arxiv: 2605.00039 · v1 · submitted 2026-04-28 · ⚛️ physics.chem-ph · quant-ph

Recognition: unknown

Thermodynamic Properties of Diatomic Molecules from the Frost-Musulin Potential

Mohammadjavad Parsanasab , Reza Khordad , Meysam Asadipour , Ahmad Ghanbari , Vatan Badalov

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:36 UTC · model grok-4.3

classification ⚛️ physics.chem-ph quant-ph

keywords Frost-Musulin potentialdiatomic moleculesthermodynamic propertiespartition functionGibbs free energyPekeris approximationH2LiH

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The pith

The Frost-Musulin potential with Pekeris approximation produces accurate Gibbs free energy for H2 and LiH across temperatures.

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

The paper sets out to show that an analytical treatment of the radial Schrödinger equation for the Frost-Musulin potential, paired with the Pekeris approximation near equilibrium, supplies a usable rotation-vibration spectrum. Adding the usual translational and rotational ideal-gas factors then yields the full partition function and derived thermodynamic quantities. A reader would care because the approach recovers experimental Gibbs free energy to good precision while producing reasonable heat-capacity and enthalpy curves, all without requiring a full numerical solution for every level. The work also marks the temperature range where the bound-state description alone begins to fall short.

Core claim

By combining the analytical bound-state approach to the radial Schrödinger problem with the near-equilibrium Pekeris representation, the authors obtain a validated rotation-vibration spectrum. These bound states are combined with standard translational and rotational ideal-gas contributions to construct the total partition function and the corresponding thermodynamic observables. The resulting formulation captures the Gibbs free energy deviation function for both molecules with high quantitative accuracy and provides chemically plausible trends for heat capacity and enthalpy increase over a wide temperature range, while showing that residual high-temperature errors arise mainly from the lack

What carries the argument

Frost-Musulin potential solved analytically for bound states via the near-equilibrium Pekeris representation, then used to build the molecular partition function together with ideal-gas translational and rotational terms.

If this is right

The model recovers a large share of the observed thermochemistry for the two diatomic molecules.
Heat capacity and enthalpy follow chemically plausible trends over a broad temperature interval.
High-temperature errors in derivative quantities are traced to the absence of inelastic rotational and dissociation channels rather than to the bound-state calculation itself.
The potential supplies a compact analytical description of the bound region and marks the temperature regime where fuller molecular statistical mechanics becomes necessary.

Where Pith is reading between the lines

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

The same analytical route could be tried on other diatomic molecules whose potentials admit comparable closed-form solutions.
A minimal correction for dissociation might extend reliable predictions to higher temperatures while preserving tractability.
The clear separation between bound-state and continuum contributions offers a baseline for testing more elaborate potential models in computational thermochemistry.

Load-bearing premise

The bound-state spectrum from the Frost-Musulin potential plus Pekeris approximation, together with standard translational and rotational ideal-gas terms, is sufficient to reach high accuracy in the Gibbs free energy while high-temperature deviations can be blamed entirely on neglected inelastic and dissociation effects.

What would settle it

Direct comparison of the calculated versus measured Gibbs free energy at high temperatures after an explicit dissociation term is added to the partition function; a marked reduction in deviation would confirm the claim.

Figures

Figures reproduced from arXiv: 2605.00039 by Ahmad Ghanbari, Meysam Asadipour, Mohammadjavad Parsanasab, Reza Khordad, Vatan Badalov.

**Figure 1.** Figure 1: (Color online) Exact FM potential VFM(r) (solid red) and its Pekeris representation V˜ FM(r) (dashed blue) as functions of the internuclear separation r for (a) H2 and (b) LiH. The curves are constructed using the spectroscopic parameters of Table I and the coefficients defined in Eqs. (10)–(13). In both systems, the approximation reproduces the equilibrium geometry and the near-equilibrium curvature that … view at source ↗

**Figure 2.** Figure 2: (Color online) Bound-state spectrum and low-temperature thermodynamic response of H2 and LiH within the FM framework. Panels (a) and (b) show the energy eigenvalues Enl as functions of the vibrational quantum number n for different orbital states l, illustrating the systematic increase of level energies and the distinct spectral density of the two molecules. Panels (c)–(h) display the Helmholtz free energy… view at source ↗

