Exactness of Symmetry-Broken Self-Interaction Correction in the Strongly-Correlated or Classical Limit: Harmonium as a Demonstration

Chen Li; Cody Woods; John Perdew; Wenqing Yao; Yunzhi Li

REVIEW 3 major objections 4 minor 30 references

Exactness of Symmetry-Broken Self-Interaction Correction in the Strongly-Correlated or Classical Limit: Harmonium as a Demonstration

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Symmetry-broken self-interaction correction makes density functional theory exact in the classical limit of two-electron harmonium.

desk verdict A real numerical demonstration that broken-symmetry PZ-SIC captures the classical limit in harmonium, but the r2SCAN quadrature failure, not the Gaussian ansatz, is the main caveat. read the letter →

arxiv 2607.22488 v1 pith:F5LDI2A4 submitted 2026-07-24 cond-mat.str-el

Cody Woods , Yunzhi Li , Wenqing Yao , Chen Li , John Perdew This is my paper · ORCID

classification cond-mat.str-el

keywords harmoniumself-interactioncorrectionstrongcorrelationclassicallimitsymmetrybreakingdensityfunctionaltheoryPerdew-Zungersemiclassical

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 tries to establish that the Perdew–Zunger self-interaction correction, applied to broken-symmetry localized orbitals, gives the exact ground-state energy of harmonium in the classical limit as Planck's constant tends to zero, and remains accurate for all values between zero and the physical value. The significance is that standard approximate density functionals such as LDA, PBE, and r2SCAN diverge or fail in this semiclassical strong-correlation limit, while self-interaction correction restores the correct finite limit. The paper also shows that spherically averaging the broken-symmetry density reproduces the exact symmetric density, supporting a time-centered interpretation of symmetry breaking.

What carries the argument

The central machinery is the Perdew–Zunger self-interaction correction formula, which removes one-electron self-interaction orbital by orbital; the variational broken-symmetry ansatz of two Gaussian orbitals located at ±R with an optimized width parameter α; and the identification of the classical limit (ℏ→0) with the strongly correlated limit, where electrons localize at the classical equilibrium separation.

What would settle it

A calculation that optimizes a more flexible broken-symmetry orbital (e.g., a multi-Gaussian or numerically represented localized orbital) for harmonium at small ℏ and finds a PZ-SIC energy below the classical minimum 0.616908 Ha, or a limiting energy different from U_cl, would falsify the exactness claim.

Watch

Extended reading notes

Core claim

The authors demonstrate numerically that for two-electron harmonium, variationally optimizing a broken-symmetry pair of Gaussian orbitals centered at ±R and applying PZ-SIC yields energies that approach the exact classical minimum U_cl = 0.616908 Ha as ℏ→0, for LSDA, PBE, and r2SCAN alike. They argue this follows because PZ-SIC is exact for any collection of non-overlapping, spin-polarized one-electron densities, and in the classical limit the two electrons become localized on opposite sides of the trap with vanishing orbital overlap. The same broken-symmetry PZ-SIC energy remains close to the exact energy for all ℏ between 0 and 1, with residual deviations at intermediate ℏ attributed to th

Load-bearing premise

The claim relies on the two-Gaussian variational ansatz (only two optimized parameters R and α) being flexible enough to reach the true minimum of the PZ-SIC energy in the ℏ→0 limit; the authors note this ansatz cannot span the full exact Hilbert space, so the approach to the exact classical value could be coincidental.

Editorial extensions

If this is right

If the central claim is correct, symmetry-broken PZ-SIC provides a practical route to the exact strong-correlation limit for any density functional approximation, not just for harmonium.
The spherical-averaging result implies that symmetry-broken localized densities can be averaged to recover symmetry-preserving exact densities, offering a way to reconcile broken-symmetry calculations with exact symmetric ground states.
The demonstrated accuracy from ℏ=0 to ℏ=1 suggests that PZ-SIC could serve as a reliable correction across correlation regimes, including weakly correlated systems.
The classical-limit exactness supplies a new exact constraint that density functional approximations should satisfy, and the paper motivates developing scaled-down SIC beyond the local spin density approximation.
For two-electron singlets, the Weizsäcker kinetic energy is exact, so the remaining DFA error is dominated by exchange–correlation; SIC's success here isolates self-interaction error as the key failure mechanism in the semiclassical regime.

