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REVIEW 4 major objections 5 minor 1 references

Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that switching on electron–electron repulsion changes electronic energy almost linearly, so a zero-repulsion starting state can estimate the interacting energy.

desk verdict Essentially first-order perturbation theory in disguise; the only empirical observation is limited to STO-3G, and the 'improved' formula is an in-sample fit. read the letter →

arxiv 1910.02772 v1 pith:TO26QJN3 submitted 2019-08-16 physics.chem-ph

Sandor Kristyan This is my paper · ORCID
classification physics.chem-ph
keywords couplingstrengthparameterquasi-linearenergycurvetotallynon-interactingreferencesystemHohenberg-Kohntheoremelectron-electronrepulsionHellmann-FeynmanHund'sruleKoopmans
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 studies the family of electronic Hamiltonians $H(a)=H_{\rm kin}+H_{\rm ne}+aH_{\rm ee}$ obtained by scaling the electron–electron repulsion by a coupling parameter $a$, with $a=1$ the physical case and $a=0$ a fully non-interacting system. It claims that the ground-state electronic energy $E(a)$ is quasi-linear in $a$: the derivative $dE/da$ is nearly constant on the interval $[0,1]$. If that is right, the exact identity $E_{\rm electr,0}=e_{\rm electr,0}+(N(N-1)/2)\langle\Psi_0|r_{12}^{-1}|Y_0\rangle/\langle\Psi_0|Y_0\rangle$ collapses to a parameter-free first-order estimate, $E_{\rm electr,0}\approx e_{\rm electr,0}+(N(N-1)/2)\langle Y_0|r_{12}^{-1}|Y_0\rangle$, using only the non-interacting Slater determinant $Y_0$. This matters because the $a=0$ calculation is a single cheap SCF step, yet the estimate would give a useful starting point for the full correlated energy and a concrete bridge between the non-interacting and physical ground states.

What carries the argument

The load-bearing object is the scaled Hamiltonian $H(a)=H_{\rm kin}+H_{\rm ne}+aH_{\rm ee}$ and its ground-state energy as a function of the coupling strength $a$. The Hellmann–Feynman theorem fixes the slope of that curve as $dE/da=\langle y_0(a)|H_{\rm ee}|y_0(a)\rangle$, so the quasi-linearity claim is precisely the statement that this slope barely changes over $[0,1]$. The second piece of machinery is the totally non-interacting reference system (TNRS), the $a=0$ endpoint where the wavefunction is an exact single Slater determinant $Y_0$ and the one-electron equations decouple; combined with the exact identity $E_{\rm electr,0}=e_{\rm electr,0}+(N(N-1)/2)\langle\Psi_0|r_{12}^{-1}|Y_0\rangle/\langle\Psi_0|Y_0\rangle$, it turns into the linear transfer Eq. (20). The paper also introduces a real symmetric correlation factor $w$ through $y_0(a)=wY_0$ to describe evolution between the endpoints, and an $L$th-order algebraic expansion in Eq. (33) for more accurate corrections on top of the same transfer.

What would settle it

Compute $E(a)$ for a small atom or molecule at $a=0,0.25,0.5,0.75,1$ with a large-basis correlated method, so that basis error no longer hides curvature; if the intermediate points deviate from the straight chord between $a=0$ and $a=1$ by more than chemical tolerance, the quasi-linearity behind Eq. (20) fails. The paper's own two-electron model $z=\exp(r_{12}/2)$ offers a place where this comparison can be made against an exact analytic solution.

