Molecular geometry and symmetry from a differential geometry viewpoint

Abstract

ABSTRACT:Relations between an earlier generalization of molecular symmetry called symmorphy and a molecular equivalence based on diffeomorphisms of electron Ž density functional graphs the so-called DFG equivalence introduced in our previous . work are analyzed. Any two DFG-equivalent electron density functions can be derived from one another by a suitable transformation of the spatial coordinates and the electronic charge density scale; the classes of DFG equivalence are the orbits of a group of linear operators operating in the space of electron density functions. Within the symmorphy framework, the symmetry group is derived from the symmorphy group by taking an intersection of a subgroup of the symmorphy group and the group of isometries for a natural choice of the Riemannian metric tensor. The Riemannian metric properties provide a choice for a suitable reference electron density function for each class of equivalent densities. Such reference densities serve as tools for a systematic classification of the infinite family of electron densities of molecular conformations.

Mathematical Geometry and Groups for Low-Symmetry Metal Complex Systems

“…The Riemannian metric property allows us to choose a suitable reference electron density function for each class of equivalent density. Such reference densities serve in combination with quantum chemical calculations as a tool for the systematic classification of infinite families of electron densities of molecular conformation [15].…”