Incorporating Memory into Propagation of 1-Electron Reduced Density Matrices

TL;DR

This work addresses how memory effects govern the dynamics of $1$-electron reduced density matrices ($Q(t)$) in time-dependent configuration interaction (TDCI). It develops a self-contained linear time-delayed propagation framework by deriving a $4$-index tensor $B$ that connects full CI density matrices to $Q(t)$, and proves a delay equation that preserves Hermitian symmetry and trace while incorporating physical constraints. Numerical tests on $\mathrm{H}_2$ and $\mathrm{HeH}^+$ in STO-3G and 6-31G basis sets show that, with sufficient memory length and appropriate stride, the method yields highly accurate $1$-RDMs and reveals a physically meaningful memory timescale. The approach provides insight into memory-dependence in electron dynamics and offers a pathway to improving memory-inclusive exchange-correlation models in TDDFT, with implications for scalable, memory-aware simulations of molecular dynamics under time-dependent fields.

Abstract

For any linear system with unreduced dynamics governed by invertible propagators, we derive a closed, time-delayed, linear system for a reduced-dimensional quantity of interest. This method does not target dimensionality reduction: rather, this method helps shed light on the memory-dependence of $1$-electron reduced 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 $1$-electron reduced density possesses memory-dependence, the precise nature of this memory-dependence has not been understood. We derive a symmetry/constraint-preserving method to propagate reduced TDCI electron density matrices. In numerical tests on two model systems ($\text{H}_2$ and $\text{HeH}^+$), 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 implement our method, we derive the $4$-index tensor that relates reduced and full TDCI density matrices. Our derivation applies to any TDCI system, regardless of basis set, number of electrons, or choice of Slater determinants in the wave function.

Figures (7)

• Figure 1: For the molecule $\text{HeH}^+$ in the STO-3G atomic orbital basis set, we apply the methods described in Section \ref{['sect:procedure']} to compute $1$-electron reduced density matrices $Q(t)$. From the left plot, we conclude that the eigenvalues of $Q(t)$ do not stay constant in time, implying that $Q(t)$ cannot satisfy the Liouville-von Neumann equation for any choice of Hamiltonian $H(t)$. The right plot shows that $\mathop{\mathrm{trace}}\nolimits(Q(t)) = 2$. Since $\text{HeH}^+$ has $N=2$ electrons, this agrees with $\mathop{\mathrm{trace}}\nolimits(Q(t)) = N$, which we prove in Appendix \ref{['appendix:trace']}.
• Figure 2: For the molecule $\text{HeH}^+$ in the STO-3G basis with $\Delta t = 0.008268$ a.u., we apply the TDCI procedure from Section \ref{['sect:procedure']} to compute the time-dependent coefficients $\mathbf{a}(t)$ for 20,000 time steps. Because $a_1(t) \equiv 0$, in the associated full TDCI density matrix $P(t) = \mathbf{a}(t) \mathbf{a}(t)^\dagger$, the second column and second row vanish identically. The consequences of this are explained in Section \ref{['sect:symm']}.
• Figure 3: For each of four molecular systems we tested, with sufficiently large total memory ($k \ell \Delta t$), the 1RDMs produced by our propagation scheme (\ref{['eqn:Qpropsymm']}) agree closely with ground truth 1RDMs at all times $t$. Each plot shows $\text{MAE}(t)$ defined by (\ref{['eqn:MAE']}) versus physical time in atomic units (a.u.) From left to right, top to bottom, the maximum MAE values are bounded by $4 \times 10^{-6}$, $1.5 \times 10^{-9}$, $4 \times 10^{-7}$, and $10^{-5}$; the corresponding values for total memory are $5.3$, $6.6$, $0.6$, and $13$ a.u., respectively. Note that strides of $k \geq 2$ were used for all systems except $\text{H}_2$ in STO-3G. For details of all other parameters, consult the text in Section \ref{['sect:results']}.
• Figure 4: For each molecule in the STO-3G basis set, we fix the stride $k=1$. For each of two choices of $\Delta t$, we plot the RMSE (\ref{['eqn:RMSE']}) as a function of total memory $k \ell \Delta t$. Note that for both molecules, the RMSE approaches zero at similar rates, regardless of $\Delta t$. This indicates that our scheme (\ref{['eqn:Qpropsymm']}) identifies a physical time scale of 1RDM memory-dependence.
• Figure 5: For each of the four molecular systems we tested, we use less total memory than in Figure \ref{['fig:MAE_total_results_with_striding']}, and still find good agreement between modeled and true 1RDMs. In particular, note that the red (model) and black (true) curves in Figure \ref{['fig:5b']} are nearly indistinguishable. In Figure \ref{['fig:5b']}, the vertical blue line shows the time at which the electric field is turned off. Plots for the same molecular system were produced using the same parameter $\ell$; for these values and other details, see Section \ref{['sect:results']}.

Theorems & Definitions (25)

• Definition 1: Coordinates, Spin, and Orbitals
• Definition 2: Slater Determinants
• Definition 3: CI Basis
• Definition 4: Full Density Operator
• Definition 5: Marginalization and Reduced Density Operator
• Definition 6: Reduced Quantities
• Proposition 1
• Proposition 2
• Definition 7: Basis for Space of Hermitian Matrices
• Lemma 1