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Extended reading notes

Core claim

The central claim has two parts. First, the electron–electron repulsion contribution $v_{\rm ee}(a)=a(N(N-1)/2)\langle y_0(a)|r_{12}^{-1}|y_0(a)\rangle$ is a quasi-linear function of $a$, so the ground-state energy curve is almost straight; the curvature is controlled by the small quantity $\langle y_0(a)|r_{12}^{-1}|\partial y_0(a)/\partial a\rangle$. Second, the first Hohenberg–Kohn theorem extends across the coupling strength: the non-interacting ground state $Y_0(a=0)$ and the physical ground state $\Psi_0(a=1)$ carry equivalent information about the nuclear framework, summarized as $\Psi_0(a=1)\leftrightarrow H_{\rm ne}\leftrightarrow Y_0(a=0)$. The transfer formula, Eq. (20), writes the physical electronic energy as the non-interacting energy plus $(N(N-1)/2)\langle Y_0|r_{12}^{-1}|Y_0\rangle$, a Coulomb/exchange term evaluated entirely with the $a=0$ determinant. The paper reports numerical support from 149 G3-benchmark molecules and derives generalized versions of the virial theorem, Koopmans' theorem, and Hund's rule on the same $a$-ladder.

Load-bearing premise

The whole transfer depends on the assumption that the non-interacting $a=0$ wavefunction is representative of the physical $a=1$ wavefunction, so the energy curve is nearly straight; the paper states this as the smallness of $\langle y_0(a)|r_{12}^{-1}|\partial y_0(a)/\partial a\rangle$, and notes that larger basis sets increase the curvature and weaken the approximation.

Editorial extensions

If this is right

  • A single $a=0$ SCF calculation, one diagonalization step with no iterative convergence, yields a parameter-free first-order estimate of the full electronic energy through the term $\langle Y_0|H_{\rm ee}|Y_0\rangle$.
  • The extended Hohenberg–Kohn statement $\Psi_0(a=1)\leftrightarrow H_{\rm ne}\leftrightarrow Y_0(a=0)$ implies that the non-interacting density and LCAO coefficients carry the same nuclear-frame information as the physical density, allowing DFT functionals to be re-expressed around the exactly known $a=0$ endpoint.
  • From the generalized virial relation, the non-interacting kinetic energy $t$ exceeds the physical kinetic energy $T$, so any correction built on the TNRS must subtract a large kinetic excess when transferring to $a=1$.
  • Koopmans' theorem holds for arbitrary $a$ and is trivial at $a=0$, so orbital energies from the decoupled $a=0$ calculation can be transferred to $a=1$ after the Eq. (20) correction.
  • The fitted $L$th-order expansion of Eq. (33) reduces the average absolute deviation from G3 molecular energies from about 3.5 hartree for minimal-basis $a=1$ HF to about 1.6 hartree, showing that the TNRS transfer is a useful skeleton for empirical corrections, though not yet chemical accuracy.

Reading between the lines

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

  • Editorial inference: because the paper reports that curvature grows with basis-set size, the most favorable domain for Eq. (20) is minimal or rigid basis sets; with flexible basis sets one would need an explicit second-order term proportional to $\partial y_0/\partial a$ to maintain accuracy.
  • Editorial inference: the same coupling-strength ladder suggests a practical warm-start recipe, using the converged $a=0$ orbitals as the initial guess for an $a=1$ SCF calculation; the paper mentions this in passing but does not develop it as a numerical method.
  • Editorial inference: the $a=0$ density is an attractive reference for building exchange-correlation approximations along the adiabatic connection, since at that endpoint both the wavefunction and density are known exactly within basis error; the paper's Hohenberg–Kohn extension points toward such a construction without formulating a specific functional.
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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

4 major / 5 minor

Summary. The paper studies the family of non-relativistic electronic Hamiltonians H(a) = Hkin + Hne + aHee. It derives exact energy relations connecting the physical system (a=1) with the totally non-interacting reference system (TNRS, a=0), most notably Eqs. (4), (13), and (18). The central claim is that enrgelectr,0(a) is quasi-linear in a, so that the physical ground-state energy can be estimated from the a=0 wavefunction via Eq. (20), Eelectr,0 ≈ eelectr,0 + (N(N-1)/2)<Y0|r12^-1|Y0>. The paper presents STO-3G numerical evidence for quasi-linearity (Fig. 1), an empirical correction formula (Eq. 33) fitted to G3 energies, and discusses extensions of the first Hohenberg-Kohn theorem, Koopmans' theorem, Hund's rule, and the virial theorem to general coupling strength a.

