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Periodic Coulomb model, fonctionnal density, Thomas-Fermi-von Weizsaker
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\newtheorem{lemma}{Lemma}[section]
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\newtheorem{remark}{Remark}[section]
\newtheorem{cor}{Corollary}
\newtheorem{corol}{Corollary}[section]
\newtheorem{prop}[lemma]{Proposition}


%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
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\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_{i<j}G(x_i-x_j)
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%
where the $ x_i \in \Omega$ denote the fermion positions, 
$V_{ext}(x) = -\sum_{l=1}^{P}Z_{l}G(x-R_l)$, $Z_l$ and $R_l$ are
respectively the charges and the positions of the $P$ nuclei.
Notice that the $1/x$ function of the Coulomb interaction
has been replaced by the "periodized version" $G(x)$ of e.g.  \cite{LS1}, which
is the periodic solution of the equation
$
   - 4\pi \Delta G = \sum_{k\in L\Z^3}\delta(\cdot - k) - 1/L^3 
$,
satisfying $\int_\Omega G=0$.

\vspace{0.5 cm}
\noindent 
As the Hamiltonian is periodic, 
the $N$-particle wave-functions are Bloch functions, i.e. a shift for a lattice 
vector $\gamma$
results in a mere phase factor :
%
  $\Psi(x+\gamma)=e^{i\mT \cdot \mg}\Psi(x)$
%
for all $x\in \R^{3N}$ and $\mg \in (L\Z^3)^{N}$ and for some
$\Theta$ fixed in $[0,2\pi/L[^{3N}$. On the other hand,  
the $N$-particle wave-functions have to be antisymmetric by the
Pauli principle.%
\footnote {In order to simplify the presentation we do 
not explicitly consider the spin.}
These conditions lead us to consider the following space
of wave functions
\beno \label{L1}
  \mL:= \bigoplus_{\mt \in [0,2\pi/L[^{3}}  \mL_\mt
\eeno
where
\beno
  \mL_{\mt} :=  \big\{ \Psi(x_1,..., x_N) = e^{i\mt \cdot (x_1+\cdots+x_N)} 
  \Phi(x_1,..., x_N) ,\, \Phi \in H^1_a ((\R^3 / L\Z^3)^N) \big\}\ .
\eeno
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(The index "a" means that the functions $\Phi$ are antisymmetric.)

\noindent
For technical reasons (cf. Lemma~\ref{Tf:isometry})
we will also consider the space of continuous
wave functions in $\mL$, i.e.
\beno % \label{L3}
  \mL^c:= \mL \ \cap \ C^0(\R^{3N})\ .
\eeno

\noindent
The fundamental energy of this $N$ fermion system is then given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\be \label{E0}
\begin{array}{lll}
  E_0 &:= & \inf \{ \langle H_{per} \Psi,\ \Psi \rangle\mid \ \Psi \in \mL,\
   ||\Psi||_{L^2}=1\}\\ 
   &= & \inf \{ \langle H_{per} \Psi, \ \Psi \rangle\mid\ \Psi \in \mL^c,\
   ||\Psi||_{L^2}=1\}\ ,
\end{array}
\ee
where $\langle , \rangle$ denotes the Hermitian scalar product in $L^2 
(\Omega^N)$.

\medskip 

\noindent
The aim of the density functional theory is to replace the minimization
problem (\ref{E0}) by a minimization problem with respect to the
density $\rho$ associated to the wave function $\Psi$ :
\be \label{rho}
  \rho_{\Psi}(x):=
    N\int_{\Omega^{N-1}} |\Psi(x, x_2,\dots,x_N)|^2 dx_2\dots dx_N.
\ee

\noindent 
This procedure reduced the number of variables from $3N$ to $3$.

