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

The k . p expressions for the gradient and curvature land 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.