A Linear Time-Delay Scheme to Propagate Reduced Electron Density Matrices

For any linear system where the unreduced dynamics are governed by unitary propagators, we derive a closed, time-delayed, linear system for a reduced-dimensional quantity of interest. We apply this method to understand the memory-dependence of reduced $1$-electron density matrices in time-dependent configuration interaction (TDCI), a scheme to solve for the correlated dynamics of electrons in molecules. Though time-dependent density functional theory has established that the reduced $1$-electron density possesses memory-dependence, the precise nature of this memory-dependence has not been understood. We derive a self-contained, symmetry/constraint-preserving method to propagate reduced TDCI electron density matrices. In numerical tests on two model systems (H$_2$ and HeH$^+$), we show that with sufficiently large time-delay (or memory-dependence), our method propagates reduced TDCI density matrices with high quantitative accuracy. We study the dependence of our results on time step and basis set. To derive our method, we calculate the $4$-index tensor that relates reduced and full TDCI density matrices. Our calculation applies to any TDCI system, regardless of basis set, number of electrons, or choice of Slater determinants in the wave function. This calculation enables a proof that the trace of the reduced TDCI density matrix is constant and equals the number of electrons.