Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization

A. V. Volotka; D. A. Glazov; V. K. Ivanov

REVIEW 3 major objections 5 minor 52 references

Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A modified Sturmian basis with a free scale parameter reproduces the Dirac spectrum and QED properties of hydrogen-like ions with finite nuclear size, including the all-order vacuum polarization density.

desk verdict A practical Sturmian basis with solid benchmarks on standard observables, but the headline vacuum-polarization result needs a convergence study and an independent reference overlay before I'd trust it. read the letter →

arxiv 2506.11988 v1 pith:LUSHAHB3 submitted 2025-06-13 physics.atom-ph

V. K. Ivanov , D. A. Glazov , A. V. Volotka This is my paper · ORCID

classification physics.atom-ph

keywords SturmianbasisDiracequationfinitenuclearsizehydrogen-likeionsvacuumpolarizationWichmann-Krollhyperfinesplittinggfactor

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 a relativistic Sturmian basis, with its single scale parameter $\lambda$ left free, is a practical tool for Dirac-equation calculations with a finite-size nucleus. The authors construct basis functions for the large and small radial components with the correct behavior at $r\to 0$ for a non-singular nuclear potential, by choosing the angular indices $l_L$ and $l_S$ appropriately. Using Rayleigh-Ritz diagonalization, they reproduce benchmark binding energies, static dipole polarizability, hyperfine splitting, the bound-electron $g$ factor, and nuclear magnetic shielding for hydrogen-like ions. They also compute the all-order Wichmann-Kroll vacuum polarization charge density, a quantity that had resisted B-spline treatments, and find agreement with Gaussian-basis and Green's-function results. If the basis is as complete as these tests suggest, it offers a linearly stable finite-basis route to properties that depend on wave functions inside the nucleus.

What carries the argument

The load-bearing object is the modified relativistic Coulomb-Sturmian basis: $\pi_L(r)=S_{n,l_L}(2\lambda r)$ and $\pi_S(r)=S_{n,l_S}(2\lambda r)$, where $S_{nl}$ is the Coulomb-Sturmian radial function (a Laguerre polynomial times $r^{l+1}e^{-\lambda r}$) and $l_{L,S}=|\kappa\pm 1/2|-1/2$. Because these functions reproduce the Dirac radial asymptotics at the origin, they can represent wave functions inside the finite-size nucleus. Rayleigh-Ritz reduction of the radial Dirac equation produces a symmetric generalized eigenvalue problem whose matrix elements are evaluated analytically from Laguerre integrals; the free parameter $\lambda$ lets the basis concentrate at the nuclear scale ($\lambda\sim 1/r_n$ for vacuum polarization) without changing the matrix structure.

What would settle it

Compute the $|\kappa|\ge 4$ partial-wave contributions to the Wichmann-Kroll vacuum polarization density with increasing basis size $n$: if the basis is complete in the nuclear interior these contributions must converge to the known Green's-function results, whereas a $\lambda$-plateau that shifts with $n$, or a failure to converge at short distances, would show the interior is not fully resolved.

Watch

Extended reading notes

Core claim

The central claim is that Coulomb-Sturmian functions can be adapted to a finite nuclear charge distribution by replacing the single orbital index $l$ with two indices, $l_L$ and $l_S$, for the large and small components, and by treating the Sturmian scale $\lambda$ as a free parameter instead of tying it to the ground-state energy. The resulting basis has the correct $r\to 0$ asymptotics of the Dirac radial functions for a non-singular potential, and so is designed to resolve the nuclear interior. With this basis and the Rayleigh-Ritz method, the authors reproduce the finite-nuclear-size corrections to binding energies, static dipole polarizability, first-order hyperfine splitting, the bound-electron $g$ factor, and nuclear magnetic shielding in hydrogen-like ions, at the level of dual-kinetic-balance B-spline calculations and analytical formulas. For the Wichmann-Kroll vacuum polarization density, the basis performs on par with, and for the $|\kappa|=2$ partial wave slightly better than, a machine-precision Gaussian basis, while avoiding the Gaussian basis's growing linear dependence.

Load-bearing premise

The load-bearing premise is that the modified Sturmian functions, chosen only to have the right shape near $r=0$ and at infinity, still form a complete enough set to represent Dirac wave functions inside a finite-size nucleus; the paper checks this numerically at each step but does not prove it.

