Differential equations for ground-state electron density and Slater sum in atoms and molecules with and without external fields

Abstract

The ground-state density amplitude {l_brace}{rho}(r){r_brace}{sup 1/2} for atoms and molecules satisfies a Schroedinger equation in which the customary one-body potential energy V(r) of density functional theory is supplemented by the addition of the Pauli potential V{sub p}(r). Since neither the exchange-correlation potential V{sub xc} or V{sub p} are presently known as functionals of the electron density {rho}(r), approximations are currently unavoidable. Here, widespread use is made of semiclassical approximations, within a self-consistent field framework both with and without magnetic fields. The importance of low-order gradient quantities {nabla}{sup 2}{rho}/{rho} and ({nabla}{rho}/{rho}){sup 2} is one focal point, while a generalized low-density approximation is another. New relativistic differential equations are given. Then, the arguments are generalized to embrace the so-called Slater sum P(r, {beta}): {beta} = (k{sub {beta}}T){sup {minus}1}, of statistical mechanics, generated by the one-body potential V(r). This is a generalized partition function, and differential equations are set up for this quantity P(r, {beta}) with and without external fields. Finally, some potentially fruitful directions for treating cylindrically symmetric inhomogeneous electron liquids are outlined, following the very recent work of Amovilli and March. These include modeling the Slater sum along the electric field direction for the Stark effect in a hydrogen like atom.