Energy spectra and thermal properties of diatomic molecules in the presence of magnetic and AB fields with improved Kratzer potential

A.N.Ikot; C.O.Edet; G.J.Rampho; U.S.Okorie

REVIEW 3 major objections 4 minor 49 references

Energy spectra and thermal properties of diatomic molecules in the presence of magnetic and AB fields with improved Kratzer potential

T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read Closed-form energy formula covers molecules in magnetic and AB fields

desk verdict A solvable-potential paper with a good Kratzer limit benchmark but a central zero-field degeneracy violation that makes the reported spectra internally inconsistent. read the letter →

arxiv 2009.09294 v1 pith:A6H7L3CO submitted 2020-09-19 quant-ph

G.J.Rampho , A.N.Ikot , C.O.Edet , U.S.Okorie This is my paper · ORCID

classification quant-ph

keywords improvedscreenedKratzerpotentialmagneticfieldABfluxNUFAmethoddiatomicmoleculesenergyspectrathermodynamicpropertiessusceptibility

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

This paper derives a closed-form expression for the bound-state energies of a charged particle moving under the improved screened Kratzer potential while exposed to a uniform magnetic field and an AB flux. Using the NUFA method, the two-dimensional radial wave equation is reduced to a hypergeometric equation, yielding explicit formulas for the energies and wave functions. The paper then feeds those energies into a partition function built with a standard summation formula and derives the thermal and magnetic properties of H2, HCl, and LiH. If the single-particle model is faithful, these analytic expressions give a direct way to predict how molecular energy levels and thermodynamic response change with magnetic field, AB flux, and the potential's control parameter.

What carries the argument

The carrying machinery is the NUFA method, a functional-analysis technique for second-order differential equations of hypergeometric type. After choosing the azimuthal vector potential so that the magnetic and AB contributions merge into a single effective angular-momentum parameter, the radial equation is simplified with a standard centrifugal approximation and the substitution $s=e^{-(\alpha+\delta)r}$. This turns the radial equation into a hypergeometric equation, whose parameters yield the energy formula (28) and whose solutions give the hypergeometric wave function (30).

What would settle it

A direct numerical integration of the radial wave equation using the same potential, the same vector potential, and the Table 1 spectroscopic constants would settle the derivation: the computed eigenvalues must match Eq. (28) to numerical precision. Any systematic discrepancy would show that the closed-form spectrum is not the true spectrum of the model Hamiltonian.

Watch

Extended reading notes

Core claim

The central claim is that the two-dimensional wave equation for the improved screened Kratzer potential, with the vector potential $A = [B r/(e^{(\alpha+\delta)r}-1) + \Phi_{\mathrm{AB}}/(2\pi r)]\hat{\phi}$, is solvable in closed form: the bound-state energies $E_{nm}$ are given by Eq. (28) and the wave functions by Eq. (30). The magnetic field and AB flux enter through a combined angular-momentum parameter, so the spectrum depends explicitly on $B$, $\Phi_{\mathrm{AB}}$, the magnetic quantum number $m$, and the potential control parameter $c$. The paper checks the formula against numerical Kratzer ground states and against Morse-potential vibrational levels for H2, then uses the spectrum to build a partition function and compute free energy, entropy, internal energy, specific heat, magnetization, and magnetic susceptibility for H2, HCl, and LiH.

Load-bearing premise

The model's weakest premise is that a real diatomic molecule can be treated as one charged particle in this two-dimensional central potential with the chosen vector potential; if that Hamiltonian does not describe the actual molecule, the derived energies and thermal properties do not apply to H2, HCl, and LiH.

Editorial extensions

If this is right

With both fields switched off the spectrum shows pseudo-degeneracy; switching on either the magnetic field or the AB flux lifts it, and both together produce the largest shift in bound-state energies.
Increasing the control parameter $c$ from $-1$ through $0$ to $1$ shifts the energy levels upward, pushing the molecule toward the continuum.
The closed-form partition function (40) delivers all standard thermal functions (free energy, entropy, internal energy, specific heat) and magnetic functions (magnetization, susceptibility) as explicit functions of temperature, $B$, $\Phi_{\mathrm{AB}}$, and $c$.
The AB flux alone deepens the binding (lower energies), while the magnetic field alone raises the energies; the paper's tables tabulate both single- and combined-field effects.
The comparison with existing Kratzer ground-state energies and Morse vibrational levels for H2 shows agreement in the decoupled limit, supporting the reduction of the improved potential to known models.

