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Bound state solutions of the Schr\"odinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods

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Pith's one-line read The paper establishes that approximate energies for atoms in the radial screened Coulomb potential can be obtained in closed form—using Coulomb, Kratzer, variational, and Hellmann-Feynman methods—with relative errors below 0.63% for the fir

desk verdict Solid analytical approximation paper for a niche potential; the math checks out and the sub-0.63% benchmark is credible, but the bias interpretation in §2.3 contradicts their own tables and the excited-state variational claim is empirical rather than proven. read the letter →

arxiv 2607.19197 v1 pith:RTB6XXNV submitted 2026-07-21 quant-ph

Fatma Zohra Khaled , Mustafa Moumni , Mokhtar Falek This is my paper
classification quant-ph
keywords radialscreenedCoulombpotentialboundstatesKratzereffectiveangularmomentummodifiedBesselfunctionsvariationalmethodHellmann-Feynmantheorempositronium
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 aims to show that the bound-state spectrum of hydrogen-like atoms in the radial screened Coulomb potential—a potential that is nonsingular at the origin and Coulombic at large distances—can be captured by simple analytical formulas instead of requiring a full numerical solution of the Schrödinger equation. The authors derive closed-form energy expressions in terms of modified Bessel functions by evaluating the exact potential in Coulomb and Kratzer reference states, then refine the results with a one-parameter variational scaling and with the Hellmann-Feynman theorem. Benchmarked against high-precision numerical data, the Kratzer-based formulas achieve relative errors below 0.63% for the first ten s-states at screening parameter c=0.1, and the variational version improves the ground-state error to about 0.4%. The same construction extends to positronium, demonstrating that the approach works for arbitrary reduced-mass systems. A sympathetic reader would care because these formulas make rapid, reasonably accurate predictions for plasma-embedded atoms and for spectral diagnostics without heavy computation.

What carries the argument

The central object is the Kratzer reference Hamiltonian -1/r + c/r^2, whose eigenfunctions are Coulomb-like functions with a shifted effective angular momentum ν_l = -1/2 + sqrt((l+1/2)^2 + 2c). This absorbs the leading short-distance screening correction exactly, making the residual potential small. The master integral ∫ x^{ν-1} e^{-β/x - γx} dx = 2(β/γ)^{ν/2} K_ν(2√(βγ)) converts all expectation values into finite double sums of modified Bessel functions of the second kind. A single unified expression with an interpolation parameter k and a scaling parameter β generates the Coulomb-reference, Kratzer-reference, and variational methods as special cases.

What would settle it

A converged numerical solution of the radial Schrödinger equation for the 1s state at c=0.1 can settle the claim: if the Kratzer expectation value deviates from the exact energy by more than the stated 0.63%, or if the variational energy falls below the exact ground-state energy, the central accuracy claim fails.

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Extended reading notes

Core claim

The central claim is that the energy eigenvalues of the radial screened Coulomb potential V(r) = -e^{-c/r}/r can be approximated analytically with sub-percent accuracy. The key move is to use the Kratzer potential -1/r + c/r^2 as a reference: because the leading short-distance screening term c/r^2 has the same radial form as a centrifugal barrier, it is absorbed exactly into an effective angular momentum ν_l = -1/2 + sqrt((l+1/2)^2 + 2c). Evaluating the exact RSCP Hamiltonian in Kratzer eigenstates produces closed-form energies as double sums of modified Bessel functions of the second kind, valid for arbitrary n and l. For the first ten s-states at c=0.1, the relative error against high-prec

Load-bearing premise

The accuracy claims rest on the assumption that a Kratzer-shaped trial wave function with one adjustable scale β lies close to the true eigenfunction; for excited states the variational step is not a rigorous upper bound because the trial states are not orthogonalized against lower states.

