Manifold curvature and Ehrenfest forces with a moving basis

TL;DR

This paper explores how the geometric properties of the quantum state manifold, specifically its curvature, explain the force terms in Ehrenfest dynamics involving a moving basis, linking physics with differential geometry.

Contribution

It provides a geometric interpretation of Ehrenfest forces, connecting velocity-dependent and Pulay terms to intrinsic and extrinsic curvature of the quantum state manifold.

Findings

Velocity-dependent Ehrenfest forces relate to intrinsic curvature.

Pulay terms are associated with extrinsic curvature.

Geometric perspective clarifies force origins in quantum-classical dynamics.

Abstract

Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.