Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron Coulomb integrals

Sandor Kristyan

REVIEW 3 major objections 5 minor 1 references

Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron Coulomb integrals

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that molecular electronic energies can be obtained from the non-interacting a=0 reference by one or two diagonalizations, bypassing SCF iteration and correcting basis-set and correlation errors.

desk verdict A coherent but unvalidated proposal to replace the SCF reference with the a=0 non-interacting Hamiltonian; the headline accuracy rests on in-sample fitted parameters, no molecular tests, and errors far above chemical accuracy. read the letter →

arxiv 1910.02942 v1 pith:GTOLGEVA submitted 2019-08-16 physics.chem-ph

Sandor Kristyan This is my paper · ORCID

classification physics.chem-ph

keywords totallynon-interactingreferencesystemcouplingstrengthparametergeneralizedMoller-PlessetsquareofHamiltonianconfigurationinteractionavoidingSCFconvergenceelectron-electronrepulsionbasisseterror

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 a cheaper route to the non-relativistic electronic Schrödinger equation: solve the artificial case in which all electron–electron repulsion is switched off ($a=0$), then generate the physical $a=1$ energies from that reference. At $a=0$ the Hamiltonian is a sum of independent one-electron operators, so one diagonalization of the matrix in Eq. (7) yields all needed molecular orbitals and Slater determinants with no SCF convergence. The physical energies are then recovered either from the first-order expression $e_0+\langle Y_0|H_{ee}|Y_0\rangle$, from the square-of-Hamiltonian formula Eq. (10), or from a configuration-interaction matrix in the $a=0$ determinant basis that needs at most a second diagonalization. The paper demonstrates on 185 atomic ions that the second-level moment approximation gives a mean deviation of 0.52 hartree from benchmark CI, close to the 0.40 hartree of conventional HF-SCF. If true, this would remove a common bottleneck in quantum chemistry—the iterative SCF cycle—and replace it with fixed-cost linear algebra.

What carries the argument

The central object is the totally non-interacting reference system (TNRS), the $a=0$ Hamiltonian $H_{\rm kin}+H_{ne}$, whose one-electron eigenproblem $h_1\phi_i=\varepsilon_i\phi_i$ (Eq. 2) is solved once by a standard symmetric eigensolver. Its Slater determinants $Y_k$ form an orthonormal basis in which the physical Hamiltonian has the simple matrix representation of Eqs. 13–15, so a second eigensolve of that matrix yields ground- and excited-state energies; the square-of-Hamiltonian moment identity Eq. 10 with the empirical $c_k,w_k$ parameters serves as a lower-cost enhancement of the diagonal estimate.

What would settle it

Evaluate $\langle Y_0|H_{ee}^2|Y_0\rangle$ directly (via the exact wave-function formula Eq. 17) for a small molecule such as H2O or CH4 in the same 6-31G* basis, and compare the resulting Eq. (10) energy with a full-CI or high-level coupled-cluster energy in that basis; also compare $\langle Y_0|H_{ee}^2|Y_0\rangle$ with the empirical $c_2\langle Y_0|H_{ee}|Y_0\rangle^{w_2}$. A significant deviation would show the atomic-fit constants do not transfer to molecules, and the TNRS shortcut loses its accuracy.

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Extended reading notes

Core claim

The central claim is that the eigenstates of $H(a)=H_{\rm kin}+H_{ne}+aH_{ee}$ at $a=1$ can be built from the $a=0$ eigenstates $Y_k$, which are single Slater determinants because $H(0)=\sum_i h_i$. In the $\{Y_k\}$ basis the full Hamiltonian matrix separates cleanly: the diagonal elements are $e_k+\langle Y_k|H_{ee}|Y_k\rangle$ and the off-diagonal elements are pure Coulomb integrals $\langle Y_{k'}|H_{ee}|Y_k\rangle$ (Eqs. 13–15), so the usual SCF rotation of orbitals is replaced by one eigensolve of $h_1\phi_i=\varepsilon_i\phi_i$ followed by one eigensolve of the CI matrix. For the ground state, the paper also derives the square-of-Hamiltonian identity $E_0^2\approx e_0^2+2e_0\langle Y_0|H_{ee}|Y_0\rangle+\langle Y_0|H_{ee}^2|Y_0\rangle$ (Eq. 10), approximates the last term empirically as $c_2\langle Y_0|H_{ee}|Y_0\rangle^{w_2}$, and fits the constants to 185 atomic ions, reducing the average error from 1.78 to 0.52 hartree relative to benchmark CI.