Reading between the lines

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

The two-Gaussian variational ansatz has only two free parameters (R and α), and the authors admit it cannot span the full Hilbert space; a more flexible localized orbital might produce a still lower PZ-SIC energy in the ℏ→0 limit, meaning the exactness could be partly coincidental.
The mechanism suggests a testable extension: for any two-electron system with a classical analog and a Hamiltonian bounded from below, symmetry-broken PZ-SIC should reach the corresponding classical minimum; harmonium is the first demonstration, and other traps could be checked.
The r2SCAN-PZSIC numerical instability reported around ℏ=0.03 in the appendix indicates that meta-GGA self-interaction corrections may need special handling in strongly localized regimes, which could affect the generality of the 'any density functional approximation' claim.
The spherical-averaging argument implies a constructive principle: one could compute broken-symmetry localized densities from SIC and orientationally average to obtain symmetry-preserving densities, potentially bypassing the need for symmetric self-consistent calculations.

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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

What you should know: this paper does exactly what its title says — it demonstrates, for two-electron harmonium, that broken-symmetry PZ-SIC energies approach the classical minimum U_cl as Planck's constant tends to zero, for LSDA, PBE, and r2SCAN. The demonstration is clean: they use the recent exact or near-exact densities from Li and Li, optimize a simple two-Gaussian broken-symmetry ansatz, and then show spherical averaging of the broken-symmetry density reproduces the exact symmetric density. That last part is a nice confirmation of the standard symmetry-breaking/restoration picture, and it is genuinely new as a numerical benchmark.

Credit where due: the variational parameters (R, alpha) are optimized against the PZ-SIC energy, not against the exact energy, so the central result is not fitted. The paper is also transparent about its own limitation — the r2SCAN instability at hbar=0.03 is documented in Table I and Appendix 2, with grid-convergence data showing the meta-GGA SIC evaluation fails. That is honest, reproducible reporting.

Now the soft spots. The stress-test note is right: the limited Gaussian ansatz is not the critical weak point. The delta-function limit is contained in the ansatz (alpha to infinity, R to r0), and PZ-SIC exactly cancels the divergent U[n_i] + Exc[n_i,0] terms for any semilocal functional. The real load-bearing step is the numerical evaluation of that cancellation. Table I shows exactly this breakdown: at hbar=0.03, r2SCAN-PZSIC gives an energy spuriously near zero and does not converge with radial grid, so the point was omitted. The paper's broad claim — "any density functional approximation" — is therefore not actually demonstrated for one of the three tested functionals at the smallest hbar. The authors call it a numerical singularity and exclude it; that's defensible, but the conclusion should be softened or the quadrature fixed.

Second, the benchmark itself is partly a fit: Eq. (8) uses a fitted coefficient 5.0 rather than the analytical HOA value 4.732. The classical limit U_cl is exact, so the main conclusion doesn't depend on the fit, but the intermediate-hbar "exact" guide is not exact. Minor.

Third, this is a numerical demonstration, not a proof. The exactness claim rests on the variational energy approaching U_cl, plus the earlier formal arguments in Ref. [15]. That's fine for a demonstration, but the conclusion states exactness as a general theorem with numerical support. A careful referee should ask for either a more rigorous limiting analysis or a more flexible basis to rule out accidental agreement.

Who is this for? People working on self-interaction correction, strong correlation, and semiclassical limits of DFT. It's a useful benchmark and a clear illustration of a known principle. It deserves a serious referee — the r2SCAN-PZSIC numerical instability is a real issue that needs airing, and the paper's own appendix already invites scrutiny. I'd send it to review with a request to either fix the r2SCAN point or qualify the "any DFA" claim, and to release code and data.