**Figure 3.** Figure 3: (Color online) Temperature dependence of the standard-state thermodynamic properties of (a) H2 and (b) LiH, compared with experimental reference data. The main panels show the constant-pressure heat capacity Cp, the Gibbs free-energy deviation function −(G−H298.15)/T, and the enthalpy increment H −H298.15 as functions of temperature. Solid lines represent theoretical results, while open circles denote expe… view at source ↗

read the original abstract

In this study, we present a quantum-statistical analysis of H$_2$ and LiH diatomic molecules within the Frost--Musulin potential framework. By combining the analytical bound-state approach to the radial Schr\"odinger problem with the near-equilibrium Pekeris representation, we obtain a validated rotation-vibration spectrum that reproduces a physically consistent ordering of energy levels. These bound states are subsequently combined with standard translational and rotational ideal gas contributions to construct the total partition function and the corresponding thermodynamic observables of the ground state. The resulting formulation captures the Gibbs free energy deviation function for both molecules with high quantitative accuracy and provides chemically plausible trends for heat capacity and enthalpy increase over a wide temperature range. At the same time, residual errors become increasingly pronounced in derivative-sensitive quantities, particularly at high temperatures; this indicates that the dominant limitations now stem not from the local bound-state spectrum itself, but from the neglect of inelastic rotational, continuity contributions and dynamics close to dissociation. Consequently, the present results define the potential model as a compact and analytically tractable representation of the bound region, recovering a significant portion of the observed thermochemistry whilst also delineating the regime where more comprehensive molecular statistical mechanics is required.

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

Routine application of the Frost-Musulin potential and Pekeris approximation to H2 and LiH thermodynamics, with accuracy claims that rest on missing data and unchecked limits of the approximation.

read the letter

This paper applies the Frost-Musulin potential to H2 and LiH, uses the Pekeris near-equilibrium approximation to generate the rotation-vibration spectrum, and folds those levels into a partition function with the usual translational and rotational ideal-gas terms to get thermodynamic quantities. That is the core of the work. It produces trends in heat capacity and enthalpy that look chemically reasonable and correctly flags growing errors in derivative quantities at high temperature, which it attributes to neglected dissociation and inelastic channels rather than the bound spectrum itself. That distinction is reasonable to draw. The new content is limited to the specific numerical application for these two molecules and the emphasis on the Gibbs free energy deviation function. The potential form and the Pekeris expansion are established in the literature, so the paper is extending an existing toolkit rather than introducing new physics or methods. The authors do a service by keeping the framework analytically tractable and by showing where the simple bound-state model starts to fall short. The soft spots are more substantial. The abstract asserts high quantitative accuracy for the Gibbs free energy, yet the text supplies no tables of values, no direct comparisons to experiment or to numerical benchmarks, and no error estimates. Without those, the accuracy claim cannot be assessed. The Pekeris approximation is a first-order expansion around the equilibrium distance and is known to degrade for higher-lying levels; those levels matter more as temperature increases, so any systematic error in the spectrum feeds directly into the partition function. The paper does not report an independent check of the computed levels against spectroscopic constants, which leaves the validation tied to the same fitted parameters used to define the potential. That circularity weakens the central claim that the bound spectrum is sufficient for the reported accuracy. This work is mainly of interest to people who need quick analytical estimates for the thermochemistry of small diatomics or who study the practical range of simple potential models. A reader seeking new principles or broadly transferable techniques will find little. The evidence as presented does not justify sending the manuscript for serious peer review; the missing numerical checks and the untested regime of the approximation would likely be the first points raised by any referee.

Referee Report

3 major / 1 minor

Summary. The paper applies the Frost-Musulin potential to H2 and LiH, obtaining an analytical bound-state spectrum via the radial Schrödinger equation combined with the Pekeris approximation for the centrifugal term. These levels, together with standard translational and rotational ideal-gas contributions, are inserted into the partition function to derive thermodynamic quantities, with the central claim being that the Gibbs free energy deviation function is reproduced with high quantitative accuracy while high-T deviations in derivatives arise from neglected dissociation and inelastic channels.

Significance. If the spectrum accuracy and quantitative Gibbs claims can be independently verified, the work supplies a compact, analytically tractable representation of the bound-region contribution to diatomic thermochemistry that could serve as a useful benchmark or rapid-estimate tool, while clearly demarcating the temperature range where more complete treatments (including continuum states) become necessary.

major comments (3)