Load-bearing premise

The load-bearing premise is that the four empirical constants fitted to 185 isolated atomic ions (one nucleus) stay essentially unchanged for molecules with several nuclei, so that the same a=0-based formulas deliver accurate a=1 energies for molecular systems.

Editorial extensions

If this is right

Ground- and excited-state energies for a=1 follow from the a=0 orbitals by one or two diagonalizations: first the one-electron matrix of Eq. (7), then, if desired, the CI matrix of Eq. (13).
The a=0 orbital set is independent of the number of electrons N, so ground states, excited states, and different charge states of the same nuclear frame are all built from the same fixed set of orbitals.
In the TNRS CI basis, single excitations can be used directly because Brillouin's theorem does not apply at a=0, so fewer determinants may suffice for the same accuracy as standard CI.
For L = 1, 2, or 3 excited determinants, the CI secular equations reduce to low-order algebraic equations solvable with only the first eigensolver.
The square-of-Hamiltonian estimate of Eq. (10) lies below the first-order estimate of Eq. (9) and remains variational when the exact two-electron integrals of Eq. (17) are used.

Reading between the lines

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

If the atomic-fitted constants survive molecular tests, the TNRS route would make post-Hartree-Fock-quality energies a fixed linear-algebra cost, which would be most valuable for large molecules where SCF convergence is slow or unreliable.
Because the a=0 orbitals are occupation-independent, the TNRS determinant basis behaves like a multireference space; a natural extension the paper does not test is using it for bond-breaking or strongly correlated open-shell systems, where single-reference SCF is known to struggle.
A direct way to isolate the claimed benefit: run a standard a=1 SCF starting from the one-step a=0 orbitals and count how many iterations are saved; the paper does not report such a comparison, only the final energies.
The fitted w2 ≈ 2 and c2 ≈ 1 suggest the second moment of the Coulomb repulsion nearly equals the square of the first moment across a wide range of atomic ions; if that approximate moment identity holds for molecules, it could become a parameter-light correlation estimate, but the present evidence is purely empirical.

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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

Colleague,

The short version: the paper proposes replacing the usual HF-SCF reference with the a=0 non-interacting Hamiltonian, then building CI and moment-based corrections from that reference. The one-electron a=0 problem is genuinely simpler—one diagonalization, no SCF—and the CI basis built from those determinants is a legitimate idea. But the claimed accuracy improvements come from empirical parameters fitted to 185 atomic ions and evaluated on the same set, with no molecular test. Even in-sample, the best average error is 0.52 hartree, worse than plain HF-SCF (0.395) and nowhere near chemical accuracy. The central selling point—that this transfers to molecules—is assumed, not demonstrated.

What's good: the algebra for and the CI matrix elements is laid out carefully, and Eq.9 as a first-order estimate is correct. The observation that SCF is unnecessary for a=0 is true. The paper is honest that the moment approximation is empirical and that the fit 'tends to avoid' w3=3.

The soft spots, in order: (1) The claim that sqrt() is better than is asserted, not proven. It is not generally true for approximate wavefunctions; you need a variance argument, and none is given. (2) The fitted parameters are the only source of improvement, and they are in-sample fits. The paper explicitly hopes they are 'quasi-independent of molecular frame' but shows no molecule. That is the load-bearing gap. (3) The accuracy is far from chemical accuracy; even perfect parameter transfer would not make this a practical method yet. (4) The title overstates; this is an approximation, not a solution.

