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Atomic Confinement Potentials and the Generation of Numerical Atomic Orbitals

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The valence orbitals of Mg and Ca are nearly independent of which soft confinement potential is used to generate them.

desk verdict A useful and mostly sound systematic comparison of NAO confinement potentials with a genuinely new exponential potential, but the final BSTE analysis mislabels a confinement deformation energy as a basis-set truncation error. read the letter →

arxiv 2505.09540 v3 pith:QA5VJ6YZ submitted 2025-05-14 physics.comp-ph

Hugo {AA}ström , Susi Lehtola This is my paper · ORCID
classification physics.comp-ph
keywords numericalatomicorbitalsconfinementpotentialsdensityfunctionaltheoryfiniteelementmethodexponentialpotentialbasis-settruncationerrorshard-walllimitMgandCaatoms
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 asks whether the specific soft potential used to confine an atom matters for the orbitals that come out. Working on Mg and Ca, it finds that the valence 3s and 4s orbitals are nearly the same for finite-barrier, polynomial, exponential, and singular confinement potentials, as long as the potential switches on outside the core. The paper also introduces a new exponential confinement potential that makes orbitals decay faster than the standard polynomial potential with the same exponent, and shows that any soft potential approaches the hard-wall limit smoothly when made steeper. The practical payoff is that numerical atomic orbital (NAO) basis sets can be built with whatever smooth potential is most convenient for integral evaluation, then cut off with a hard-wall boundary at negligible energy cost.

What carries the argument

The machinery is a set of radial confinement potentials appended to the atomic Kohn-Sham Hamiltonian, solved with a high-order finite element method in which the radial grid ends at a practical infinity $r_\infty$ that acts as a hard wall. The new exponential potential, $V_c(r) = N![\exp(r/r_0) - \sum_{k=0}^{N-1}(r/r_0)^k/k!]$, has the same small-$r$ Taylor behavior as the polynomial potential $(r/r_0)^N$ but grows exponentially at large $r$, which is what gives faster decay. Hermite interpolating polynomials let the calculation force the orbital derivative to zero at $r_\infty$, making the truncation check parameter-free. The key comparison tool is the norm distance between a soft-confined orbital and the best-fit hard-wall orbital, which quantifies how smoothly each potential approaches the hard-wall limit.

What would settle it

Run the same confinement comparison for an open-shell atom such as O or Fe with unrestricted, non-spherically averaged DFT (or unrestricted Hartree-Fock) and compare the 2p or 3d radial orbitals under polynomial, exponential, and finite-barrier confinement at matched truncation error; if the orbital shapes differ by more than a small tolerance, the central insensitivity claim fails.

Watch

Extended reading notes

Core claim

The central claim is that ground-state atomic orbital shapes under soft confinement are largely independent of the confinement potential's functional form. For the Mg 3s and Ca 4s orbitals, the radial functions obtained with the four potential families coincide to the eye once the confinement onset radius is large enough that the core is untouched; differences appear only for strong, small-cavity confinement. The paper demonstrates this by direct fully numerical Kohn-Sham calculations, by showing that all potentials approach the hard-wall orbital systematically as their steepness grows, and by fitting hard-wall orbitals to soft-confined orbitals with small L2 norms. As a novel alternative, the exponential potential of eq. (14) is shown to localize the 3s orbital at smaller truncation radii than the polynomial potential with the same $N$, improving basis sparsity; and fixing basis-set truncation errors to a target value yields periodic-table-wide confinement radii that fluctuate periodically with atomic number $Z$.

Load-bearing premise

All results rest on DFT calculations with spherically averaged electron densities; if that averaging distorts valence orbital shapes for open-shell or nonspherical atoms, the insensitivity claim and the reported truncation radii may not carry over to real NAO generation.

