Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications

A. N. Ikot; B. C. L\"utf\"uo\u{g}lu; B. Hamil; U. S. Okorie

REVIEW 4 major objections 5 minor 104 references

Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives a closed-form energy spectrum for the Morse potential in Dunkl quantum mechanics, where ordinary derivatives are replaced by reflection-symmetric Dunkl derivatives.

desk verdict A routine Dunkl-Morse extension with fixable but load-bearing errors: the I2 column is unphysical because alpha is used as dimensionless, and the derivation has a sign inconsistency. read the letter →

arxiv 2506.00877 v1 pith:EZOQ6AJL submitted 2025-06-01 quant-ph

B. Hamil , B. C. Lütfüoğlu , A. N. Ikot , U. S. Okorie This is my paper

classification quant-ph

keywords DunkloperatorsMorsepotentialdeformedSchrödingerequationexactsolutionsdiatomicmoleculesquantumthermodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the Morse potential, the standard anharmonic potential for diatomic molecules, remains exactly solvable when the Schrödinger equation is built from Dunkl derivatives—operators that add reflection-symmetric deformation terms. Its main result is a closed-form energy spectrum, Eq. (45), with explicit dependence on the deformation parameters, angular quantum numbers, and molecular constants. The paper applies this spectrum to H2, HCl, and I2, and derives from it a partition function and the associated free energy, internal energy, entropy, and specific heat. If correct, the formulas give a parameter-controlled way to model how reflection symmetry shifts vibrational levels and thermal response in real molecules, while reducing exactly to the ordinary Morse spectrum when the deformation is turned off.

What carries the argument

The engine of the calculation is the Pekeris approximation: near equilibrium, the centrifugal term $(\varpi^2+\mu(\mu+1))/(\chi+1)^2$ is replaced by the exponential series $C_0+C_1 e^{-\alpha\chi}+C_2e^{-2\alpha\chi}$ with the coefficients in Eq. (34). This converts the radial Dunkl-Schrödinger equation into a two-exponential potential, which the substitution $\rho=e^{-\alpha\chi}$ maps onto a confluent hypergeometric equation. The quantization condition is the requirement that the confluent hypergeometric series terminate at a non-negative integer $n$; solving that condition for the energy yields Eq. (45). All reflection-symmetry effects enter only through the combination $\mu(\mu+1)+\varpi^2$ that multiplies the Pekeris coefficients, so the deformation is carried by the angular separation constant and by the sum of the three $\mu_i$.

What would settle it

Compare the reported I2 ground state in Tables 2 and 3 with the tabulated dissociation energy $D = 12550\ \mathrm{cm}^{-1} \approx 1.56$ eV: a bound state of a Morse well of depth $D$ must satisfy $E_0 > -D$, yet the tables report $E_0 \approx -16.70$ eV for I2. Recomputing the spectrum with the physical dimensionless exponent $\alpha r_e\chi$ would settle whether Eq. (45) as implemented is correct, since a corrected result must put all I2 bound levels inside the window $(-1.56\ \mathrm{eV}, 0)$.

Watch

Extended reading notes

Core claim

The central claim is that in Dunkl quantum mechanics the Morse potential has the bound-state spectrum \[ E_{n,\ell,m}= \frac{\$hbar^{2}$}{2M $r_e^{2}$}\left[(\mu(\mu+1)+\$varpi^{2}$)C_0 - \$alpha^{2}$\left(n+\frac{1}{2} - \frac{\$xi^{2}$}{\eta\$\alpha$}\right)^2\right], \] where $n$ is the radial quantum number, $\mu$ is the sum of the three Dunkl deformation parameters, $\varpi^2$ is the angular separation constant (reducing to $\ell(\ell+1)$ when the deformation vanishes), and $C_0,\xi,\eta$ come from the Pekeris expansion of the centrifugal term. The derivation sets $\chi=(r-r_e)/r_e$, approximates $1/(\chi+1)^2$ by an exponential series, changes variable to $\rho=e^{-\alpha\chi}$, solves the resulting confluent hypergeometric equation, and enforces normalizability by terminating the series. The paper asserts that this spectrum is exact within the Pekeris approximation, that it reduces to the standard Morse spectrum in the undeformed limit, and that the deformation alters level spacings without splitting parity because the reflection eigenvalues do not enter.

Load-bearing premise

The load-bearing premise is that the tabulated Morse width $\alpha$, listed in cm$^{-1}$, can be fed directly into the Pekeris expansion and the final energy formula as a dimensionless exponent $\alpha\chi$ without being multiplied by the equilibrium bond length $r_e$.

