Calculation of ground state energy of Lithium and Beryllium based on variational method

Mei-Qin Deng; Ren-hong Fang

REVIEW 3 major objections 5 minor 20 references

Calculation of ground state energy of Lithium and Beryllium based on variational method

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that a Slater-determinant trial wavefunction with a single effective charge reproduces the ground-state energies of lithium and beryllium to within 2.5% and 3.1% of experiment.

desk verdict Routine but correct variational calculation for Li and Be; the final energies check out, and the printed exchange-integral typo is the main thing to fix. read the letter →

arxiv 2505.05455 v3 pith:CCFI75NK submitted 2025-05-08 physics.atom-ph

Mei-Qin Deng , Ren-Hong Fang This is my paper · ORCID

classification physics.atom-ph

keywords variationalmethodgroundstateenergylithiumatomberylliumantisymmetricwavefunctionSlaterdeterminantatomicshellstructureeffectivecharge

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

The paper sets out to show that a deliberately small variational calculation, forced to respect the indistinguishability of electrons and the atomic shell structure, reproduces the ground-state energies of lithium and beryllium to within a few percent. The trial wavefunction is a Slater determinant built from hydrogenic 1s and 2s orbitals with one shared screening charge $\lambda$, and minimizing the energy over $\lambda$ gives $-198.345$ eV for lithium and $-386.663$ eV for beryllium, against experimental values of $-203.48$ eV and $-399.14$ eV. The significance of the claim is that the two qualitative facts about many-electron atoms, antisymmetry and shell structure, do most of the work even when every other detail is kept intentionally simple.

What carries the argument

The central object is a one-parameter antisymmetrized trial wavefunction: a Slater determinant of hydrogenic 1s and 2s spin-orbitals, $\psi_{100} \propto e^{-\lambda r/a_B}$ and $\psi_{200} \propto (1 - \lambda r/(2a_B)) e^{-\lambda r/(2a_B)}$, all sharing the same effective charge $\lambda$. The argument is carried by two tools: the virial theorem fixes the kinetic-energy and electron-nucleus expectation values directly in terms of $\lambda$, and the electron-electron repulsion is reduced to six-dimensional integrals evaluated by expanding $1/r_{12}$ in Legendre polynomials and integrating over angles. Because the resulting energy function is the quadratic $E(\lambda) = a\lambda^2 - b\lambda$, the minimization is closed form: $\lambda_0 = b/(2a)$ and $E_{\min} = -b^2/(4a)$.

What would settle it

Repeat the minimization with two independent variational parameters, $\lambda_{1s}$ for the 1s electrons and $\lambda_{2s}$ for the 2s electrons. If the energy falls substantially below $-198.345$ eV for lithium and $-386.663$ eV for beryllium, the shared-$\lambda$ assumption is the limiting simplification; if it barely moves, the one-parameter ansatz is exonerated.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the antisymmetrized trial wavefunction $\Psi = (1/\sqrt{N!})\sum_P \delta_P P[\varphi_a(1)\varphi_b(2)\cdots]$, with $\varphi_a = \psi_{100}\chi_+$, $\varphi_b = \psi_{100}\chi_-$, $\varphi_c = \psi_{200}\chi_+$, and (for beryllium) $\varphi_d = \psi_{200}\chi_-$, leads to a closed-form energy function $E(\lambda) = a\lambda^2 - b\lambda$ in units of $\alpha^2 mc^2$. For lithium, $a = 9/8$ and $b = 33401/5832$, giving $\lambda_0 = 33401/13122 \approx 2.5454$ and $E \approx -198.345$ eV; for beryllium, $a = 5/4$ and $b = 3146107/373248$, giving $\lambda_0 = 3146107/933120 \approx 3.3716$ and $E \approx -386.663$ eV. The antisymmetry contributes an explicit exchange integral to the electron-electron interaction, and the shell structure decides which orbitals enter the determinant; the paper reports the resulting errors as 2.5% and 3.1% relative to experiment.

Load-bearing premise

The load-bearing premise is that every electron, in the 1s shell and in the 2s shell, feels the same effective nuclear charge, so one number $\lambda$ controls all screening; if the inner and outer electrons screen the nucleus differently, the trial wavefunction is too rigid to capture that difference.