\noindent Using (\ref{Hperdef}), we can decompose the
energy of a wave function $\Psi$ 
as follows:
\be \label{Edecomp}
  E(\Psi):=\langle H_{per} \Psi,\ \Psi \rangle
   = E_{kin}(\Psi) 
     + \int_{\Omega} V_{ext}(x) \rho_{\Psi} (x) dx + E_{ee}(\Psi).
\ee
In formula (\ref{Edecomp}) $E_{kin}$ denotes the kinetic energy,
\be \label{Ekin}
  E_{kin}(\Psi) := \int_{\Omega ^N}|\nabla \Psi(x)|^2 dx,
\ee
while $E_{ee}$ denotes the inter-electron energy,
\be \label{Eee}
  E_{ee}(\Psi) := \int_{\Omega^2}\rho_{2,\Psi}(x, x') G(x,x') dx dx',
\ee
where
\be \label{2densite}
  \rho_{2,\Psi}(x, x'):=\frac{N(N-1)}{2}\int_{\Omega^{N-2}}
  \left|\Psi(x,x', x_3, \dots , x_N)\right|^2\  dx_3\cdots dx_N\: .
\ee

\noindent 
Following (\cite{Lieb}, Theorem 1.2), it can be shown that the map
$\Psi \mapsto  \rho_{\Psi}$ maps $\mL^c$ onto
\be \label{D}
  D:=\{ \rho \in C^{0}(\R^3/L\Z^3),\, \rho \geq 0,\, \int_{\Omega}\rho =N 
  ,\, \sqrt \rho \in H^1_{loc}(\R^3) \}.
\ee
%
Therefore we can define the density functionals corresponding to
the kinetic and the global energy term by
%
\be \label{calE} 
  \Ekin(\rho) & := & \inf \{ E_{kin}(\Psi) |
                    \Psi \in \mL^c \mbox{ and } \rho_{\Psi} = \rho \} \\
  \mathcal{E}(\rho) & := & \inf \{ E(\Psi) |
                        \Psi \in \mL^c \mbox{ and } \rho_{\Psi} = \rho \},\nonumber
\ee
and we have
$$
  E_0 = \inf \{ \mathcal{E}(\rho) \mid \rho \in D \}.
$$

\medskip

\noindent
We shall use Schauder estimates in H\"older spaces, 
this leads us to consider the space of 
densities
$D_{\ma} \subset D$, with $0<\ma <1$
$$
  D_{\ma}:=\{ \rho \in C^{0,\ma}(\R^3/L\Z^3),\, \rho>0,\, \int_{\Omega}\rho =N 
  ,\, \sqrt \rho \in H^1_{loc}(\R^3) \}
$$
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<j\leq 3} \det \left(
 \begin{array}{ll}
   \partial_i v_i & \partial_j v_i \\
   \partial_i  v_j & \partial_j v_j
 \end{array}\right).
\ee
By integration by part (assuming $v$ has $C^2$ regularity) and using the periodicity
of $v$ we have
$$
  \int_\Omega \det \left(
  \begin{array}{ll}
  \partial_i v_i & \partial_j v_i \\
  \partial_i v_j & \partial_j v_j
  \end{array}\right)
  =-\int_\Omega \det \left(
  \begin{array}{ll}
   v_i &\partial_i \partial_j v_i \\
   v_j & \partial_i\partial_j v_j
  \end{array}\right)
  =\int_\Omega \det \left(
  \begin{array}{ll}
  \partial_j v_i & \partial_i v_i \\
  \partial_j v_j & \partial_i v_j
  \end{array}\right).
$$
Thus 
$$
  \int_\Omega \det \left(
  \begin{array}{ll}
  \partial_i v_i & \partial_j v_i \\
  \partial_i v_j & \partial_j v_j
  \end{array}\right)=0
$$
and similarly,
$$
  \int_\Omega \det (Dv) =0\ .
$$
Hence by density of $C^2$-functions in $\cY$, we get
$$
  \int_\Omega Q(Dv) =0 \, , \mbox{ for any } v \in \cY.
$$
Since $\int_\Omega \epsr =0$, one deduces that 
$\cN$ maps $\cY$ into $\cX$ and thus $\cL\cN$ maps $\cY$ into $\cY$.