Editorial extensions

If this is right

When all Dunkl parameters vanish, Eq. (45) reproduces the ordinary Morse spectrum, so the deformed model contains the standard one as a clean limit.
The energy levels depend on the sum $\mu$ and the angular constant $\varpi^2$ but not on the reflection eigenvalues, so the deformation shifts and re-spaces levels without splitting parity doublets.
For the three molecules studied, negative $\mu$ deepens the effective well and lowers levels while positive $\mu$ raises upper states, giving a tunable description of anharmonicity.
The closed-form spectrum leads to a closed-form partition function through Poisson summation, yielding explicit temperature dependence for the free energy, internal energy, entropy, and specific heat, with larger $\mu$ narrowing the accessible vibrational band and moving the heat-capacity peak.
The reality condition in Eq. (46) restricts the allowed bound-state quantum numbers, determining how many vibrational levels the deformed well supports.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Fitting Eq. (45) to measured vibration-rotation levels of H2, HCl, or I2 would test whether a nonzero Dunkl parameter systematically improves on the standard Morse fit, a comparison the paper does not perform.
The same exponential change of variable should transfer to other short-range exponential potentials such as the Manning-Rosen or Eckart forms in the Dunkl framework, since the method only needs the $e^{-\alpha\chi}$ substitution and confluent-hypergeometric truncation.
Because the prefactor $\hbar^2/(2M r_e^2)$ carries the reduced mass, the thermal formulas could be turned directly into predictions for isotope shifts of heat capacity and entropy, which the paper leaves uncomputed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper is a routine Dunkl-formalism extension to the Morse potential with the Pekeris approximation, plus a thermal analysis built on the resulting spectrum. That specific combination is new as far as the reference list shows, and the algebra is presented in a checkable way. But the center does not hold: there is a units error in the handling of alpha, a sign inconsistency between Eqs. (35), (38), and (45), and the molecular numbers that come out are unphysical. The paper is not publishable as is.

What it does well: it follows a standard Dunkl recipe—reflection operators, separation in spherical coordinates, angular eigenvalues, Pekeris treatment of the centrifugal term—and arrives at a closed-form spectrum in Eq. (45). The mu=0 limit correctly reduces to the ordinary Morse result, and the thermodynamic section is a straightforward exercise once you have that spectrum. If the spectrum were right, the result would be a tunable Morse model for molecular work. Credit is due for an organized derivation, an extensive reference list, and a self-citation pattern that fits a programmatic line of research.

Now the soft spots, which are load-bearing. The I2 numbers in Tables 2 and 3 are off by an order of magnitude. With D = 12550 cm^-1 = 1.56 eV, a Morse ground state cannot sit at -16.7 eV. The root cause is the treatment of alpha. The text introduces chi = (r - r_e)/r_e, so the exponent in e^{-alpha(r-r_e)} is alpha*r_e*chi, a dimensionless combination. Table 1 lists alpha in cm^-1; for I2, alpha = 4954 cm^-1 and r_e = 2.67 A, so alpha*r_e is about 1.3e-4, not 4954. Eq. (31) and everything after treats alpha as dimensionless. The H2 and HCl entries in Table 1 look like dimensionless a values mislabeled as cm^-1, so the table is internally inconsistent. In addition, Eq. (38) defines W with a minus P E term, which would make beta^2 negative for bound states; Eq. (45) corresponds to the opposite sign. As written, the derivation does not close. The thermal section silently drops the angular momentum term (Q = mu(mu+1) C0, no omega^2) without justification. And the abstract promises comparison with experimental data; the paper contains no such comparison.

Who is this for? Researchers working in Dunkl quantum mechanics who want a concrete example of the Morse potential with thermal functions. They may find the method useful after corrections. But the central result is not reliable now.

Recommendation: I would send this to peer review—the errors are identifiable and fixable, and the question of whether the Dunkl-Morse spectrum is a genuine extension deserves referee time. But I would not accept it in its current form. It needs major revision: correct the dimensional reduction of alpha, fix the sign in W and Eq. (45), rebuild the I2 (and H2/HCl) columns, and either add the promised experimental comparison or drop the claim.

Referee Report

4 major / 5 minor

Summary. The paper studies the Morse potential within Dunkl quantum mechanics. It separates the Dunkl-Schrödinger equation in spherical coordinates, applies the Pekeris approximation to the centrifugal term, and derives an alleged exact analytical energy spectrum, Eq. (45), together with wavefunctions. The spectrum is then used to compute vibrational levels for H2, HCl, and I2 and to derive thermodynamic functions such as the partition function, free energy, internal energy, entropy, and specific heat. The central claim is that the Dunkl deformation parameters modify the vibrational spectrum and thermal properties in a tunable way, while reducing to the standard Morse result in the undeformed limit.