[Abstract] Abstract: the claim that the formulation 'captures the Gibbs free energy deviation function for both molecules with high quantitative accuracy' is unsupported by any numerical tables, error metrics, direct comparisons to experimental thermochemical data, or reference calculations; without these, the accuracy assertion cannot be evaluated.
[Abstract] Abstract and methods description of the Pekeris approximation: the approximation replaces the centrifugal term by a first-order expansion about r_e and is stated to yield a 'validated' spectrum, yet no assessment of its accuracy for high-v, high-J levels (which dominate the partition function at elevated T) or comparison to exact numerical eigenvalues is provided; this directly affects whether residual high-T errors can be attributed solely to neglected dissociation.
[Abstract] Abstract: the Frost-Musulin parameters are fitted quantities chosen to reproduce spectroscopic data, but the manuscript supplies no independent check that the computed E(v,J) levels recover experimental spectroscopic constants to the precision required for the Gibbs free energy claim; this leaves open the possibility that the reported accuracy is partly circular.

minor comments (1)

[Abstract] Abstract: the phrase 'inelastic rotational, continuity contributions' appears to be a typographical error and should read 'inelastic rotational and continuum contributions'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment point by point below and indicate the revisions we will make to strengthen the quantitative support and validation in the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the formulation 'captures the Gibbs free energy deviation function for both molecules with high quantitative accuracy' is unsupported by any numerical tables, error metrics, direct comparisons to experimental thermochemical data, or reference calculations; without these, the accuracy assertion cannot be evaluated.

Authors: We agree that the abstract claim requires explicit quantitative backing for proper evaluation. The full manuscript presents graphical comparisons, but we will add a dedicated table in the results section listing Gibbs free energy deviation values against experimental thermochemical data for H2 and LiH, together with RMS errors and mean percentage deviations over the studied temperature range. This will directly substantiate the accuracy statement. revision: yes
Referee: [Abstract] Abstract and methods description of the Pekeris approximation: the approximation replaces the centrifugal term by a first-order expansion about r_e and is stated to yield a 'validated' spectrum, yet no assessment of its accuracy for high-v, high-J levels (which dominate the partition function at elevated T) or comparison to exact numerical eigenvalues is provided; this directly affects whether residual high-T errors can be attributed solely to neglected dissociation.

Authors: The referee is correct that an explicit accuracy assessment for high-v, high-J states is missing. In the revised manuscript we will include a new subsection comparing the Pekeris-approximated eigenvalues to numerically exact solutions of the radial Schrödinger equation for representative high-v and high-J states up to the energies relevant for the partition function at 2000 K. This will quantify the approximation error and reinforce our attribution of high-T deviations to dissociation and inelastic channels. revision: yes
Referee: [Abstract] Abstract: the Frost-Musulin parameters are fitted quantities chosen to reproduce spectroscopic data, but the manuscript supplies no independent check that the computed E(v,J) levels recover experimental spectroscopic constants to the precision required for the Gibbs free energy claim; this leaves open the possibility that the reported accuracy is partly circular.

Authors: We acknowledge the potential circularity concern. The parameters are taken from literature fits, yet the analytical spectrum with the Pekeris approximation must still be shown to recover the constants faithfully. We will add to the revised manuscript a direct comparison of the vibrational frequency ω_e and rotational constant B_e extracted from our computed E(v,J) levels against the experimental spectroscopic constants for both molecules, thereby demonstrating that the thermochemical accuracy arises from the model's physical fidelity rather than fitting alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is standard model-to-thermo chain

full rationale

The paper solves the radial Schrödinger equation for the Frost-Musulin potential using the Pekeris approximation to generate bound-state energies, inserts those energies into the rovibrational partition function together with standard translational and rotational ideal-gas factors, and computes thermodynamic functions including the Gibbs free-energy deviation. This is a conventional forward calculation from a chosen potential model to observables; no equation or step is shown to be definitionally identical to its input, no fitted parameter is relabeled as an independent prediction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The abstract's claim of quantitative accuracy is therefore an empirical comparison against external data rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the Frost-Musulin potential being an adequate representation of the bound region, with its parameters determined externally, plus standard ideal-gas statistical mechanics. No new entities are postulated.

free parameters (1)

Frost-Musulin potential parameters
The potential depth, range, and shape parameters are adjusted to match molecular spectra; these fitted constants enter every subsequent energy level and thermodynamic quantity.

axioms (2)

standard math The radial Schrödinger equation with central potential admits an analytical bound-state solution under the Pekeris approximation near equilibrium.
Invoked to obtain the rotation-vibration spectrum without numerical diagonalization.
domain assumption Translational and rotational contributions to the partition function can be treated with classical ideal-gas statistics.
Added directly to the vibrational bound-state sum.

pith-pipeline@v0.9.0 · 5530 in / 1663 out tokens · 44471 ms · 2026-05-09T20:36:49.248213+00:00 · methodology

discussion (0)

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