{"id":"f48f1892-2990-41b2-8473-eaf82652c3b9","arxiv_id":"1908.02882","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors solve a Jastrow-similarity-transformed electronic Hamiltonian with FCIQMC, achieving millihartree accuracy for first-row atoms in small basis sets.","lead":"A quantum chemistry method writes the electron wavefunction with an extra pair-distance factor, turning the Schrödinger equation into a non-Hermitian problem solved by stochastic sampling. It reaches near-exact energies for first-row atoms using small basis sets, potentially reducing the cost of accurate electronic structure calculations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1 mH accuracy claim depends on unverified sub-mH accuracy of the numerical quadrature for the K and L matrix elements; no grid-convergence data are reported.","rationale":"I agree with the reader's weakest-assumption analysis: the quadrature is the least documented numerical ingredient. The mathematical derivation of the similarity transformation is sound, and the reuse of SM17 Jastrow parameters for the cations is a fair transferability test. However, the published text and supplementary material assert convergence only qualitatively. The singularity in the Laplacian of u and the large K/L expectation values make a hidden few-mH quadrature error plausible. This does not warrant rejection; it is fixable by adding a convergence table. It does, however, justify the CONDITIONAL verdict. The reader's verdict should remain CONDITIONAL, so no change is needed.","tokens_in":9259,"tokens_out":9423,"duration_ms":108844,"concrete_test":"Re-run the ST-FCIQMC Ne cc-pVQZ calculation with at least two finer grids (e.g., double the Treutler-Ahrichs radial points and increase the Lebedev angular order from about 17 to about 29 or 35), and report the change in the total energy and in the individual <K> and <L> expectation values. If the energy shifts by more than 0.1 mH, the paper's mH-level accuracy claim is not supported by the reported numerical setup. A pass (sub-0.1 mH change) would clear the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical claim (MAE about 1 mH for total energies, 0.50 mH for IPs with SM17/cc-pVQZ) is only as solid as the matrix elements inserted into the non-Hermitian effective Hamiltonian. Those matrix elements are all computed on a direct-product Treutler-Ahrichs/Lebedev grid; the paper states convergence is rapid but gives no grid sizes and no convergence series. This matters because the integrands contain cusp-related terms: K^(2)_rs involves the Laplacian of u(r1,r2), which for the Boys-Handy/SM17 form has a 1/r12 singularity when the expansion contains terms linear in r12. The authors avoid direct evaluation by integration by parts (Supplementary Eq. S5), but the resulting integrand still contains the discontinuous gradient of u. For Ne the reported Hartree-Fock expectation values are <K> = -382 mH and <L> = +109 mH; a small relative quadrature error in either term easily exceeds the claimed 2 mH total error unless the errors cancel almost perfectly. The paper offers no numerical evidence for such cancellation. Since total energies are quoted to 0.01 mH, undocumented quadrature error is a load-bearing assumption, not a cosmetic issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript presents ST-FCIQMC, a method that writes the electronic wavefunction as Ψ = e^τ Φ with a Jastrow factor τ = Σ_{i<j} u(r_i,r_j) and solves for Φ as a full multi-determinant expansion in the non-Hermitian similarity-transformed Hamiltonian H̃ = e^{−τ}H e^{τ}. The commutator expansion truncates exactly at second order, producing additional two-body (K) and three-body (L) operators; matrix elements are evaluated by numerical quadrature on atom-centered Treutler–Ahlrichs/Lebedev grids, and the eigenproblem is solved projectively with the initiator FCIQMC implementation in NECI. Using the Boys–Handy Jastrow forms SM7 (7 parameters) and SM17 (17 parameters) from the Schmidt–Moskowitz VMC study, the authors report total energies and ionization potentials for the first-row atoms Li–Ne in cc-pVDZ/TZ/QZ bases, with a central claim of about 1 mH mean absolute error (MAE) for total energies and 0.50 mH for IPs using SM17/cc-pVQZ. Additional results for Ne^{q+} cations show total energies within a few mH of exact nonrelativistic values without a core-correlation basis set.","tokens_in":9514,"tokens_out":17231,"duration_ms":185298,"significance":"If the numerical accuracy withstands scrutiny, this is a significant methodological advance: it combines explicit correlation with a full multi-determinantal solution in a