Editorial extensions

If this is right

  • NAO generators can choose confinement potentials for smoothness, integral ease, or strict support without changing the atom's essential valence shape.
  • The new exponential potential yields smaller truncation radii than polynomial confinement at equal $N$, so basis sets made from it should give sparser Hamiltonian matrices and cheaper polyatomic calculations.
  • Adding a hard-wall cutoff at a radius where the soft-confined orbital is already negligible (1 $\mu E_h$ energy error) gives strictly localized, smooth NAO basis functions and removes numerical noise.
  • Fixing the basis-set truncation error to a target ($10^{-2}$, $10^{-3}$, or $10^{-4}$ $E_h$) produces consistent confinement radii across the periodic table, with periodic fluctuations that suggest where computational savings are largest.
  • For singular potentials with denominator exponent $n=1,2,3$, the exponent $n=3$ gives the smallest radii and $n=2$ the largest, though all differences stay below about 0.5 $\AA$.

Reading between the lines

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

  • If the spherically averaged density premise holds, the insensitivity result should extend to other closed-shell atoms and to open-shell atoms treated with fractional occupations; a natural next test is unrestricted non-spherical calculations.
  • The exponential potential's faster decay could make it useful outside NAO generation, for example in models of quantum dots or pressure-confined atoms where exponential localization is physically motivated.
  • The fixed-truncation-error calibration scheme suggests a path to automated, functional-by-functional NAO basis generation across the periodic table, reducing the manual parameter tuning currently needed in codes that use singular potentials.
  • The failed convergence of the $n=4$ singular potential, despite its promising asymptotic faster decay, points to a numerical-stabilization problem worth solving, since tighter localization would directly cut polyatomic integral costs.
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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

2 major / 4 minor

Summary. This paper reviews soft confinement potentials used in numerical atomic orbital (NAO) generation and studies four families of such potentials for the Mg and Ca atoms with fully numerical DFT calculations in the HelFEM finite-element program. The potentials considered are the finite-barrier potential of Eq. (6), the polynomial potential of Eq. (9), a newly introduced exponential potential of Eq. (14), and the singular potentials of Eqs. (10)-(12) with n = 1, 2, 3. The authors report that the valence orbital shapes are qualitatively insensitive to the form of the confinement potential, that all soft potentials approach the hard-wall limit smoothly and systematically, and that the new exponential potential yields smaller truncation radii than the polynomial potential at the same parameters. The paper also presents an analysis in Section IVC3 that is described as an assessment of NAO basis-set truncation errors (BSTEs) for the singular potentials, and it proposes fixing the confinement parameters by setting this quantity to a target value for atoms H-Xe.

Significance. The review and systematic numerical comparison of confinement potentials are valuable for the NAO community, and the proposed exponential confinement potential of Eq. (14) is a useful addition that appears to provide genuinely improved localization in the tested cases. The calculations are variational, converged to a stated 1 microhartree criterion, systematically scanned over parameters, and performed with the open-source HelFEM code, which strengthens confidence in the qualitative findings. The orbital-contraction and hard-wall-limit results for Mg and Ca are well supported by the figures and tables. However, the final BSTE analysis in Section IVC3 rests on a quantity that is not a basis-set truncation error, so the associated conclusions about computational savings are not supported as written.

major comments (2)
  1. [Section IVC3, Eq. (21)] The quantity defined in Eq. (21) is not a basis-set truncation error. Since the FEM calculations are converged to the complete-basis limit (Section III), no finite NAO basis is constructed or truncated, and Eq. (21) reduces to <psi_c|H_0|psi_c> - E_unconfined, i.e., the energy of the unconfined atom evaluated at the confined ground-state density. This is a confinement-induced deformation energy, not an error caused by truncating an NAO basis. Therefore the values labelled BSTE in Figs. 16-18 and the proposal to fix this quantity when choosing r_i do not control a basis-set truncation error of generated NAOs, and the r_c^6 computational-savings conclusion in Section IVC3 is unsupported. Please rename this quantity and revise the abstract, Section IVC3, and Section V accordingly, or alternatively carry out an actual calculation with a finite NAO basis to quantify the true truncation error.
  2. [Section IVC3, Figs. 17-18; Section V] Because the plotted quantities are deformation energies rather than basis-set truncation errors, the proposed fixed-BSTE parameter-selection scheme is not a validated method for controlling NAO basis quality. The periodic trends in r_i shown in Figs. 17 and 18 reflect how strongly each atom responds to confinement, not how incomplete a generated NAO basis would be. The concluding statement in Section V that considerable computational savings may be achievable from this scheme therefore needs to be re-evaluated once the quantity is correctly interpreted.
minor comments (4)
  1. [Section III, Eq. (19)] The radial Schrodinger equation as written is inconsistent with Eq. (4): the standard form should read [-1/2(d^2/dr^2 + (2/r)d/dr) + l(l+1)/(2r^2) + V(r)]R(r) = E R(r). The signs of the centrifugal and potential terms and the missing factor 1/2 do not affect the subsequent continuity argument about R''(r), but they should be corrected.
  2. [Abstract and Section III] The phrase 'the the finite element method' contains a duplicated definite article and should read 'the finite element method'.
  3. [Section V] The word 'adressed' is a typo and should be 'addressed'.
  4. [Table II] The table caption says 'the 3s orbital of the Mg atom in the finite-barrier potential at r = 4 a0'; this should be 'with r0 = 4 a0' to avoid confusion with a radial coordinate value.

