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

The k . p expressions for the gradient and curvature land hence effective m asses) of electronic energy bands in reciprocal space are presented for cal culations involving nonlocal and non-normconserving Vanderbilt ultrasoft ps eudopotentials [D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)]. The expressio ns have been implemented, and comparison with numerically calculated gradie nts and curvatures show that they are practical, accurate, and useful in th e calculation of electronic densities-of-states.