Raman tensor calculated from the2n+1theorem in density-functional theory

In this work we present a method to determine the Raman coefficients of tetrahedrally bonded semiconductors directly in density-functional theory. For this purpose we apply the $2n+1$ theorem to derive an analytical expression for the Raman tensor. To the best of our knowledge, this is the first ab initio application of the $2n+1$ theorem involving mixed third-order derivatives of the total energy in terms of phonon displacements and electric fields. Numerical results are given for Si, Ge, and various III-V compounds. Furthermore we compare our results with those obtained by frozen-phonon-like calculations and experimental data where available. Our approach can be easily extended to the calculation of Raman tensors for larger systems with other symmetries.