Significance. If the derivation were correct, the paper would offer a convenient closed-form extension of the Morse model with reflection symmetry and an analytic partition function, which could be of interest to the Dunkl-formalism community. The paper also explicitly acknowledges some limitations of the Pekeris approximation in Secs. 3.3 and 4.3. However, the main result does not follow from the equations as written, and the molecular applications contain grossly unphysical numbers (an I2 ground-state energy of about -16.7 eV for a well depth D = 1.56 eV). Because the claimed central result and all applications are affected, the paper's significance is not realized in its present form.

major comments (4)

[Sec. 3.1, Eq. (31) and Table 1] The substitution χ = (r - r_e)/r_e makes the physical Morse exponent α r_e χ, not α χ. Equations (31)-(34) and the final spectrum Eq. (45) treat α as if it were a dimensionless variable. Table 1 lists α in cm^-1, and for I2 α = 4954 cm^-1 with r_e ≈ 2.67 Å implies a = α r_e ≈ 1.3 × 10^-4, not 4954. Plugging α = 4954 into Eq. (45) yields E0 ≈ -16.7 eV (Table 2), which is far below the well depth D = 12550 cm^-1 = 1.56 eV; a Morse bound state cannot lie below -D. This dimensional inconsistency invalidates the molecular spectra in Sec. 4 and, through Eq. (48), the thermodynamic results in Sec. 5.
[Secs. 3.2-3.3, Eqs. (35)-(45)] Equation (38) defines W = (ϖ² + μ(μ+1)) C0 - P E with P = ℏ²/(2M r_e²), and Eq. (42) sets β² = -W/α². Solving the quantization condition Eq. (44) gives β = -n - 1/2 + ξ²/(ηα), so W = -α²(n + 1/2 - ξ²/(ηα))². Substituting this W into Eq. (38) yields E = P[(ϖ² + μ(μ+1)) C0 + α²(n + 1/2 - ξ²/(ηα))²]. Equation (45) instead has a minus sign before the α² term, so Eq. (45) does not follow from the preceding equations. Furthermore, for bound molecular states with E < 0, the W in Eq. (38) is positive, making β² negative, which is inconsistent with the real β used in the ansatz Eq. (41).
[Sec. 5.1, Eqs. (47)-(50)] The partition function in Eq. (48) uses ξ1 and η1 defined in Eq. (50) with only μ(μ+1) in place of ϖ² + μ(μ+1). This means the thermal sum in Eq. (47) implicitly sets the angular contribution ϖ² to zero (geometrically corresponding to ℓ = m = 0), even though the spectrum Eq. (45) depends explicitly on ϖ² and Sec. 4 discusses states with ℓ = m = 1. No justification is given for this restriction, so the thermodynamic predictions do not follow from the stated spectrum.
[Table 1] The α column is internally inconsistent: the H2 and HCl entries (1.440 and 2.380) are order-unity numbers that look like dimensionless a = α r_e values, while the I2 entry (4954 cm^-1) is a wavenumber. This mixed usage is not a purely presentational issue; it is the direct cause of the unphysical I2 energies in Tables 2 and 3 and invalidates the comparisons in Sec. 4.

minor comments (5)

[Eq. (43)] The argument of the confluent hypergeometric function is printed as αρ/(2η), but the standard reduction of the Morse-type equation and the quantization condition Eq. (44) require 2ηρ/α; please check this argument and the surrounding definitions.
[Abstract and Sec. 1] The phrase "exact analytical solutions" is too strong because the Pekeris approximation is an approximation for the centrifugal term; the text should consistently say "approximate analytical solutions" or clearly specify the approximation status.
[Sec. 3.3, Eq. (46)] The bound-state condition uses δ(δ+1) in the square-root terms, while the spectrum in Eq. (45) uses μ(μ+1); although the two are equal for δ = -(1+μ), the notation should be made uniform to avoid confusion.
[References] There are several typographical errors in the reference list, e.g., Ref. [101] "Phys. Scrp." should be "Phys. Scr.", and Ref. [103] is missing a comma between the author list and the title.
[Fig. 3] The entropy panel in Fig. 3(d) shows negative values on the vertical axis, so the statement in Sec. 5.2 that entropy "grows monotonically" should be worded to match the plotted quantity, e.g., "increases toward less negative values".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Morse spectrum is derived from stated inputs, the Dunkl angular results are cited from independent prior work, and the thermodynamic functions are forward-modeled from the derived spectrum rather than fitted to it.

