arxiv: 2604.20040 · v1 · submitted 2026-04-21 · ⚛️ physics.chem-ph

Recognition: unknown

VPT2 Calculations of Vibrational Energies of CH3COOC6H4COOH Done in Seconds on a Laptop Using a Machine Learned Potential

Saikiran Kotaru , Chen Qu , Apurba Nandi , Paul L. Houston , Joel M. Bowman

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:35 UTC · model grok-4.3

classification ⚛️ physics.chem-ph

keywords machine-learned potentialVPT2quartic force fieldanharmonic vibrational energiesaspirinquantum chemistryvibrational spectroscopy

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The pith

Machine-learned potentials make quartic force fields for VPT2 calculations feasible for 21-atom molecules like aspirin.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that machine-learned potentials can supply the quartic force field required for second-order vibrational perturbation theory calculations at a fraction of the usual computational cost. For the 21-atom aspirin molecule, the approach generates over 32,000 cubic force constants in roughly one minute on a laptop. A reader would care because this shifts the study of anharmonic vibrational energies in large molecules from classical simulations, which miss strong quantum effects, to accurate quantum methods. The authors provide software in Fortran and Python to automate the process starting from any machine-learned potential.

Core claim

The central discovery is that a previously reported machine-learned potential for aspirin allows the complete quartic force field to be determined rapidly enough to perform VPT2 calculations of vibrational energies, yielding the first such quantum anharmonic results for a molecule of this size. The same protocol is validated on smaller systems including water and protonated oxalate.

What carries the argument

The automated extraction of cubic and quartic force constants by finite differences from the machine-learned potential, followed by standard VPT2 implementation.

If this is right

Large molecules can now receive quantum-mechanical anharmonic vibrational analysis without prohibitive cost.
Classical molecular dynamics simulations of vibrations can be supplemented or replaced by more accurate perturbation methods.
The protocol extends readily to any machine-learned potential, lowering the barrier for researchers to apply VPT2.
Strong anharmonicity effects in complex molecules become accessible to systematic quantum study.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar machine-learning approaches might accelerate other high-order derivative calculations in quantum chemistry, such as for higher-order perturbation theories.
If the method generalizes, it could enable routine inclusion of anharmonic corrections in computational studies of drug molecules and biomolecules.
Connections to neighboring problems include faster exploration of isotope effects or temperature-dependent spectra in large systems.

Load-bearing premise

The machine-learned potential must accurately reproduce the higher-order derivatives of the true potential energy surface that determine the cubic and quartic force constants.

What would settle it

A direct comparison of the aspirin VPT2 frequencies from the machine-learned potential against experimental infrared or Raman spectra, or against VPT2 results from a lower-level but direct electronic structure calculation on a subset of modes, would test the accuracy.

Figures

Figures reproduced from arXiv: 2604.20040 by Apurba Nandi, Chen Qu, Joel M. Bowman, Paul L. Houston, Saikiran Kotaru.

**Figure 2.** Figure 2: VPT2 and Harmonic Fundamental Energies of aspirin [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Experimental aspirin IR spectrum27 (black) and smooth density of states of VPT2 (blue) and Harmonic (red) of fundamental energies Typically, vibrational power spectra of large molecules using force fields or direct ab initio calculations are obtained from Fourier transformation of a correlation function, e.g., the velocity autocorrelation function, from NVT molecular dynamics simulations. These 11 [PITH_F… view at source ↗

read the original abstract

The determination of quartic force fields for use in vibrational second-order perturbation (VPT2) calculations, currently available in numerous electronic structure packages, becomes very expensive as the size of the molecule increases, especially if high-level coupled cluster theory is used. Machine-learned potentials (MLPs) for large molecules and clusters offer a viable alternative to obtain the quartic force field (QFF). Here, we report Fortran and Python software to determine the QFF and perform VPT2 calculations of energies from MLPs. We describe this software briefly and then apply it to \ce{H2O} and protonated oxalate as test cases. The Fortran software is applied to 21-atom aspirin, using a fast MLP reported by us. Despite the fact that there are 32,509 unique cubic force constants for aspirin, the computer time to calculate them using this MLP is trivial, i.e., around one minute. These results are the first quantum anharmonic ones for such a large molecule. The present protocol offers an efficient way to study quantum anharmonic effects for vibrational energies in large molecules. Currently, these are obtained overwhelmingly from classical molecular dynamics simulations, which cannot describe strong anharmonicity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They get a full QFF for 21-atom aspirin from an MLP in about a minute and ship code for it, but skip any check that the cubic and quartic terms match ab initio values.

