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An investigation on the Jacobi-Legendre polynomial expansion for potential-energy-surface fitting of multi-species triatomic systems

This study investigates the applicability of the Jacobi-Legendre polynomial cluster-expansion approach for fitting potential energy surfaces of multi-species triatomic systems, demonstrating that while the method struggles with long-range interactions, it achieves high accuracy in the short-to-medium range for systems like HeLiH+ and HSO, as validated by bound-state calculations.

Original authors: Ajay Mohan Singh Rawat, Antonino Polimeno, Sergio Rampino

Published 2026-09-10

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CC BY 4.0

Original authors: Ajay Mohan Singh Rawat, Antonino Polimeno, Sergio Rampino

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand how molecules behave, scientists often rely on a concept called the potential energy surface. Imagine a vast, invisible landscape where every point represents a specific arrangement of atoms within a molecule. The height of the land at any given spot tells you how much energy that arrangement holds. When atoms move across this landscape, they roll downhill toward lower energy, which dictates how they vibrate, how they collide, and how they react with one another. This map is the essential blueprint for predicting everything from the colors of distant stars to the chemistry of our own atmosphere. However, creating an accurate map is difficult because the energy changes in complex, non-linear ways as atoms move closer or farther apart. For decades, researchers have tried to build mathematical formulas that can trace these contours with high precision, but the task becomes especially tricky when dealing with systems made of different types of atoms, where the interactions can vary wildly depending on the distance and angle between them.

In a recent study, researchers at the University of Padua set out to test a new mathematical tool designed to draw these energy maps for three-atom systems. They focused on a method based on Jacobi-Legendre polynomials, a technique originally developed to model carbon materials but never before rigorously tested on mixed-species molecules. The team chose two very different test cases to see if the method could handle the job: a system made of helium, lithium, and hydrogen ions, and another composed of hydrogen, sulfur, and oxygen. The first system is relatively simple, featuring a single, shallow dip in its energy landscape, while the second is far more complex, with deep valleys and multiple peaks that represent different stable arrangements. By feeding the computer a massive set of pre-calculated energy points, the researchers asked the new method to learn the shape of the terrain and then checked how well its version of the map matched the original.

The results showed that the new approach is remarkably effective for the regions where chemical reactions actually happen. In the short and medium distances between atoms—where the atoms are close enough to feel each other's presence strongly—the method produced a nearly perfect copy of the known energy landscapes. The differences between the new map and the original were tiny, often amounting to less than one-thousandth of an electron volt, a scale so small it is barely perceptible in the world of molecular physics. The researchers found that the new maps correctly identified the deepest points of energy, known as stationary points, and matched their locations and depths with excellent precision. For the complex sulfur-oxygen-hydrogen system, the method successfully captured the existence of two deep, symmetric wells, each holding about eight electron volts of energy, just as the original model did.

However, the study also revealed a clear limitation. When the atoms are pulled far apart, into the long-range region where they barely interact, the new method began to show small, artificial ripples in the energy landscape. These ripples are a known side effect of using this specific type of polynomial expansion, which struggles to remain perfectly flat when the forces between atoms become negligible. Despite this, the researchers determined that the method is entirely suitable for studying bound states, which are the stable, vibrating configurations of molecules that exist before they fly apart. To prove this, they used the new map of the helium-lithium-hydrogen system to calculate the specific energy levels of its vibrations. The results matched the calculations from the original, trusted map almost exactly, with the energy levels differing by less than one wave number on average.

This work suggests that while the Jacobi-Legendre polynomial approach may need a small adjustment to handle the very edges of the molecular world, it is a powerful and efficient tool for the core regions where chemistry takes place. By converting a difficult, non-linear fitting problem into a simpler linear one, the method offers a fast and accurate way to generate the energy maps needed to understand molecular behavior. The researchers confirmed that for systems like the ones they tested, this approach can reproduce the essential features of the energy landscape with high fidelity, providing a solid foundation for future studies into how molecules vibrate, rotate, and interact in the quantum realm.

Technical Summary: An Investigation on the Jacobi-Legendre Polynomial Expansion for Potential-Energy-Surface Fitting of Multi-Species Triatomic Systems

Problem Statement
In quantum-mechanical studies of few-atom systems, the Potential Energy Surface (PES) is the fundamental component governing molecular dynamics, spectroscopy, and thermochemistry. While high-level ab initio calculations can determine electronic energies, they are computationally prohibitive for dynamical simulations. Consequently, PESs are typically represented by analytical functions fitted to a representative set of electronic energy data. Although numerous fitting methods exist—ranging from traditional physics-driven forms to modern machine-learning (ML) approaches—there is a need to evaluate the efficacy of specific linear ML schemes, particularly the Jacobi-Legendre polynomial (JLP) cluster expansion, when applied to multi-species triatomic systems. Previous applications of the JLP method were limited to single-species carbon systems; this work investigates whether the method can accurately model the distinct topographical features of multi-species systems without the use of switching mechanisms to other functional forms at long ranges.