Editorial extensions

If this is right

  • Energies for any bound state (n, l) and any screening parameter c are available as closed-form Bessel-function expressions, so parametric scans over screening strength need no numerical Schrödinger solver.
  • The effective-angular-momentum construction works uniformly across l, so states with non-zero angular momentum are approximated with errors comparable to the s-states.
  • The extension to positronium shows the method transfers to any hydrogen-like reduced-mass system, giving mass-dependent screening shifts.
  • The Hellmann-Feynman result implies ∂E/∂c > 0, so energies rise monotonically with screening and s-states remain bound for all finite c—there is no critical screening in this potential, unlike the Yukawa case.
  • Combining the Coulomb reference (which underbinds) with the Kratzer reference (which overbinds) brackets the true energy, providing an error estimate without a separate high-precision calculation.

Reading between the lines

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

  • Applying the same Kratzer-reference trick to other screened potentials, such as the Yukawa or exponential-cosine screened Coulomb potential, would test whether the effective-angular-momentum absorption generalizes; the paper does not do this.
  • A Gram-Schmidt orthogonalized version of the scaled-Kratzer variational basis would restore a rigorous upper-bound property for excited states and likely improve the variational energies further.
  • Because the Coulomb reference underbinds and the Kratzer reference overbinds at moderate c, the arithmetic mean of the two reference energies is a cheap estimator whose error could be quantified systematically against exact data.
  • The predicted absence of critical screening at large c for s-states is a qualitative signature that could be checked in simulations or experiments of strongly screened plasma-embedded atoms.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops three analytical approximation schemes for bound states of the radial screened Coulomb potential (RSCP), V(r) = -(1/r)e^{-c/r}, for hydrogen-like atoms: (i) expectation values evaluated with Coulomb and Kratzer reference eigenfunctions, (ii) a one-parameter variational optimization using a scaled Kratzer basis, and (iii) a Hellmann–Feynman route obtained by integrating the derivative of the expectation value. Closed-form expressions in terms of modified Bessel functions are derived for arbitrary n and l, with a unified master formula in Appendix A. The results are benchmarked against GPS data at c=0.1, reporting relative errors below 0.63% for the Kratzer expectation-value method and below 0.41% for the variational method for the first ten s-states; the extension to Positronium is presented in Section 7.

Significance. The RSCP is not exactly solvable analytically, and the paper provides a transparent, self-contained set of closed-form approximations that could be useful for plasma diagnostics and parametric spectral studies. The algebraic core appears sound: I re-derived the master integral (13), the general expressions (14), (15), (A.18)–(A.20), and the c→0 limits, and they are internally consistent. A strength is that no parameter is fitted to the GPS benchmark; β_opt is chosen by minimizing the expectation value of the exact RSCP Hamiltonian. However, the manuscript makes two load-bearing interpretive claims that are not supported by its own evidence: the excited-state variational calculation lacks a variational upper bound and the stated 'complementary biases' of the two reference methods are contradicted by Tables 1–2. These issues do not invalidate the energy formulas themselves but they do affect the significance of the claimed improvements and the error-estimation rationale.