Referee Report

3 major / 4 minor

Summary. This paper studies the two-electron harmonium model with a tunable Planck constant, building on the semiclassical exact densities of Li and Li (Ref. [10]). It evaluates LSDA, PBE, and r2SCAN on those densities, first without and then with Perdew-Zunger self-interaction correction. In a symmetry-preserving (restricted) implementation, PZ-SIC substantially improves the strong-correlation energetics but leaves residual deviations at intermediate ℏ. The central part of the paper is a symmetry-broken two-Gaussian ansatz, Eqs. (10)-(11), with two variational parameters, R and α; the PZ-SIC energy is minimized for each ℏ, and the resulting energy approaches the classical minimum U_cl = 0.616908 Ha as ℏ→0 for HF, LSDA, PBE, and r2SCAN. Spherical averaging of the broken-symmetry density at ℏ=0.1 reproduces the exact spherically symmetric density. The conclusion states that symmetry-broken PZ-SIC applied to any density functional approximation is exact in the strongly-correlated limit and usefully accurate for all 0<ℏ<1.

Significance. If fully supported, the central claim would provide a practical route to strong correlation in DFT: symmetry-broken PZ-SIC restores the classical limit of semilocal functionals and gives a physical interpretation of the symmetric density as an orientational average. The harmonium model is exactly solvable, and the paper uses the exact densities of Ref. [10] rather than self-consistent approximate densities, which isolates functional-driven error. The comparison across three rungs of the semilocal ladder is informative and the restricted-versus-broken-symmetry contrast is clearly drawn. However, the exactness claim is stated more categorically than the evidence warrants: the numerical demonstration uses a two-parameter ansatz, and for r2SCAN the PZ-SIC evaluation fails to converge in the very limit where the claim is made. The paper is therefore valuable as a benchmark and a proof-of-principle, but the broad 'any density functional approximation' conclusion needs either a rigorous argument (e.g., from Ref. [15]) made explicit, or a numerical demonstration that is stable for all tested functionals.

major comments (3)

[Appendix 2 and Table I] The broad claim that symmetry-broken PZ-SIC applied to 'any density functional approximation' yields the exact energy in the strongly-correlated limit is not supported by the numerical evidence for r2SCAN, one of the three tested functionals. At ℏ=0.03 the r2SCAN-PZSIC energy remains anomalously near zero and does not converge as R_POINTS is increased from 900 to 2000; the point is therefore omitted as a 'numerical singularity'. This is precisely the limit in which the cancellation of divergent Hartree and XC self-interaction terms must be resolved. Without a stable evaluation scheme for meta-GGAs, the exactness claim is demonstrated only for LSDA and PBE, not for r2SCAN. Please provide a convergent r2SCAN value or restrict the conclusion.
[Section III.D and Conclusion] The exactness statement is stronger than the variational evidence. The two-Gaussian ansatz of Eqs. (10)-(11) has only two parameters, and the authors state that it 'cannot span the full exact Hilbert space even when the variational optimization is fully converged'. The numerics show only that the optimized energy of this restricted ansatz approaches U_cl; the actual PZ-SIC minimum over a more flexible broken-symmetry orbital space could, in principle, differ. If the exactness is meant to follow from the formal argument of Ref. [15], that argument should be stated explicitly and connected to the numerical limit. If it is meant as a numerical demonstration, a basis-convergence study (e.g., multiple Gaussians per center) is needed to show that the two-parameter result is not an artifact.
[Eq. (8) and figures] The reference line called 'exact energy' is a fit: E(ℏ)=U_cl+(1/2)ℏ(5.0)ω, where the coefficient 5.0 is 'fitted at the endpoints' and differs from the harmonic-oscillator-approximation value 4.732. Using this fitted curve to assess the claim that broken-symmetry PZ-SIC is 'usefully accurate for all values between 0 and the physical value' is partly circular. Please report the actual reference energies from the exact construction of Ref. [10] (or the endpoint constraints used in the fit) and compare against those. The limiting exactness at ℏ=0 is unaffected, but the intermediate-ℏ accuracy claim needs a non-fitted benchmark.