Reading between the lines

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

The same NUFA reduction could be applied to other central potentials whose radial equation is castable in the same hypergeometric form, giving closed-form spectra and thermal functions for a family of screened molecular potentials.
Because the spectrum depends explicitly on $B$ and $\Phi_{\mathrm{AB}}$, the energy-level shifts predicted here could be probed spectroscopically in high-field experiments on trapped molecules or quantum dots, though the model omits many-body, spin, and rotational effects.
The truncation of the summation formula in the partition function means the thermal formulas are approximate; direct numerical summation of Eq. (28) would provide an independent check of the thermodynamic curves.
If the single-particle mapping to diatomic molecules is accepted, the predicted non-monotonic susceptibility curves (e.g., H2 vs HCl vs LiH) give a qualitative fingerprint that thermodynamic measurements could seek.

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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

Colleague,

The key thing you should know: the central energy formula, Eq. (28), cannot be right as written because the zero-field tables break the m ↔ −m degeneracy required by any central potential. For H2, c=−1, B=0, Φ_AB=0, Table 2 gives E_{00,m=1}=0.015776 eV and E_{00,m=−1}=0.016833 eV. That 0.001 eV difference is not round-off. With B=0 and no AB flux, the m-dependence should enter only through m², so E(m) must equal E(−m). Tables 3, 5, and 8 show the same asymmetry. Either Eq. (28) contains a linear-m term that should vanish when η and ξ_AB are zero, or the tables were generated from something other than Eq. (28). In both cases the paper's central result—and everything built on it, including the thermodynamics and magnetic response—is not self-consistent.

What is genuinely new: the improved screened Kratzer potential (ISKP) is a new solvable model, and this is the first NUFA treatment of it with magnetic and AB fields. The closed-form spectrum and wave function are explicit, and the Kratzer-limit benchmark (Table 11) reproduces Baoa and Shizgal to six significant figures. That is real, reproducible evidence that the NUFA machinery works in a limiting case. The thermodynamic and magnetic properties are standard applications of the spectrum, though the printed partition-function derivation (Eqs. 35–40) is too compressed to reproduce without the original Mathematica session.

Other soft spots, in proportion: the vector potential A1 = [B r/(e^{(α+δ)r}−1)] φ̂ does not describe a uniform magnetic field, so the parameter "B" is not the usual field strength and this is never discussed. The H2 comparison against Morse-potential states (Table 12) shows deviations growing to roughly 0.4 eV at higher levels; the paper reports these without reconciliation. Both are fixable in revision. The degeneracy violation is not.

Who this is for: people working on exactly solvable quantum models and the NUFA method. The paper has enough formal content and one clean benchmark that a referee can do something useful with it—pin down where the m-linear term enters, demand corrected tables, and require a reproducible partition function. I would not publish it as is. But I would send it to peer review rather than desk reject, because the flaw is pinpointable and the benchmark shows real competence.

Send it back for major revision or reject after review.

Referee Report

3 major / 4 minor

Summary. The paper claims to solve the two-dimensional Schrödinger equation for the improved screened Kratzer potential (ISKP) in the presence of external magnetic and Aharonov-Bohm fields using the Nikiforov-Uvarov Functional Analysis (NUFA) method, yielding the closed-form energy spectrum in Eq. (28) and the wave function in Eq. (30). The spectrum is then used to compute partition functions and thermal and magnetic properties for H2, HCl, and LiH for various values of the control parameter c, magnetic field B, and AB flux Φ_AB. The manuscript also includes comparisons with the Kratzer limit and with known H2 vibrational states.

