Before treating the full supercell system, let us consider the Coulomb energies of the particles in an isolated simulation cell. 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 .