Editorial extensions

If this is right

Finite-nuclear-size corrections to hydrogenic binding energies, polarizabilities, hyperfine structure, and $g$ factors can be obtained in one basis, without kinetic-balance construction, matching DKB B-spline and analytical benchmarks.
The Wichmann-Kroll vacuum polarization density, previously a challenge for finite-basis methods, is accessible with this basis at both short and large distances from the nucleus.
The absence of linear dependence lets the basis be enlarged to improve accuracy without the precision collapse that limits Gaussian-basis calculations.
The construction depends only on spherical symmetry and the correct $r\to 0$ asymptotics, so the same machinery extends in principle to arbitrary binding potentials.
For each quantity the free-$\lambda$ plateau supplies an internal convergence check and an uncertainty estimate, as shown for the hyperfine and shielding factors.

Reading between the lines

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

A natural extension, not pursued in the paper, is to apply the same $\lambda$-tuning to one- and two-loop QED diagrams such as electron self-energy, where the short-distance part of the electron propagator is decisive and linear independence of the basis could remove a practical obstacle.
The $\lambda$-plateau behavior could be developed into an automatic uncertainty procedure: scan $\lambda$, locate the plateau or extremum, and use its width as a numerical error band for any computed matrix element.
If completeness in the nuclear interior is confirmed by stronger tests or a proof, the same recipe of fixing large- and small-component asymptotics separately may transfer to relativistic molecular calculations, where exponential-type bases and linear independence are both attractive.

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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

The paper does two things worth separating. First, it proposes a concrete modification to Coulomb-Sturmian bases — using different orbital indices lL and lS for the large and small Dirac components — so the basis has the correct finite-nucleus asymptotics at r = 0. Second, it applies this basis to a batch of hydrogen-like-ion observables, ending with the all-order (Wichmann-Kroll) vacuum polarization density. The non-VP results are credible. Energies, static dipole polarizability, hyperfine splitting, g factor, and nuclear magnetic shielding all agree with DKB B-spline calculations and analytic formulas to the precision shown. The basis is simple, avoids kinetic balance, and does not develop linear dependence — that alone is a real practical advantage over Gaussians for this problem class.

The genuinely new piece is the combination of the lL/lS substitution with the VP application. No previous Sturmian basis handled the finite-nucleus interior in this way, and this is the first Sturmian-based Wichmann-Kroll calculation. That fills a genuine niche: B-splines fail for VP and Gaussians hit linear dependence before convergence, so a linearly stable alternative is worth developing.

Now the soft spots, in proportion. The completeness of the modified basis is numerically tested but not proven. The paper is honest about this, and for the standard observables the numerical evidence is convincing. The bigger problem is the VP section, which is the paper's headline. The text says the curves can be compared with the Green's-function reference [33], but no such overlay is shown. There is no n-convergence study for the VP density, no tabulated values or error bars, and λ is chosen by hand (λ ~ 1/rn). The two λ values in Figs. 3–6 give somewhat different curves at intermediate distances, and a reader cannot tell how much is genuine λ sensitivity versus truncation error. Since VP is the quantity most sensitive to the nuclear interior, this is a missing-support problem, not an observed inconsistency, but it is exactly where a referee should push.

The manual λ selection also appears throughout the paper. Table VI shows the plateau is only a plateau for sufficiently large n and over a limited λ window; an automated convergence criterion would make the method much more usable. No code or detailed VP data is released, which also weakens reproducibility for a method paper.

Who is this for? People doing relativistic finite-basis calculations, especially for VP or nuclear-size-sensitive observables in heavy ions. The non-VP benchmarks make the basis modification worth taking seriously. But the VP claim needs the missing convergence analysis and an independent overlay before I would cite it as a demonstrated result.

Send it to peer review, but with major revision. The referee should require the n-convergence study for the VP density, tabulated VP values with uncertainties, and an explicit comparison with the independent reference.

Referee Report

3 major / 5 minor

Summary. The paper proposes a relativistic Coulomb-Sturmian-like basis set for solving the radial Dirac equation with a finite-size nucleus. The basis functions are obtained from ordinary Coulomb Sturmians by replacing the orbital index with the asymptotic exponents l_L and l_S appropriate for the large and small components in the finite-nucleus interior. The authors apply the basis in a Rayleigh-Ritz procedure to compute binding energies, static dipole polarizability, first-order hyperfine splitting, the bound-electron g factor, and the nuclear magnetic shielding factor, comparing with analytical formulas and DKB B-spline results. As the main new application, they compute the all-order (Wichmann-Kroll) vacuum polarization charge density for hydrogen-like uranium and compare it with their earlier Gaussian-basis results. The paper argues that the Sturmian basis avoids linear-dependence problems and is well suited for finite-nucleus problems, with the scale parameter λ treated as a free parameter.