minor comments (4)

[Throughout] Typos and inconsistencies: 'Plank's constant' should be 'Planck's constant'; 'the the sum' appears in Section II.B; 'DF A' is written with an inconsistent space in several places; 'PZSIC' and 'PZ-SIC' are used interchangeably.
[Eq. (2)] Several symbols in the exact density formula are not defined in this manuscript (B, C, c, γ, G_k). Please define them or refer explicitly to the corresponding equations in the supplemental material of Ref. [10].
[Figures 1-3] The captions describe the black dashed line as 'the exact energy of Eq. (8)', but Eq. (8) is a fitted interpolation, not an exact expression. Re-labeling it as 'fit to reference energies' or 'semiclassical fit' would avoid misleading readers.
[Section III.B] The phrase 'truncated approach' is introduced before it is defined in Appendix A. Consider moving or expanding the definition to the first occurrence.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the harmonium PZ-SIC limit is not fitted to the exact energy, and the self-citations supply context rather than the derivation.

full rationale

The central claim is that symmetry-broken PZ-SIC reaches Ucl = 0.616908 Ha as ℏ→0. The variational parameters (R, α) are minimized against the PZ-SIC energy itself, not against Ucl, so the limit is not a fit to the target: 'For each ℏ, the localization distance R and the Gaussian exponent parameter were optimized variationally by minimizing the total PZ-SIC energy.' The exactness in the localized limit follows from the PZ-SIC construction plus the non-overlap cancellation property stated in the paper: 'Perdew–Zunger self-interaction correction is exact for any one-electron density and for any collection of non-overlapping spin-polarized one-electron densities.' As α→∞ and R→r0, the two-Gaussian ansatz contains the non-overlapping delta-function limit, and the semilocal E_xc contributions split into separate one-electron pieces, so the reduced energy is exactly the classical potential energy. This is a theorem by construction, not a circular reuse of the benchmark. The classical benchmark Ucl is independently obtained from Eq. (7), and Eq. (8) is explicitly an endpoint fit used only as a visual guide; it is not an input to the PZ-SIC minimization. The spherical-averaging comparison is also an output check, not an input. The prior works cited for the general statement (Li and Li [10]; Perdew [15]) are from the same group, but the present harmonium computation is an independent falsifiable demonstration, and the general theorem is not contradicted by the paper's own equations. The admitted r2SCAN numerical instability at ℏ=0.03 (Table I, Appendix 2) is a numerical limitation of the demonstration, not a circularity; it does not make the derivation equivalent to its inputs. No load-bearing step reduces to a fitted parameter or a definition of the target quantity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the exact harmonium benchmark from prior work, the classical-limit identification, the PZ-SIC non-overlap exactness argument, and two variational parameters optimized for the calculation. No new physical entities are introduced. The fitted coefficient in Eq. (8) is a minor but notable free parameter in the benchmark.

free parameters (3)

Gaussian exponent alpha (α_opt) = varies with ℏ, see Fig. 4
Variationally optimized for each ℏ by minimizing the total PZ-SIC energy; this is a free variational parameter that controls orbital width.
Localization distance R_opt = varies with ℏ, see Fig. 5
Variationally optimized for each ℏ; determines the separation of the two Gaussian orbitals.
Coefficient 5.0 in Eq. (8) = 5.0
The paper states the energy variation 'can be fitted at the endpoints to E(ℏ)=U_cl+(1/2*ℏ(5.0)ω)', while the HOA normal-mode estimate is 4.732. This fitted coefficient is used as part of the exact-energy guide.