major comments (3)
  1. [Section 3, Eqs. (28)–(34), Table 3] The variational calculation for excited states is not protected by the variational theorem. The trial functions (28) for different n_r are not orthogonalized against lower states, so minimizing E_var(β) for n_r>0 can lower the energy artificially by admixing lower-state character. The manuscript acknowledges this in the text after Eq. (28), yet Section 5.2 and the abstract present the variational results as providing the 'best accuracy' for the first ten s-states. Since the reported improvement over the Kratzer reference is small (roughly 0.06–0.40% in Table 1), this is exactly the scale at which non-orthogonality contamination could masquerade as improvement. A concrete test would be to Gram–Schmidt orthogonalize each excited trial function against the optimized lower states and re-minimize β; if the improvement persists, the empirical claim is robust. As it stands, the improvement is a
  2. [Section 2.3, bullet 'Kratzer reference', Tables 1–2] The stated complementary bias is contradicted by the paper's own numerical results. For every listed state, the Kratzer reference energy is above E_GPS (less bound), not below it. For example, Table 1 gives E_Kr(1s) = -0.3769734 Ha versus E_GPS(1s) = -0.3793464 Ha, and E_Kr(2s) = -0.1079848 Ha versus E_GPS(2s) = -0.1083227 Ha; Table 2 shows the same sign at c=1.0 and c=10.0. Thus the Kratzer reference does not overestimate binding; it underestimates binding, and the statement in Section 3 that the true RSCP energy lies between the Coulomb and Kratzer predictions is not supported. Consequently, the 'complementary biases ... robust error estimation' claim in the Abstract and Conclusions is not supported. This does not affect the energy formulas themselves, but it invalidates the error-estimation interpretation and should be corrected.
  3. [Section 7, Table 4] Table 4's caption states that relative errors are computed with respect to the reference values of Ref. [34], but no [34] Positronium values are listed; the table's last column is labeled S-H (2021) [32]. Moreover, the parenthetical percentages are not consistent with the S-H column if that is the reference: for the 1s variational energy, |(-0.2131429) - (-0.2134)|/0.2134 ≈ 0.120%, not 0.106% as tabulated. Without the actual [34] reference energies, the claimed sub-0.1% accuracy for Positronium cannot be verified. Please either quote the reference values in the table or add a dedicated reference column.
minor comments (5)
  1. [Abstract and Introduction] Several typographical and grammatical errors should be corrected: 'RSCP. using' in the abstract, 'prvide' in the Introduction, 'numeriacl', 'significally', and 'efficeint' later in the text.
  2. [Section 5.3, Table 3] The text claims 'β_opt > 1 for all states and screening parameters', but Table 3 only reports β_opt for c=0.1. Either provide data for additional screening parameters or rephrase the claim to refer to the computed range.
  3. [Section 6, Eq. (42)] The observation that Eq. (42) coincides with the bare Kratzer eigenvalue in Eq. (24) is important and should be stated earlier, in Section 2.2, where the Kratzer reference is introduced, not only in the comparison with Ref. [34].
  4. [Figure 2] The legend and symbol definitions in Figure 2 are crowded and ambiguous: '1s Kratzer■Var.' and '2s◦Coulomb▲HF-Kr' mix methods and symbols without clear visual separation. Please use distinct linestyles or separate panels.
  5. [Eq. (25)] The symbol I_RSCP is introduced in Eq. (25) but defined only in Appendix A. A pointer to Eq. (A.21) would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all energy formulas are derived from self-contained integrals, the variational parameter is minimized against the exact RSCP Hamiltonian, and the benchmark comparison is external.

full rationale

The paper's derivations are self-contained. The Coulomb- and Kratzer-reference energies (Eqs. (12), (14), (24)) are obtained by evaluating exact expectation values ⟨Ψ_ref|Ĥ_RSCP|Ψ_ref⟩, with the full unscreened RSCP potential retained; the master integral (13) and Laguerre expansions are standard mathematics. The variational parameter β_opt is determined by minimizing E_var(β) for the exact Hamiltonian (Eqs. (29)–(34), (A.20), (A.25)), not by fitting to the GPS benchmark, so the reported sub-0.63%/0.41% errors are genuine external comparisons. The overlap with Xu et al.'s asymptotic formula is explicitly identified as the k=1, β=1 Kratzer limit (§6), not disguised as new. No load-bearing step reduces to a self-citation: the cited GPS data [33,34], Stachura–Hancock [32], and Hellmann–Feynman references [35,36] are external, with no author overlap. The paper also honestly flags a known limitation: the variational upper-bound property for excited states requires orthogonality constraints (§3, Eq. (27) discussion), so the excited-state variational improvement is empirical rather than theorem-guaranteed — a correctness caveat, not circularity. A separate internal inconsistency exists: §2.3 claims the Kratzer reference overestimates binding, but Tables 1–2 show E_Kratzer > E_GPS (less bound) for every listed state; this undermines the 'complementary biases' error-estimation narrative but does not affect the independent derivation of the energy formulas themselves.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central approximations rest on the heuristic Taylor expansion of e^{-c/r} to define reference Hamiltonians, and on the adequacy of the single-parameter scaled-Kratzer trial family. No new physical entities are introduced; the only adjustable parameter is β_opt per state, chosen by energy minimization rather than by fitting to benchmark data.