Editorial extensions

If this is right

For any atom whose ground configuration is two filled $s$ shells, the same determinant construction gives a closed-form quadratic energy function, so the calculation remains fully analytic.
By the variational theorem, the quoted energies are upper bounds to the exact nonrelativistic energies, so part of the apparent closeness to experiment reflects a cancellation between neglected correlation and neglected relativistic or QED shifts.
The optimal effective charges, $\lambda_0 \approx 2.55$ for lithium and $3.37$ for beryllium, quantify how strongly the outer electrons screen the nuclear charge.
The method needs only a handful of Coulomb integrals and no numerical diagonalization, so it can be repeated by hand in a graduate quantum mechanics course and extended to lithium-like and beryllium-like ions by changing $Z$.

Reading between the lines

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

A testable extension would replace the shared $\lambda$ with independent variational charges $\lambda_{1s}$ and $\lambda_{2s}$; if the energy minimum drops significantly below the quoted values, the single-parameter form is the main source of the residual error.
Applying the same construction along the lithium and beryllium isoelectronic sequences would produce predicted curves $\lambda_0(Z)$ and error trends that the paper does not report, offering a cheap check of whether the ansatz remains adequate as the nuclear charge grows.
Because the Hamiltonian in Eq. (4) contains only nonrelativistic Coulomb terms while the comparison values are experimental, the reported agreement may involve a partial cancellation between missing correlation energy and missing relativistic corrections; disentangling the two is beyond the paper.

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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

Short answer: this is a standard variational calculation for lithium and beryllium, and the central numbers are right. I re-derived the coefficients: for Li, the Slater determinant gives V_ee = 5965/5832 λ, λ0 = 33401/13122, and -198.345 eV; for Be, V_ee = 586373/373248 λ, λ0 = 3146107/933120, and -386.663 eV. Those agree with experiment to about 2.5% and 3.1%, and they are rigorous upper bounds because the trial wavefunction is normalized and the energy is minimized variationally.

What the paper does well: it is a self-contained, explicit treatment of antisymmetry and shell structure, with the six-dimensional integrals worked out in an appendix. It does not oversell the method, and the comparison to experiment is honest. It is not new — refs. [11-17] already cover variational calculations for these atoms — but as a pedagogical derivation it is clear and reproducible in principle.

Soft spots, in decreasing order of importance. First, Eq. (18) and Eq. (A8) print the exchange integral I3 as 16/272; the correct value, used later in Eq. (15), is 16/729. As printed, the derivation cannot be followed, even though the final energy is unaffected. Second, Eqs. (15) and (28) have prefactor displays ("3×", "6/4!") that do not match the bracketed integrands; again, the final coefficients are fine, but the printed steps are sloppy. Third, the trial wavefunction uses a single effective charge λ for both 1s and 2s electrons. That is a real modeling restriction — the 1s and 2s shells would generally screen differently — and it biases the answer. It is not a correctness flaw, because the variational principle still gives a valid upper bound, but the choice is motivated by simplicity rather than physics.

The citation pattern is unremarkable and there is no self-citation. The paper makes no inflated claims. For a teaching-oriented journal, this is a suitable manuscript after the typos are corrected. I would not cite it in a research paper, but a referee can verify the calculation quickly and the errors are mechanical, not conceptual. Send it to peer review if the venue values worked pedagogical examples.

Referee Report

3 major / 5 minor

Summary. The paper presents variational calculations of the ground-state energies of lithium and beryllium. The trial wavefunctions are Slater determinants built from hydrogenic 1s and 2s orbitals that share a single effective charge λ, chosen to minimize the energy expectation. Using the virial theorem for the one-electron terms and direct/exchange Coulomb integrals for the electron-electron repulsion, the authors obtain -198.345 eV for Li and -386.663 eV for Be, compared with experimental values of -203.48 eV and -399.14 eV (relative errors 2.5% and 3.1%). The manuscript concludes that the variational method, together with the antisymmetry requirement and atomic shell structure, gives good agreement with experiment.

Significance. If the printed derivation is corrected, the central results are valid variational upper bounds and the agreement with experiment is a genuine, falsifiable test of the one-parameter screened-hydrogenic model. The paper's main strength is that the final energies are independently reproducible from the stated integrals (once the misprints in Eqs. (15), (18), and (28) are fixed), and the calculation is self-contained rather than fitted to the target energies. The physical content is, however, elementary: the single-λ ansatz is a known approximation, and the paper does not go beyond standard textbook material. Its value for a research journal is therefore primarily pedagogical.

major comments (3)

