Density-functional perturbational theory for dielectric tensors in the ultrasoft pseudopotential scheme

We introduce a density-functional perturbational scheme based on ultrasoft pseudopotentials for calculating dielectric tensors of periodic systems. We obtain a variational functional for the second-order derivative of the energy with respect to an electric field. Our scheme makes use of the correspondence between all-electron and pseudo-wave-functions introduced in the projector augmented wave method. While we here specifically focus on ultrasoft pseudopotentials, our scheme also covers the case of norm-conserving ones. In the latter case, our formulation coincides with earlier ones but highlights the implied approximations. By construction, our scheme also applies to the all-electron projector augmented wave method. We first assess the validity of our scheme by calculating polarizability tensors and Raman intensities of small molecules $({\mathrm{H}}_2\mathrm{O},{\mathrm{CH}}_4,{\mathrm{NH}}_3).\mathrm{}$ We find good agreement with both experimental data and previous all-electron results. We then illustrate the potential of our scheme for treating systems of relatively large size through an application to a disordered model structure of vitreous silica.