Circularity Check

1 steps flagged · score 6.0 of 10

The 'BSTE' section defines its central quantity by Eq. (21) as a confinement deformation energy, so that final contribution is circular by construction; the orbital insensitivity and localization claims are independent and non-circular.

  1. renaming known result [Section IVC3, Eq. (21)]
    "This part of the study thus measures how well the NAOs generated with the various potentials reproduce the exact solution. ... For each functional, we compute the truncation error that would arise in an atomic calculation with the generated NAO basis functions as ΔE(ri) = Econfined(ri) − Econfinement(ri) − Eunconfined (21) where Econfined(ri) is the self-consistent total energy of the atom in confinement, Econfinement(ri) is the confinement energy included in the previous term, and Eunconfined is the energy of the unconfined atom."

    By the definitions in Eq. (21), Econfined = ⟨ψc|H0+Vc|ψc⟩ and Econfinement = ⟨ψc|Vc|ψc⟩, so ΔE = ⟨ψc|H0|ψc⟩ − Eunconfined. All terms come from the fully converged FEM solution; no finite NAO basis is constructed, truncated, or solved. The quantity labeled 'basis-set truncation error' is therefore, by construction, the confinement-induced deformation energy of the exact confined solution relative to the free atom. The reported BSTEs and the fixed-BSTE radii in Figs. 16–18 are this deformation energy relabeled, so the paper's claim to have assessed NAO truncation errors reduces to its own definition rather than to any actual NAO-basis calculation.

full rationale

The main physical content of the paper is self-contained numerical FEM experiments: orbital shapes under finite-barrier, polynomial, exponential, and singular potentials are obtained by solving the stated radial Schrödinger equation, and the insensitivity, decay, and hard-wall-limit statements follow from those calculations rather than from any fitted parameter or imported uniqueness theorem. Self-citations to the HelFEM methodology and to the authors' hard-wall study are references to an open, independently implementable method and to a previous systematic study; they are not load-bearing circular premises. The exponential potential's localization advantage is a computed consequence for the same N, not an enforced equality. However, Section IVC3 is different. Eq. (21) names a 'basis-set truncation error' but algebraically reduces to ⟨ψc|H0|ψc⟩ − Eunconfined, i.e., the energy cost of deforming the exact solution by the confinement potential, with no NAO basis or basis truncation anywhere in the expression. Consequently the BSTE values, the fixed-BSTE radii, and the associated computational-savings conclusions are the confinement deformation relabeled, making that specific final contribution circular by construction. The qualitative orbital-insensitivity and faster-exponential-localization claims remain unaffected by this defect.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The confinement potentials themselves are model inputs, not derived from data; their parameters are scanned or taken from program defaults. The new exponential and n=3 potentials are the only genuinely new constructs. The hard-wall fit radius r-infinity is an analysis parameter fitted to each orbital. The main hidden modeling assumption is spherical density averaging and the exact representation of hard-wall boundaries by FEM grid truncation.