\smallskip \noindent 
Now we want to solve Equation \fer{v} by 
the contraction principle. Let
$$
  \cB:=\left\{ u\in \cY\, \big| \ \noaps{Du}\leq 2 K\,\naps{\epsr}{0} \right\},
$$
and let $C$ be a positive constant independent of $L$ and $\ma$ such that for
any $w_1$ and $w_2$ in $C^{0,\ma}$ with $\nlp{w_1}{\infty}$, 
$\nlp{w_2}{\infty} \leq 1$,
\be 
  \label{estim:Q}
  \naps{Q(w_1)-Q(w_2)}{0} \leq C(\nlp{w_1}{\infty}+\nlp{w_2}{\infty}) 
  \naps{w_1-w_2}{0}
\ee
($C$ exists since $Q(w)$ is a sum of terms which are quadratic and cubic with
respect to the components of $w$).

\medskip
 
\noindent
We claim that for 
$$
  L^{\ma}\naps{\epsr}{0} 
    < \eta_{\ma}:=
  \min\{\frac{1}{2(\sqrt3)^{\ma} K(\ma)},\frac{1}{4(\sqrt3)^\ma CK(\ma)},
  \frac{1}{4(\sqrt3)^{\ma} CK(\ma)^2}\},
$$
$\cL\cN$ is a contraction mapping
on $\cB$ with respect to the norm $\nlp{.}{\cY}$.\\
Indeed, if $v\in \cB$ then using \fer{estim:L},
\beno
\nlp{\cL \cN v}{\cY}& = &\naps{D(\cL \cN v)}{0} \leq K \naps{\cN v}{0}\\
&\leq & K(\naps{\epsr}{0} + \naps{Q(Dv)}{0}).
\eeno
As $L^{\ma}\naps{\epsr}{0} < \eta_{\ma}$  and in view of \fer{normes}
one has for $v\in \cB$,
$$
\nlp{Dv}{\infty}\leq 2 (\sqrt3 L)^{\ma}K\naps{\epsr}{0} < 1.
$$

\noindent
Thus, using \fer{estim:Q} with $w_1=v$ and $w_2=0$,
\beno
\nlp{\cL \cN v}{\cY}&\leq & K(\naps{\epsr}{0} + C 
\nlp{Dv}{\infty}\naps{Dv}{0})\\
&\leq &K(1 + 4 (\sqrt3 L)^{\ma}KC\naps{\epsr}{0})\naps{\epsr}{0}\\
&\leq & 2K \naps{\epsr}{0}
\eeno
where we used in the last inequality that
$ L^{\ma}\naps{\epsr}{0}\leq \eta_{\ma} \leq \frac{1}{4(\sqrt3)^\ma CK}$.\\
Analogously, if $u$ and $v$ are in $\cB$, we have similarly
\beno
\nlp{\cL \cN v - \cL \cN u}{\cY}&\leq & K \naps{\cN v -\cN u }{0}\\
&\leq &K\naps{Q(Dv) -Q(Du) }{0}\\
&\leq &KC (\nlp{Dv}{\infty} + \nlp{Du}{\infty})\nlp{v-u}{\cY}\\
&\leq &2(\sqrt3 L)^{\ma}K^2C\naps{\epsr}{0}\nlp{v-u}{\cY}\\
&\leq & \frac{1}{2} \nlp{v-u}{\cY}
\eeno
where we used 
$ L^{\ma}\naps{\epsr}{0}\leq \eta_{\ma} \leq \frac{1}{4(\sqrt3)^\ma CK^2}$.\\

\medskip

\noindent
Therefore by the contraction principle, equation \fer{v} has a unique solution
$v$ in $\cY$. Hence $f:=Id+v$ is a $C^{1,\ma}$~function which
solves the Jacobian equation $J_f(x)=\frac{\rho(x)}{\rho_0}$ and satisfies
$$
\naps{D(f-Id)}{0}\leq 2 K \naps{\epsr}{0}.
$$
Furthermore as $v$ is periodic,
$f(x) = x + v(x)$ satisfies $f(x+L\,m) = f(x) + L\,m$ for 
any $m\in \Z^3$ and $x\in\R^3$.\\ 
That $f$ is a diffeomorphism is a consequence of  $\nlp{D(f-I)}{\infty} < 1$.\\
Finally, in view of \fer{normes},
we deduce \fer{estif} from \fer{estif2}.
\carre

