Prediction of Raman spectra with ultrasoft pseudopotentials

A formalism to calculate Raman susceptibilities as well as nonlinear optical susceptibilities with ultrasoft pseudopotentials is presented. Based on the $2n+1$ theorem, the third-order energy derivatives are computed from the first-order wave functions, in which a uniform electric field is treated by taking a long-wavelength limit analytically. For this treatment, two non-self-consistent second-order Sternheimer equations are introduced. Since they need to be solved only once, an additional computational effort is quite small as compared with those for the linear response calculations, in particular, for large systems. The method is applied for α-SiO₂ and YMn₂H₆ to validate the formalism.