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\begin{document}
{\bf Total energy and Kohn-Sham Hamiltonian of a crystal within DFT}
\par
Let us consider a crystal with $ N\rightarrow\infty$ unit cells
of volume $\Omega $, periodically repeated, with lattice vectors
$\R$. (Pseudo--)Atoms of type $\mu$ and ionic charge $Z_\mu$
are located at $\d_\mu$ in the unit cell. The system contains 
$ N \sum_\mu Z_\mu $ electrons. Its electron states are described
by $ N $ points $\k$ in the Brillouin Zone. Assuming for simplicity
a local electron-ion potential $\hat V^\mu$:
\begin{eqnarray}
E_{tot} & = & E_{kin} + E_{ion-el} + E_{Hartree} + E_{xc} + E_{ion-ion} \\
        & = & - {\hbar^2 \over 2m} \sum_{ \k,v } 
              \int \psi^*_{ \k,v } (\r)
              \nabla ^2 \psi_{ \k,v }(\r) d\r
+ \sum_{ \k,v,\mu,\R}\int \psi^*_{ \k,v } (\r)
  \hat V^\mu(\r-\d_\mu-\R) \psi_{ \k,v } (\r) d\r \nonumber \\
       &  & \mbox{}
+ {e^2 \over 2} \int {n(\r) n(\r') \over \mid \r-\r' \mid} d\r d\r'
+ \int n(\r)\epsilon_{xc}[n(\r)]d\r 
+ {e^2 \over 2} \sum'_{\mu,\nu,\R,\R'} {Z_\mu Z_\nu \over 
\mid \d_\mu+\R-\d_\nu-\R' \mid }
\end{eqnarray}
where the electron charge density $n(\r)$ is given by
\begin{equation}
n(\r) = \sum_{\k,v} {\mid \psi_{ \k,v }(\r) \mid }^2
\end{equation}
(the sum is over the lowest $ \sum_\mu Z_\mu $ occupied states for 
a semiconductor or insulator, up to the Fermi surface for a metal).
Integrals extend on all space. The primed sum appearing in the ion-ion 
term excludes terms with $\d_\mu+\R-\d_\nu-\R'=0$. 

The Kohn-Sham equation is
\begin{equation} \biggl [
- {\hbar^2 \over 2m}  \nabla ^2 
+ \sum_{\mu,\R} \hat V^\mu(\r-\d_\mu-\R) 
+ e^2 \int { n(\r') \over \mid \r-\r' \mid } d\r'
+ V_{xc}(\r)
   \biggr ] \psi_{ \k,v }(\r) 
= \epsilon_{ \k,v } \psi_{ \k,v }(\r) \end{equation}
where the exchange-correlation potential 
$ V_{xc}(\r) = (\delta E_{xc}/\delta n(\r))$.
For the LDA case only:
\begin{equation}
E_{xc}[n(\r)] = \int n(\r) \epsilon_{xc}(n(\r)) d\r, \qquad
V_{xc}(\r) = {d \over d n} \bigl(n \epsilon_{xc}(n) \bigr)_{n=n(\r)}
\end{equation}