assumptions (4)

domain assumption The exact harmonium density and energy are available from the special-ℏ construction of Li and Li (Eq. (2)) and the harmonic oscillator approximation (Eq. (3)) for ℏ<0.2.
Invoked in Section III.B and the Appendix to generate benchmark densities and energies; if these exact constructions are inaccurate at low ℏ, the comparison is corrupted.
domain assumption The uniform scaling relations (Eq. (4)) exactly map the variable-ℏ system to a unit-ℏ system.
Used throughout to evaluate functionals with LibXC; this is standard coordinate scaling but is assumed exact for the harmonium model.
domain assumption PZ-SIC is exact for any collection of non-overlapping spin-polarized one-electron densities (Perdew's argument).
This is the theoretical basis for expecting exactness as orbital overlap vanishes; the paper relies on this unproven (but widely accepted) argument rather than proving it.
domain assumption The classical limit ℏ→0 is equivalent to the strongly correlated limit for harmonium.
Taken from Ref. [10]; the paper uses this to identify the target limit.

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Cite this review

Pith. "Pith review of Exactness of Symmetry-Broken Self-Interaction Correction in the Strongly-Correlated or Classical Limit: Harmonium as a Demonstration." pith.science (2026). https://pith.science/paper/F5LDI2A4

@misc{pith2026260722488,
  author       = {Pith},
  title        = {Pith review of: Exactness of Symmetry-Broken Self-Interaction Correction in the Strongly-Correlated or Classical Limit: Harmonium as a Demonstration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5LDI2A4}},
  note         = {Machine review of arXiv:2607.22488}
}

read the original abstract

Strong electron correlation is an important challenge to both wavefunction and density functional theory. It has been argued that the Perdew-Zunger 1981 self-interaction correction to any density functional approximation, after symmetry breaking, can correctly describe the ground-state energy in the strongly-correlated limit in which each electron is described by a highly-localized and non-overlapped one-electron spin orbital. It has also been argued that the classical limit, in which Planck's constant tends to zero, is the strongly-correlated limit of quantum mechanics, where standard density functionals fail badly, as demonstrated by the exactly-solvable problem of harmonium (two Coulomb-interacting electrons bound by a spherically-symmetric harmonic-oscillator external potential). Here we combine these two ideas and demonstrate that, for harmonium, symmetry-broken self-interaction correction is exact in the limit where Planck's constant tends to zero, and usefully accurate for all values between 0 and the physical value (1 in atomic units). We also show that the Planck-constant-dependent symmetric ground-state density can be restored by spherical averaging of the broken-symmetry density.

Figures

Figures reproduced from arXiv: 2607.22488 by the authors.

**Figure 2.** highlights the performance of the symmetryunbroken PZ-SIC approach relative to the exact energy. Energy is evaluated on exact or near exact symmetric ground state densities n(r) and spin-orbital densities nσ(r) = n(r) 2 . Note here that the spin orbitals are orthogonal in spin space, not in position space. Since [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 1.** that even more sophisticated functionals (e.g. r2SCAN), which include kinetic energy density dependence, inherit the same type of asymptotic trends because their constraints were not designed to enforce the semiclassical harmonium limit. Across the full ladder of approximations, no commonly used DFA respects the exact semiclassical scaling constraint. This makes harmonium a particularly valuable tool for identify… view at source ↗

**Figure 3.** FIG. 3. Implementation of SIC correction (symmetry bro [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Variationally-optimized Gaussian exponent parame [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 6.** Figure 6: FIG. 6. Spherical average test at [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. Radial probability density for [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: FIG. 8. Combined graphs of radial densities [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Works this paper leans on

30 extracted references · 2 canonical work pages

[15]

Zunger, J.-X

A. Zunger, J.-X. Xiong, and J. P. Perdew, Symmetry breaking transforms strong to normal correlation and false metals to true insulators (2025), arXiv:2512.18236 [cond-mat.mtrl-sci]

arXiv 2025
[10]

A. J. Cohen, P. Mori-S´ anchez, and W. Yang, Challenges for density functional theory, Chemical Reviews112, 289 (2012)

2012
[1]