[II, Eq. (18) and Appendix A, Eq. (A8)] The exchange integral I3 is printed as 16/272 × λ/a_B, but the correct value is 16/729 × λ/a_B. This value is load-bearing: the coefficient 5965/5832 in Eq. (15) and Eq. (19) equals 5/8 + 2×(17/81) − 16/729, so the printed 16/272 is not the value used. As printed, a reader who follows Eq. (18) obtains a different electron-electron energy and hence a different minimized energy. The denominator and the intermediate algebra in Eq. (A8) must be corrected.
[II, Eq. (15) and III, Eq. (28)] The prefactors are internally inconsistent. In Eq. (15), the bracket expression integrates to the total electron-electron energy J_1s1s + 2J_1s2s − K_1s2s, so the leading factor '3×' would triple-count the interaction; the final coefficient 5965/5832 corresponds to the bracket without the '3×'. In Eq. (28), the factor 6αℏc×(1/4!) times the bracket equals the correct total J_1s1s + J_2s2s + 4J_1s2s − 2K, so the initial '3×' is again spurious. These prefactors do not enter the final coefficients, but their presence makes the derivation non-reproducible as printed.
[II, Eq. (6) and III, Eq. (22)] The trial wavefunction assumes that the 1s and 2s electrons share a single effective charge λ. This imposes equal screening on core and valence electrons and prevents the wavefunction from describing different effective charges for the two shells. The paper should explicitly state this restriction and discuss its likely effect on the variational energy (allowing independent λ_1s and λ_2s would lower the upper bound). The absence of this discussion is a gap, though not a correctness error, because the variational principle guarantees the computed value is an upper bound.

minor comments (5)

[Appendix B] The line '(4312)^2 + 4312(−1342 − 4213 + 1243)' appears twice, and the permutation 4213 is missing from its expected place in the list; this should be corrected to give the complete 24-term expansion.
[II, Eq. (15) and III, Eq. (28)] The angle-bracket notation in Eqs. (15) and (28) is undefined; please clarify whether ⟨e²/(4πε0 r12)⟩ denotes the expectation in Ψ and whether the subsequent integrals are single-pair or total pair contributions.
[II and III] The experimental values −203.48 eV and −399.14 eV are cited without a reference; please provide a source for these values.
[References] The reference list consists almost entirely of Chinese-language teaching journals; adding a standard textbook or review reference for the variational treatment of many-electron atoms would help readers assess the novelty and context.
[II, Eq. (6)] The notation a'_B = 2a_B in Eq. (6) is introduced without explanation; a brief comment that this is the scaled Bohr radius for the hydrogenic 2s orbital with effective charge λ would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variational calculation is self-contained and the single parameter is minimized, not fitted to experiment.

full rationale

The central results (Eqs. (21) and (32)) are obtained by minimizing the expectation value of the Hamiltonian with respect to the variational parameter λ; the experimental ground-state energy enters only at the final comparison stage and is not used to determine any parameter. The trial wavefunction and the Coulomb Hamiltonian are stated independently, and the required integrals are evaluated directly in Appendix A. The single-λ screening ansatz is a modeling simplification, not a quantity fitted to the target result. There is no load-bearing self-citation: reference [20] is a standard mathematical-physics textbook used for the Legendre expansion, and the other references are external textbooks or pedagogy. The printed typos (e.g., I3=16/272 in Eq. (18) instead of 16/729, and the inconsistent prefactors in Eqs. (15) and (28)) are presentational errors that do not affect the final numerical values and do not indicate circularity. The derivation is therefore self-contained against the external experimental benchmarks.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The calculation rests on standard quantum mechanics plus the specific choice of a one-parameter trial wavefunction. The effective charges are variational parameters, not fitted to the experimental energies. No entities are invented.

free parameters (2)

effective charge lambda for lithium = lambda0 = 33401/13122 ≈ 2.5454
Variational parameter in the trial wavefunction, chosen to minimize the energy expectation, not fitted to the experimental energy.
effective charge lambda for beryllium = lambda0 = 3146107/933120 ≈ 3.3716
Variational parameter in the trial wavefunction, chosen to minimize the energy expectation, not fitted to the experimental energy.

assumptions (5)

standard math The variational principle: the expectation value of the Hamiltonian in any normalized trial state is an upper bound on the ground state energy (Eq. (2)).
Invoked in the introduction to justify minimizing the energy over lambda.
domain assumption Electrons are identical fermions, so the trial wavefunction must be antisymmetric under exchange (Eq. (3)).
Used to build the Slater determinant trial functions.
domain assumption The trial wavefunction is a Slater determinant of hydrogenic 1s and 2s orbitals with a common effective charge lambda (Eqs. (5)-(6)).
This is the key modeling assumption; it neglects shell-dependent screening and correlation.
standard math The virial theorem is used to evaluate kinetic and nuclear-potential expectation values from the hydrogenic energy (Eqs. (13)-(14), (26)-(27)).
Applies for a Coulomb potential; used to avoid explicit integral evaluation for one-body terms.
standard math The multipole expansion of 1/r12 (Eq. (A3)) is used to evaluate the two-electron integrals.
Standard expansion used in Appendix A.