The paper is worth a serious referee because the core idea is coherent and the derivations are mostly sound, but as written it needs major revision: out-of-sample tests on molecules, a real comparison of H^2 vs energies for the same wavefunction, and honest positioning of the accuracy. I'd bring it to a reading group only to discuss the reference-state idea, not as a demonstrated method. I wouldn't cite it in my own work yet.

Referee Report

3 major / 5 minor

Summary. The paper proposes a 'totally non-interacting reference system' (TNRS) approach to the electronic Schrödinger equation. The authors solve the a=0 Hamiltonian (kinetic plus nuclear–electron attraction, with electron–electron repulsion switched off) by a single diagonalization of the one-electron Hamiltonian, then estimate the physical a=1 ground-state energy by adding the electron–electron repulsion expectation value (Eq. 9), by a second-moment formula built from <Y0|H^2|Y0> (Eq. 10), by empirical moment corrections with parameters c2, c3, w2, w3 fitted to 185 atomic ions (Table 1), and by configuration interaction in the basis of a=0 Slater determinants (Eqs. 12–15). The abstract claims that one or two eigensolver applications bypass SCF convergence while also correcting basis-set error and correlation effects. The manuscript presents explicit formulas, LCAO coefficient tables for argon ions, and a fitted accuracy table, but contains no polyatomic test and no out-of-sample validation of the fitted parameters.

Significance. If the central claim were validated, the idea of obtaining molecular ground-state energies from a non-interacting reference by one or two diagonalizations would be of practical interest, particularly for systems where SCF convergence is difficult. The paper contains useful algebraic material: Eq. 9 is a legitimate first-order estimate, the CI-from-TNRS matrix elements in Eqs. 13–15 are clearly written, and the Appendix gives explicit decompositions of <Hee^2> (Eqs. 17–21). However, the main advertised benefit—replacing SCF for general molecules while correcting basis-set and correlation errors—is not supported by the evidence presented. The only quantitative test is in-sample fitting to 185 atomic ions, the best fitted moment error (0.5171 hartree) is larger than the HF-SCF/6-31G*/a=1 baseline error (0.3951 hartree) reported in the same table, and no molecular calculation is shown. The paper also asserts without proof that the negative square root of <H^2> is a better estimate than the linear expectation value. The significance is therefore conditional: the framework is worth exploring, but the current manuscript does not establish the promised practical advantage.

major comments (3)

[Table 1 and 'Empirical treatment/improvement of Eq.10'] The claim that E0 ≈ -sqrt(<Y0|H(a=1)^2|Y0>) is 'better' than E0 ≈ <Y0|H(a=1)|Y0> is asserted but not proved, and it is not generally true. For a normalized trial function ψ, the linear expectation value <H> is an upper bound to the ground-state energy, while -sqrt(<H^2>) is a lower bound; either quantity can be closer to E0 depending on the variance and excited-state contamination. For example, with an excited state only slightly above the ground state and a modest component in the trial function, the negative square root can be farther from E0 than the linear expectation. The numerical improvement in Table 1 is in-sample and does not establish a general variational improvement. The manuscript should either prove a bound of the form |E0 + sqrt(<H^2>)| ≤ |E0 - <H>| under stated conditions, or reframe the claim as an empirical observation rather than a general property.
[Table 1 and 'Empirical treatment/improvement of Eq.10'] The parameters c2, c3, w2, and w3 are fitted to the same set of 185 atomic ions on which the deviations in Table 1 are computed, so the reported improvement from 1.7791 hartree (first moment) to 0.5171 hartree (second moment) is an in-sample fit, not a predictive test. The text explicitly assumes these constants are 'quasi-independent of molecular frame,' but no polyatomic calculation is reported anywhere in the manuscript. Since the abstract promises 'correction for basis set error and correlation effect' for general molecular systems, the absence of any out-of-sample test—either molecular calculations or at least a training/test split on atomic ions—is a load-bearing gap. Additionally, the best fitted moment error (0.5171) is larger than the HF-SCF/6-31G*/a=1 deviation (0.3951) in the same table, so the table does not support the claim that the TNRS moment approach improves on the SCF baseline it seeks to replace.
[Section 'Configuration interactions (CI) from TRNS', Eqs. (12)–(15)] The abstract's claim that 'one or two eigensolver applications buys off the needs of SCF convergence' is not quantified and is potentially misleading. The CI step in Eq. (13) requires the full set of two-electron integrals <Y_k'|Hee|Y_k>; for a CI space large enough to correct basis-set and correlation errors (L > 3, multiple virtual orbitals), the dimension of the Hamiltonian matrix grows factorially with the number of determinants, and diagonalizing that matrix is computationally comparable to or more expensive than a standard SCF. The only CI example presented is a 2x2 secular equation (L=1). The manuscript should provide a complexity comparison between the proposed two-eigensolver route and standard HF-SCF for a realistic molecular basis set, or restrict the claim to the small-L regime where the computational advantage is concrete.