Significance. The exact identities in Eqs. (4), (13), and (18) are correct and potentially useful; they quantify the interaction-energy contribution in an internally consistent way and connect the non-interacting and physical ground states through an exact adiabatic integral. If the quasi-linearity conjecture were established with controlled accuracy, the TNRS transfer in Eq. (20) would be a parameter-free first-order estimate of correlation-free but interaction-inclusive energies, which is an attractive idea. The manuscript also provides a concrete computational protocol (HF-SCF/basis/a=0) that is faster and more stable than standard a=1 SCF. However, the paper's central quantitative claims currently rest on a minimal-basis empirical observation and an in-sample fit; the main approximation is not proven and is admitted to worsen with larger basis sets. The correct exact identities are the strongest part of the contribution.

major comments (4)
  1. [§Calculating ground state with a=0 (Eq. 20 and the discussion following Eq. 17)] The central approximation, Eq. (20), replaces the exact integrand in Eq. (13) by its a=0 value, which is equivalent to assuming that <y0(a)|r12^-1|y0(a)> is constant on [0,1]. The manuscript's justification is the smallness of <y0|r12^-1|∂y0/∂a> in Eq. (17), but no error bound or quantitative measure of this smallness is provided. The paper itself states that the curve is less linear with increasing basis set, and Fig. 1 shows only STO-3G results for seven molecules. Since the basis-set limit is the relevant regime for chemical predictions, the load-bearing assumption is unsupported exactly where it matters most.
  2. [§Computation properties of TNRS (Eq. 33)] The coefficients in Eq. (33) are obtained by least-squares fitting to the same 149 G3 ground-state molecular energies that are then used to report the average and maximum absolute deviations (1.615905 h and 7.015398 h for L=2). This is an in-sample fit, not a prediction; the reported 'improvement' is therefore not evidence for the validity of the TNRS transfer or for the correction formula. An out-of-sample test (e.g., cross-validation or a separate test set) is needed before Eq. (33) can be presented as a meaningful calibration.
  3. [§Calculating ground state with a=0 (paragraph beginning 'Importantly, ∂enrgelectr,0(a)/∂a ≈ const.')] The second empirical assumption, that 'the LCAO coefficients are close to each other between Y0 and Ψ0' (i.e., quasi-independence of the LCAO coefficients on a), is used to justify both Eq. (20) and the w-based representation in Eqs. (26)-(29), but no systematic measurement, error estimate, or basis-set study is given. This assumption is not derived and is at least as consequential as the quasi-linearity of the energy; the paper should either quantify its accuracy or explicitly mark it as a conjecture.
  4. [§Calculating ground state with a=0 (Eqs. 24-25)] The proposed extension of the first Hohenberg-Kohn theorem, Y0(a=0) ⇔ Hne ⇔ Ψ0(a=1), relies on the statement that the a=0 ground-state density determines the nuclear framework. This requires conditions of non-interacting v-representability and non-degeneracy that are not discussed. If the theorem is intended as a rigorous extension, those conditions must be stated; if it is intended as a practical mapping, the limitations should be acknowledged.
minor comments (5)
  1. [Introduction, Eq. (2)] The coupling strength parameter is denoted 'a' in the Hamiltonian, but Eq. (33) uses 'a' for expansion coefficients; this notational collision makes the text confusing and should be resolved.
  2. [Figure 1 caption] The caption admits that 'larger basis set yields slightly larger curvature (not shown)'; showing at least one larger-basis-set curve would allow the reader to assess the extent of curvature growth and is important for the quasi-linearity claim.
  3. [§Computational protocol] The technical modification to the Gaussian SCF algorithm (changing 1/rij to a/rij with a single line) is described only in words; for reproducibility, the authors should provide the precise patch, the exact version of Gaussian, and the SCF convergence settings used.
  4. [References] Reference [1] is a bare link to an author's arXiv/chemrxiv page rather than a citable work; a specific published paper should be cited. Several other references are self-citations; the authors should ensure that the most relevant independent literature on adiabatic connection and coupling-strength integration is cited.
  5. [Table 1] The table lists CI energies for C, N, and O in the third column, but the basis set and method for those CI values are not specified in the table or the text; this makes the comparison difficult to evaluate.

