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 {Δ}n(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 {Δ}n(r)=(4{π})^-1{{{∇}}}²[(δG/δn{)}system-(δG/δn{)}ₐₜₒₘₛ], where the kinetic-exchange-correlation potential of the full system and the summed isolated atoms are (δG/δn{)}systemand (δG/δn{)}ₐₜₒₘₛ$, respectively. G is the integrand of the kinetic-exchange-correlation energy. The second-order perturbation solution leads to a Fredholm equation relating (1) the second functional derivative, (${{{δ}}}²G/δ{n}²)system, with summed atomic densities; (2) the induced electrostatic energy; and, (3) the difference between ({δ}G/{δ}n)system and ({δ}G/{δ}n)ₐₜₒₘₛ. Numerical examples are shown for the Ni, O, and Na atoms embedded in jellium. {} 1996 The American Physical Society.