minor comments (5)

[Throughout] The manuscript contains numerous typos and inconsistent notations, including 'hermetic' for 'Hermitian', 'TRNS' for 'TNRS' in Eq. (10), and the confusing convention that orbital indices are i=0,1,... in the text but i=1,2,... in the LCAO matrices and Eq. (7); these should be standardized.
[Table 1] The column heading 'Deviation from CI/ hartree' does not specify whether the reported values are mean absolute deviations, root-mean-square deviations, or maximum deviations. The table should report the error metric definition, sample standard deviations, and the number of ions for which each method is defined (e.g., whether the 3rd moment is evaluated on all 185 ions or only those where z0 > 0).
[Eq. (9)] The sentence 'In the vicinity of stationary points Eelectr,0 - eelectr,0 ≈ N(N-1)/5 h' is presented without derivation or citation; if it is intended as an empirical rule, it should be either derived or removed, because it is not used elsewhere in the paper and is not obviously general for molecules.
[References] Reference [4] is a self-citation to 'https://arxiv.org/ and https://chemrxiv.org for kristyan' without a title or identifier; reference [5] gives a journal name, volume, and pages but no article title. Both should be supplied with full bibliographic details.
[LCAO matrices and Table 2] The full LCAO coefficient matrices for Ar6+ and Ar8+ in STO-3G are not essential to the argument and interrupt the narrative; they could be moved to an appendix or supplementary material. Table 2's use of dashes, full, and arrows is not explained in the caption and is difficult to parse; a legend should be added.

Circularity Check

1 steps flagged · score 6.0 of 10

The reported accuracy gain is an in-sample fit of the moment parameters; no out-of-sample molecular test supports the transferability assumed by the central claim.

fitted input called prediction [Section 'Empirical treatment/improvement of Eq.10 ...', Eq.10 and Table 1]
"Empirical parameter fit for ck and wk was done using the set of 185 atomic ions (1≤ N, ZA ≤18, M=1, Eelectr,0(CI)= -0.5 to -527.544 hartree for H to Ar atoms, resp.), its statistics is listed in Table 1 to test the TNRS method using HF-SCF/6-31G*/a=0 for Y0."

The same 185-ion set is used both to fit c2, w2, c3, w3 and to report the deviations of the 1st/2nd/3rd level moments. The reduction from 1.7791 to 0.5171 hartree is therefore an in-sample optimization, not an independent prediction. The paper's conditional 'quasi-independent of molecular frame' is not tested on any molecule, so the abstract's claim that the method 'provid[es] the correction for basis set error and correlation effect' rests on fitted constants whose transferability is assumed. The fitted 3rd-level result (0.6977) being worse than the fitted 2nd-level result (0.5171) further shows that the reported improvement is a fit statistic rather than a convergent moment expansion.

