Time Dependent Density-Functional Theory - Linear Response

The formal derivation of the linear response of time-dependent density-functional theory as shown by E. K. U. Gross is presented. The transformation of formal theory to the working linear response equations in the form of Casida's eigenvalue equation is demonstrated, and the results are applied to small monatomic, diatomic and triatomic systems. The application of different operators to the perturbed density is discussed, with the most attention being given to the dynamic polarizabilities. The dynamic polarizabilities and excitation spectra for N₂ [nitrogen gas] are then analyzed. The first excitation energy is noted to be in line with Koopmans' theorem. Finally three orbital localization algorithms and their implementation are detailed, with comparisons between the one-sided and two-sided Jacobi implementations present. The performance of a serial and then a parallel algorithm are shown. The poor performance of parallel algorithm is explained.