On the density matrix based approach to time-dependent density functional response theory

The formulation of time-dependent Kohn–Sham (TDKS) response theory based on the noninteracting one-particle density matrix is reanalyzed in detail. A transparent derivation starting from a von-Neumann-type equation of motion for the TDKS one-particle density matrix is presented. The resulting scheme has a simple structure and leads to compact expressions for frequency-dependent response properties. A systematic treatment of excited states is inferred from a pole analysis of the frequency-dependent density matrix response. A variational principle for excitation energies is established. Excited state properties are straightforward by analytical derivative techniques. The theory provides a particularly suitable starting point for linear scaling implementations. Magneto-optic properties such as rotatory strengths and the rotatory dispersion are accessible from the TDKS current-density response. The formalism is gauge-invariant. Various new sum rules within the adiabatic approximation (AA) are derived. It is shown that there is no “assignment problem” for excited states in the density matrix based formulation; the common density based approach is included as a special case. Merits and limitations of the AA are discussed.