Circularity Check

1 steps flagged · score 6.0 of 10

The TNRS transfer (Eq.20) is an uncontrolled first-order approximation, not circular; the clearly circular step is Eq.33, whose coefficients are least-squares fitted to the same 149 G3 energies later quoted as the average and maximum deviations.

  1. fitted input called prediction [Section 'COMPUTATION PROPERTIES OF TNRS (a=0)', Eq.33 and the following coefficient/deviation table]
    "2nd order (L=2) coefficients in Eq.33 (by least square fitting to 149 ground state G3 molecular energies to minimize the average absolute deviation) are ... The average absolute deviation in h and % and the maximum absolute deviation in h from G3 values are L=2 in Eq.33 : 1.615905 h or 1.02 %, 7.015398 h"

    The coefficients a_j, b_j, c_j in Eq.33 are obtained by least-squares fitting to the same 149 G3 molecular energies that are then used to report the average and maximum absolute deviations of Eq.33. The quoted 'improvement' relative to HF-SCF/STO-3G/a=1 is therefore a training-set error, not an out-of-sample prediction or independent validation. The reported accuracy of Eq.33 is forced by construction to be small on that fitted set, so presenting these deviations as evidence for the method's predictive capability is circular.

full rationale

The central derivation chain is not circular: Eqs.12-14 are the exact Hellmann-Feynman/adiabatic-connection identities, Eq.19 is an exact algebraic relation between eigenfunctions at different coupling strengths, and Eq.20 is the explicit linear-in-a approximation obtained by replacing the integrand with its a=0 value. This is an uncontrolled first-order perturbation estimate, and the paper itself notes that curvature grows with basis set, but that is a correctness/robustness concern rather than a reduction to inputs. The HK extension in Eqs.24-25 follows directly from the standard first Hohenberg-Kohn theorem applied at different values of a; it does not depend on self-citation. The self-citations in the reference list are not load-bearing for the main identities. The genuine circularity is localized to the numerical demonstration around Eq.33: the coefficients are least-squares fitted to the same 149 G3 energies used to compute the quoted average and maximum deviations, so that part of the claimed improvement is a fitting statistic rather than a prediction. Because this affects a secondary but explicitly claimed numerical improvement, while the exact identities and the main approximation remain independent of that fit, the overall circularity is partial rather than total.

Assumptions & free parameters 16 free parameters · 6 assumptions · 0 invented entities

The practical claims rest on an assumed quasi-linearity (an ad hoc empirical conjecture), on fitted coefficients in Eq.33, and on standard DFT/HF background theorems. The exact identities (Eqs.4, 18, Hellmann-Feynman) are not circular, but the fitted correction in Eq.33 is in-sample and should be treated as a fit, not a predictive derivation.