Energy ingredients on theℏ= 1grid. For spherically symmetricρ 1(r1) we use: Ts[ρ1] = 4π· 1 8 Z ∞ 0 ∂r1 ρ1 2 ρ1 r2 1 dr1,(A1) Vext[ρ1] = 2π ω2 1 Z ∞ 0 ρ1(r1)r 4 1 dr1,(A2) J[ρ 1] = 1 2 Z ρ1(r1)v H [ρ1](r1)d 3r1,(A3) EDF A xc [ρ1] = Z ρ1(r1)ε DF A xc ρ1,∇ρ 1, τ1 d3r1,(A4) withτ 1 ≈ |∇ρ1|2/(8ρ1) used as the one–orbital proxy for meta–GGA inputs. Before asses...
[2]

This value was omitted because the meta-GGA self- interaction correction becomes numerically unstable in the strongly localized, low-density limit(See Table I)

A note about instability For the symmetry unbroken PZ-SIC r2SCAN calcula- tion, there is a numerical instability aroundℏ= 0.03. This value was omitted because the meta-GGA self- interaction correction becomes numerically unstable in the strongly localized, low-density limit(See Table I) . Al- though the kinetic, external-potential, and Hartree con- tribut...

2000
[3]

Hohenberg and W

P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Physical Review136, B864 (1964)

1964
[4]

Kohn and L

W. Kohn and L. J. Sham, Self-consistent equations in- cluding exchange and correlation effects, Physical Review 140, A1133 (1965)

1965
[5]

J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Physical Review B23, 5048 (1981)

1981
[6]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996)

1996

Show all 30 references

[7]

J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, and J. Sun, Accurate and numerically efficient r 2scan meta-generalized gradient approximation, The Journal of Physical Chemistry Letters11, 8208 (2020)

2020
[8]

J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly con- strained and appropriately normed semilocal density functional, Physical Review Letters115, 036402 (2015)

2015
[9]

For a discussion on some common DFT challenges, see [8, 9]
[11]

A. J. Cohen, P. Mori-S´ anchez, and W. Yang, Insights into current limitations of density functional theory, Science 321, 792 (2008)

2008
[12]

Li and C

Y. Li and C. Li, Exact constraint of density functional ap- proximations at the semiclassical limit., Journal of Chem- ical Physics 10.1063/5.0256369 (2025)

2025 doi
[13]

W. Yao, Z. Yin, and C. Li, Formally exact and practi- cally useful analytic solution of harmonium, ACS Omega 10.1021/acsomega.4c06679 (2024)

2024 doi
[14]

J. P. Perdew, A. Ruzsinszky, J. Sun, N. K. Nepal, and A. D. Kaplan, Interpretations of ground-state symme- try breaking and strong correlation in wavefunction and density functional theories, Proceedings of the National Academy of Sciences118, e2017850118 (2021)

2021
[16]

J. P. Perdew, Symmetry breaking with a proper approxi- mate density functional: Exact ground-state energy as a close lower bound on the computed energy, Fizyka Nizkikh Temperatur / Low Temperature Physics52, to appear (2026), also available on ChemRxiv

2026
[17]

J. P. Perdew, Scan meta-gga, strong correlation, sym- metry breaking, self-interaction correction, and semi- classical limit in density functional theory: Hidden con- nections and beneficial synergies?, APL Computational Physics1, 010903 (2025)

2025
[18]

Shahi, R

C. Shahi, R. Maniar, J. Ning, R. K. Sah, M. R. Ped- erson, A. Ruzsinszky, J. E. Peralta, K. A. Jackson, and J. P. Perdew, Local spin density approximation strongly improved by a better-informed local scaling of its self- interaction correction, Journal of Chemical Theory and C...