projective non-Hermitian framework, avoids strong-orthogonality projections, and demonstrates that flexible e-e-n Jastrow factors substantially compactify the configuration problem. The derivation of H̃ is exact, short, and easy to verify; the full retention of the three-body terms distinguishes it from earlier transcorrelated work; and the use of the same Jastrow parameters for the cations (Table II) provides a genuine transferability test. The Ne-cation series (Table III), showing core-correlation effects captured without tight core functions, is notable. The paper also states its own limitations honestly (the ~100-orbital three-body storage bottleneck; the missing p-wave cusp condition).","major_comments":[{"comment":"The paper's central numerical claims — MAE ≈ 1 mH for SM17/cc-pVQZ total energies (Table I) and MAE = 0.50 mH for IPs (Table II) — require that the numerical quadrature of the K and L matrix elements be accurate well below the mH scale, but no grid-size or convergence data are reported anywhere. The main text asserts only that 'convergence of the integrals with grid size is rapid.' This omission is load-bearing rather than cosmetic: the Boys–Handy Jastrow contains terms linear in r12, so ∇u is discontinuous at r1 = r2, and the K^(2) integrand has a 1/r12 singularity that the manuscript treats by integration by parts (a sensible choice), but the gradient discontinuities remain in K^(1) (Eq. S2) and in the three-body integrand (Eq. S8). The paper's own Ne numbers show the scale of the risk: ⟨K⟩ ≈ −382 mH and ⟨L⟩ ≈ +109 mH at the HF level, against a claimed net error of about 2 mH, so a relative quadrature error of only about 0.3% in either term already exceeds the claimed total accuracy. I request a convergence table (selected K and L matrix elements and the resulting total energy versus grid size) for at least Ne and Li, together with the grid specifications actually used.","section":"Supplementary, Eqs. (S5)–(S8); main text, p.3"},{"comment":"All FCIQMC energies are quoted without stochastic error bars, walker counts, or initiator-parameter information. FCIQMC is a stochastic projective method, and the initiator approximation introduces a systematic bias that depends on the threshold and walker population; the 1 mH and 0.5 mH MAE claims, and the small differences between basis sets in Tables I–II, are only meaningful if the combined statistical and initiator uncertainties are far below 1 mH. The statement that 'a very small degree of non-variationality (less than 1 mH)' is observed in a few cases also needs statistical support, since without error bars it is impossible to tell whether the energies reported below the exact values (Li and Be in Table I) are below exact within noise or systematically. Please report error bars (or otherwise justify their absence) and provide the walker numbers and initiator threshold used for the Table I–III results.","section":"Tables I–III; p.4"}],"minor_comments":[{"comment":"There are several typos: 'Gutzwiller similatrity transformations' (p.2), 'N^2_grid N^2_bas steos' (Supplementary, after Eq. (S5)), and 'Work is underway to to alleviate' (final paragraph of the main text).","section":"p.2; Supplementary after Eq. (S5); final paragraph"},{"comment":"The phrase 'the formal unitary invariance of our Φ' is confusing: the similarity transformation e^{−τ}H e^{τ} is not unitary, and the property that removes the need for orbital optimization is the invariance of a full-CI expansion under orbital rotations; please reword.","section":"p.2, after Eq. (2)"},{"comment":"The typeset equations for K̂ and L̂ contain unbalanced parentheses and do not clearly separate multiplicative terms from differential-operator terms acting on the orbital products; please reformat the equations so the action of the gradient operators is unambiguous.","section":"Eq. (2) and following definitions of K̂ and L̂"},{"comment":"Please state explicitly whether the CCSD(T) and CCSD(T)-F12 calculations are frozen-core. As written, the comparison in Table I mixes the all-electron ST-FCIQMC treatment with what are presumably valence-only coupled-cluster calculations, which inflates the apparent MAE difference; the paper's own Table III discussion acknowledges that CCSD(T)-F12 needs core-valence basis sets.","section":"Table I; main text p.3"},{"comment":"The reference row in Table I is labeled 'Expt' even though the main text says these are experimental values corrected for relativistic effects; please relabel the row (e.g., 'Estimated exact nonrelativistic') and identify explicitly which atoms are non-variational at cc-pVQZ — from Table I these appear to be Li and