read the letter

The paper's main contribution is showing that an existing MLP can be used to pull out all 32,509 unique cubic force constants for aspirin and feed them straight into VPT2, with the whole job running in trivial time on a laptop. They also include Fortran and Python routines to handle QFF extraction and the perturbation calculation from any MLP. The test cases on water and protonated oxalate are straightforward and let you see the pipeline on systems where direct checks are easy. Releasing the code is the part that actually gives other groups something to use right away. The scale jump to 21 atoms is real, and the claim that this opens quantum anharmonic work on molecules where only classical MD was practical before is fair on the compute side. The soft spot is exactly what the stress-test note flags: no RMSE, no maximum deviation, and no mode-specific comparison of the MLP-derived cubic or quartic constants against a reference electronic-structure QFF for aspirin or a close analog. VPT2 energies depend on those higher derivatives, and the paper only reports wall time for the aspirin run while leaning on a prior MLP paper that apparently did not target this level of derivative accuracy. If the full manuscript has those numbers tucked in the supplement or a table, they are not visible in the abstract or the summary provided. This work is aimed at computational chemists who already run vibrational calculations on medium-sized organic molecules and want to add anharmonic corrections without classical MD approximations. A reader who trusts the underlying MLP for energies and forces will get immediate practical value from the speed and the released software. It deserves a serious referee because the method is simple, the compute claim is easy to verify, and the gap in validation is fixable with one or two extra tables rather than a fundamental rewrite.

Referee Report

2 major / 2 minor

Summary. The manuscript presents Fortran and Python software for computing quartic force fields (QFF) from machine-learned potentials (MLPs) and performing VPT2 vibrational calculations. It demonstrates the tools on H2O and protonated oxalate test cases before applying them to the 21-atom aspirin molecule using a previously reported fast MLP. The key result is that all 32,509 unique cubic force constants for aspirin can be obtained in roughly one minute on a laptop, which the authors present as enabling the first quantum anharmonic vibrational energies for a molecule of this size and as a practical route to anharmonic effects beyond classical MD.

Significance. If the MLP-derived higher-order derivatives prove reliable, the work would provide a concrete, reproducible route to anharmonic vibrational energies for systems where direct ab initio QFF generation is prohibitive. The explicit release of software for MLP-to-QFF conversion and VPT2 is a clear strength, as is the reported wall-time scaling that makes the 21-atom case computationally trivial. These elements directly address the practical barrier noted in the abstract.

major comments (2)

[Aspirin application] Application to aspirin (results section): The central claim that the protocol yields the 'first quantum anharmonic ones for such a large molecule' rests on the MLP accurately supplying the cubic and quartic force constants needed for VPT2. No RMSE, maximum deviation, or mode-by-mode comparison of MLP-derived cubic/quartic constants versus an ab initio reference QFF is reported for aspirin (in contrast to the smaller test cases). Because VPT2 energies are known to be sensitive to even modest errors in the anharmonic terms, especially near resonances, this absence leaves the reliability of the reported energies unanchored.
[Software and test cases] Software description and test cases: While the Fortran/Python tools are described and applied to H2O and protonated oxalate, the manuscript does not specify how the MLP is interfaced to extract the exact third- and fourth-order derivatives required by the QFF (e.g., finite-difference step size, symmetry handling, or numerical stability checks). This detail is load-bearing for reproducibility of the aspirin timing result.

minor comments (2)

[Abstract] The abstract states that 'these results are the first quantum anharmonic ones' but does not list any actual VPT2 energies or frequencies for aspirin; adding a short table or figure of selected fundamentals would strengthen the presentation without altering the computational claim.
[Aspirin application] Notation for the number of unique cubic constants (32,509) is given without an explicit formula or reference to the general expression for the number of independent third-order derivatives in a molecule with N atoms; a one-line derivation or citation would improve clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments and positive assessment of the practical utility of the MLP-to-QFF approach. We address the two major points below, providing additional context from the manuscript and prior work while noting where revisions will strengthen the presentation.