\medskip

\noindent
We will need later the following technical estimate.

\begin{corol} \label{cor:div}
Let $\rho$ and $f$ be as in Lemma~\ref{lem:def}. Then
$$
  \mathrm{div}(f - Id) = \epsr + O (L^{2\ma}\naps{\epsr}{0}^2)\ .
$$
\end{corol}
\noindent
{\em Proof.}\ 
We use $ \mathrm{div} (f - Id) = \epsr + Q(Dv) $,
the fact that $Q$ has quadratic and cubic terms (see formula \fer{Q2}),
and the estimates \fer{normes}, \fer{estif}. \carre

\medskip

\noindent  
Notice that, with our definition, the image of the basic cell 
$\mO $ by a periodic deformation
is not necessarily equal to $\mO $. However, a periodic deformation
is a diffeomorphism of the torus $\T_L$, hence $f(\T_L)=\T_L$ and
we have 

\begin{lemma}
\label{lem:int}
Let $f$ be a periodic deformation of the cube $\mO$
and $g$ be an integrable periodic function. Then
$$\int_{f(\mO )}  g(x) dx = \int_{\mO } g(x) dx.$$
\end{lemma}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Kinetic energy estimates}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{Rewriting the kinetic energy using deformations}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\noindent
Using the deformation $f$ of Lemma~\ref{lem:def}, we replace the 
minimization problem \fer{calE},
for a given density $\rho$ by the same minimization problem for the constant 
density $\rho_0$ and we control the error.
Let $0<\ma <1$ and $\eta (\ma )$ as in Lemma~\ref{lem:def}.

\begin{lemma}\label{lem:Ekin}
For any $\rho \in D_{\ma}$, one has uniformly for 
$L^\ma \nepsr < \eta (\ma )$: 
\be \label{Ekin-replace2}
 \Ekin(\rho) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx
 \, +\, \Ekin(\rho_0)(1 + O(L^\ma \nepsr)).
\ee

\end{lemma}
%
{\em Proof. }\
Let $\rho$ be a density in  $D_{\ma}$ such that
$\nepsr< \eta (\ma )$. By Lemma~\ref{lem:def} there exists 
a periodic deformation,
$f$ , satisfying $J_f = \frac{\rho}{\rho_0}$.
Using Lemma~\ref{Tf:isometry} and \fer{Ekin}, 
\be \label{Ekin-replace1} 
\Ekin(\rho)= \inf
  \left\{\int_{\mO^N}|\nabla (T_f\Psi)(x)|^2 dx,\mid\ \Psi \in \mL^c
 \, \mbox{and} \,\rho_{\Psi}(x)=\rho_0\right\} .
\ee