Be, by about 0.4 and 0.1 mH, respectively.","section":"Table I caption; p.4 text"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for a general quantum-chemistry journal, and the two requested additions (a quadrature convergence series and FCIQMC statistical reporting) are straightforward within the scope of a revision. One point for the editor: the claimed superiority over CCSD(T)-F12 in Table I should be read with the caveat that the CC references are presumably frozen-core while ST-FCIQMC is all-electron; the authors should be asked to make this explicit. Overlap with prior work (Boys–Handy, Ten-no, Luo–Alavi, Dobrautz–Luo–Alavi) is properly cited, and the combination of a flexible e-e-n Jastrow with a full multi-determinant projective solution appears genuinely new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a read. The paper does what the transcorrelated community has been circling for years: it keeps all three-body terms in the similarity-transformed Hamiltonian and solves the resulting non-Hermitian problem with FCIQMC, using a flexible Boys-Handy Jastrow. That is a real step beyond the single-determinant transcorrelated work of Boys-Handy and Ten-no, and it builds directly on the authors' earlier Hubbard-model study.\n\nThe derivation is clean and the commutator truncation is properly justified: only the kinetic energy fails to commute with tau, so the double commutator is the last term. The numerical results are impressive. With the SM17 Jastrow and cc-pVQZ, total energies for first-row atoms land within about 1 mH of exact nonrelativistic values, and ionization potentials within 0.5 mH—beating CCSD(T)-F12 at the same basis. Re-using the same Jastrow for cations without re-optimization is a genuine transferability test, and the core-electron results in Table III are striking.\n\nThat said, the paper under-reports numerics in two spots. No stochastic error bars are given for the FCIQMC energies. Even if the noise is below 0.01 mH, the reader should not have to infer it. More importantly, the quadrature is asserted to converge rapidly, but no grid sizes or convergence series are reported. The K and L matrix elements involve singular or discontinuous integrands; for Ne the Hartree-Fock expectation values are <K> = -382 mH and <L> = +109 mH, and the final energy comes from a delicate cancellation to within 2 mH. A small relative quadrature error in either term can shift the total by more than the claimed precision. This is a load-bearing assumption, though likely fixable. The Jastrow parameters also come from a prior VMC fit on the same atoms; the IP results mitigate this, but the total energies are not fully ab initio.\n\nOverall: the method is sound and the results are strong enough to deserve publication, but the numerical evidence needs completion before the 1 mH claim is bulletproof. I would send it to a serious referee and ask for grid-convergence data, stochastic error bars, and an explicit quadrature-accuracy statement. That is a revision, not a rejection.","headline":"Solid extension of transcorrelated methods to full three-body terms in FCIQMC; the 1 mH claim needs quadrature-convergence evidence before I'd trust it fully.","tokens_in":10048,"tokens_out":2059,"would_cite":true,"duration_ms":22564,"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":"A Jastrow-factorised similarity transformation of the electronic Schrödinger equation, solved with stochastic full configuration interaction, reproduces first-row atomic total energies to about 1 mH and ionization potentials to 0.50 mH in…","keywords":["Jastrow factor","similarity transformation","full configuration interaction quantum Monte Carlo","transcorrelated Hamiltonian","three-body integrals","non-Hermitian Hamiltonian","explicitly correlated methods","first-row atoms"],"falsifier":"Recalculate the ST-FCIQMC energies for, say, Ne or F with the quadrature grid doubled (or with the $K$ and $L$ integrals evaluated by an independent high-accuracy method) and compare; a shift in total energy larger than about 0.2 mH would show the reported accuracy is not yet established. Alternatively, apply the same procedure to a two-electron system where the matrix elements can be computed essentially exactly.","tokens_in":9081,"feed_emoji":"⚛️","tokens_out":9817,"duration_ms":99300,"temperature":0.7,"pith_summary":"By factorising the electronic wavefunction as $\\Psi = e^{\\tau}\\Phi$, with $\\tau = \\sum_{i<j}u(\\mathbf{r}_i,\\mathbf{r}_j)$ a symmetric pair correlation function, the paper derives a similarity-transformed Hamiltonian $\\tilde H = e^{-\\tau}\\hat H e^{\\tau}$ whose ground-state eigenfunction