Hohenberg-Kohn density-functional theory as an implicit Poisson equation for density changes from summed fragment densities

An implicit Poisson equation is derived for the change in density {Delta}{ital n}({bold r}) from summed atomic or atomiclike densities within the Hohenberg-Kohn formulation of density-functional theory. Iterative and perturbation solutions are detailed. The first-order perturbation and first iterative solutions are equivalent, yielding {Delta}{ital n}({bold r})=(4{pi}){sup {minus}1}{nabla}{sup 2}[({delta}{ital G}/{delta}{ital n}){sub system}{minus}({delta}{ital G}/{delta}{ital n}){sub atoms}], where the kinetic-exchange-correlation potential of the full system and the summed isolated atoms are ({delta}{ital G}/{delta}{ital n}){sub system} and ({delta}{ital G}/{delta}{ital n}){sub atoms}, respectively. {ital G} is the integrand of the kinetic-exchange-correlation energy. The second-order perturbation solution leads to a Fredholm equation relating (1) the second functional derivative, ({delta}{sup 2}{ital G}/{delta}{ital n}{sup 2}){sub system}, with summed atomic densities; (2) the induced electrostatic energy; and, (3) the difference between ({delta}{ital G}/{delta}{ital n}){sub system} and ({delta}{ital G}/{delta}{ital n}){sub atoms}. Numerical examples are shown for the Ni, O, and Na atoms embedded in jellium. {copyright} {ital 1996 The American Physical Society.}