free parameters (5)
  • Finite-barrier height V0 = 0.1 to 10^4 Eh (scanned)
    Eq. (6). Hand-chosen scan range to interpolate between free atom and hard-wall limit; not fitted to external data.
  • Finite-barrier onset radius r0 = 2.0, 3.0, 4.0 a0 (scanned)
    Eq. (6). Chosen so core orbitals are unaffected; independent variable of the study.
  • Polynomial/exponential strength r0 and exponent N = r0 in {2,3,4} a0 (plus shifted values down to 0.1 a0), N in {1..10} (scanned)
    Eqs. (9), (14), (15). Parameters of the soft potentials; scanned, not fitted.
  • Singular potential parameters V0, ri, rc, n = V0 in {1,20,250} Eh; FHI-aims defaults for ri and rc; n in {1,2,3}; rc-ri down to 0.01 Å
    Eq. (12). Taken from literature defaults (FHI-aims) or scanned; n=3 explored as new.
  • Hard-wall fit radius r-infinity = Tables II and V (e.g., 4.01 to 4.43 a0 for finite barrier; 4.07 to 5.40 a0 for shifted potentials)
    Eq. (20). Fitted for each soft orbital by minimizing the norm against the hard-wall orbital; used as a diagnostic for approach to the hard-wall limit.
assumptions (5)
  • domain assumption The Kohn-Sham equation with a spherically averaged density adequately represents the atoms for NAO generation purposes.
    Abstract: 'we perform fully numerical density functional calculations with spherically averaged densities, as is usual in NAO studies'; Section III. All results depend on this.
  • domain assumption FEM calculations converge variationally to the complete-basis limit; a 1 microhartree convergence criterion yields the exact confined-atom solution.
    Section III: 'calculations were converged such that the energy changes less than 1 µEh upon the addition of further elements.' Accuracy of FEM for atoms is cited from refs. 85 and 93, not re-derived here.
  • domain assumption The FEM domain truncation at r-infinity exactly implements a hard-wall boundary because the basis functions vanish at that point.
    Section III: 'All basis functions are built to vanish at this point'; 'the correct discretization is obtained with r∞ = rc.' Used throughout to claim equivalence of grid truncation and hard-wall confinement.
  • domain assumption In the asymptotic analysis, the singular confinement potential dominates the Coulomb, exchange-correlation, and centrifugal terms near r = rc, so the simplified equation (A2) captures the decay.
    Appendix eq. (A2): 'the confinement potential thus dominates, and we can study the asymptotic behavior with a simplified equation.' The dropped terms are assumed subdominant for n >= 1.
  • standard math Orbital energy shifts by a constant do not change the wave function, so a repulsive confinement potential yields the same orbitals as an attractive one offset by a constant.
    Section II, eqs. (3)-(4) and surrounding discussion. Standard property of the Schrodinger equation used to justify generating unoccupied orbitals with positive orbital energies.
invented entities (2)
  • Exponential soft-confinement potential (eq. 14)
    purpose: Designed to yield exponential localization of NAO radial functions with a single strength parameter r0 and exponent N; claimed to localize faster than the polynomial potential of eq. (9).
    Introduced in Section II. Its benefit (smaller r-infinity for the same N, Tables III and IV) is demonstrated only by the authors' own FEM calculations; no independent external benchmark or falsifiable prediction outside this paper is provided.
  • Singular confinement potential with n=3 (eq. 12)
    purpose: New member of the singular-potential family (n=1 Junquera, n=2 Blum); tested for orbital contraction and basis-set truncation errors.
    Introduced in Section II as a generalization; the n=3 asymptotic form eq. (A5) is derived in the Appendix, but no external verification is provided. Calculations for n=4 failed to converge, and n=3 failed for rc-ri <= 0.5 Å, limiting its practical use.

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Pith. "Pith review of Atomic Confinement Potentials and the Generation of Numerical Atomic Orbitals." pith.science (2026). https://pith.science/paper/QA5VJ6YZ

@misc{pith2026250509540,
  author       = {Pith},
  title        = {Pith review of: Atomic Confinement Potentials and the Generation of Numerical Atomic Orbitals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QA5VJ6YZ}},
  note         = {Machine review of arXiv:2505.09540}
}
abstract