free parameters (16)
  • Eq33 L2: a1 = -0.761233
    Least-squares coefficient in the L=2 power series (Eq.33) fitted to 149 G3 molecular energies.
  • Eq33 L2: b1 = -0.448435
    Least-squares coefficient in Eq.33 (L=2) fitted to G3 energies; scales the nuclear attraction term.
  • Eq33 L2: c1 = 0.430207
    Least-squares coefficient in Eq.33 (L=2) fitted to G3 energies; scales the electron-electron repulsion term.
  • Eq33 L2: a2 = 2.270220E-004
    Quadratic kinetic-energy coefficient in Eq.33 (L=2), fitted to G3 energies.
  • Eq33 L2: b2 = -5.068453E-005
    Quadratic nuclear-attraction coefficient in Eq.33 (L=2), fitted to G3 energies.
  • Eq33 L2: c2 = 1.678742E-004
    Quadratic electron-repulsion coefficient in Eq.33 (L=2), fitted to G3 energies.
  • Eq33 L3: a1 = -0.853118
    Least-squares coefficient in the L=3 power series (Eq.33) fitted to G3 energies.
  • Eq33 L3: b1 = -0.519268
    Least-squares coefficient in Eq.33 (L=3) fitted to G3 energies.
  • Eq33 L3: c1 = 0.289831
    Least-squares coefficient in Eq.33 (L=3) fitted to G3 energies.
  • Eq33 L3: a2 = 5.224182E-004
    Quadratic kinetic-energy coefficient in Eq.33 (L=3), fitted to G3 energies.
  • Eq33 L3: b2 = -1.321651E-004
    Quadratic nuclear-attraction coefficient in Eq.33 (L=3), fitted to G3 energies.
  • Eq33 L3: c2 = 6.744563E-004
    Quadratic electron-repulsion coefficient in Eq.33 (L=3), fitted to G3 energies.
  • Eq33 L3: a3 = -2.026111E-007
    Cubic kinetic-energy coefficient in Eq.33 (L=3), fitted to G3 energies.
  • Eq33 L3: b3 = -2.221198E-008
    Cubic nuclear-attraction coefficient in Eq.33 (L=3), fitted to G3 energies.
  • Eq33 L3: c3 = -4.823247E-007
    Cubic electron-repulsion coefficient in Eq.33 (L=3), fitted to G3 energies.
  • HF/6-31G** scaled coupling a = 0.99353272
    Empirical scaling of the electron-electron interaction that improves HF/6-31G** average deviation from G3; fitted to the benchmark set.
assumptions (6)
  • domain assumption The ground state of H(a) is non-degenerate for a in [0,1], so the Hellmann-Feynman and HK bijections apply.
    The derivations assume a well-defined ground-state density and wavefunction map; degeneracies would complicate the HK statements (Eqs.24-25).
  • domain assumption At a=0 the ground state Y0 is a single Slater determinant (or can be chosen so).
    Used in 'CALCULATING GROUND STATE WITH a=0' and Eq.26; holds for non-degenerate ground states of one-electron Hamiltonians.
  • domain assumption G3 energies are treated as accurate references for the 149 molecules.
    Used in Fig.2 and in the Eq.33 fits; G3 is an approximate composite method, not exact experiment.
  • ad hoc to paper The second derivative of E(a) with respect to a is negligible (quasi-linearity).
    Assumed after Eq.17 and used to derive Eq.20; supported only by Fig.1 in a minimal basis, not by a theorem.
  • ad hoc to paper LCAO coefficients are quasi-independent of a.
    Stated in 'COMPUTATION PROPERTIES OF TNRS'; underpins the transfer of Y0 to Psi0 and the w-based reformulation in Eq.26.
  • standard math The virial theorem holds for atoms and stationary points for any a (Eq.22).
    Standard virial theorem applied to Coulomb potentials; used in Eq.23.

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Pith. "Pith review of Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation." pith.science (2026). https://pith.science/paper/TO26QJN3

@misc{pith2026191002772,
  author       = {Pith},
  title        = {Pith review of: Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TO26QJN3}},
  note         = {Machine review of arXiv:1910.02772}
}
read the original abstract

The non-relativistic electronic Hamiltonian, Hkin + Hne + aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the well-known virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter a. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input a. The most important findings are, 1, vee(a) is quasi-linear function of a, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1)<=>Hne<=>Y0(a=0)) and its consequences in relation to a. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

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Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    2.: S.Kristyan: Computational and Theoretical Chemistry 975, 20-23 (2011) 3.: W.Koch, M.C.Holthausen: A Chemist’s Guide to Density Func

    1.: S.Kristyan: https://arxiv.org/ and https://chemrxiv.org for kristyan. 2.: S.Kristyan: Computational and Theoretical Chemistry 975, 20-23 (2011) 3.: W.Koch, M.C.Holthausen: A Chemist’s Guide to Density Func. Theory, 2001, 2nd Ed., Wiley-VCH Verlag GmbH 4.: A.Szabo, N.S.Ostlund: Modern Quant. Chem.: Intro. Adv. Electr. Struct. Theory, 1982, McMillan, NY...

    work page 2011

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