full rationale

The non-circular core is genuine: Eq.9 is the exact first-order expectation value of Hee with respect to the a=0 determinant; Eq.10 is the exact algebraic identity <Y0|H(1)^2|Y0> = e0^2 + 2e0<Y0|Hee|Y0> + <Y0|Hee^2|Y0>; and the CI in Eqs.12-15 is a legitimate basis expansion using the a=0 determinants. None of these steps reduces by construction to its own input. The circularity is confined to the empirical moment improvement: the parameters c2, c3, w2, w3 are fitted to the 185 atomic ions and the same 185 ions are then presented as the test set in Table 1. Thus the central accuracy gain attributed to the 2nd-level moment approximation is an in-sample fit, not an out-of-sample prediction. The manuscript itself states the transferability condition ('quasi-independent of molecular frame') but reports no molecular calculation, leaving the abstract's general 'correction for basis set error and correlation effect' claim unvalidated outside the fitting set. The self-citations [4] and [5] are present but are not load-bearing in the main derivation; the load-bearing gap is the fitted, untested moment parameters.

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

The central claim rests on four assumptions: closeness of a=0 and a=1 determinants, transferability of fitted moment parameters, validity of density-product approximations for two-electron integrals, and the superiority of the squared-Hamiltonian estimate. Only the first is given supporting numerical evidence, and that evidence is limited to two argon ions.

free parameters (4)

c2 = 1.1556
Fitted to minimize deviation from CI energies for 185 atomic ions; used in the 2nd level moment approximation.
w2 = 2.0
Fitted weight in the 2nd level moment approximation; value remains at the natural value 2.
c3 = 0.9668
Fitted for the 3rd level moment approximation.
w3 = 0.838
Fitted for the 3rd level moment approximation; deviates from the natural value 3.

assumptions (4)

domain assumption The a=0 single Slater determinant Y0 has LCAO coefficients close to the a=1 HF determinant S0, so the density and one-electron properties are similar.
Supported by Ar6+ and Ar8+ examples, but no general proof is given.
ad hoc to paper The moment parameters c_k and w_k are quasi-independent of the nuclear frame.
Stated as a hope in the section on empirical improvement; not tested on molecules.
domain assumption The density-product approximations in Eqs.18-19 adequately represent the exact two-electron integrals <y0|Hee|y0> and <y0|Hee^2|y0>.
These are standard DFT approximations, but they introduce uncontrolled error, especially for exchange.
ad hoc to paper The negative square root of <H^2> is a better estimate of the ground-state energy than <H>.
Asserted without proof; not generally true if positive-energy components contribute to the wavefunction.

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Cite this review

Pith. "Pith review of Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron Coulomb integrals." pith.science (2026). https://pith.science/paper/GTOLGEVA

@misc{pith2026191002942,
  author       = {Pith},
  title        = {Pith review of: Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron Coulomb integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTOLGEVA}},
  note         = {Machine review of arXiv:1910.02942}
}

read the original abstract

The non-relativistic electronic Hamiltonian, H(a)= Hkin + Hne + aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Chem.: Intro

1.: W.Koch, M.C.Holthausen: A Chemist’s Guide to Density Functional Theory, 2001, 2nd Ed., Wiley-VCH Verlag GmbH 2.: R.G.Parr, W.Yang: Density - Functional Theory of Atoms and Molecules, 1989, Oxford University Press, New York 3.: A.Szabo, N.S.Ostlund: Modern Quant. Chem.: Intro. Adv. Electronic Structures Theory, 1982, McMillan, NY. 4.: S.Kristyan: https...

work page 1997

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Chem.: Intro

1.: W.Koch, M.C.Holthausen: A Chemist’s Guide to Density Functional Theory, 2001, 2nd Ed., Wiley-VCH Verlag GmbH 2.: R.G.Parr, W.Yang: Density - Functional Theory of Atoms and Molecules, 1989, Oxford University Press, New York 3.: A.Szabo, N.S.Ostlund: Modern Quant. Chem.: Intro. Adv. Electronic Structures Theory, 1982, McMillan, NY. 4.: S.Kristyan: https...

work page 1997