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A generalized q-deformed Eckart model for the spectroscopic and thermodynamic properties of diatomic molecules

This paper presents a generalized q-deformed Eckart potential model parameterized for diatomic molecules that successfully reproduces spectroscopic data and thermodynamic properties with high accuracy, while also highlighting specific limitations in the centrifugal approximation and equilibrium predictions.

Original authors: E. S. Eyube, M. Ferdjaoui, R. D. Musa, P. O. Osuhor, Y. Dahiru, A. Zakariya, G. S. Perez, J. B. Ayuba

Published 2026-10-29

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CC BY 4.0

Original authors: E. S. Eyube, M. Ferdjaoui, R. D. Musa, P. O. Osuhor, Y. Dahiru, A. Zakariya, G. S. Perez, J. B. Ayuba

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

Atoms do not exist in isolation; they reach out and bind together to form molecules, held in a delicate balance by invisible forces. To understand how these tiny structures behave, scientists must map the energy landscape that governs them. Imagine a valley where a ball can roll back and forth; the shape of that valley determines how the ball moves, how fast it vibrates, and how much energy it takes to push it out of the valley entirely. In the microscopic world, this valley is the potential energy between two atoms, and its shape dictates everything from the color of light the molecule absorbs to how it reacts with other substances as the temperature changes. For decades, researchers have tried to write simple mathematical descriptions of these valleys, but many of these descriptions fail when the atoms are pulled far apart or when they vibrate with high energy. The challenge lies in creating a single, flexible model that can accurately predict both the precise vibrational notes a molecule sings and the broad thermodynamic properties it exhibits across a wide range of temperatures.

A team of researchers has now tested a new, highly flexible model designed to describe these molecular valleys with greater precision. They focused on a specific mathematical shape known as the generalized q-deformed Eckart potential, a formula that can be tuned to fit the unique characteristics of different pairs of atoms. Rather than simply guessing the numbers that go into the formula, the team derived them directly from the physical reality of the molecules themselves. They used four key pieces of information available from experiments: how much energy is required to pull the atoms apart completely, the exact distance between the atoms when they are most stable, how stiff the bond is when the atoms are nudged slightly, and how the molecule's rotation and vibration interact with one another. By anchoring the model to these real-world measurements, they created a customized map for several diatomic molecules, including bromine, chlorine, iodine chloride, and rubidium cesium.

The researchers then solved the fundamental equations of quantum mechanics to see how a particle would move within these newly mapped valleys. This allowed them to calculate the specific energy levels the molecules could occupy, which correspond to the vibrational frequencies observed in spectroscopy. When they compared their calculated energy levels against high-precision reference data, the results were remarkably close for most of the molecules tested. The model reproduced the energy landscape of iodine chloride with an average error of less than one-quarter of a percent, a level of accuracy that suggests the formula captures the essential physics of the bond very well. However, the model was less successful with rubidium cesium, where the errors were larger, particularly on the side of the valley where the atoms are close together. This discrepancy highlights a limitation in the mathematical approximation used to handle the spinning motion of the molecule, suggesting that while the model works beautifully for many systems, it struggles when applied to certain types of heavy atoms or when the atoms are pushed to the extremes of their range.

Beyond just mapping the energy levels, the team used their results to predict how these molecules behave in a gas as the temperature rises from a cool room to the scorching heat of a star's atmosphere. They calculated standard thermodynamic quantities, such as entropy, which measures disorder, and enthalpy, which tracks heat content, for iodine chloride, bromine chloride, and sulfur monoxide. To ensure their calculations aligned with established scientific records, they adjusted a single, constant factor in their equations, a step that accounts for subtle statistical details often missed in simplified models. After this adjustment, the predicted entropy and energy values matched the reference data with high fidelity, typically deviating by less than one percent. However, the model showed a distinct weakness at very high temperatures for sulfur monoxide, where the predicted heat content drifted significantly from the accepted values, differing by over nine kilojoules per mole at six thousand Kelvin. This suggests that the model's description of the highest energy states for this specific molecule is incomplete.

The study also examined how these molecules participate in chemical reactions, specifically calculating the equilibrium constants that determine how much product forms when reactants mix. Here, the results were mixed. While the model performed reasonably well at high temperatures, it produced large errors at low temperatures for all three molecules tested. The researchers noted that these errors were particularly pronounced in the logarithmic scale used to report the results, meaning that small numerical differences translated into large multiplicative differences in the actual reaction balance. This indicates that while the model is a powerful tool for understanding molecular structure and high-temperature behavior, it requires further refinement to accurately predict chemical equilibrium in cooler environments. Ultimately, the work demonstrates that by carefully tying a mathematical model to the fundamental physical constraints of a molecule, scientists can create a tool that bridges the gap between the quantum vibrations of a single bond and the macroscopic properties of a gas, even if the tool is not yet perfect for every possible scenario.

Technical Summary: A Generalized q-Deformed Eckart Model for Diatomic Molecules

Problem Statement
Accurate interaction-energy models are essential for interpreting spectroscopy, chemical bonding, and temperature-dependent thermodynamic properties of diatomic molecules. While quantum-chemical methods (e.g., density functional theory, coupled-cluster) provide high accuracy, they are computationally demanding, particularly when generating complete potential-energy curves or handling multiple species and temperature points. Analytical potential-energy functions offer a complementary route, expressing interaction energy directly in terms of internuclear distance. However, existing analytical potentials often lack a complete spectroscopic parameterization that connects dissociation limits, equilibrium conditions, potential well curvature, and higher-order vibration-rotation relations. Specifically, the generalized q-deformed Eckart (GqDE) potential had not been established as a diatomic interaction-energy function through such a comprehensive parameterization, nor had its predicted curves and energies been verified against independent spectroscopic and electronic-structure reference data.

