Incorporating the Virial Field into the Hartree-Fock Equations

Abstract

The underlying operational observation of chemistry — that of a functional group exhibiting a characteristic set of properties — requires that the density distribution of a group and hence its properties, be relatively insensitive to changes in its neighbouring groups. However, the fields that are used in the determination of the wave function and of the density in DFT are not short-range, but instead reflect the long-range nature associated with the individual e-n and e-e Coulombic fields. The observation upon which the theory of atoms in molecules is founded concerns the paralleling transferability of the electron density with all ‘dressed’ property densities. The virial field V(r), the virial of the Ehrenfest force on an electron, describes the energy of interaction of an electron at some position r with all of the other particles in the system, averaged over the motions of the remaining electrons. When integrated over all space it yields the total potential energy of the molecule, including the nuclear energy of repulsion. Because of this inclusion, V(r) yields the most short-range description possible of the potential interactions in a many-electron system. V(r), as well as the kinetic energy density G(r) and their sum E _e (r), all demonstrably parallel the short-range behaviour underlying the transferable nature of p(r). All three energy fields are determined by the one-matrix whose diagonal elements in addition determine p(r). Thus it is the one-matrix that is short-ranged and responsible for the observation of functional groups with characteristic, transferable properties in chemistry. This paper describes a method for incorporating the virial field into the self-consistent field calculation to obtain an exact prescription of the ‘average field’ experienced by a single electron in a many- electron system.