Calculation of Berry curvature using non-orthogonal atomic orbitals

We present a derivation of the full formula to calculate Berry curvature on non-orthogonal numerical atomic orbital (NAO) bases. Because usually, number NAOs is larger than that Wannier bases, we use contraction method reduce basis sizes, which can greatly improve calculation efficiency without significantly reducing accuracy. benchmark by calculating ferroelectric BaTiO3and bcc Fe, as well anomalous Hall conductivity for Fe. The results are in excellent agreement with finite-difference and previous literature. find there corrections terms Kubo curvature. For NAO base, differences between two methods negligibly small, but reduced bases sets, correction become larger, may not be neglected some cases. developed this work readily applied generalized functions.