Methodology
The study employs a JLP-based functional form adapted from the cluster expansion proposed by Domina et al. [27]. The total potential energy ( $E$ ) is expressed as a sum of one-, two-, and three-body contributions.

Mathematical Formulation: The two-body terms are constructed using "single-vanishing" Jacobi polynomials ( $\tilde{P}$ ), while three-body terms utilize "double-vanishing" Jacobi polynomials and Legendre polynomials ( $P$ ). These polynomials are functions of transformed interatomic distances ( $x_{ij}$ ) and angles ( $\theta_{ijk}$ ), mapped to the range $[-1, 1]$ .
Boundary Conditions: The use of vanishing polynomials ensures that the two-body contribution vanishes at a cutoff distance ( $r_{cut}$ ) and exhibits repulsive behavior at short distances, while the three-body contribution vanishes at both $r_{cut}$ and a minimum distance ( $r_{min}$ ). This design forces the potential to approach zero at long range and isolates two-body interactions at very short range.
Linear Regression: The formulation renders the fitting problem linear with respect to the expansion coefficients. The coefficients are determined via Singular Value Decomposition (SVD) using training data.
Test Systems: Two multi-species triatomic systems were selected to test the method against existing analytical PESs:
1. HeLiH+: A system with a simple topography (one shallow well, ~0.2 eV). Training data was generated from the RAGM PES [30] with 57,785 points.
2. HSO: A system with a complex topography (multiple stationary points, two deep wells ~8.0 eV). Training data was generated from the MNV PES [33] with 40,878 points.
Validation: The fitted PESs were evaluated using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and maximum absolute error (MAX). Additionally, bound-state calculations were performed on the HeLiH+ system using the DVR3D code to compare vibrational energy levels against the reference PES.

Key Results

Fitting Accuracy:
- HeLiH+: The JLP fit achieved an RMSE of $1.287 \times 10^{-3}$ eV (10.38 cm $^{-1}$ ) over the full dataset. Within the short-to-medium range (distances < 7.0 $a_0$ , energies < 1.0 eV), the RMSE improved to $8.20 \times 10^{-4}$ eV. The global minimum geometry and energy matched the reference (RAGM) PES with excellent agreement (e.g., well depth of -0.204 eV for both).
- HSO: The JLP fit achieved an RMSE of $3.381 \times 10^{-3}$ eV (27.269 cm $^{-1}$ ). Despite the complex topography, the fitted surface reproduced the deep wells (~8 eV) and saddle points with high fidelity. The stationary point geometries and energies showed very good agreement with the reference (MNV) PES.
Long-Range Limitations: As anticipated, the pure JLP approach exhibited oscillatory behavior in the long-range region (beyond ~7.0–7.5 $a_0$ ), leading to artificial stationary points. However, the magnitude of these oscillations was small and did not significantly degrade the quantitative RMSE metrics calculated over the entire dataset.
Bound-State Calculations: For HeLiH+, bound-state energy levels were calculated for both the JLP and reference PESs. The results showed a strict one-to-one quantitative correlation, with an average deviation of only 0.208 cm $^{-1}$ . Both surfaces supported the same number of vibrational states.

Significance and Claims
The paper demonstrates that the Jacobi-Legendre polynomial expansion is a viable and accurate method for fitting the PES of multi-species triatomic systems, provided the application focuses on the short-to-medium range interactions.

Accuracy: The method produces smooth surfaces with RMSE values in the order of a few meV, sufficient for high-precision modeling of bound states, vibrational/rotational spectra, and tunneling effects.
Linearity: By converting the non-linear fitting problem into a linear one through feature space transformation, the method avoids the computational costs associated with non-linear optimization algorithms.
Limitations: The authors explicitly note that while the JLP method excels in the interaction region, it requires long-range corrections (e.g., switching to asymptotic forms) for modeling atom-diatom or full-dissociation limits where van der Waals and dipole interactions dominate.
Conclusion: The study confirms that JLP-based PESs are suitable for quantum-mechanical studies of bound states in multi-species systems, offering a robust alternative to existing fitting schemes for the short and medium ranges.

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