Methodology
The study develops a parameterized GqDE potential and applies it to calculate spectroscopic and thermodynamic properties.

Potential Parameterization: The GqDE potential is defined using hyperbolic q-deformed functions controlled by strength, range, and deformation parameters. The parameters are derived by satisfying four physical constraints:
- Dissociation limit: The potential approaches the dissociation energy ( $D_e$ ) at infinite separation.
- Equilibrium condition: The first derivative vanishes at the equilibrium bond length ( $r_e$ ).
- Curvature: The second derivative at $r_e$ matches the force constant derived from the harmonic vibrational wavenumber ( $\omega_e$ ).
- Spectroscopic shape: The third derivative is constrained using the Varshni relation, linking the potential's shape to the rotational constant ( $B_e$ ) and vibration-rotation interaction constant ( $\alpha_e$ ).
  These conditions allow all potential parameters to be expressed in terms of experimentally accessible molecular constants.
Energy Spectrum Calculation: The radial Schrödinger equation is solved for the GqDE potential. For non-zero rotational quantum numbers ( $\ell \neq 0$ ), the centrifugal term is approximated using a quadratic Pekeris expansion in the same variable used for the potential. This transformation allows the equation to be reduced to a Gauss hypergeometric form, yielding an analytical expression for the vibration-rotation energy spectrum.
Thermodynamic Calculations: The derived energy spectrum is used to construct the molecular partition function, comprising vibrational, rotational, and translational contributions.
- The vibrational partition function is calculated by summing over discrete bound states up to a maximum quantum number ( $n_{max}$ ), determined by the condition where the energy equals the dissociation limit.
- A temperature-independent partition-function offset ( $\eta$ ) is introduced and calibrated for each molecule against NIST-JANAF Gibbs-energy data to correct for statistical contributions not explicitly modeled.
- Standard molar entropy ( $S$ ), enthalpy increments ( $\Delta H$ ), Gibbs-energy functions ( $\Phi$ ), and pressure-based equilibrium constants ( $K_p$ ) are derived from the partition function.
Validation: The model is tested against reference data for several diatomic molecules:
- Spectroscopic Validation: Potential-energy curves and vibrational energies for $Br_2$ , $Cl_2$ , $ICl$ , and $RbCs$ are compared with spectroscopically reconstructed turning-point data (RKR) and reference vibrational term values.
- Thermodynamic Validation: Calculated molar properties for $ICl$ , $BrCl$ , and $SO$ are compared with NIST-JANAF thermochemical data over the temperature range of 300–6000 K.

Key Results

Spectroscopic Accuracy: The model reproduces potential-energy curves and vibrational energies with high accuracy, though performance varies by molecule.
- Potential Energy: Percentage average absolute deviations (PAAD) were 1.2266% ( $Br_2$ ), 0.6461% ( $Cl_2$ ), 0.2457% ( $ICl$ ), and 2.5430% ( $RbCs$ ). $ICl$ showed the best agreement, while $RbCs$ exhibited the largest deviation, primarily on the inner (short-range) branch.
- Vibrational Energies: PAAD values were 1.4933% ( $Br_2$ ), 1.0259% ( $Cl_2$ ), 0.5743% ( $ICl$ ), and 1.6687% ( $RbCs$ ). The deviations generally increase with vibrational quantum number, indicating that the model reproduces the potential minimum curvature well but diverges slightly at higher energies where the wavefunction extends further from equilibrium.
- Centrifugal Approximation: The Pekeris approximation accurately reproduces the centrifugal factor near $r_e$ but shows increasing relative deviation at larger internuclear distances, particularly for $RbCs$ .
Thermodynamic Properties:
- After calibrating the partition-function offset, the model showed close agreement with NIST-JANAF data for entropy and Gibbs-energy functions. Mean deviations were generally low (e.g., 0.0998% and 0.1443% for $BrCl$ entropy and Gibbs function, respectively).
- Enthalpy: The enthalpy increments showed larger discrepancies at high temperatures. For $SO$ , the enthalpy increment differed from the reference by $-9.027 \text{ kJ mol}^{-1}$ at 6000 K.
- Equilibrium Constants: Calculations for the formation of $ICl$ , $BrCl$ , and $SO$ showed substantial discrepancies at low temperatures (300–900 K), with percentage deviations in $\log_{10} K_p$ exceeding 100% in some cases. However, agreement improved significantly at higher temperatures (above 3000 K for $ICl$ and $BrCl$ , and above 900 K for $SO$ ).

Significance and Claims
The paper claims to provide the first complete spectroscopic parameterization of the generalized q-deformed Eckart potential for diatomic molecules. Its primary contribution is connecting the potential's parameters directly to dissociation energy, equilibrium bond length, curvature, and Dunham vibration-rotation relations, thereby enabling the prediction of both vibrational spectra and thermodynamic properties from a single analytical framework.

The authors emphasize that the study successfully links spectroscopic parameterization with molecular thermodynamic calculations. However, they maintain a modest tone regarding the predictive power of the model:

The agreement in thermodynamic properties is achieved after calibrating a temperature-independent offset; thus, the results assess the consistency of the energy spectrum rather than serving as an independent validation of absolute partition-function normalization.
The large deviations in equilibrium constants at low temperatures and the specific high-temperature enthalpy errors for $SO$ highlight limitations in the current model, particularly regarding the centrifugal approximation for highly excited states and the specific shape of the potential at large separations.
The authors conclude that while the model is effective for reproducing reference data within specific bounds, further verification is required regarding reaction definitions, elemental reference states, and the validity of the partition-function approximations against discrete summations before stronger conclusions about thermodynamic predictive accuracy can be drawn.

Technical Summary: A Generalized q-Deformed Eckart Model for Diatomic Molecules

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