Signed Phases and Fields Associated with Degeneracies

TL;DR

This paper derives expressions for the sign of the topological angle around degeneracies in molecular energy surfaces and explores the associated pseudo-fields in conical intersections, linking theoretical models with computational results.

Contribution

It introduces new formulas for the sign of the topological angle and analyzes the pseudo-fields in conical intersections, connecting topological properties with nonadiabatic couplings.

Findings

Expressions for the sign of the topological angle involving coupling Hamiltonian derivatives.

Both curl and tensorial fields are aligned with the seam directions and are zero outside the seam.

The angular dependence of NACTs matches recent computational results for C2H.

Abstract

In the first part, expressions are given for the {\it sign} of the topological angle that is acquired upon making a loop around a degeneracy ("conical intersection") point of two molecular energy surfaces. The expressions involve the partial derivatives (with respect to the nuclear coordinates) of the matrix elements of the coupling Hamiltonian. Examples are given of a few studied cases, such as of excited states that have topological angles with a sign opposite to those in the ground states. In the second part, the two dimensional (or two parameter) situation that characterizes a conical intersection (ci) between potential surfaces in a polyatomic molecule is constructed as a limiting case of the three dimensional Dirac-monopole situation. For an electron occupying a twofold state, we obtain both the "magnetic-field" (or curl-field) and the tensorial (or Yang-Mills-) field (which is the sum of a curl and of a vector- product term). These pseudo- fields represent the reaction of the electron on the nuclear motion via the nonadiabatic coupling terms (NACTs). We find that both fields are aligned with the orthogonal, (so called) seam directions of the ci and are zero everywhere outside the seam, but they differ as regards the flux that they produce. In a two-state situation, the fields are representation dependent and the values of, e.g., the fluxes depend on the state that the electron occupies. The angular dependence of the NACTs and the fields calculated from a general linearly coupled model agrees with recently computed results for $C_2 H$ [A.M. Mebel, M. Baer and S.H. Lin, J.Chem. Phys. {\bf 115} 3673 (2001)]. An effective-Hamiltonian formalism is proposed for experimentally observing and distinguishing between the different fields.