{"id":"cb116083-d709-4429-9c10-702f206224bd","arxiv_id":"2506.04895","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A deterministic finite-basis method for optimizing Jastrow factors is derived and validated, yielding noise-free Jastrows with energies close to VMC energy-optimized results.","lead":"This paper introduces a deterministic, noise-free way to optimize Jastrow factors in electronic structure calculations, by minimizing the variance of the transcorrelated reference energy in a finite basis set. It derives analytic gradients for the required integrals and shows the method yields energies close to variational Monte Carlo energy minimization while avoiding stochastic noise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Cost function Eq. (26) is a finite-basis/xTC proxy calibrated only on Li; its fidelity for B–Ne, molecules, and especially H-containing systems is unestablished.","rationale":"The paper's mathematical core—analytic derivatives and L-BFGS minimization—is credible and implemented sufficiently for the benchmarks. The independent support is the Li calibration and the internal consistency of the VMC-versus-deterministic trends. However, the abstract's strongest claim is specifically about the quality of the resulting Slater-Jastrow wavefunctions as measured by VMC energy and variance, and that quality is only as good as the cost function's fidelity to the true variance. The cost function is not the VMC variance; it is sigma2_ref computed in a finite basis from the xTC-Hamiltonian. The paper demonstrates the finite-basis proxy can be made to work for one parameter in Li, but does not provide the analogous check for the actual many-parameter optimizations that go into Table IV, nor for hydrides where it knowingly violates its own core-valence requirement. The xTC approximation adds a second layer of unbenchmarked sensitivity. These are empirical gaps rather than demonstrated errors; therefore a conditional verdict with a request for this calibration is appropriate, not rejection. The reader's weakest-assumption statement captures the same core issue, so no change to the existing conditional verdict is needed.","tokens_in":17140,"tokens_out":13463,"duration_ms":163983,"concrete_test":"For H2O and HF, run deterministic Jastrow optimization twice: once with the paper's mixed basis (cc-pCVTZ on O/F, cc-pVTZ on H) and once with cc-pCVQZ on O/F and cc-pVQZ on H, then evaluate both optimized Jastrows by VMC in cc-pVTZ and compare EVMC and sigma2_VMC with VMC variance-optimized Jastrows. If the optimal parameters or the sigma2_VMC ranking change materially, the H-basis compromise and the cc-pCVTZ proxy are not sufficient for the central claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central VMC claim requires Eq. (26)—the finite-basis sum of squared off-diagonal xTC-Hamiltonian elements in cc-pCVTZ—to be a faithful proxy for the variance that VMC would minimize. Fig. 1 shows this proxy is extremely basis-sensitive: the variance ranges over three orders of magnitude between basis sets, valence-only cc-pVTZ underestimates it and shifts the minimum, and only cc-pCVTZ matches VMC; but this calibration is performed for Li only, with one optimizable BH parameter. No such comparison is reported for B–Ne, for molecules, or for the hydrides in Tables II–III. In Sec. IV D the authors state that no core-valence basis is available for H and use cc-pVTZ on H, asserting without demonstration that this \"does not introduce significant error\"—yet Fig. 1 identifies valence-only basis sets as precisely the regime where the deterministic variance fails. A second unvalidated layer is that Eq. (26) is evaluated with the xTC Hamiltonian, from which genuine three-body terms have been removed; variance is more sensitive than the energies for which xTC has been benchmarked. If either proxy fails outside Li, the optimized Jastrow parameters need not minimize the true variance, and the Table IV ordering underpinning the abstract's claim is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a deterministic alternative to stochastic VMC optimization of Jastrow factors, based on minimizing the variance of the transcorrelated (TC) reference energy computed in a finite one-particle basis. The authors derive analytic gradients of the TC Hamiltonian matrix elements with respect to Jastrow parameters, implement them in the TCHInt library, and benchmark the scheme on first-row atoms, diatomic molecules, and hydrides. They show that deterministic optimization, often seeded by a VMC guess, produces Jastrow parameters with lower variance in subsequent VMC/xTC-CCSD(T) energies, and that the resulting Slater–Jastrow