Significance. If the central spectrum were correct, the paper would provide a compact analytic description of bound states of a screened Kratzer-type potential under magnetic and AB fields, with direct applications to thermodynamic and magnetic properties of diatomic molecules. The Kratzer-limit benchmark in Table 11 matches the published values of Baoa and Shizgal to the quoted precision, which is a genuine positive check on part of the NUFA algebra. The paper is also explicit in giving closed forms for the energy, wave function, and partition function, and it discusses falsifiable trends in the thermodynamic quantities. However, the central application fails internal consistency in the field-free limit and the molecular comparison in Table 12 shows large deviations, so the significance of the claimed results is not realized.

major comments (3)

[Section 3, Eq. (28) and Tables 2–10] In the limit B=0 and Φ_AB=0 the Hamiltonian of Eq. (17) is a central potential, so the energy must depend on the magnetic quantum number m only through m^2; energies for m and -m must coincide. The tabulated zero-field energies violate this necessary condition: for example, Table 4 (H2, c=1, B=0, Φ_AB=0) lists E_{0,1}=-6.63770 and E_{0,-1}=-6.62423, a difference of 0.013 eV, and similar asymmetries appear in Tables 2, 3, 5, 8, and 10. This is far too large to be round-off. The implication is that Eq. (28) either contains a term linear in m that does not vanish when η=0 and ξ_AB=0, which would be impossible for a central potential, or the tabulated energies were not generated from Eq. (28), which would make the results irreproducible. Either way, the central energy formula and the thermodynamic results built on it are not self-consistent.
[Section 3, Eq. (18) and the statement '∇×A_1 = B'] The chosen vector potential A_1 = B r/(e^{(α+δ)r}-1) φ-hat does not have curl equal to the uniform field B. Its curl is B_z = (1/r) ∂/∂r [B r^2/(e^{(α+δ)r}-1)], which is position-dependent and even diverges as (α+δ)r → 0. Thus the Hamiltonian in Eq. (17) does not describe a charged particle in a uniform magnetic field of strength B, and the interpretation of the parameter B as the external magnetic field is not supported. Since all magnetic-field dependence of the energies and all magnetization and susceptibility results in Section 4 rest on this identification, this is a load-bearing modeling error.
[Table 12 and Section 5] The paper's own comparison of H2 vibrational energies contradicts the claim that Eq. (28) describes the selected diatomic molecules. In Table 12, for n=1 the present value 0.8225 eV differs from the quoted literature values (0.9935–1.0032 eV) by roughly 0.17 eV, and the discrepancy grows with n (n=2: present 1.092 vs 1.443–1.4615 eV; n=3: present 1.335 vs 1.862–1.8917 eV). These differences are orders of magnitude larger than the benchmark agreement in Table 11. Consequently, the numerical results do not support the stated application to H2, HCl, and LiH.

minor comments (4)

[Throughout] The manuscript contains many typographical and rendering errors, including two different equations both numbered (19), duplicate reference numbers [42], and the spelling 'Aharanov-Bohm' in the abstract; a thorough editing pass is needed.
[Reference [14]] The improved screened Kratzer potential is defined by reference [14], which is cited as 'Submited for publication'; the paper is therefore not self-contained in its central model definition.
[Eq. (37)] The maximum quantum number n_max is introduced as n_max = -Ω + Q ± √(Q - R) without derivation or a statement of when the square root is real, although R must be non-negative for this expression to be meaningful.
[Eq. (40)] The final partition function expression is extremely convoluted and appears to contain malformed error-function terms; given that the underlying spectrum is already invalidated, this expression should be re-derived and checked for typographical consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the closed-form spectrum is derived from an explicit Hamiltonian and NUFA condition, and Tables 11-12 anchor the result externally.

full rationale

The derivation chain is self-contained at the level relevant to circularity. Eq. (28) follows from substituting the explicitly stated vector potential (18), the ISKP (1), the Greene-Aldrich approximation (19b), and the NUFA energy condition (12) into a single radial equation; the target energies are not among the inputs. The self-references are to the model potential (Ref. [14], submitted) and to the NUFA method (Ref. [40]); neither is load-bearing because Eq. (1) defines the potential in full and Section 2 states the method equations used. External comparisons in Tables 11 and 12 (Baoa-Shizgal and Morse-potential results) provide independent anchors, so the central formula is not forced by a self-citation chain. The zero-field m=+1 versus m=-1 asymmetries in Tables 2-10 contradict the m^2 dependence that Eq. (28) must have at B=0 and Phi_AB=0; this is an internal numerical inconsistency and a correctness risk, but it is not circular because the tables are inconsistent with the formula rather than equivalent to it by construction. Likewise, the abstract's statement that the fields remove degeneracy is a direct consequence of the m-linear field terms in Eq. (29), i.e., a physical consequence of the input Hamiltonian, not a prediction that was inserted as an input. The unpublished status of Ref. [14] and the duplicated Ref. [42] are bibliographic-support issues, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central calculation rests on a new model potential from an unpublished self-citation, a solvability-oriented gauge choice, and an undocumented value of δ. The only external anchor is the Kratzer-limit benchmark, which is a special case and does not validate the molecular application.

