Time-dependent density-matrix-functional theory

Although good progress has been made in the calculation of correlation energies from total energy expressions which are implicit functionals of the one-particle reduced density matrix, and explicit functionals of the natural orbitals (NOs) and their occupation numbers, a formulation of the calculation of excitation energies in this so-called density-matrix-functional theory (DMFT) is still lacking. In this paper we propose a time-dependent density-matrix-functional theory (TDDMFT). It is based on the equation of motion (EOM) for the 1-matrix P-(s)(t) in the representation of the stationary NOs. In the final form of the EOM, the rate of change of the P-(s)(t), partial derivative P-(s)(t)/partial derivative t, is determined by the commutator of the generalized time-dependent Fock matrix F-(s)(t) with P-(s)(t) plus an additional term D-(s)(t). The matrix F-(s)(t) determines the evolution of the NOs in the time-dependent one-electron Schrodinger equations, while D-(s)(t) determines the time evolution of the NO occupations. With the neglect of the electron Coulomb correlation, the time-dependent one-electron equations for the NOs reduce to those for the Hartree-Fock (HF) orbitals of time-dependent HF (TDHF) theory. The coupled-perturbed equations of TDDMF response theory (TDDMFRT) are derived for the linear response of the 1-matrix delta P-(s)(t) to a time-dependent perturbation delta v(ext)(t) of the external potential. The frequency-dependent changes delta P-(s),P-ij(omega) and delta P-(s),P-kl(omega) are coupled through the coupling matrix K-ijkl(omega), which is produced with the derivatives of F-(s)(t) and D-(s)(t) with respect to P-kl(t(')). Based on the response equations, TDDMFRT eigenvalue equations are derived for the electron excitations omega(q).