Before treating the full supercell system, let us consider the Coulomb energies of the particles in an isolated simulation cell[3]. This is exactly the situation one would be faced with when studying clusters of atoms within QMC.

The cell contains *N* electrons each with charge -1 at positions
and *M* nuclei with charges at positions
. When the Born-Oppenheimer approximation is used,
the positions of the nuclei act only as parameters in the electronic
Hamiltonian. This Hamiltonian can be written as

For an isolated simulation cell, the term *U* is simply a superposition
of the Coulomb energies for each particle,

The Coulomb energy for each particle is the result of interactions
with all the other charges. There is no self-interaction and so the
electrostatic potentials, , which appear in
the equation for *U*, are the full Coulomb potentials, ,
minus the Coulomb potential of the particle situated at

The full Coulomb potential, , may be calculated by solving Poisson's equation,

where is the charge density, and the boundary condition is that the potential tends to zero as .

Tue Nov 19 17:11:34 GMT 1996