Significance. If the vacuum polarization results are fully validated, the proposed basis set would be a useful tool for relativistic atomic-structure and QED calculations, offering an alternative to B-splines and Gaussian bases while being linearly independent and providing analytic matrix elements. The benchmark comparisons for energies, polarizability, hyperfine splitting, and g factor are credible and agree with independent results to the stated precision. The resolution-of-identity test and the absence of spurious states are positive indicators. However, the strongest new claim—that the basis reproduces the all-order vacuum polarization density—is not yet backed by the same level of convergence evidence as the other observables; the paper lacks an n-convergence study, error bars, and an overlay of an independent reference for that quantity. The overall approach is promising, but the central VP claim needs strengthening before the paper can be accepted.

major comments (3)

[Sec. IV.E, Figs. 3–8] The vacuum polarization density, which is the paper's headline application, is presented without a convergence study in the basis size n. The text states that the quality of results reaches a plateau as n increases, but no table or figure analogous to Table VI documents this for ρ(r). Since the VP density is dominated by the nuclear interior where the modified basis differs most from the exact Sturm-Liouville functions, a plateau-like insensitivity to λ could mask systematic truncation error. The authors should provide a quantitative n- and λ-convergence analysis for the VP density, with either tabulated values at selected radii or a figure showing the dependence on n.
[Sec. IV.E, Figs. 3–8] The text claims that the VP curves 'can be compared' with the Green's-function results of Ref. [33] and the finite-basis results of Ref. [30], but no such overlay is shown. The only comparison presented is with the authors' own Gaussian-basis results from Ref. [31]. Adding an overlay of an independent calculation (e.g., the Green's function integration from Ref. [33]) would directly test the VP claim and address the concern that both basis-set calculations might share a common systematic error. Without this, the statement that the CS basis provides 'slightly better' results is not well quantified.
[Sec. IV.D, Figs. 1–2 and Table VI] The selection of the free parameter λ is done manually for each observable and each ion, based on identifying a plateau or a maximum in the λ-dependence. The paper states that the plateau width implies an uncertainty, but this is a heuristic criterion. For the VP calculation, only two λ values (λ=94 and λ=54) are shown, without demonstrating that they lie in a converged plateau for the VP density. The authors should either provide a more formal convergence criterion (e.g., monitoring the plateau as a function of n and λ) or at least present a convergence table for the VP quantity analogous to Table VI, so that the reader can judge whether the chosen λ values are justified.

minor comments (5)

[Abstract and throughout] The manuscript would benefit from a careful proofreading pass; for example, 'The tin hydrogen-like ion' (Sec. IV.A) should be 'The hydrogen-like tin ion', and 'Strumian' appears in the conclusion instead of 'Sturmian'.
[Eq. (52)] In the sentence following Eq. (52), 'where where for brevity' contains a duplicated 'where'.
[Eq. (41)] In the definition of the shell-nucleus potential, the notation 'r > r n' has an unwanted space; it should be 'r > r_n'.
[Sec. IV.E] The parameter choice for the VP calculation is stated as 'λ ~ 1/r_n', but the two values actually used (λ=94 and λ=54) are not derived from this estimate for Z=92 with r_n = 5.8507 fm; a brief explanation of how these values were chosen would improve reproducibility.
[Sec. III] The text surrounding Eq. (45) refers to 'B-orthogonality' and the footnote reference to Ref. [31], but the definition of the B matrix is not given in the present paper; a self-contained definition would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the basis is benchmarked against independent DKB B-splines, analytical formulas, and Green's-function results; λ is a convergence parameter, not fitted to the reported observables.

full rationale

The derivation chain is self-contained. The new ingredient is the modified Sturmian basis (22), whose large/small-component indices lL,S are set by the finite-nucleus asymptotic relations (18)-(20); this is a basis construction, not a result derived from the target quantities. The free scale λ is selected by convergence-plateau scans (Figs. 1-2, Table VI) and is not optimized to reproduce any of the tabulated observables. The central tests are external: binding energies are compared with DKB B-splines and analytic point-nucleus energies (Table I); static polarizability with Szmytkowski's analytic formulas (Table II); hyperfine A(Zα) with Shabaev's formula and DKB (Table III); g factor with analytic Eqs. (62)-(63) and DKB (Table IV); shielding S(Zα) with Eq. (65) and Moskovkin et al. (Table V). The VP density is computed from the spectral formula (66)-(72) with the same regularization used in earlier finite-basis work; the paper cites Mohr et al.'s independent Green's-function method [33] as the comparison reference. The only self-references ([31] for Gaussian-basis VP curves and a B-orthogonality footnote) are non-load-bearing numerical benchmarks, not inputs to the derivation. The manuscript's own admission that exact finite-nucleus Sturmians would require Whittaker functions (Sec. II.C) is a stated design choice, not circularity. A verification gap remains: the VP figures do not overlay [33] and no separate n-convergence table is given for the VP density, but this is an evidentiary/completeness issue, not a case where a prediction reduces to its input by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The method's extra machinery is the free scale parameter lambda, the manual plateau-selection rule, the delta step in the VP subtraction, the truncation of the VP kappa-sum, and the untested completeness assumption for the modified basis.

