Magnetic-Translational Sum Rule and Approximate Models of the Molecular Berry Curvature

TL;DR

This paper explores the role of Berry curvature in molecular systems under magnetic fields, demonstrating how it arises from translational symmetry and proposing efficient approximations for computational modeling.

Contribution

It introduces a physical interpretation of Berry curvature as nuclear shielding and develops approximate models using Mulliken fragmentation to reduce computational cost.

Findings

Finite basis sets of London orbitals capture Berry curvature accurately.

Standard Gaussian basis sets do not fully capture Berry curvature.

Proposed approximations recover up to 95% of the exact Berry curvature.

Abstract

The Berry connection and curvature are key components of electronic structure calculations for atoms and molecules in magnetic fields. They ensure the correct translational behavior of the effective nuclear Hamiltonian and the correct center-of-mass motion during molecular dynamics in these environments. In this work, we demonstrate how these properties of the Berry connection and curvature arise from the translational symmetry of the electronic wave function and how they are fully captured by a finite basis set of London orbitals but not by standard Gaussian basis sets. This is illustrated by a series of Hartree-Fock calculations on small molecules in different basis sets. Based on the resulting physical interpretation of the Berry curvature as the shielding of the nuclei by the electrons, we introduce and test a series of approximations using the Mulliken fragmentation scheme of the electron density. These approximations will be particularly useful in ab initio molecular dynamics calculations in a magnetic field, since they reduce the computational cost, while recovering the correct physics and up to 95% of the exact Berry curvature.