  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.