free parameters (3)

lambda (Sturmian scale parameter) = varies per case; e.g., 2.74 for 1s energy (Table I), 94 for VP density (Fig. 3), 0.2 to 20 for HFS (Table III)
The basis has a single freely variable length scale lambda. It is chosen per ion, per observable, and per nuclear model by scanning lambda and selecting a plateau or maximum in the computed value; the reported uncertainty is the plateau spread. This is the main tunable input of the method.
delta (VP subtraction step) = 10^-6
Small parameter used in the numerical derivative in the C-symmetrized VP subtraction (Eq. 72); fixed by hand and inherited from Refs. [30,31]. Not converged against, but expected to be small enough.
kappa cutoff = |kappa| <= 3
The VP density sum is truncated at |kappa| = 3 because higher terms are stated to be negligible for the first few components. This is a truncation, not a fit, but it is a controlled approximation.

assumptions (5)

standard math Coulomb Sturmian functions are complete for the point-Coulomb radial Schrodinger problem (and their relativistic analogues for the point-nucleus Dirac problem).
Invoked in Sec. II.A and II.C as the starting point for constructing the basis; completeness is expected from Refs. [26,27] but is not re-derived here.
domain assumption The finite-nucleus Dirac wavefunction has zero-distance asymptotics P ~ r^{l_L+1}, Q ~ r^{l_S+1} as given by Eqs. (19)-(20), following Grant [19, Sec. 5.4].
The basis functions are built specifically to match these asymptotics. If the true asymptotics differ for a different nuclear charge distribution or potential, the basis construction loses its justification.
domain assumption The generalized Rayleigh-Ritz eigenvalue problem Hv = epsilon Cv for the Dirac Hamiltonian yields the physical spectrum without variational collapse or spurious states.
The paper claims 'spurious states do not appear in our calculations' (Sec. IV.A) but provides no proof; this is a standard expectation for kinetically balanced or appropriately constructed bases.
ad hoc to paper A flat plateau or a maximum in the computed value as a function of lambda marks the converged basis-set limit.
Section IV.D and Figs. 1-2. The correct values of A(Zalpha) and S(Zalpha) are selected by identifying this feature; the uncertainty is the plateau spread. This criterion is not derived and is not automated.
ad hoc to paper Equation (72), with C-symmetrization (71) and delta=10^-6, isolates the (Zalpha)^{>=3} Wichmann-Kroll vacuum polarization density.
Borrowed without modification from Refs. [30,31]. The validity of this subtraction is essential for the VP results, and the paper does not re-derive it.

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Pith. "Pith review of Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization." pith.science (2026). https://pith.science/paper/LUSHAHB3

@misc{pith2026250611988,
  author       = {Pith},
  title        = {Pith review of: Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LUSHAHB3}},
  note         = {Machine review of arXiv:2506.11988}
}

abstract

We investigate the application of the Sturmian basis set in relativistic atomic structure calculations. We propose a simple implementation of this approach and demonstrate its ability to provide various quantities for hydrogen-like ions, including binding energies, static dipole polarizability, $g$ factor, hyperfine splitting, and nuclear magnetic shielding. Finally, we calculate the all-order (Wichmann-Kroll) vacuum polarization charge density, which was a challenge for the finite-basis-set approach until recently. Comparison of the obtained results with the previously published numerical and analytical calculations is presented. All calculations are performed with the finite size of the nucleus and can in principle be extended to arbitrary binding potentials.

Figures

Figures reproduced from arXiv: 2506.11988 by the authors.

**Figure 1.** The dependence of the computed value of Afs(Zα) on λ for Z = 50 and rn = 4.643 fm and different sizes of the basis. The VP charge density can be expressed as [33] ρ(x) = e Tr[SF (x, x′ )γ0]|x′→x = e 2 X En>0 ϕ † n (x)ϕn(x) − X En<0 ϕ † n (x)ϕn(x) ! , (66) where SF (x, x′ ) is the electron propagator in Furry’s picture. The limit x ′ → x is assumed to be the mean value of the limits from “left” and “right”. The VP d… view at source ↗

**Figure 2.** The dependence of the computed value of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** VP induced charge density, calculated with CS [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 7.** Figure 7: A comparison between VP densities, acquired via [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 5.** Figure 5: VP induced charge density, calculated with CS [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

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