Characterizing the E⊗e Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools

Abstract

The term ‘mathematical chemistry’ is mostly associated with applications of graph theory in topological issues of 3D chemical structures, thought of as a collection of atoms as dots and bonds as lines. We propose here new directions in this field, coming from the side of theoretical chemistry approached with modern computational tools. Possible challenges are proposed in using ancillary tools of differential geometry for examining the potential energy surfaces of certain specific structural prototypes. Concretely, we describe here the geodesics on the surfaces related to the potential energy functions of the so-called E⊗e Jahn–Teller effect, a spontaneous symmetry-breaking phenomenon also known as a case of conical intersection. To illustrate the case, first-principles (ab initio) quantum chemical calculations are performed on the cyclo-propenyl molecular radical C3H3.

A Note on a Well-Defined Sectional Curvature of a Semi-Symmetric Non-Metric Connection

“…To our knowledge, similar models of semi-symmetric non-metric connections are not given. Then, we expect to develop models and applications of semi-symmetric non-metric connections in some other fields, e.g., in quantum and theoretical chemistry, directions in which the first author of this paper and her co-workers took preliminary steps, using the tools of differential geometry (see [4], [12]).…”

“…For instance, one may devise applications in the topology of multidimensional surfaces representing the energy dependence of chemical edifices as function of symmetry-classified changes in molecular geometries, with relevance in modeling physical properties and chemical transformations of molecules and materials. In this sense, a special situation, which deserves further attention, with the help of analytic tools outlined here, would be the case of spontaneous breaking of molecular symmetry in systems with energy surfaces containing the so-called conical intersections (see [4]). We foresee that the use conformal changes and semi-symmetric connections could open a new geometric perspective on such problems, pertaining to molecular symmetry and chemical dynamics.…”

Symmetry in Quantum and Computational Chemistry: Volume 2

“…is a subproblem when predicting the structure of matter, particularly the structure of a crystal, regarding information that makes it possible to predict almost all of its properties, and the design of mathematical models for the closest symmetrical packing is an important and a challenging task in the practical application of optimization theory in theoretical chemistry). In [2], a characterization of the E⊗e Jahn-Teller (a spontaneous symmetry-breaking phenomenon, also known as a case of conical intersection) potential energy surfaces by differential geometry tools was presented. In [3], a synthesis of 2-pyridyltellurenyl bromide via Br 2 oxidative cleavage of the Te-Te bond of dipyridylditelluride was reported, and a single-crystal X-ray diffraction analysis of 2-pyridyltellurenyl bromide demonstrated that the Te atom of 2-pyridyltellurenyl bromide was involved in four different noncovalent contacts (Te• • • Te, Te• • • Br, and Te• • • N), forming a 3D supramolecular symmetrical framework.…”