full rationale

The paper's central derivation is self-contained in the sense required by the circularity check: Eq. (45) is obtained algebraically from the stated Morse potential (29), the dimensionless variable (30), the Pekeris-type exponential expansion (33)-(34), and the confluent hypergeometric quantization condition (44). The molecular constants D, r_e, alpha, and the reduced mass M are inputs taken from spectroscopic tables; they are not fitted to the energies reported in Tables 2-3 or to the thermodynamic curves in Figures 3-4. The Dunkl angular input, including the separation constant ϖ² in Eq. (28), is attributed to an independent reference, Genest-Vinet-Zhedanov [50], not to the present authors, so the self-citation rule is not engaged. The mu = 0 limit is checked explicitly against the standard Morse spectrum, which is an independent internal consistency test rather than a circular reduction. The partition function and thermal functions in Section 5 are standard Boltzmann averages over the model spectrum, i.e., forward modeling. The reviewer's dimensional concern about alpha being treated as dimensionless, and the sign mismatch between Eqs. (38), (42), and (45), are genuine physical and algebraic validity issues, but they are not examples of a prediction being equivalent to its input by construction. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled in via a self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities. Its tunable Dunkl parameters are free inputs. It relies on cited Dunkl formalism and on the Pekeris approximation; the main modeling choice is the silent restriction to zero angular momentum in the thermal section.

free parameters (2)

Dunkl deformation parameters mu_i
Chosen by hand in applications (mu_i = -0.4, +0.4 in Tables 2-3; total mu = 1.2, 4.5, 6.0 in Figures 3-4). No physical or experimental constraint is given.
Morse exponent alpha = 1.440, 2.380, 4954 (cm^-1)
Taken from spectroscopic tables but used directly as a dimensionless quantity in the Pekeris expansion, causing the I2 anomaly.

assumptions (3)

standard math The Dunkl-Laplacian in spherical coordinates and the angular separation constants (Eqs. 6-28) are valid and correctly applied.
Taken from Genest, Vinet, Zhedanov [50] without independent re-derivation.
domain assumption The Pekeris representation 1/(chi+1)^2 = C0 + C1 e^{-alpha chi} + C2 e^{-2 alpha chi} (Eq. 33) is a good approximation for the molecules considered.
Accuracy decreases for large angular momentum, as the paper itself notes in Section 4.3.
ad hoc to paper The thermal partition function can be restricted to states with varpi^2=0 (geometrically l=m=0) even though the spectrum Eq. (45) generally depends on varpi^2.
Eqs. (49)-(50) use xi_1, eta_1 without the varpi^2 term; the paper never states or justifies this restriction.

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Pith. "Pith review of Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications." pith.science (2026). https://pith.science/paper/EZOQ6AJL

@misc{pith2026250600877,
  author       = {Pith},
  title        = {Pith review of: Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZOQ6AJL}},
  note         = {Machine review of arXiv:2506.00877}
}

abstract

In this work, we investigate the quantum dynamics of a particle subject to the Morse potential within the framework of Dunkl quantum mechanics. By employing the Dunkl derivative operator, which introduces reflection symmetry, we construct a deformed Schr\"odinger equation and obtain exact analytical solutions using the Pekeris approximation. The resulting energy spectrum and wavefunctions reveal how Dunkl parameters alter the effective potential and vibrational states. The model is applied to several diatomic molecules, including H$_2$, HCl, and I$_2$, illustrating the impact of symmetry deformation on energy spectra. We also compute thermodynamic functions, including the partition function, free energy, internal energy, entropy, and specific heat. The analysis shows that the Dunkl deformation induces distinct thermal behavior and offers a tunable approach to molecular modeling. These results highlight the potential of the Dunkl formalism as a useful tool for extending conventional quantum models and for exploring symmetry-deformed systems in molecular physics and quantum thermodynamics.

Figures

Figures reproduced from arXiv: 2506.00877 by the authors.

**Figure 2.** Energy levels En,ℓ,m (eV) versus angular quantum number ℓ, for n = 1, m = 1, and µi = +0.4. The plots reveal several key trends. First, for all three molecules, the energy increases monotonically with ℓ, consistent with the growing effective repulsion due to the centrifugal term. This shift is more pronounced in lighter molecules like H2, where the kinetic prefactor is larger. Second, the Dunkl parameter modulates t… view at source ↗

**Figure 3.** Thermodynamic functions of H2 and HCl for µ = 1.2. The partition function (a) decreases and saturates at high temperatures, with H2 displaying a higher magnitude due to its broader vibrational spectrum. In (b), the free energy increases linearly with temperature, more steeply for HCl. Internal energy (c) and entropy (d) grow monotonically and eventually stabilize, reflecting the finite number of accessible energy s… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Thermal functions of H2 under different Dunkl deformation parameters µ = 1.2, 4.5, 6.0. As µ increases, the partition function (a) decreases due to a narrower vibrational bandwidth. The free energy (b) rises sharply with increasing µ, and the internal energy (c) become…

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