Many physical systems can be described, at least approximately, by interacting bosons. Examples can be found in interacting Bose-Einstein condensates [1–3], low-energy nuclear models [4–8], vibron and cluster models [9] and theories of molecular structure [10–16]. For example, in the interacting s-boson model for nuclei, valence nucleon pairs are treated approximately as s and d bosons. In this case, the Hamiltonian of the system is constructed in terms of U ( 6 ) generators. Generally, as an extension, the largest dynamical symmetry group generated by s and d boson operators is U ( 2 L + 2 ) when the total number of bosons is a conserved quantity. If only one- and two-body interactions are introduced, and the angular momentum of the s-boson system is conserved, the Hamiltonian of the model can be expressed in terms of a linear combination of the first- and second-order Casimir operators of subalgebras contained in all possible chains of the reduction U ( 2 L + 2 ) → O ( 3 ) in Formula 1. H = a 1 C ̂ 1 U 2 L + 2 + a 2 C ̂ 2 U 2 L + 1 + a 3 C ̂ 3 O 2 L + 2 + ⋯ + b C ̂ 1 O 2 L + 1 + c C ̂ 2 O 3 . (1)

When the Hamiltonian is constructed out of a linear combination of the first- and second-order Casimir operators of U ( 2 L + 2 ) , O ( 2 L + 2 ) , U ( 2 L + 1 ) , O ( 2 L + 1 ) , and O ( 3 ) , other possible chains in the reduction are not considered. The focus of recent branching rules has been on the U ( n ) → S O ( n ) by [17], so branching rules for the irreps of the algebras provide the classification of states for these models. However, the pairing models of multi-level are also characterized by an overlaid U ( n 1 + n 2 + ⋯   ) algebraic structure, described further in Ref. [18], either by Formula 2 U ∑ i = 1 k n i ⊃ S O ∑ i = 1 k n i ⊃ S O n i ⊗ ⋯ S O n k ⊃ S O i … k 3 . (2)

Exact diagonalization of Formula 1 in the transitional region is more challenging compared to the limits, particularly when the configuration space dimension is large [19]. Nonetheless, the Hamiltonian Formula 1 can still be numerically diagonalized using the generators of the SU(1,1) Lie algebra as described in references [20–22], [24–26]. Recently, it has been shown that exact solutions for the interacting sd-boson model in the U (5) to O (6) transitional region can also be achieved using an algebraic Bethe ansatz within an infinite-dimensional Lie algebraic framework [27]. A similar approach, focusing on the dominance of bosons in the interacting boson system, was explored in Ref. [28]. While numerical matrix diagonalization of the Hamiltonian Formula 1 is feasible, the algebraic Bethe ansatz method is particularly valuable for analogous quantum many-body problems, especially those where direct diagonalization is impractical.

Recent advances in the vibron model, including its application to both molecular and nuclear systems, have been comprehensively reviewed and extended in Refs. [29–31], providing updated theoretical and computational frameworks. Recent work by [32] introduced a model intended to describe the energy spectra of homonuclear diatomic molecules using the Bethe Ansatz Equation (BAE). However, several aspects of their application of the Bethe Ansatz method and other model assumptions warrant further examination and refinement to achieve more accurate predictive capabilities. Their approach, inspired by an algebraic technique initially proposed by [27, 33], marks a significant step towards solvable models for molecular vibrational spectra. In this paper, we extend the foundation laid by [32] by critically examining key components such as the BAE, Dunham expansion terms, and the parameters used in their fitting procedures, including the Root Mean Square deviation ( σ ) . By reevaluating these components and addressing the impact of the q-deformed approach within this context, we aim to clarify the physical content of the model and identify opportunities for more precise parameterization. Our objective is to refine the vibron model by introducing exact solutions that enhance its alignment with physical parameters, addressing specific inaccuracies in the calculations. This study presents adjusted equations and parameter sets to rectify prior inconsistencies and achieve more reliable results, thereby providing a revised framework for the study of homonuclear molecular spectra.

As described in Refs. [34–39], several earlier works indicate that the S U ( 1,1 ) technique in both deformed and non-deformed formulations can be applied to predict energy spectra. Energy spectra of homonuclear molecules such as H 2 and N 2 have been analyzed using quantum deformed and non-deformed models within the algebraic framework for diatomic molecules [32]. However, a direct extension of this method to the q-deformed case is questionable. It appears that the analysis in [32] was conducted without fully verifying that a q-deformed model does not necessarily yield better results than the non-deformed models upon closer examination. Given that several previous studies have already been published in this field [34–36], we find it challenging to identify any substantial novelty or unique contributions in the spectra obtained through the q-deformed approach.