is the remaining many-body function $\\Phi$. The transformed Hamiltonian is non-Hermitian and contains two- and three-body terms; the paper evaluates those integrals numerically and solves the eigenvalue problem projectively with full configuration interaction quantum Monte Carlo (FCIQMC). The central quantitative claim is that for the first-row atoms Li–Ne, this ST-FCIQMC approach with the 17-term SM17 Jastrow factor reaches total energies within about 1 mH of the exact nonrelativistic limits using only a cc-pVQZ basis, and ionization potentials with a mean absolute error of 0.50 mH. The importance is that explicit electron correlation can be combined with multi-determinantal wavefunctions without projection operators, so flexible Jastrow functions are effective and modest basis sets can reach the basis-set limit.","feed_headline":"Atomic energies within 1 mH from a Jastrow-transformed Hamiltonian","feed_subtitle":"Folding electron correlation into a Jastrow factor lets small basis sets match exact nonrelativistic atomic energies.","key_machinery":"The central machinery is the similarity-transformed Hamiltonian $\\tilde H = e^{-\\tau}\\hat H e^{\\tau}$ arising from the Jastrow factorisation $\\Psi = e^{\\tau}\\Phi$, where the commutator expansion truncates exactly at second order because only the kinetic-energy operator fails to commute with $\\tau$. This produces a non-Hermitian two-body operator $\\hat K$ and a Hermitian three-body operator $\\hat L$, whose matrix elements $K^{pq}_{rs}$ and $L^{pqr}_{stu}$ are evaluated by numerical quadrature on atom-centred radial–angular grids directly in the molecular-orbital basis. The non-Hermitian eigenvalue problem is then solved projectively by FCIQMC, which permits the full multi-determinantal form of $\\Phi$ without orbital optimisation. This chain is what allows the flexible SM17 Jastrow factor to absorb much of the correlation, compactifying the configurational expansion and making small basis sets sufficient.","core_discovery":"The paper's central claim is that the full similarity-transformed Hamiltonian obtained from the Jastrow factorisation $\\Psi = e^{\\tau}\\Phi$ can be treated without approximation in FCIQMC: $\\tilde H = e^{-\\tau}\\hat H e^{\\tau}$ is non-Hermitian and contains additional two-body ($\\hat K$) and three-body ($\\hat L$) terms, and retaining all three-body terms is both feasible and essential. Using the 17-term SM17 Jastrow factor with parameters from prior variance-minimised variational Monte Carlo optimisation, the method reproduces total energies of the first-row atoms with a mean absolute error near 1 mH in a cc-pVQZ basis, and ionization potentials with a mean absolute error of 0.50 mH. The flexible SM17 factor, which includes electron–electron–nucleus terms, markedly outperforms the electron–electron-only SM7 form. Because $\\Phi$ is a full multi-determinantal expansion solved in the presence of the Jastrow-induced potentials, the redundancy between Jastrow and configurational degrees of freedom is handled without the strong-orthogonality projectors used in F12 methods, and core-electron correlation is captured by the Jastrow factor itself, so tight core-correlating basis functions are not needed.","pith_inferences":["A natural test of the method's generality beyond atoms would be to apply the same rigid Jastrow parameters to molecules; the paper's framework is molecular but only atomic results are reported.","If the three-body integral storage bottleneck, which currently limits calculations to about 100 orbitals, is removed by the tensor-decomposition route the paper suggests, the method could address strongly correlated open-shell molecules where a multi-determinantal $\\Phi$ should have a systematic advantage.","The paper attributes the small over-correlation of same-spin pairs to the missing p-wave cusp; a spin-dependent Jastrow factor that satisfies this condition could plausibly push errors below 1 mH, but that is an extension beyond what is demonstrated here."],"forward_implications":["For first-row atoms, the reported mean absolute errors of about 1 mH in total energies and 0.50 mH in ionization potentials at cc-pVQZ mean the method is close to the exact nonrelativistic basis-set limit in small basis sets.","At the same basis set, CCSD(T)-F12 shows larger errors (20 mH for total energies at cc-pVQZ and 0.96 mH for ionization potentials), so the similarity-transformed Jastrow approach is a competitive explicitly correlated alternative.","Because core correlation is captured by the Jastrow factor, the method reproduces total energies of Ne through Ne$^{7+}$ within a few mH without a