Computing excited states of molecules using normalizing flows

TL;DR

The paper tackles the challenge of computing highly excited vibrational states by recognizing that coordinate choice crucially governs basis-set convergence. It introduces a general nonlinear coordinate parametrization based on normalizing flows, implemented as an invertible iResNet, and optimized via the variational principle to minimize the vibrational energies. The method yields dramatic gains in accuracy (up to several orders of magnitude over conventional coordinates) and transfers effectively across basis truncations, enabling cost-efficient calculations for high-dimensional systems and improving assignment of approximate quantum numbers. It also demonstrates promising extensions to electronic-structure problems, suggesting broad applicability of learned vibrational and electronic coordinates for rapid, accurate spectrum predictions.

Abstract

Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that leverages normalizing flows -- parametrized invertible functions -- to learn optimal vibrational coordinates that satisfy the variational principle. This approach produces coordinates tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis-set convergence of the calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/HNC. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian and hence allows for an effective assignment of approximate quantum numbers. We demonstrate that the optimized coordinates are transferable across different levels of basis-set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.

Figures (12)

• Figure 1: Schematic diagram of the computational workflow of the normalizing-flow function. As wrapping functions we used $L=\tanh(x)$. A fixed scaling procedure was applied to map the initial coordinate ranges to an identical domain. The linear scaling parameters $a$ and $b$ were optimized together with the MLP $\theta$-parameters.
• Figure 2: Convergence of H$_2$S, HCN and H$_2$CO vibrational energy levels. Plotted are the 100 lowest energy discrepancies using standard (blue squares) and normalizing-flows (red circles) coordinates. Light and dark colors represent truncations corresponding to a smaller and larger number of basis functions, respectively. a Energy discrepancies for H$_2$S at $P_\text{max}=12$ (140 basis functions) and $20$ (506). b Energy discrepancies for HCN at $P_\text{max}=12$ (140) and $32$ (1785). c Energy discrepancies for H$_2$CO at $P_\text{max}=9$ (1176) and $12$ (3906).
• Figure 3: Convergence of HCN/HNC vibrational energy levels. Shown are the lowest 200 energies for using Jacobi coordinates (squares) and normalizing-flow coordinates (circles). The energy discrepancies ($\Delta E_i$) relative to our converged benchmark reference are shown for several basis sets, truncated at $P_\text{max}=12$ (140 basis functions), 16 (285), 20 (506), 24 (811), and 28 (1200). Vibrational states assigned to the HCN isomer, the HNC isomer, and states with an energy above the isomerization barrier (delocalized) are differentiated by color. All states are slightly offset along the $P_\text{max}$ axis for visual clarity.
• Figure 4: Two-dimensional cuts of the HCN/HNC potential energy surface.a Cut along the Jacobi coordinates $R$ and $\alpha_{\angle \text{R-CN}}$. b Cut along the optimized normalizing-flow coordinates. The optimization was performed for a basis set truncated at $P_\text{max}=16$ (285 basis functions) with the loss function defined as the sum of the 100 lowest vibrational state energies. An effective decoupling of the surface in the normalizing-flows coordinates is patent.
• Figure 5: Convergence of the first 100 vibrational energies of H$_2$S. Convergence of the first 100 energy levels of H$_2$S using valence (blue squares), and optimized normalizing-flows (red circles) coordinates, as a function of $P_\text{max}$. Results obtained with normalizing-flow coordinates optimized for $P_\text{max}=12$ (green up-triangles), $P_\text{max}=16$ (cyan down-triangles), and $P_\text{max}=20$ (orange plus signs) and applied to calculations with larger $P_\text{max}$ are also shown. Thick horizontal lines indicate the average energy-level error for each $P_\text{max}$, which is minimized through training. Data points are slightly offset along the $P_\text{max}$ axis for improved visual clarity.