Many-body quantum dynamics by the reduced density matrix based on the time-dependent density functional theory

We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of G"{o}rling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant $\lambda$ naturally arises. We then find the reduced density matrix $\rho_\lambda({\bf r},{\bf r}‘,t)$,which,byconstruction,hasthe$\lambda$-independent diagonal elements ${\rho }_{\lambda }(\mathbf{r},\mathbf{r},t)=n(\mathbf{r},t)$, $n(\mathbf{r},t)$ being the particle density. The off-diagonal elements of $\rho_\lambda({\bf r},{\bf r}’,t)$ contribute importantly to the processes, which cannot be treated via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in $\lambda$, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi’s golden rule formula.