\noindent
It remains to compute the kinetic energy of the deformed wave function given by 
\fer{def:Tf}:
%
\beno % \label{1}
% \begin{array}{lll}
\int_{\Omega ^N} |\nabla (T_f\Psi)(x)|^2 dx  & = &
  N\int_{\Omega ^N} |\nabla_1 (T_f\Psi)(x)|^2 dx    \\
 &=&
  N\int_{\Omega ^N}\big|\nabla(J_f(x_1))^{1/2} \prod_{j=2}^{N}(J_f(x_j))^{1/2} 
\Psi(f(x_1),\dots,f(x_N))   \\
 & & + \prod_{j=1}^{N}(J_f(x_j))^{1/2} 
    [Df (x_1)]^T \nabla_1\Psi(f(x_1),\dots,f(x_N))\big|^2 dx
% \end{array}
\eeno
where $\nabla_1$ denotes the gradient with respect to the
first variable.
%
By the change of variable $y_j = f(x_j)$ for $j=2,\dots ,N$,
 we get using Lemma~\ref{lem:int},
\be \label{ABC}
 \int_{\Omega ^N}
  |\nabla (T_f\Psi)(x)|^2 dx=
  A+B+C
\ee
where, denoting $y'=(y_2,\dots , y_N)$,
\be \label{A}
A := N\int_{\Omega^N}\big|\nabla(J_f(x_1))^{1/2}  
\Psi(f(x_1),y')\big|^2 dx_1 dy',
\ee 
%
\be \label{B}
B:= N\int_{\Omega^N}\big|(J_f(x_1))^{1/2} 
[Df (x_1)]^T \nabla_1\Psi(f(x_1),y')\big|^2 dx_1 dy'
\ee
and
\be \label{C}
C:= N\int_{\Omega ^N}(J_f(x_1))^{1/2} \nabla(J_f(x_1))^{1/2} 
\cdot \nabla_1|\Psi(f(x_1),y')|^2 dx_1 dy'.
\ee
By the change of variable $y_1 = f(x_1)$ in \fer{B} we obtain
denoting $y=(y_1,\dots , y_N)$
%
\be \label{B2}
B= N\int_{\Omega ^N}\big| 
[Df (f^{-1}(y_1))]^T \nabla_1\Psi(y_1,y')\big|^2  dy.
\ee
Integrating with respect to $y'$ and using $\rho_\Psi =\rho_0$, 
\fer{A} leads to
\be \label{A2}
A= \int_{\Omega}\big|\nabla(J_f(x_1))^{1/2}\big|^2  
\rho_0(f(x_1)) dx_1 =\int_{\mO} |\nabla\sqrt{\rho(x_1)}|^2 dx_1
\ee 
while \fer{C} leads to
\be \label{C2}
C= \int_{\Omega }(J_f(x_1))^{1/2} \nabla(J_f(x_1))^{1/2} \cdot
\nabla\rho_0(f(x_1)) dx_1 = 0
\ee
since $\rho_0$ is a constant function.
Thus combining \fer{ABC}, \fer{B2}, \fer{A2} and \fer{C2} we conclude
%
\be \label{Ekin-eq1}
 \int_{\Omega ^N}
  |\nabla (T_f\Psi)(x)|^2 dx = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx
 \, +\, S(f,\Psi),
\ee
%
where 
%
\be \label{Sdef}
S (f,\Psi) := 
N \int_{\Omega^N} \big|[Df (f^{-1}(x_1))]^T \nabla_{1}\Psi(x)\big|^2 dx.
\ee
%
\noindent
Finally using estimate \fer{estif2} of Lemma~\ref{lem:def} we deduce
%
\be \label{S-eq1}
\begin{array}{lll}
S(f,\Psi)&=&
 N \int_{\Omega ^N} | \nabla_{1}\Psi(x)|^2 dx\: (1 + O(L^{\ma}\nepsr)) \\
 &=& \int_{\Omega ^N} | \nabla\Psi(x)|^2 dx\: (1 + O(L^{\ma}\nepsr))
.
\end {array}
\ee
%
\noindent
Combining (\ref{Ekin-replace1}), (\ref{Ekin-eq1}) and (\ref{S-eq1}),
Lemma~\ref{lem:Ekin} is proved.
\carre

\begin{remark}\label{rem:4.1}
In the same way it can be proved 
that for any $\Psi$ minimizing $\cE_{kin}(\rho)$ 
$$
 \Ekinloc(\Psi)(x) = |\nabla\sqrt{\rho}(x)|^2 
  + \frac{1}{L^3} E_{kin}(\Psi)\,(1 + O(L^{\ma}\nepsr))\ .
$$
This estimate leads to the local estimate \fer{eq:Tloc2}.
\end{remark}


\medskip

\noindent  
When restricting our analysis to wave functions that are deformations of plane
waves we can improve the error term in Lemma~\ref{lem:Ekin}
This will be used to prove Theorem~\ref{theo1'}. 

