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Structural homeomorphism between the electron density and the virial field

Abstract:

The virial field V(r) is defined by the local statement of the quantum mechanical virial theorem, as the trace of the Schrödinger stress tensor. This field defines the electronic potential energy density of an electron at r and integrates to minus twice the electronic kinetic energy. It is the most short-ranged description possible of the local electronic potential energy and it exhibits the same transferable behavior over bounded regions of real space (corresponding to the functional groups of chemistry) as does ρ(r). This article establishes a structural homeomorphism between - V(r) and ρ(r), showing that the two fields are homeomorphic over all of the nuclear configuration space. The stable or unstable structure defined by the gradient vector field ∇ρ(r; X) for any configuration X of the nuclei can be placed in a one-to-one correspondence with a structure defined by the field - ∇V(r;X′). In particular, a molecular graph for ρ(r) defining a molecular 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 energy density in the latter. The homeomorphism is also geometrically faithful, an equilibrium geometry in general, exhibiting equivalent structures in the two fields. The demonstration that the virial field, whose integrated value equals twice the total energy, is essentially just a locally scaled version of the electron density is suggestive of possible new approaches in density functional theory. © 1996 John Wiley & Sons, Inc.