Natural-Orbital-Based Neural Network Configuration Interaction

TL;DR

This work addresses the efficiency bottleneck of selective configuration interaction by introducing approximate natural orbitals as an optimal single-particle basis. NOs are generated from intermediate NNCI solutions and used to rotate the orbital basis before continuing determinant selection, yielding more compact CI expansions and faster convergence. Across benchmarks on C3H8, NH3, H2O, and CO, NO-based bases reduce the number of determinants required for a target $E_ ext{corr}$ and improve the best energies, with a one-shot NO update capturing most gains and a second update benefitting larger systems. The study provides practical guidelines for integrating orbital optimization into ML-assisted SCI workflows, showing that approximate NOs offer a simple, effective boost to computational efficiency.

Abstract

Selective configuration interaction methods approximate correlated molecular ground- and excited states by considering only the most relevant Slater determinants in the expansion. While a recently proposed neural-network-assisted approach efficiently identifies such determinants, the procedure typically relies on canonical Hartree-Fock orbitals, which are optimized only at the mean-field level. Here we assess approximate natural orbitals - eigenfunctions of the one-particle density matrix computed from intermediate many-body eigenstates - as an alternative. Across our benchmarks for H$_2$O, NH$_3$, CO, and C$_3$H$_8$ we see a consistent reduction in the required determinants for a given accuracy of the computed correlation energy compared to full configuration interaction calculations. Our results confirm that even approximate natural orbitals constitute a simple yet powerful strategy to enhance the efficiency of neural-network-assisted configuration interaction calculations.

Figures (4)

• Figure 1: Color map for the correlation energy $E_\mathrm{corr}$ of C3H8 as a function of the number of determinants (vertical axis) and the number of basis orbitals (horizontal axis) with thin black iso-contour lines. (a) molecular orbital basis and (b) natural orbitals obtained from an intermediate many-body solution (green symbol in (a)) and (c) natural orbitals obtained from a costly many-body solution (gray symbol in (a)). The white dashed line indicates full-CI benchmarks from Ref. Gao2024. The inset displays the absolute values of the 1-RDM at the best converged point in each basis, its off-diagonal weight $W_\mathrm{off}$ (see Eq. \ref{['eq:off_diag_density_norm']}) indicates the quality of the basis as $W_\mathrm{off}\rightarrow 0$ for the exact solution.
• Figure 2: Color map for the correlation energy $E_\mathrm{corr}$ of NH3 as a function of the number of determinants (vertical axis) and the number of basis orbitals (horizontal axis) with thin black iso-contour lines. (a) molecular orbital basis and (b) natural orbitals obtained from an intermediate many-body solution (green symbol in (a)) and (c) natural orbitals obtained from a costly many-body solution (gray symbol in (a)). The white dashed line indicates full-CI benchmarks from Ref. Gao2024. The inset displays the absolute values of the 1-RDM at the best converged point in each basis, its off-diagonal weight $W_\mathrm{off}$ (see Eq. \ref{['eq:off_diag_density_norm']}) indicates the quality of the basis as $W_\mathrm{off}\rightarrow 0$ for the exact solution.
• Figure 3: Color map for the correlation energy $E_\mathrm{corr}$ of H2O as a function of the number of determinants (vertical axis) and the number of basis orbitals (horizontal axis) with thin black iso-contour lines. (a) molecular orbital basis and (b) natural orbitals obtained from an intermediate many-body solution (green symbol in (a)). The white dashed line indicates full-CI benchmarks from Ref. Gao2024. The inset shows the absolute values of the 1-RDM at the best converged point in each basis, its off-diagonal weight $W_\mathrm{off}$ (see Eq. \ref{['eq:off_diag_density_norm']}) indicates the quality of the basis as $W_\mathrm{off}\rightarrow 0$ for the exact solution.
• Figure 4: Color map for the correlation energy $E_\mathrm{corr}$ of CO as a function of the number of determinants (vertical axis) and the number of basis orbitals (horizontal axis) with thin black iso-contour lines. (a) molecular orbital basis and (b) natural orbitals obtained from an intermediate many-body solution (green symbol in (a)). The white dashed line indicates full-CI benchmarks from Ref. Gao2024. The inset shows the absolute values of the 1-RDM at the best converged point in each basis, its off-diagonal weight $W_\mathrm{off}$ (see Eq. \ref{['eq:off_diag_density_norm']}) indicates the quality of the basis as $W_\mathrm{off}\rightarrow 0$ for the exact solution.