We aim to develop novel reusable open source infrastructure [Lehtola, J. Chem. Phys. 159, 180901 (2023)] for numerical atomic orbitals (NAOs). Soft confinement potentials are typically used to force the NAO radial basis functions ${\psi}_{nl}(r)$ to vanish smoothly in increasing $r$ and to generate localized unoccupied states; we review such potentials and other commonly-used techniques in NAO generation as a follow-up to our recent study on atoms in hard-wall confinement [{\AA}str\"om and Lehtola, J. Phys. Chem. A 129, 2791 (2025)]. In addition to NAO generation, confinement potentials are also employed to simulate environmental effects in other research areas, such as studies of (i) atoms in solids, (ii) quantum dots, and (iii) high-pressure chemistry. As in our earlier work, we perform fully numerical density functional calculations with spherically averaged densities, as is usual in NAO studies. Our calculations employ the the finite element method (FEM) implemented in the HelFEM program, yielding variational energies and enabling the use of various boundary conditions. We consider four families of potentials to study the Mg and Ca atoms, which are textbook examples of extended electronic structures. We show that the resulting ground-state orbitals are surprisingly insensitive to the employed form of the confinement potential, and that the orbitals decay quickly under confinement. We study increasingly steep potentials and examine how they approach the hard-wall limit. Finally, we assess NAO basis set truncation errors for types of singular potentials that are now broadly used in the NAO literature.

Figures

Figures reproduced from arXiv: 2505.09540 by the authors.

Figure 1
Figure 1. The Woods–Saxon potential of eq. (8) with various values of a. The case a = ∞ coincides with eq. (6). Pašteka et al.70 considered values of N up to N = 20, but these calculations were limited to Gaussian basis sets, which, as discussed by Pašteka et al.70 and in ref. 81 are likely unreliable as Gaussian basis functions have the same asymptotic form as the solutions for N = 2, only. As the orbitals resulting from a c… view at source ↗
Figure 2
Figure 2. Radial part of the 3s orbital of Mg in finite-barrier confinement with varying V0 and r0 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The radial part of the 3s orbital of the Mg [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Radial density of the 3s orbital of Mg without [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The radial part of the 3s orbital of the Mg [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The polynomial (fig. 6a) and exponential (fig. 6b) confinement potentials considered in this work. 10−6 10−4 10−2 1 102 104 106 108 Vc(r) 1 2 3 4 5 6 7 8 9 10 r/r0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 (a) Polynomial confinement potential of eq. (9) as a function of r/r0…
Figure 7
Figure 7. Figure 7: Plots of the polynomial (fig. 7a) and exponential (fig. 7b) confinement potentials considered in this work, now in semilogarithmic scale instead to the linear scale used in fig. 6. 2. Truncating the radial grid Next, we study the truncation of the radial grid in the ca…
Figure 8
Figure 8. Figure 8: The Mg 3s orbital in polynomial confinement [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 10
Figure 10. Figure 10: The radial part of the 3s orbital of the Mg [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Radial density of the 3s orbital of Mg without cutoff (upper) and with cutoff (lower) in polynomial [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The radial part of the 3s orbital of the Mg [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 14
Figure 14. Figure 14: The radial part of the 3s orbital of the Mg [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 16
Figure 16. Figure 16: Truncation errors of atoms with the PBE functional with the “light”, “intermediate”, “tight” and “really tight” defaults in FHI-aims. 0 10 20 30 40 50 Z 1 2 3 4 5 6 7 8 r i ˚[A] ∆E = 10−2 ∆E = 10−3 ∆E = 10−4 (a) Spherical symmetric DFT. 0 10 20 30 40 50 Z 1 2 3 4 5 6 …
Figure 15
Figure 15. Figure 15: The radial part of the 3s orbital of Mg confined by the singular potential for n = 1 resulting in eq. (10) (solid lines), n = 2 resulting in eq. (11) (dashed lines), and n = 3 (dash-dotted lines) and various values of rc − ri in Å as well as the hard-wall at r∞ = ri .…
Figure 17
Figure 17. Figure 17: The parameter ri of eq. (11) corresponding to fixed BSTE (eq. (21)). The parameter rc = ri + 2.0 Å in all calculations. PBE values are indicated with circles, PW92 values with triangles, and r 2SCAN values with squares [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: The parameter ri obtained from the singular potentials with various exponents and by setting the truncation error of eq. (21) to 10−3 Eh for the PBE functional. Finalising the analysis for the singular potentials, we set the target BSTE to 10−3 Eh and calculate the va…
Figure 19
Figure 19. Figure 19: Asymptotic behavior of confined orbitals as [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]

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