Spectral and Fermi surface properties from Wannier interpolation

We present an efficient first-principles approach for calculating Fermi surface averages and spectral properties of solids, and use it to compute the low-field Hall coefficient of several cubic metals and the magnetic circular dichroism of iron. The first step is to perform a conventional first-principles calculation and store the low-lying Bloch functions evaluated on a uniform grid of $k$ points in the Brillouin zone. We then map those states onto a set of maximally localized Wannier functions, and evaluate the matrix elements of the Hamiltonian and the other needed operators between the Wannier orbitals, thus setting up an ``exact tight-binding model.'' In this compact representation the $k$-space quantities are evaluated inexpensively using a generalized Slater-Koster interpolation. Owing to the strong localization of the Wannier orbitals in real space, the smoothness and accuracy of the $k$-space interpolation increases rapidly with the number of grid points originally used to construct the Wannier functions. This allows $k$-space integrals to be performed with ab initio accuracy at low cost. In the Wannier representation, band gradients, effective masses, and other $k$ derivatives needed for transport and optical coefficients can be evaluated analytically, producing numerically stable results even at band crossings and near weak avoided crossings.