Analytic hyperpolarizability and polarizability derivative with fractional occupation numbers for large extended systems.

The analytic hyperpolarizability and polarizability derivative with fractional occupation numbers are derived using Wigner's 2n + 1 rule. The formulation contains no terms that blow up for quasi-degenerate systems. The density-functional tight-binding method is used for implementation, which makes it possible to compute these third-order derivatives for systems containing up to one thousand atoms within 8 h using 24 CPU cores. A comparison between analytic and numerical non-resonance Raman activity spectra indicates that the numerical differentiation approach can give a significant deviation unless the strength of perturbative electric field is carefully chosen. With extremely high electronic temperatures, the polarizability and hyperpolarizability should converge to zero.