{"id":"508b4f9f-5252-4a1f-9d68-afef59e6f7a6","arxiv_id":"1910.02942","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Ground-state electronic energies are estimated from a non-interacting reference using first-order perturbation, squared-Hamiltonian moments with empirical parameters, and configuration interaction, avoiding SCF convergence.","lead":"This paper proposes a way to compute electronic energies by first solving a reference system with electron-electron repulsion turned off, then adding the repulsion back through perturbation, a squared-Hamiltonian moment estimate, and configuration interaction. It claims this avoids self-consistent field convergence while correcting for basis-set and correlation errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim hinges on unvalidated transfer of atom-fitted moment parameters to molecules; Table 1 shows only in-sample accuracy far from chemical accuracy.","rationale":"The reader's weakest-assumption identification matches my independent reading. The algebraic core of the paper, diagonalizing the a=1 Hamiltonian in the basis of a=0 Slater determinants, is standard and formally sound; exact evaluation of <Y0|Hee^2|Y0> via Eq.17 is also available, so the method is not internally inconsistent. However, the numerical evidence that the method 'buys off SCF convergence' with useful accuracy is confined to atoms and depends on fitted moment parameters whose frame independence is explicitly hoped for rather than demonstrated. Table 1 also shows that even in-sample the best moment approximation deviates from CI by about 0.52 hartree, so the published data do not support chemical-accuracy claims for molecules. A concrete molecular test would settle whether the central claim has validity beyond the training set. I therefore keep the reader's conditional verdict rather than escalating it.","tokens_in":19878,"tokens_out":5302,"duration_ms":54414,"concrete_test":"For H2, LiH, H2O and CH4 in the same 6-31G* basis, compute the exact <Y0|Hee^2|Y0> from Eq.17 and compare it with c2 * z0^2 using the fitted c2=1.1556 (z0=<Y0|Hee|Y0>). If the ratio differs from 1.1556 by more than 20%, the empirical moment model is frame-dependent. Separately, compute E from Eq.10 with the fitted c2,w2 and compare to full CI/CCSD(T) in the same basis; an average error above ~1.6 mEh per molecule would refute the claimed molecular correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's practical route to 'correction for basis set error and correlation effect' is the empirical moment approximation for <Y0|Hee^k|Y0> in Eq.10, with c2, w2, c3, w3 fitted to 185 atomic ions in Table 1. The text explicitly assumes these constants are 'quasi-independent of molecular frame', but no polyatomic calculation is reported. If c2 or w2 change with the nuclear frame and density shape, the improved second/third moment estimates become uncontrolled, and the only demonstrated accuracy gain (from 1.78 to 0.52 hartree average deviation) is both in-sample and still three orders of magnitude above chemical accuracy. Since the abstract's 'one or two eigensolvers' claim is intended for general molecular systems, the absence of any out-of-sample molecular test, for either the moment parameters or the truncated {Y_k} CI expansion, is the load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20242,"tokens_out":5481,"duration_ms":56312,"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":[{"comment":"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.","section":"Table 1 and 'Empirical treatment/improvement of Eq.10'"},{"comment":"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":"Table 1 and 'Empirical treatment/improvement of Eq.10'"},{"comment":"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.","section":"Section 'Configuration interactions (CI) from TRNS', Eqs. (12)–(15)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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).","section":"Table 1"},{"comment":"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.","section":"Eq. (9)"},{"comment":"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.","section":"References"},{"comment":"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.","section":"LCAO matrices and Table 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a promotional tone relative to the evidence: the abstract advertises a general SCF-free route for molecules, but the only numerical support is an in-sample fit to atomic ions, with errors above chemical accuracy and above the HF-SCF baseline in the same table. The central originality is modest—Eq. (9) is straightforward first-order perturbation theory, and the moment approximation with fitted parameters resembles empirical density-functional corrections. I would encourage the editor to require, before publication, either (a) genuine out-of-sample molecular benchmarks showing that the moment parameters transfer, or (b) a substantial revision that limits the claims to the demonstrated atomic-ion regime and to the CI-from-TNRS formalism without the empirical moment corrections. The H^2 superiority claim should be proved or removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe 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.\n\nWhat's good: the algebra for <Y|H^2|Y> 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.\n\nThe soft spots, in order: (1) The claim that sqrt(<H^2>) is better than <H> 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.\n\nThe 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 <H> 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.","headline":"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.","tokens_in":20571,"tokens_out":3464,"would_cite":false,"duration_ms":34926,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["totally non-interacting reference system","coupling strength parameter","generalized Moller-Plesset","square of Hamiltonian","configuration interaction","avoiding SCF convergence","electron-electron repulsion","basis set error"],"falsifier":"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.","tokens_in":19684,"feed_emoji":"⚛️","tokens_out":12012,"duration_ms":101100,"temperature":0.7,"pith_summary":"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.","feed_headline":"No SCF loop: two diagonalizations give molecular energies","feed_subtitle":"Starting from the electron-repulsion-free reference, one or two eigensolves recover ground and excited states.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Slater determinant algebra, the Møller–Plesset perturbation expansion, and the CI machinery whose adaptation to the a=0 reference is the paper's core construction.","marker":"[3]"},{"why":"Supplies the STO/GTO integral-evaluation and one-step a=0 eigensolver procedures used in the numerical tests.","marker":"[4]"},{"why":"Justifies the CI expansion by establishing that excited determinants (including LUMO occupations) are necessary for estimating the ground state.","marker":"[5]"},{"why":"Provides the benchmark CI energies for the 185 atomic ions used to fit the empirical moment parameters and to measure the method's deviation.","marker":"[6]"}],"fun_headline_variants":["Two eigensolves replace the entire HF-SCF cycle","Switch off repulsion, solve, switch on: no SCF needed","Method kills SCF: one-particle solve plus CI diagonalize","Skip SCF: solve a=0, then one CI matrix for all states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two eigensolves replace the entire HF-SCF cycle","Switch off repulsion, solve, switch on: no SCF needed","Method kills SCF: one-particle solve plus CI diagonalize","Skip SCF: solve a=0, then one CI matrix for all states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001018,"raw_usage":{"total_tokens":4331,"prompt_tokens":1014,"completion_tokens":3317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":3241}},"tokens_in":630,"tokens_out":3317,"duration_ms":22423,"temperature":1.0,"reasoning_tokens":3241,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:00:03.612578+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}