In this study, by applying the non-deformed model, we demonstrate that the energy spectra values can be accurately reproduced without the introduction of the q-deformed model. Furthermore, we argue that since the q-deformed model does not yield superior results compared to other models, its introduction is unnecessary. The transition’s characteristics and boundaries are highly sensitive to the selection of control parameters and the specific region of the model’s parameter space. Our analysis demonstrates that the transition behavior cannot be generalized in this manner without explicit qualification [19]. The many-body configuration can exhibit both vibrational and rotational modes (see Figure 1), characterized by quantum numbers associated with particles 1 and 2. In this study, we do not examine bending and twisting modes of many-body systems, as these require higher-dimensional treatment. The pure configuration problem illustrated in Figure 1 is slightly complicated by the “internal” degrees of freedom associated with particles 1 and 2. In this work, we apply the solvable model to account for both the geometric and internal excitations of the vibron model.

Despite the effectiveness of the algebraic approach, challenges arise when attempting to accurately reproduce the Dunham energies. In this study, we demonstrate that an improper selection of quantum numbers can result in significant discrepancies, leading to misleading interpretations of the energy spectra.

The one-dimensional vibron model serves as a valuable framework in molecular spectroscopy, particularly for describing stretching vibrations of molecules. Within this model, the U ̄ ( 2 ) algebra is realized through bilinear combinations of two bosonic creation and annihilation operators, labeled as s and t bosons, which span the fundamental representation. The S U ( 1,1 ) dynamical algebra emerges naturally from the pairing structure of these bosons, where quadratic combinations of the creation and annihilation operators form the generators [40, 41] These S U ( 1,1 ) generators arise from the pairing operators and commute with all elements of the O ( 2 L + 1 ) group, establishing the complementary algebraic structure essential for the BAE solution. While U ( 2 ) describes the single-boson excitations, the S U ( 1,1 ) algebra captures the two-boson pairing correlations, providing the dual algebraic framework required for exact solvability of the vibron Hamiltonian.

Upon re-evaluating the calculation process, we refined the energy spectra by determining optimal parameters, including the most suitable pairing strength in the transition region. By incorporating the BAE method into the model, we identified a correct set of control parameters that enhance the accuracy of the predictions. In this study, we address this specific issue by employing a modified approach to the control parameters and solution methods for the BAE. As the previous results did not align with the expected Dunham expansion energies, we utilized the original data for the Dunham expansion from Table 1 of Ref. [44] for H 2 and Table 78 of Ref. [45] for N 2 . Following the methodology outlined in Ref. [36], we reproduced the energy spectra based on these parameters. The updated results for H 2 and N 2 molecules are presented in Tables 1, 2, showing the recalculated spectra. We observed a noticeable discrepancy between these recalculated spectra and the previously reported results in Ref. [32]. Through a comparison of control and fitting parameters, it is clear that differences in the parameterization have led to divergence in the calculated spectra for both H 2 and N 2 . Our findings indicate that solving the BAE with all roots included is crucial in accurately determining the energy spectra. This approach provides a more consistent framework for modeling these molecular systems.

Additionally, we observed some typographical and notational issues in the manuscript that may contribute to discrepancies in the calculations. For instance, it appears that the revised form of Equation 2.16 from Ref. [32] differs from the expected structure, and Equation 2.19 would be more accurately expressed following the form of Formulae 12 in Ref. [36]. Another notable issue arises with Equation 2.20, which contains an inconsistency. A particular point of clarification is the formula for the root mean square deviation, σ . The appropriate expression for σ should be as Formula 22: σ = 1 N tot ∑ i , tot | E exp ( i ) − E cal ( i ) | 2 1 2 , (22) where N tot denotes the total number of energy levels included in the extraction process. The optimal set of Hamiltonian parameters can be derived using the Dunham expansion data, rather than the experimental data alone, to ensure consistency. We also noted a typographical error in Table 1, where an energy level of 1,394.8 appears out of sequence, showing a decrease instead of the expected increase in the energy spectrum. This has been corrected in our revised Table 2 to maintain the continuity and accuracy of the energy progression.

In their conclusion, the authors [32] made substantial claims regarding the superiority of the q-deformed model over the results of F. Pan et al., who initially introduced this model. However, there is limited discussion to substantiate how the q-deformed model provides a marked improvement over previous findings. Given these considerations, a straightforward extension of the q-deformed model remains open to question. It also appears that the authors used the root mean square deviation, σ , as a criterion for model evaluation, with lower σ values indicating a closer fit to the energy spectra. To further investigate this assertion, we recomputed the spectra and deviations σ for three different transitions: U ( 1 ) − O ( 2 ) , U ( 3 ) − O ( 4 ) , and U q ( 3 ) − O q ( 4 ) . Upon comparing the σ values for these transitions, we found no substantial preference for the q-deformed model. In fact, we were able to reproduce the energy spectra and σ values in our Tables 1, 2, showing very similar levels of agreement across models. This suggests that alternative parameterizations or models can achieve comparable accuracy without necessarily resorting to q-deformation. The energies listed under E U ( 1 ) − O ( 2 ) , E U ( 3 ) − O ( 4 ) , and E U q ( 3 ) − O q ( 4 ) are total model energies that include a large constant contribution (zero point energy and rotational term). They are not directly comparable to the Dunham vibrational energies E ( D u n h a m ) ; the meaningful comparison is between the relative level spacings and the deviations σ given below, which are computed after removing an overall constant shift. The absolute energies calculated by the algebraic model are offset by a constant term that depends on the zero point energy and the fixed rotational quantum number. This constant is not included in the Dunham expansion listed in the first column. Therefore, the large numerical differences between E ( D u n h a m ) and the model energies do not reflect a failure of the model; rather, they highlight the need to include a constant term when comparing with experimental data, as is done in the Dunham expansion E ( ν , J ) = const + Y 10 ( ν + 1 / 2 ) + …   . The accuracy of the model is assessed via σ , which is computed after aligning the spectra by a constant shift.

