Interaction
Having identified that almost all the exchange and correlation
interaction between electrons occurs over a short range, it seems
sensible to use the exact Coulomb interaction, i.e. at
short range, to capture this exchange and correlation as accurately as
possible. The exchange and correlation energy can then be
written as
where
and the limit s on an integral describes an integral over the
supercell and the limit a describes an integral over all space. The
first term describes each electron interacting via the full
Coulomb interaction with all other electrons within its
Wigner-Seitz cell. This is similar to the evaluation of the new
Jastrow function described in chapter 
where each
electron-electron separation vector is reduced into the Wigner-Seitz
cell centred on the electron being considered by subtracting
reciprocal lattice vectors from the electron-electron separation
vector as illustrated in figure . The second term in
Eq.() is required to cancel out the contribution from
the first term to the Hartree energy. The Ewald interaction still
correctly describes the Hartree energy,
Combining Eqs.( and ) produces a
general expression for the electron-electron energy
