Transferability and interpretability of vibrational normalizing-flow coordinates

TL;DR

This work addresses how to choose vibrational coordinates to improve accuracy and efficiency in variational vibrational calculations. It introduces normalizing-flow coordinates, realized via an iResNet invertible network, and optimized by a variational loss to tailor coordinates to a given molecule and basis, thereby enhancing Hamiltonian separability and basis-set convergence. The study demonstrates that these coordinates can interpretably shift the average density center toward the eigenbasis, capture anharmonicity through nonlinear mappings, and transfer effectively across basis truncations, isotopologues, and chemically related molecules—with notable energy-accuracy gains over traditional linear or fixed coordinates. The findings suggest that learned coordinate systems may reveal intrinsic vibrational structure and offer a practical path toward generalizable, physically meaningful representations of molecular vibrational motion, with future work aimed at scaling to larger systems, incorporating symmetry, and embedding molecular descriptors into the coordinate maps.

Abstract

The choice of vibrational coordinates is crucial for the accuracy, efficiency, and interpretability of molecular vibrational dynamics and spectra calculations. We explore the recently proposed normalizing-flow vibrational coordinates, which are learned molecule-specific coordinate transformations optimized for a given basis set. Much like how spherical coordinates naturally simplify the hydrogen atom by embedding physical insight into the coordinate system, normalizing-flow coordinates offload complexity from the basis functions into the coordinate transformation itself. This shift not only improves basis-set convergence, but also enhances interpretability of vibrational motions. We provide an analysis of the utility, interpretation and associated constraints of normalizing-flow vibrational coordinates. Moreover, we demonstrate that these coordinates can be generalized across different isotopologues, and even structurally related molecules, achieved with minimal fine-tuning of selected output parameters.

Figures (5)

• Figure 1: (Top panel) Morse potential expressed in fixed linear coordinates (black), optimized linear coordinates (red) and normalizing-flow coordinates (blue). The Hermite functions, $\mathcal{H}_n(q)$, with $n=0,3,...,18$ are depicted as grey dashed lines. (Bottom panel) The corresponding optimized coordinate $q = f(r;\theta)$ as a function of the displacement coordinate $r-r_e$.
• Figure 2: (Top panel) The one-dimensional bending PES of H$_2$S (red) and H$_2$O (black) with bond lengths fixed to the equilibrium values. (Bottom panel) A comparison between the H$_2$O PES (black) and the composition of the H$_2$S with two different transformation optimised to try to approximate the water potential. The linear transformation $L^{-1}$ is shown in red and the nonlinear $h^{-1}$ in blue. The linear approach extends the potential beyond the physical domain, marked by the light-blue vertical line at $\theta=\pi$, while the nonlinear transformation keeps the potential within the domain.
• Figure 3: (Top panel) Optimized normalizing-flow coordinates for varying number of target states (using the same number of basis functions as target states in each case) as a function of the physical coordinate $r$. (Middle panel) The first augmented density distribution as a function of $r$. (Bottom panel) Potential energy curve along with the first few first associated eigenfunctions used in the coordinate optimization. Each eigenfunction is vertically shifted by its corresponding eigenvalue for visual purposes.
• Figure 4: Convergence of the first 100 vibrational energy levels of (a) D$_2$S and (b) HDS using fixed linear transformation of valence coordinates (red squares), transferred normalizing-flow coordinates (green circles), and fine-tuned transferred normalizing-flow coordinates (blue triangles). The vertical axis represents the calculated energy error, $E_i - E_i^{(\text{Ref})}$, while the horizontal axis shows the basis-set truncations, $P_\text{max}$. The transferred coordinates were originally optimized for H$_2$S with $P_\text{max}=12$, and the fine-tuned parameters were obtained through re-optimization of the linear parameters in the normalizing-flows model. The black horizontal lines show the average error per energy level. The data points are slightly offset along the $P_\text{max}$ axis for visual clarity.
• Figure 5: Convergence of the first 100 energy levels of H$_2$O for $P_\text{max}=12$ (140 basis functions) and $P_\text{max}=20$ (506) using fixed linear transformed valence coordinates (red and orange squares) compared to fine-tuned (FT.) normalizing-flow coordinates transferred from H$_2$S calculations, optimized for $P_\text{max}=12$, for $P_\text{max}=12$ and $P_\text{max}=20$ (green and blue circles). The vertical axis represents the calculated energy error, $E_i - E_i^{(\text{Ref})}$, while the horizontal axis shows the reference energy, $E_i^{(\text{Ref})}$. The converged energy values for H$_2$O were computed using optimized normalizing-flow coordinates with $P_\text{max}=24$.