free parameters (3)

Control parameter c = -1, 0, 1
A parameter of the ISKP model that selects the potential shape. It is scanned rather than fitted, but it directly changes the energy spectra, so it is a free parameter of the model.
Screening parameter δ = not stated
The ISKP contains two screening parameters α and δ. Table 1 lists α for each molecule but δ is never given; the numerical results depend on this undocumented choice.
Magnetic field B and AB flux Φ_AB values in tables = B=2, Φ_AB=2 (units unspecified)
The tabulated energies and plotted thermodynamics use specific field strengths whose units are not defined.

assumptions (4)

ad hoc to paper The diatomic molecule can be modeled as a single charged particle in a 2D cylindrical potential V(r), with the radial coordinate r treated as the internuclear distance.
The 2D Schrödinger equation with a central potential is applied to diatomic molecules; this identification is not derived from molecular structure and is a strong modeling assumption.
domain assumption Greene-Aldrich approximation: 1/r^2 ≈ (α+δ)^2/(1 - e^{-(α+δ)r})^2 for the centrifugal term.
Used to make the radial equation solvable; valid only for small (α+δ)r and introduces approximation error for the short-range Kratzer potential.
ad hoc to paper The vector potential is chosen as A = (B r/(e^{(α+δ)r}-1) + Φ_AB/(2πr)) φ-hat, which couples the magnetic field to the screening parameter in a way that makes the equation analytically solvable.
This gauge choice is not dictated by the physics of a diatomic molecule; it is chosen for solvability.
standard math Euler-Maclaurin summation can be truncated at a finite n_max with the error-function expressions of Eqs. (39)-(40).
The partition function is evaluated by Euler-Maclaurin formula; the specific truncation and integral evaluation are asserted.

invented entities (1)

Improved Screened Kratzer Potential (ISKP)
purpose: Model potential for diatomic molecules, defined by Eq. (1) and used as the central Hamiltonian.
Proposed by the same group in an unpublished paper (Ref. [14]); its validity for real molecules is not established here. The H2 comparison with Morse spectra in Table 12 shows large deviations for excited states, so it has no independent falsifiable confirmation in this paper.

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Cite this review

Pith. "Pith review of Energy spectra and thermal properties of diatomic molecules in the presence of magnetic and AB fields with improved Kratzer potential." pith.science (2026). https://pith.science/paper/A6H7L3CO

@misc{pith2026200909294,
  author       = {Pith},
  title        = {Pith review of: Energy spectra and thermal properties of diatomic molecules in the presence of magnetic and AB fields with improved Kratzer potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A6H7L3CO}},
  note         = {Machine review of arXiv:2009.09294}
}

read the original abstract

In the present study, the improved screened Kratzer potential (ISKP) is investigated in the presence of external magnetic and Aharanov-Bohm (AB) fields within the framework of non-relativistic quantum mechanics. The Schrodinger equation is solved via the Nikiforov-Uvarov Functional Analysis (NUFA) method and the energy spectra and the corresponding wave function for the ISKP in the presence of external magnetic fields are obtained in a closed form. The obtained energy spectra are used to study three selected diatomic molecules (H2, HCl and LiH). It is observed that the present of the magnetic and AB fields removes the degeneracy for different values of the control parameter. The thermodynamic and magnetic properties of the ISKP in the present of the magnetic and AB fields are also evaluated. The effects of the control potential parameter on the thermodynamic and magnetic properties of the selected diatomic molecules are discussed.

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

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