\noindent 
Let $K := \{k_1,\dots,k_N\}$ be a subset of $\Z^3$,
and let $\psi_j(x)$ be the "free electron plane waves" defined by
%
\be \label{psijdef}
\psi_j(x):=\frac{1}{\sqrt{L^3}}
  \exp(\frac{2i\pi}{L}\, k_j\cdot x) \quad , \:  j=1,\dots,N.
\ee
%
For this set of orbitals the following Slater determinant (cf \fer{slatdet-def})
%
\be \label{eq:plane-waves}
  \Psi_K :=[\psi_j]
\ee
%
is a plane wave of density $\rho_0$. 
The corresponding deformed orbitals are 
\be\label{eq:dpw0}
   \psi_j^f(x):= (J_f(x))^{1/2} \psi_j(f(x)) .
\ee
The corresponding deformed N-wave-function is given by
\be \label{eq:dpw}
 \Psi_K^f := T_f \Psi_K = [\psi_j^f]
\ee
and has density $\rho$ in the case where $f$ is solution of the 
Jacobian equation $J_f=\rho/\rho_0$.

\begin{defi}
\label{Ksym} 
We say that a set $K\subset \Z^3$ is symmetric if there exists $r \in \R$
such that $K$ is equal to the intersection of $\Z^3$ with the ball in  $\R^3$
of center $0$ and radius $r$.
\end{defi}

\noindent 
For  symmetric $K$ we have:
\begin{itemize}
\item
For any $k=(k^1,k^2,k^3) \in K$ and for any $\eps_1 =\pm 1$,
$\eps_2=\pm 1$ and $\eps_3=\pm 1$  we have that \linebreak
             $(\eps_1 k^1, \eps_2 k^2, \eps_3 k^3)$ is again in $K$.
\item 
For any $k=(k^1,k^2,k^3) \in K$ and for any permutation $\sigma$ we have that \linebreak
             $(k^{\sigma(1)},k^{\sigma(2)},k^{\sigma(3)})$
 is again in  $K$.
\end{itemize}

\noindent 
The following Lemma gives a second order (in $\epsr$)
approximation of $ E_{kin}(\Psi_K^f)$ in term of $E_{kin}(\Psi_K)$
and $\int_{\mO} |\nabla\sqrt{\rho}|^2 dx$ when $K$ is symmetric:
%-----------------------------
\begin{lemma} \label{lem:Ekin2}
%-----------------------------
Let $K$ be a symmetric subset of $\Z^3$
whose cardinality is $N$. 
Let $0<\ma <1$ and for any $\rho \in D_{\ma}$ let $f$ 
be the deformation defined in Lemma~\ref{lem:def}. 
Then we have for the kinetic energy of the deformed plane waves :
$$
  E_{kin}(\Psi_K^f) = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx
  \, +\, E_{kin}(\Psi_K)(1 + O(L^{2\ma}\nepsr^2)) .
$$
\end{lemma}
%------------------------------
{\em Proof. }\
As $(\psi_j^f)_{j=1,\dots,N}$ is a set of orthonormal functions in 
$L^2(\mO)$, we have 
$$
  E_{kin}(\Psi_K^f) = \int_{\mO^N} |\nabla \Psi_K^f|^2 = 
    \sum_{j=1}^N \int_{\mO} |\nabla \psi_j^f|^2 .
$$
Using \fer{Ekin-eq1} for $N=1$ we obtain for each $j=1,\dots,N$:
$$ 
 \int_{\mO} |\nabla \psi_j^f|^2 =
 \frac{1}{N} \int_{\mO} |\nabla\sqrt{\rho}|^2 
 + \int_{\mO} \big|[Df (f^{-1}(x))]^T \nabla\psi_j(x)\big|^2 dx.
$$
Therefore we obtain
%
\be \label{Ekin-eq1'}
  \int_{\Omega ^N}
  |\nabla (T\Psi_K^f)(x)|^2 dx = \int_{\mO} |\nabla\sqrt{\rho}|^2 dx
  \, +\, S(f,\Psi_K) 
\ee
and 
%
\be
  S(f,\Psi_K) 
  &=& \frac{1}{L^3}
  \sum_{j=1}^N\int_{\mO} \ | Df(f^{-1}(x))^T\ \frac{2\pi}{L} k_j |^2 dx
  \nonumber  \\
  &=& \frac{1}{L^3}
  \int_\mO J_f(x) \sum_{k\in K} \ | Df(x)^T\ \frac{2\pi}{L} k |^2 dx.
        \label{Tkinpsiest1}
\ee

