yang \cite{PY}, Sec.\ 6.1, p.\ 108). Furthermore, we have $\int_{\R^3\backslash (\frac{2\pi}{L}R_N\mO)} \ \frac{q(t)^2}{|t|} d^3 t = O(\frac{1}{R_N^2})$. Hence $ V_{av,0}= -3\ \frac{R_N}{L} + O(\frac{1}{R_N^2}) = -C_S \rho_0^{1/3}+ O(\frac{1}{R_N^2}) $.\\ To prove (\ref{eq:av-1}), we proceed in the same way and obtain $ V_{av,1} = \cI_{N,L} + O(\frac{L^\ma \nepsr}{R_N^2}) = \cI_{N,L} + o(\frac{1}{R_N^2}) $, where \beno \cI_{N,L} := - 3 \frac{R_N}{L} \left[ \frac{1}{\pi} \int_{\R^3} \frac{q(t)^2}{|t|} \frac{\}{\} d^3 t \right]\ . \eeno Then, we remark the following identity when we integrate on the unit sphere $S^2$ ($dw$ denotes the measure on the sphere): \be \label{eq:identity} \int_{S^2} \frac{\}{\} dw(t) = \frac{1}{3} div(X)(x_0) \int_{S^2} dw(t). \ee To see this, we develop $\=\sum_{i,j} t_i t_j \frac{\partial_i X}{\partial x_j}$. We note that for $i\neq j$ the integral $\int_{S^2} \frac{t_i t_j}{|t|^2} dw(t)$ vanishes by symmetry and for $1\leq i \leq 3$, the integrals $\cJ_i=\int_{S^2} \frac{t_i^2}{|t|^2} dw(t)$ are equal to the same value $\cJ$; in particular $\cJ=\frac{1}{3}(\cJ_1+\cJ_2+\cJ_3)=\frac{1}{3} \int_{S^2}dw(t)$. Then, using that $q$ is radial, and formula \fer{eq:identity}, we obtain \be \cI_{N,L} & = & - 3 \frac{R_N}{L} \left[ \frac{Tr(DX(x_0))}{3}\frac{1}{\pi} \int_{\R^3} \frac{q(t)^2}{|t|} d^3 t \right] \nonumber\\ & = & - 3 \frac{R_N}{L} \frac{div(X)(x_0)}{3} \nonumber \\ & = & - C_S\rho_0^{1/3}\ \frac{div(X)(x_0)}{3} \label{eq:BR1} \ee where we have used again \fer{eq:pyintegral} for the second equality. Recall also that $\mathrm{div}(X)(x_0)=\epsr(x_0)+O(L^{2\ma}\nepsr^2)$ by Corollary~\ref{cor:div}; combined with (\ref{eq:BR1}) and the previous bounds, we obtain finally (\ref{eq:av-1}). %--------------- Now we insert the estimates (\ref{eq:av-0}) and (\ref{eq:av-1}) in (\ref{eq:vav1}), and since $\rho_0^{1/3} = (\frac{4}{3}\pi)^{1/3}\ R_N/L$, we obtain \beno V_{av}(x_0,[\psi_j^f]) & = & -C_S\rho_0 ^{1/3} \left(1+\frac{1}{3}\epsr(x_0)+O(L^{2\ma}\nepsr^2), +O(\frac{1}{R_N}) \right) + O(\md_N), \eeno where $\md_N:= \md_{1,N}+ (L^\ma\nepsr)^\mb \md_{2,N}$ and $$ \md_{1,N} := \disp L^{-1} \int_\mO \frac{ |\tD_{R_N}(h)|^2 }{\rho_0}\ dh, \quad \quad \md_{2,N} := \int_\mO \frac{ |D_R(h)|^2 }{\rho_0}\frac{dh}{|h|}. $$ As shown above, we have the bounds $ \md_{1,N}=O(L^{-1})=O(\rho_0 ^{1/3}/R_N) $, and $\md_{2,N}=O(R_N/L)=O(\rho_0 ^{1/3})$. Hence \be \label{eq:vav2} V_{av}(x_0,[\psi_j^f]) & = & - C_S\,\rho_0 ^{1/3} \left(1+\frac{1}{3}\epsr(x_0)+O((L^\ma\nepsr)^\mb) + O(\frac{1}{R_N}) \right) \ee Finally, we note that $\rho(x)^{1/3}=(1+\epsr(x))^{1/3}\ \rho_0^{1/3} = (1+\frac{1}{3}\epsr(x)+O(L^{2\ma}\nepsr^2) )\ \rho_0^{1/3}$ and thus \be \label{eq:5.5} \rho_0^{1/3} (1+\frac{1}{3}\epsr(x)) = \rho(x)^{1/3}(1+O(L^{2\ma}\nepsr^2)). \ee Also, the errors terms in \fer{eq:vav2} satisfy: \be \rho_0^{1/3}(O((L^\ma \nepsr)^\mb)+O(\frac{1}{R_N}) ) = O\left(\rho(x)^{1/3}((L^\ma \nepsr)^\mb +\frac{1}{R_N})\right). \label{eq:5.6} \ee Combining (\ref{eq:5.5}) and (\ref{eq:5.6}) we obtain \beno V_{av}(x_0)= -C_S\,\rho(x_0) ^{1/3} \left(1+O((L^\ma \nepsr)^\mb) +O(\frac{1}{R_N}) \right) \ . \eeno Since $R_N = (\frac{3}{4\pi})^{1/3} N^{1/3}$ this concludes the proof of Theorem~\ref{theo2}. \carre %---------------------------------------- \subsection{Proof of Theorem~\ref{theo3}} %---------------------------------------- We deduce Theorem~\ref{theo3} from Corollary~\ref{cor:2} and Theorem~\ref{theo1'} as follows:\\ We use that $\cE(\rho)\leq E([\psi_j^f])$ where $\Psi=[\psi_j^f]$ are the deformed plane waves \fer{eq:dpw}. The deformation $f$ is chosen as in Lemma \ref{lem:def}, and $K=(k_1,\dots,k_N):=K_N$ is chosen as in Lemma~\ref{lem:2} (i.e., it is a minimizer of $T_0(N)$). In order to bound the kinetic energy, we recall the bound \fer{kin:pfk} used in the proof of Theorem~\ref{theo1'}: \be \label{5.01} E_{kin}([\psi_j^f]) % \sum_{j=1}^N \int_\mO |\nabla\psi_j^f|^2 = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx + C_F \int_\mO \rho^{5/3}\ \bigg\{1 + O (L^{2\ma}\nepsr^2) + O(\frac{1}{N^{1/2}})\bigg\} . \ee Thus in view of \fer{Edecomp}, \fer{Eee1} and Corollary~\ref{cor:2}, Theorem~\ref{theo3} is proved. \carre \medskip \noindent It remains to justify Remark~\ref{rem:glob}. The upper bound for the thermodynamic limit is a direct consequence of Theorem~\ref{theo3} (since $\rho_0$ is constant). For the "high density" limit we have to prove that the error terms in $E_{ex}([\psi_j])$ can be absorbed by the error terms of the kinetic energy bound \fer{5.01}. So, as $\rho = O(L^{-3} N)$ and $\beta=1/(1-\ma)>1$, it is enough to prove that $$ N^{4/3}\ \bigg\{\nepsr + \frac{1}{N^{1/3}}\bigg\} = O(N^{5/3})\ \left\{\nepsr^2 + \frac{1}{N^{1/2}}\right\}. $$ % This relation holds since, using that $ab\leq a^2 +b^2$, we have % \beno N^{4/3} \nepsr & = & N^{4/3}\ N^{1/6} \nepsr\frac{1}{N^{1/6}}\\ &\leq& \ N^{4/3} \bigg\{ N^{1/3}\nepsr^2 + \frac{1}{N^{1/3}} \bigg\} \\ &\leq& \ N^{5/3} \bigg\{ \nepsr^2 + \frac{1}{N^{2/3}} \bigg\}. \eeno % and Remark \ref{rem:glob} follows. \carre %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % OLD NOTES : CAS N,L -> infty : lemme 2 du fax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{lemma}[symmetric case] % For $R=R_N$ large, and $L\geq 1$, we have: % \be % A_{N,L}:= \int_\mO \frac{\left| |D_0(h)|^2 - |D_R(h)|^2 \right|}{\rho_0} % \frac{dh}{|h|} = O\big(\frac{1}{L}\big) + % O\big(\frac{1}{L^{3/2}}\frac{\Log(R)}{R^{1/2}}\big) % \ee % \end{lemma} % % % ANCIEN LEMME C3. avec (L,N) % \begin{coroll}[symmetric case]\label{c3} % In the "High Density Limit" ($L=cst$, $N\conv\infty$), % we have % \beno \int_\mO \frac{|D_0(h)|^2}{\rho_0} \frac{dh}{|h|} % & = & \int_\mO \frac{ |D_R(h)|^2 }{\rho_0}\frac{dh}{|h|} + % O\big(1 \big) \\ % & = & - C_S \rho_0^{1/3} + O\big(1 \big) % \eeno % \end{coroll} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % end of Sec.5 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ACKNOWLEDGEMENT % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{0.5cm} {\em {\bf {ACKNOWLEDGEMENT :}}} Support by the ACI "Mod\`eles Math\'ematiques en Chimie Quantique", the TMR network "Asymptotic Methods in Kinetic Theory" and the Austrian goverment START prize project "Nonlinear Shr\"odinger and Quantum Boltzmann equations" are acknowledged. 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