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.

翻译：对于任何未约化动力学由酉传播子控制的线性系统，我们推导出一个封闭的、含时滞的线性系统，用于描述约化维度的感兴趣量。我们将该方法应用于理解含时组态相互作用（TDCI）中约化单电子密度矩阵的记忆依赖性——这是一种求解分子中电子关联动力学的方案。尽管含时密度泛函理论已证实约化单电子密度具有记忆依赖性，但这种记忆依赖性的确切本质尚未被理解。我们推导出一种自洽的、保持对称性/约束的方法，用于传播约化TDCI电子密度矩阵。在两个模型系统（H₂和HeH⁺）的数值测试中，我们表明，在足够大的时延（或记忆依赖性）下，我们的方法能以高定量精度传播约化TDCI密度矩阵。我们研究了时步和基组对结果的影响。为推导该方法，我们计算了连接约化与完整TDCI密度矩阵的四阶张量。该计算适用于任何TDCI系统，无论基组、电子数或波函数中Slater行列式的选择为何。这一计算可以证明：约化TDCI密度矩阵的迹是常数且等于电子数。