Calculation of Berry curvature using nonorthogonal atomic orbitals

TL;DR

This paper derives a comprehensive formula for calculating Berry curvature using non-orthogonal atomic orbitals, introduces an orbital contraction method for efficiency, and validates the approach through benchmark calculations on materials like BaTiO3 and Fe.

Contribution

It provides a new full formula for Berry curvature calculation with non-orthogonal atomic orbitals and demonstrates an efficient basis reduction method with validated accuracy.

Findings

The derived formula matches previous results with high accuracy.

Orbital contraction improves computational efficiency significantly.

Correction terms to the Kubo formula are important for reduced bases.

Abstract

We present a derivation of the full formula to calculate the Berry curvature on non-orthogonal numerical atomic orbital (NAO) bases.Because usually, the number of NAOs is larger than that of the Wannier bases, we use a orbital contraction method to reduce the basis sizes, which can greatly improve the calculation efficiency without significantly reducing the calculation accuracy. We benchmark the formula by calculating the Berry curvature of ferroelectric BaTiO$_3$ and bcc Fe, as well as the anomalous Hall conductivity (AHC) for Fe. The results are in excellent agreement with the finite difference and previous results in the literature. We find that there are corrections terms to the Kubo formula of the Berry curvature. For the full NAO base, the differences between the two methods are negligibly small, but for the reduced bases sets, the correction terms become larger, which may not be neglected in some cases. The formula developed in this work can readily be applied to the non-orthogonal generalized Wannier functions.