Second-order k⋅p perturbation theory with Vanderbilt pseudopotentials and plane waves

The k·p expressions for the gradient and curvature (and hence effective masses) of electronic energy bands in reciprocal space are presented for calculations involving nonlocal and non-normconserving Vanderbilt ultrasoft pseudopotentials [D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)]. The expressions have been implemented, and comparison with numerically calculated gradients and curvatures show that they are practical, accurate, and useful in the calculation of electronic densities-of-states.