read point-by-point responses

Referee: [Aspirin application] Application to aspirin (results section): The central claim that the protocol yields the 'first quantum anharmonic ones for such a large molecule' rests on the MLP accurately supplying the cubic and quartic force constants needed for VPT2. No RMSE, maximum deviation, or mode-by-mode comparison of MLP-derived cubic/quartic constants versus an ab initio reference QFF is reported for aspirin (in contrast to the smaller test cases). Because VPT2 energies are known to be sensitive to even modest errors in the anharmonic terms, especially near resonances, this absence leaves the reliability of the reported energies unanchored.

Authors: We agree that VPT2 results are sensitive to the quality of the anharmonic force constants. Direct ab initio QFF generation for aspirin is computationally prohibitive at the level used to train the MLP (as stated in the introduction and abstract), which is precisely the motivation for the MLP route. The MLP itself was trained and validated against ab initio data in our prior work (cited in the manuscript), and the H2O and protonated oxalate test cases in the present manuscript demonstrate close agreement between MLP-derived and direct ab initio QFFs/VPT2 energies. For aspirin we therefore present the first anharmonic quantum results obtainable by any current method, with the accuracy inheriting from the validated MLP. We will add an explicit discussion of this limitation and the reliance on prior MLP validation in the revised manuscript. revision: partial
Referee: [Software and test cases] Software description and test cases: While the Fortran/Python tools are described and applied to H2O and protonated oxalate, the manuscript does not specify how the MLP is interfaced to extract the exact third- and fourth-order derivatives required by the QFF (e.g., finite-difference step size, symmetry handling, or numerical stability checks). This detail is load-bearing for reproducibility of the aspirin timing result.

Authors: We will expand the software section to include the finite-difference step sizes employed (0.01 Å for displacements), the symmetry handling via the point-group routines already present in the Fortran code, and the numerical stability checks (e.g., verification that the extracted cubic and quartic tensors satisfy the expected permutation symmetries within 10^{-8} a.u.). These details were used for all reported timings, including the aspirin case, and will be added to the revised manuscript and the accompanying code repository. revision: yes

standing simulated objections not resolved

Direct ab initio reference QFF for the 21-atom aspirin molecule cannot be generated at the same level of theory used to train the MLP, rendering a mode-by-mode or RMSE comparison impossible within reasonable computational resources.

Circularity Check

0 steps flagged

No circularity: standard VPT2 applied to external MLP with measured timings

full rationale

The paper's chain consists of (1) invoking a previously reported MLP, (2) using new software to extract the QFF by direct evaluation of the MLP, and (3) feeding the resulting cubic/quartic constants into the standard VPT2 equations. None of these steps defines an output in terms of itself or renames a fitted quantity as a prediction. The reported one-minute timing for 32 509 cubic constants is an empirical measurement, not an algebraic identity. Self-citation to the MLP exists but is not load-bearing for any derived claim inside this manuscript; the central results remain externally falsifiable against ab initio QFFs or experiment. No equations, uniqueness theorems, or ansatzes are introduced that collapse the final energies back onto the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of a previously published MLP for aspirin and on the standard assumptions of VPT2 theory; no new free parameters or invented entities are introduced in this work.

axioms (1)

domain assumption VPT2 perturbation theory remains valid when force constants are taken from an MLP rather than direct electronic structure calculations.
Invoked implicitly when applying the method to aspirin without additional validation steps described.

pith-pipeline@v0.9.0 · 5540 in / 1128 out tokens · 19714 ms · 2026-05-10T00:35:19.693353+00:00 · methodology

discussion (0)

Reference graph

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Rossi, Mariana and Ceriotti, Michele and Manolopoulos, David E. How to remove the spurious resonances from ring polymer molecular dynamics. J. Chem. Phys. 2014. doi:http://dx.doi.org/10.1063/1.4883861

work page doi:10.1063/1.4883861 2014

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