free parameters (1)
  • β_opt (variational scaling parameter) = 1.0502 (1s), 1.0254 (2s), ..., 1.0052 (10s) at c=0.1
    Per-state parameter chosen by minimizing the energy expectation value E(β) of the exact RSCP Hamiltonian (eqs. 28–34). It is a variational parameter, not fitted to GPS data, but it is an adjustable knob that enters the reported energies.
assumptions (5)
  • standard math Master integral: ∫_0^∞ x^{ν-1} e^{-β/x-γx} dx = 2(β/γ)^{ν/2} K_ν(2√(βγ)) for Re β, Re γ > 0.
    Used in Appendices A and B to evaluate all screened expectation values; cited to Gradshteyn & Ryzhik [39].
  • standard math Generalized Laguerre expansion (A.16) and orthogonality (A.9) hold for non-integer indices 2ν+1.
    Needed to define Kratzer eigenfunctions for non-integer effective angular momentum ν (eq. 22).
  • domain assumption The Kratzer reference potential -1/r + c/r², obtained by truncating the Taylor expansion of e^{-c/r} at first order, provides a good zeroth-order description of the RSCP.
    Section 2.3 states this is the physical reason for the method's accuracy; the convergence of the expansion is heuristic, and no error bound is given for the truncation.
  • domain assumption The RSCP is an appropriate model for plasma-embedded atoms and exotic atoms.
    Section 1 motivates via muonic hydrogen, white dwarf envelopes, and plasma diagnostics, but the paper offers no direct experimental validation of this specific potential form over alternatives.
  • ad hoc to paper The scaled Kratzer ansatz (28) with a single parameter β suffices to approximate the true eigenfunction; for excited states the variational upper bound is not guaranteed because no orthogonality to lower states is imposed.
    Section 3 acknowledges the orthogonality caveat; the empirical improvement of the variational method over fixed references is presented as fact but is not a theorem.

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Pith. "Pith review of Bound state solutions of the Schr\"odinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods." pith.science (2026). https://pith.science/paper/RTB6XXNV

@misc{pith2026260719197,
  author       = {Pith},
  title        = {Pith review of: Bound state solutions of the Schr\"odinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTB6XXNV}},
  note         = {Machine review of arXiv:2607.19197}
}
abstract

We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at $c=0.1$, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.

Figures

Figures reproduced from arXiv: 2607.19197 by the authors.

Figure 1
Figure 1. presents the relative errors as functions of the principal quantum number n for fixed c = 0.1, demonstrating the convergence behaviour of the different methods. The error scaling reveals important 1 2 3 4 5 6 7 8 9 10 10−1 100 101 × Xu et al. ◦ Coulomb ♦ HF-Kr □ Kratzer • Variational Principal quantum number n Relative error (%) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Comprehensive comparison of energy eigenvalues for the first three s-states (1s, 2s, 3s) of the hydrogen atom in the RSCP as functions of the screening parameter c. All methods are shown for c values from 0 to 1 with steps of 0.1. GPS reference values are shown at both c = 0.1 and c = 1.0 (diamonds). Symbols: solid lines = Kratzer reference; open circles (◦) = Coulomb reference; filled squares (■) = variational meth… view at source ↗
Figure 3
Figure 3. Comprehensive comparison of energy eigenvalues for higher excited s states (4s, 5s, 6s, and 7s) of the hydrogen atom in the RSCP as functions of the screening parameter c. All methods are shown for c values from 0 to 1 with steps of 0.1. GPS reference values are shown at c = 0.1 (diamonds). Symbols: solid lines = Kratzer reference; open circles (◦) = Coulomb reference; filled squares (■) = variational method; filled… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of the EVA with Kratzer basis (solid lines) with the asymptotic approximation of Xu et al. (2024) (dashed lines) for the first three s-states over the range 0 ≤ c ≤ 1. Filled squares (■) show the variational results. Diamonds (♦) indicate GPS reference value…
Figure 5
Figure 5. Figure 5: Comparison of the EVA with Kratzer basis (solid lines) with the asymptotic approximation of Xu et al. (2024) (dashed lines) for the p-states (2p, 3p, 4p) over the range 0 ≤ c ≤ 1. Filled squares (■) show the variational results. Diamonds (♦) indicate GPS reference valu…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.