The Parallel-Transported (Quasi)-Diabatic Basis

TL;DR

This paper introduces a parallel-transported diabatic basis for electronic states, facilitating analytic expansions near degeneracies and relating derivative couplings to curvature, with applications in nuclear Schrödinger equation treatments.

Contribution

It develops a new diabatic basis using parallel transport, linking it to the singular-value basis and providing a method for Taylor series expansions without singularities.

Findings

The parallel-transported basis agrees with the singular-value basis up to second order.

Taylor series expansions are achieved via the projection operator without energy denominators.

The basis relates derivative couplings to curvature, aiding analytic treatments near degeneracies.

Abstract

This article concerns the use of parallel transport to create a diabatic basis. The advantages of the parallel-transported basis include the facility with which Taylor series expansions can be carried out in the neighborhood of a point or a manifold such as a seam (the locus of degeneracies of the electronic Hamiltonian), and the close relationship between the derivative couplings and the curvature in this basis. These are important for analytic treatments of the nuclear Schr\"odinger equation in a neighborhood of degeneracies. The parallel-transported basis bears a close relationship to the singular-value basis; in this article both are expanded in power series about a reference point and they are shown to agree through second order but not beyond. Taylor series expansions are effected through the projection operator, whose expansion does not involve energy denominators or any type of singularity, and in terms of which both the singular-value basis and the parallel-transported basis can be expressed. The parallel-transported basis is a version of Poincar\'e gauge, well known in electromagnetism, which provides a relationship between the derivative couplings and the curvature and which, along with a formula due to Mead, affords an efficient method for calculating Taylor series of the basis states and the derivative couplings. The case in which fine structure effects are included in the electronic Hamiltonian is covered.