Variational methods for cellular models

Abstract

Two complementary variational principles are derived for the one-electron Schrödinger or local-density-functional equation in a closed cell of arbitrary shape, for external Neumann and Dirichlet boundary conditions, respectively. The surface operators scrR and ${\mathrm{scrR}}^{\mathrm{-}1}$, respectively, are stationary in these two variational principles. Subject to a condition of compatibility of the boundary conditions, these results are combined to give variational equations that are valid for an arbitrary cluster of atomic cells, assumed to fill space within an outer boundary. Cell interface terms agree with prior variational derivations for discontinuous functions. It is shown that structure constants of multiple scattering theory can be used within the variational formalism, giving contracted Hermitian matrix equations linearized in energy. New variational prescriptions are given for two broad classes of applications: (i) electron scattering by a cluster, or bound states using scattering theory; (ii) embedding a cluster in a substrate.