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), ∂P₍ₛ₎(t)∕∂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 Schr\"odinger 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 δP₍ₛ₎(t) to a time-dependent perturbation δvₑₓₜ(t) of the external potential. The frequency-dependent changes δP(s),ij(ω) and δP(s),kl(ω) are coupled through the coupling matrix Kᵢⱼₖₗ(ω), which is produced with the derivatives of F₍ₛ₎(t) and D₍ₛ₎(t) with respect to Pₖₗ(t^'). Based on the response equations, TDDMFRT eigenvalue equations are derived for the electron excitations ωq.