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Cite this review

Pith. "Pith review of Calculation of ground state energy of Lithium and Beryllium based on variational method." pith.science (2026). https://pith.science/paper/CCFI75NK

@misc{pith2026250505455,
  author       = {Pith},
  title        = {Pith review of: Calculation of ground state energy of Lithium and Beryllium based on variational method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCFI75NK}},
  note         = {Machine review of arXiv:2505.05455}
}

read the original abstract

With the consideration of identity principle and atomic shell structure, we calculated the ground state energy of Lithium atom and Beryllium atom based on variational method, which accords quite well with the experimental results.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

R. X. Mao and Y. X. Li, Journal of Yunnan Nationalities Uni versity 15, 228 (2006)

work page 2006
[2]

X. H. Deng, C. K. Xu, and X. W. Wang, Journal of Hengyang Nor mal University 34, 146 (2013)

work page 2013
[3]

H. J. Quan and L. X. Zheng, College Physics 33, 6 (2014)

work page 2014
[4]

W. B. Yang, Chinese Educational Technology Equipment 2, 122 (2014)

work page 2014
[5]

C. S. Zhang, T. R. Ning, and J. Lu, College Physics 34, 32 (2015)

work page 2015
[6]

Lu and H

X. Lu and H. Zheng, College Physics 38, 62 (2019)

work page 2019
[7]

W. Yin, H. H. Qiu, and J. J. Wang, Guangxi Physics 42, 17 (2021)

work page 2021
[8]

Y. M. Pi, Y. N. Mou, and G. F., Science and Technology Infor mation 16, 217 (2023)

work page 2023

Show all 20 references

[9]

B. C. Qian, Quantum Mechanics, Higher Education Press (20 06)
[10]

S. X. Zhou and H. Chen, Quantum Mechanics, Higher Educat ion Press (2022)

2022
[11]

X. X. Wang, College Physics 12, 26 (1993)

1993
[12]

C. Y. Kong, Journal of Chongqing Teachers College (Natu ral Science) 12, 37 (1995)

1995
[13]

Saleh-Jahromi and W

A. Saleh-Jahromi and W. Moebs, European journal of phys ics 19, 335 (1998)

1998
[14]

S. Z. Huang, S. X. Sun, and X. F. Zheng, Journal of Anhui No rmal University (Natural Science) 22, 300 (1999)

1999
[15]

J. S. Li, D. L. Wang, and S. Z. Huang, Journal of Anhui Norm al University (Natural Science) 25, 339 (2002)

2002
[16]

Wu and L

F. Wu and L. J. Meng, College Physics 36, 12 (2017)

2017
[17]

L. J. Dong, Journal of Shanxi radio and TV University 3, 85 (2018)

2018
[18]

K. G. Li, G. J. Liu, S. Z. Huang, and Z. F. Cui, Journal of An hui Normal University (Natural Science) 29, 134 (2006)

2006
[19]

S. F. Huang, K. Ma, and S. Z. Huang, Journal of Hefei Unive rsity (Natural Science) 17, 37 (2007)

2007
[20]

K. M. Liang, Mehtod of Mathematical Physics, Higher Edu cation Press (1995)

1995

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

R. X. Mao and Y. X. Li, Journal of Yunnan Nationalities Uni versity 15, 228 (2006)

work page 2006

[2] [2]

X. H. Deng, C. K. Xu, and X. W. Wang, Journal of Hengyang Nor mal University 34, 146 (2013)

work page 2013

[3] [3]

H. J. Quan and L. X. Zheng, College Physics 33, 6 (2014)

work page 2014

[4] [4]

W. B. Yang, Chinese Educational Technology Equipment 2, 122 (2014)

work page 2014

[5] [5]

C. S. Zhang, T. R. Ning, and J. Lu, College Physics 34, 32 (2015)

work page 2015

[6] [6]

Lu and H

X. Lu and H. Zheng, College Physics 38, 62 (2019)

work page 2019

[7] [7]

W. Yin, H. H. Qiu, and J. J. Wang, Guangxi Physics 42, 17 (2021)

work page 2021

[8] [8]

Y. M. Pi, Y. N. Mou, and G. F., Science and Technology Infor mation 16, 217 (2023)

work page 2023