Comparison between projector augmented-wave and ultrasoft pseudopotential formalisms at the density-functional perturbation theory level

Abstract

We compare the recently derived density-functional perturbation expressions within the projector augmented-wave (PAW) formalism [C. Audouze et al., Phys. Rev. B 73, 235101 (2006)] to those found in the ultrasoft pseudopotential (USPP) framework, for the perturbations of the atomic-displacement type. As a preliminary step, we re-examine the correspondence of the two models in the unperturbed case, showing precisely the different kinds of dependencies of each physical quantity on a perturbation parameter. Then, we identify the different contributions for the first- and the second-order responses of the energy in both formalisms, pointing out the additional contributions in the PAW case. We complete also the formalism by providing nonvariational and variational second-order derivative expressions for PAW, and a variational second-order derivative expression for USPP.