STRUCTURAL HOMEOMORPHISM BETWEEN THE ELECTRON-DENSITY AND THE VIRIAL FIELD

The virial field V(r) is defined by the local statement of the quantum mechanical virial theorem, as the trace of the Schrodinger stress ten ser. This field defines the electronic potential energy density of an electron at r and integrates to minus twice the electronic kinetic ene rgy. It is the most short-ranged description possible of the local ele ctronic potential energy and it exhibits the same transferable behavio r over bounded regions of real space (corresponding to the functional groups of chemistry) as does rho(r). This article establishes a struct ural homeomorphism between - V(r) and rho(r), showing that the two fie lds are homeomorphic over all of the nuclear configuration space. The stable or unstable structure defined by the gradient vector field del rho(r; chi) for any configuration chi of the nuclei can be placed in a one-to-one correspondence with a structure defined by the field - del V(r; chi'). In particular, a molecular graph for rho(r) defining a mo lecular structure is mirrored by a corresponding virial graph for V(r) and the lines of maximum density linking bonded nuclei in the former field are matched by a set of lines of maximally negative potential en ergy density in the latter. The homeomorphism is also geometrically fa ithful, an equilibrium geometry in general, exhibiting equivalent stru ctures in the two fields. The demonstration that the virial field, who se integrated value equals twice the total energy, is essentially just a locally scaled version of the electron density is suggestive of pos sible new approaches in density functional theory. (C) 1996 John Wiley & Sons, Inc.