\noindent  
Let $M(x) := Df(x)\ Df(x)^T = (M_{\ma,\mb})_{\ma,\mb = 1,2,3}$ and 
$k = (k^1, k^2, k^3)$.
Using the symmetries of $K$ we obtain
\be
 \sum_{k\in K} | Df(x)^T\cdot k |^2 &=& 
 \sum_{\ma,\mb = 1}^3  M_{\ma,\mb}
 \big( \sum_{k \in K } k^{\ma} k^{\mb} \big) \nonumber \\
 &= &\sum_{\ma,\mb = 1}^3  M_{\ma,\mb}
 \delta_{\ma,\mb} \big( \sum_{k \in K }  (k^{\ma})^2 \big) \nonumber\\
 &= &\frac{1}{3}\sum_{\alpha=1}^3  M_{\ma,\ma}
     \big( \sum_{k \in K } | k|^2 \big)  \ . \label{eq:56p}
\ee
Therefore, using \fer{Tkinpsiest1}, \fer{eq:56p}, and
$$
   E_{kin}(\Psi_K)= \sum_{k \in K } |\frac{2\pi}{L} k|^2,
$$
we obtain
\be \label{Tkinpsiest2}
  S(f,\Psi_K) =\frac{1}{3\,|\mO|} E_{kin}(\Psi_K)\int_\mO J_f(x)\  
  Tr(Df(x)\ Df(x)^T)\ dx\ .
\ee
Denoting $f = Id +v$  we get
$$
\int_\mO J_f(x)\  Tr(Df(x)\ Df(x)^T)\ dx = 
\int_\mO (1+\epsr(x) )(3 +2 \mbox{div}(v)(x) + Tr(Dv(x)\ Dv(x)^T))\ dx \ .
$$
By Corollary \ref{cor:div}, we have 
 $\mbox{div}(v) = \epsr + O(L^{2\ma}\nepsr^2)$.
Furthermore by Lemma~\ref{lem:def}, we have
$\nlp{Dv}{\infty}=  O(L^{\ma}\nepsr)$
and thus $\nlp{Tr(Dv\ Dv^T)}{\infty}=O(L^{2\ma}\nepsr^2)$.
Hence we get 
\be \label{Tkinpsiest3}
\int_\mO J_f(x)\  Tr(Df(x)\ Df(x)^T)\ dx 
&=&\int_\mO (1+\epsr(x) )(3 +2 \epsr(x) + O(L^{2\ma}\nepsr^2)\ dx
\nonumber \\
&=&\int_\mO 3\left(1 + \frac{5}{3}\,\epsr(x) + O(L^{2\ma}\nepsr^2)\right) dx\\
&=& 3L^3(1 +O(L^{2\ma}\nepsr^2)) \nonumber
\ee
where we used in the last equality, $\int_\mO \epsr(x) dx = 0$.

\noindent
Thus Lemma~\ref{lem:Ekin2} follows by
combining \fer{Ekin-eq1'}, \fer{Tkinpsiest2} and \fer{Tkinpsiest3}.
\carre




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proof of Theorem~\ref{theo1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\noindent 
In view of Lemma~\ref{lem:Ekin},
to obtain Theorem~\ref{theo1} it
suffices to prove the following statement:

%-----------------------------
\begin{prop}\label{prop:Ekin0}
%-----------------------------
$$
  \Ekin(\rho_0) = 
  C_F(1 + O(\frac{1}{ N^{1/2}})) \int_{\Omega}{\rho_0}^{5/3}dx \: .
$$
\end{prop}
%-----------------------------
\medskip

\noindent
Notice that\ \ 
$e_0 :=\inf \{ E_{kin}(\Psi), \Psi \in \mL ,\nlp{\Psi}{L^2} =1\}$ 
is reached by plane waves and that the density of a plane wave 
(c.f \fer{eq:plane-waves}) 
of $L^2$-norm 1 is equal to $\rho_0$.
Hence $e_0=\Ekin(\rho_0)$ and
the minimum $\Ekin(\rho_0)$ is also reached by plane waves.