2026
[19]

Taut, Two electrons in an external oscillator poten- tial: Particular analytic solutions of a Coulomb correla- tion problem, Physical Review A48, 3561 (1993)

M. Taut, Two electrons in an external oscillator poten- tial: Particular analytic solutions of a Coulomb correla- tion problem, Physical Review A48, 3561 (1993)

1993
[20]

Heitler and F

W. Heitler and F. London, Wechselwirkung neutraler atome und hom¨ oopolare bindung nach der quanten- mechanik, Zeitschrift f¨ ur Physik44, 455 (1927)

1927
[21]

M. A. L. Marques, M. J. T. Oliveira, and T. Burnus, 10 Libxc: A library of exchange and correlation functionals for density functional theory, Computer Physics Commu- nications183, 1907 (2012)

1907
[22]

P. W. Anderson, More is different: Broken symmetry and the nature of the hierarchical structure of science, Science 177, 393 (1972)

1972
[23]

D. R. Hartree, The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Meth- ods, Proceedings of the Cambridge Philosophical Society 24, 89 (1928)

1928
[24]

Kais and R

S. Kais and R. D. Levine, Density functional theory for Hooke’s atom, Physical Review A41, 2301 (1990)

1990
[25]

Cioslowski and K

J. Cioslowski and K. Pernal, Hooke’s law correlation en- ergy, Journal of Chemical Physics113, 8434 (2000)

2000
[26]

Katriel and E

J. Katriel and E. R. Davidson, Exact solutions for two- electron systems in harmonic potentials, Proceedings of the National Academy of Sciences of the United States of America77, 4403 (1980)

1980
[27]

Li and J

Z. Li and J. Li, Exact density and semiclassical limit of the harmonium model, Journal of Chemical Physics162, 174112 (2025)

2025
[28]

Lehtola, C

S. Lehtola, C. Steigemann, M. J. T. Oliveira, and M. A. L. Marques, Recent developments in LibXC — A comprehensive library of functionals for density func- tional theory, SoftwareX7, 1 (2018)

2018
[29]

Szabo and N

A. Szabo and N. S. Ostlund,Modern Quantum Chem- istry: Introduction to Advanced Electronic Structure The- ory(Dover, 1996)

1996
[30]

J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density gradient expansion for exchange in solids and surfaces, Physical Review Letters100, 136406 (2008)

2008

Pith tools

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

[1] [15]

Zunger, J.-X

A. Zunger, J.-X. Xiong, and J. P. Perdew, Symmetry breaking transforms strong to normal correlation and false metals to true insulators (2025), arXiv:2512.18236 [cond-mat.mtrl-sci]

arXiv 2025

[2] [10]

A. J. Cohen, P. Mori-S´ anchez, and W. Yang, Challenges for density functional theory, Chemical Reviews112, 289 (2012)

2012

[3] [1]

Energy ingredients on theℏ= 1grid. For spherically symmetricρ 1(r1) we use: Ts[ρ1] = 4π· 1 8 Z ∞ 0 ∂r1 ρ1 2 ρ1 r2 1 dr1,(A1) Vext[ρ1] = 2π ω2 1 Z ∞ 0 ρ1(r1)r 4 1 dr1,(A2) J[ρ 1] = 1 2 Z ρ1(r1)v H [ρ1](r1)d 3r1,(A3) EDF A xc [ρ1] = Z ρ1(r1)ε DF A xc ρ1,∇ρ 1, τ1 d3r1,(A4) withτ 1 ≈ |∇ρ1|2/(8ρ1) used as the one–orbital proxy for meta–GGA inputs. Before asses...

[4] [2]

This value was omitted because the meta-GGA self- interaction correction becomes numerically unstable in the strongly localized, low-density limit(See Table I)

A note about instability For the symmetry unbroken PZ-SIC r2SCAN calcula- tion, there is a numerical instability aroundℏ= 0.03. This value was omitted because the meta-GGA self- interaction correction becomes numerically unstable in the strongly localized, low-density limit(See Table I) . Al- though the kinetic, external-potential, and Hartree con- tribut...

2000

[5] [3]

Hohenberg and W

P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Physical Review136, B864 (1964)

1964

[6] [4]

Kohn and L

W. Kohn and L. J. Sham, Self-consistent equations in- cluding exchange and correlation effects, Physical Review 140, A1133 (1965)

1965

[7] [5]

J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Physical Review B23, 5048 (1981)

1981

[8] [6]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996)

1996