So, deformation parameter τ is extremely small: In our calculations we used τ = 0.0001 , which is negligible compared to other control parameters such as C ≈ 0.93 (for H 2 ) and C ≈ 0.75 (for N 2 ). Such a tiny value of τ means that the algebraic deformation is practically absent; the model remains very close to the non-deformed limit.

The improvement stems from parameter flexibility, not from deformation physics: The q -deformed Hamiltonian contains one additional free parameter ( τ ) with respect to the non-deformed one. The slight lowering of σ is therefore expected simply because a model with more parameters can fit a given dataset somewhat better, even if the underlying physics is unchanged. This does not indicate that the deformation captures new physical effects in the systems studied.

No systematic advantage across systems: For the N 2 molecule (Table 2), the σ values of the deformed and non-deformed models are very close ( ≈ 1480 cm − 1 vs. ≈ 1551 cm − 1 ), and both are comparable to the U ( 1 ) → O ( 2 ) limit. There is no consistent evidence that the q -deformed model provides a more accurate description across different molecules or spectral ranges.

Physical interpretation: A genuine deformation effect would be signalled by a value of τ that significantly deviates from zero and leads to a qualitative change in the spectrum or a marked improvement in fitting that cannot be achieved by simply adjusting the existing parameters. This is not observed here.

Therefore, we maintain our conclusion that introducing the q -deformation is unnecessary for describing the vibrational spectra of H 2 and N 2 within the vibron model framework. The small decrease in σ seen in Table 1 is a consequence of extra fitting freedom, not of a physically relevant deformation.

In this study, we have identified the real, solvable values for the BAE within the vibron model, allowing us to determine exact and precise control parameters for both vibrational and rotational limits and transition between them. By directly comparing these values with experimental data, we validated the accuracy of our approach, which consistently reproduces the observed energy spectra with a high degree of precision. This achievement marks a significant step in refining the algebraic framework for molecular energy calculations, confirming the utility of these control parameters in capturing the fundamental dynamics within the vibron model.

In summary, through numerical analysis and updated fitting data, we found that when the deformation parameter q is real, with a very small value ( τ = 0.0001 ) , the deformation effect on the energy spectra is minimal. This indicates that deformation has a limited impact on the calculated energy spectra.

F. Pan et al. first introduced both deformed and non-deformed models utilizing the S U ( 1,1 ) algebra, applying these frameworks to various nuclei and molecules [35, 36]. A critical point of discussion arises regarding the novel contributions in Ref. [32]. It is important to note that the statement, “The quantum deformation method allows for the inclusion of all higher-order terms of a particular type while introducing only a few additional parameters to the Hamiltonian, which could be viewed as a potential extension,” is directly quoted from Ref. [35]. However, contrary to this assertion, the number of fitting parameters in the q-deformed model is actually greater than in the non-deformed models. Specifically, the number of fitting parameters for transitions U ( 1 ) − O ( 2 ) , U ( 3 ) − O ( 4 ) , and U q ( 3 ) − O q ( 4 ) are 3, 4, and 5, respectively, indicating that the q-deformed model does not lead to a reduction in the number of parameters as claimed.

The full potential of the q-deformed model has yet to be fully demonstrated. However, it is believed that deformation effects may become more pronounced when applied to spectra at highly excited states [42]. Further investigation into the accuracy of this model, using established methodologies [35, 36], is essential. This ongoing research highlights the promise of the q-deformed Hamiltonian approach in the study of nuclear and molecular structures, though it remains an open problem that warrants more in-depth exploration. Our analysis has been restricted to low- and medium-lying vibrational states (up to ν ≈ 10 for H 2 and ν ≈ 21 for N 2 ), where the molecular potential is still deep and nearly harmonic. In this regime, the q -deformation does not yield a systematic improvement over the non-deformed model. However, we note that for highly excited states near the dissociation limit, where level compression and strong anharmonicities become important, the role of algebraic deformations may be more pronounced. Future work extending the present approach to such high-lying states would be valuable to fully assess the potential of q -deformed models. A natural extension of this work would be to test the q -deformed vibron model in the high-excitation regime, where deviations from harmonicity are strong and algebraic deformations may play a more decisive role.

In recent years, there has been a marked increase in scientific interest in solvable models. Building on the approach developed by [6, 35, 36, 48], who pioneered these impactful models, could further advance our understanding of this area.