\noindent
Therefore, we have 
\be\label{eq:TNdef0}
   \Ekin(\rho_0) = \big(\frac{2\pi}{L}\big)^2\ T(N), 
\ee
where 
\be \label{eq:TNdef}
  T(N) := \min \{ \sum_{k\in K} |k+\theta |^2 \mid\ \mt \in \R^3\ \mbox{and}\ 
    K \subset \Z^3,\ \# K = N \} .
\ee
Note also that 
$$
  \sum_{k\in K} |k+\theta|^2  =  
  \sum_{k\in K} |k|^2  + 2 (\sum_{k\in K} k, \ \mt) + N|\mt|^2.
$$
Thus for each $K$, the optimal choice for $\mt$ is given by 
$\mt_K:=-\frac{1}{N} (\sum_{k\in K} k)$ and the corresponding energy is 
\be \label{eq:kin001}
  \sum_{k\in K} |k+\theta_K|^2  =  
  \sum_{k\in K} |k|^2 - \frac{1}{N} (\sum_{k\in K} k)^2 .
\ee
By a translation argument on $K$ and $\mt$, we can restrict the minimization
\fer{eq:TNdef} over sets $K$ for which $\mt_K\in [-\frac{1}{2},\frac{1}{2})^3$.
Therefore  $|\sum_{k\in K} k| \leq \frac{\sqrt{3}}{2} N $ 
and \fer{eq:kin001} leads to 
$$
  \sum_{k\in K} |k+\theta_K|^2  \geq \sum_{k\in K} |k|^2 - \frac{3}{4} N \ .
$$
Hence we have proved
\be \label{eq:t00}
  T(N)=T_0(N) + O(N)
\ee
where
\be \label{eq:T0def}
  T_0(N) : = \min \{  
  \sum_{k\in K} |k|^2 \mid \ K\subset \Z^3, \# K = N\}\  .
\ee
% |k|
% |\mt|

\noindent 
Since $\int_\mO \rho_0^{5/3}=N^{5/3}/L^2$, 
Proposition \ref{prop:Ekin0} is then a consequence of \fer{eq:TNdef0},
\fer{eq:t00} and of the following Lemma:
%
%-------------------------
\begin{lemma}\label{lem:1}
%-------------------------
For any $N\in \N$,
$$
 T_0(N)  = 
  \frac{1}{(2\pi)^2} C_F \
  N^{5/3} \big( 1+ O\big( \frac{1}{N^{1/2}}\big) \big) \: .
$$
where $C_F$ is the Fermi constant 
(i.e $\frac{1}{(2\pi)^2} C_F  = \frac{3}{5} \big(\frac{3}{4\pi}\big)^{2/3}$).
\end{lemma}
%------------------------
\noindent 
Note that Lemma~\ref{lem:1} is an improvement of the more classical estimate
(see for instance \cite{FS})
$$
  T_0(N) = \frac{1}{(2\pi)^2} C_F\ 
       N^{5/3} \big( 1+ O\big( \frac{1}{N^{1/3}}\big) \big) \: .
$$
This improvement is essential for our purpose as explained 
in Remark~\ref{rem3}.

\medskip

\noindent 
{\em Proof of Lemma \ref{lem:1}}\ \
For $r > 0$  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{\<t,DX(x_0)t\>}{\<t,t\>} 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{\<t,DX(x_0)t\>}{\<t,t\>} dw(t)
  =  \frac{1}{3} div(X)(x_0) \int_{S^2} dw(t).
\ee
To see this, we develop
 $\<t,DX\ t\>=\sum_{i,j} t_i t_j \frac{\partial_i X}{\partial x_j}$.
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. 


The first author (O.B.)
also thanks I.\ Catto for signaling Ref. \cite{F1} of Friesecke.
 

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% FIN DOCUMENT TEXTE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
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