core-correlating basis, suggesting core properties may be accessible with valence-only basis sets.","The large and opposing contributions of the two- and three-body terms imply that retaining the full three-body terms, rather than approximating them away, is essential for the observed accuracy.","The initiator and projective FCIQMC treatment keeps the non-variationality below about 1 mH in the studied cases, so the non-Hermitian character of $\\tilde H$ does not prevent reliable ground-state energies."],"supporting_citations":[{"why":"Introduces the full configuration interaction quantum Monte Carlo method used as the projective eigensolver.","marker":"[13]"},{"why":"Provides the initiator approximation that makes large CI spaces tractable in FCIQMC.","marker":"[14]"},{"why":"Adapts FCIQMC to non-Hermitian similarity-transformed Hamiltonians, a prerequisite for this work.","marker":"[15]"},{"why":"Shows that three-body terms can be retained in FCIQMC and that full treatment yields essentially exact results in a model system.","marker":"[16]"},{"why":"Introduces the transcorrelated similarity-transformed Hamiltonian and the polynomial Jastrow form used here.","marker":"[17]"},{"why":"Supplies the atom-centred radial–angular quadrature grids used to evaluate the two- and three-body integrals.","marker":"[23]"},{"why":"Gives the SM7 and SM17 Jastrow parameters from variance-minimised variational Monte Carlo.","marker":"[26]"},{"why":"Provides the relativistic-corrected experimental reference energies used to define the reported errors.","marker":"[28]"},{"why":"Gives the CCSD(T)-F12 energies and ionization potentials used as comparison baselines.","marker":"[30]"}],"fun_headline_variants":["Jastrow similarity transform hits atomic energies to 1 mH","Small basis sets match exact atomic energies via Jastrow approach","Three-body terms enable mH accuracy from modest basis sets","Non-Hermitian Jastrow Hamiltonian yields mH precision","Jastrow factorization: atomic energies within 1 mH without large basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The millihartree-level accuracy claim depends on the numerical quadrature used for the two- and three-body integrals being accurate far below the mH scale, yet the paper reports no grid-size or quadrature-error convergence tests.","fun_headline_variants_meta":{"raw":{"variants":["Jastrow similarity transform hits atomic energies to 1 mH","Small basis sets match exact atomic energies via Jastrow approach","Three-body terms enable mH accuracy from modest basis sets","Non-Hermitian Jastrow Hamiltonian yields mH precision","Jastrow factorization: atomic energies within 1 mH without large basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001141,"raw_usage":{"total_tokens":4724,"prompt_tokens":921,"completion_tokens":3803,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":3714}},"tokens_in":537,"tokens_out":3803,"duration_ms":28381,"temperature":1.0,"reasoning_tokens":3714,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:30:57.883716+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recalculate the ST-FCIQMC energies for, say, Ne or F with the quadrature grid doubled (or with the $K$ and $L$ integrals evaluated by an independent high-accuracy method) and compare; a shift in total energy larger than about 0.2 mH would show the reported accuracy is not yet established. Alternatively, apply the same procedure to a two-electron system where the matrix elements can be computed essentially exactly.","supporting_citations":[{"cited_title":"Luo and A","cited_arxiv_id":null,"evidence_quote":"Adapts FCIQMC to non-Hermitian similarity-transformed Hamiltonians, a prerequisite for this work."},{"cited_title":"Dobrautz, H","cited_arxiv_id":null,"evidence_quote":"Shows that three-body terms can be retained in FCIQMC and that full treatment yields essentially exact results in a model system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the transcorrelated similarity-transformed Hamiltonian and the polynomial Jastrow form used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the atom-centred radial–angular quadrature grids used to evaluate the two- and three-body integrals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the SM7 and SM17 Jastrow parameters from variance-minimised variational Monte Carlo."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the relativistic-corrected experimental reference energies used to define the reported errors."},{"cited_title":"H¨ attig, D","cited_arxiv_id":null,"evidence_quote":"Gives the CCSD(T)-F12 energies and ionization potentials used as comparison baselines."}],"review_version":1}