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Isolated Simulation Cell

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 tex2html_wrap_inline5837 and M nuclei with charges tex2html_wrap_inline5841 at positions tex2html_wrap_inline5839 . 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, tex2html_wrap_inline6761 , which appear in the equation for U, are the full Coulomb potentials, tex2html_wrap_inline6765 , minus the Coulomb potential of the particle situated at tex2html_wrap_inline5837


The full Coulomb potential, tex2html_wrap_inline6765 , may be calculated by solving Poisson's equation,


where tex2html_wrap_inline5861 is the charge density, and the boundary condition is that the potential tends to zero as tex2html_wrap_inline6773 .

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996