wavefunctions have lower energies than variance-minimized VMC while retaining low variances. The paper also demonstrates transferability of Jastrow factors optimized in cc-pCVTZ to smaller basis sets for post-Hartree–Fock treatments.","tokens_in":17439,"tokens_out":6872,"duration_ms":85775,"significance":"If the central approximation holds, this is a valuable methodological contribution. The analytic gradient formalism is clearly presented and implements a genuinely deterministic, reproducible Jastrow optimization, removing stochastic noise that is particularly problematic for weak interactions. The paper is also commendable for explicitly investigating basis-set sensitivity (Fig. 1) and for showing transferability of the optimized Jastrow factors to cheaper post-HF basis sets. The potential impact is broad: it would benefit both transcorrelated methods and standard VMC as a new optimization route. The main risk is that the cost function, Eq. (26) evaluated in cc-pCVTZ with the xTC Hamiltonian, is verified as a faithful proxy for the true variance only for a single system (Li), leaving the B–Ne and molecular results, especially hydrides, without direct validation of the underlying approximation.","major_comments":[{"comment":"The central approximation of the paper is that the finite-basis sum in Eq. (26), evaluated in cc-pCVTZ with the xTC Hamiltonian, faithfully represents the variance that VMC would minimize. Fig. 1 calibrates this only for Li, where the variance spans three orders of magnitude across basis sets and the valence-only cc-pVTZ minimum is shifted. Section IV D then uses cc-pVTZ on H for HF and H2O, asserting without demonstration that this does not introduce significant error. Since Fig. 1 identifies exactly the valence-only regime as the failure regime, this assertion needs direct support; otherwise the hydride results do not validate the method. Please provide a similar calibration for at least one additional atom (e.g., Ne or C) and for a hydrogen-containing system, or explicitly show why the H case is different.","section":"Sec. IV A and IV D; Eq. (26)"},{"comment":"Eq. (26) is evaluated with the xTC approximation, which removes genuine three-body interaction terms from the TC Hamiltonian and folds them into two-body terms (Sec. II C). The variance is a more sensitive functional of the Hamiltonian than the energy, and the xTC approximation has been benchmarked only for energies (Ref. 54). No test in the paper compares the deterministic variance obtained with xTC against the corresponding quantity with the full TC Hamiltonian. Without such a test, the optimized parameters are not shown to minimize the quantity that VMC actually uses. Please add a comparison for a small system, e.g., Li or Be, using the full three-body integrals.","section":"Sec. II C and IV; Eq. (26)"},{"comment":"The abstract and conclusions state that deterministically optimized wavefunctions have energies 'comparable' to those from energy-minimisation VMC. In Table IV, deterministic energies are higher than the energy-minimized values by up to 0.0244 Eh (Ne) and 0.0154 Eh (F), while the variance reductions are relatively small (e.g., Ne: 8.0831 to 7.8219 Eh^2). This claim needs a quantitative criterion (e.g., within a given number of mEh) or should be softened to 'lower than variance-minimized VMC and approaching energy-minimized VMC'.","section":"Abstract and Sec. V; Table IV"}],"minor_comments":[{"comment":"In the second equality of Eq. (33), the indices appear to be swapped: φ_Q(r2) and φ_S(r1) should read φ_Q(r1) and φ_S(r2). The subsequent equations are consistent, so this is a typographical error in a central derivation, but it should be corrected.","section":"Eq. (33)"},{"comment":"In Eq. (35), the factor of 2 is missing in the final equality; ∂(∇_1 u)^2/∂f_l should equal 2∇_1 u · ∇_1 u_l, not ∇_1 u · ∇_1 u_l.","section":"Eq. (35)"},{"comment":"The caption states that the cc-pCVTZ VMC curves are only reported for the left plots, yet the text claims agreement with VMC for both the electron-electron and electron-nucleus cases. Please clarify whether the electron-nucleus case was compared to VMC or provide the missing VMC curve.","section":"Fig. 1 caption"},{"comment":"The statement that no core-valence basis sets are available for the hydrogen atom is not universally accurate; for example, some quantum chemistry packages provide cc-pCVTZ for H, albeit possibly identical to cc-pVTZ. Please clarify the situation or rephrase the justification for using cc-pVTZ on hydrogen.","section":"Sec. IV D"},{"comment":"In Table II, the H row has an empty entry under 'VMC(CVTZ)'. Please indicate why this entry is missing (e.g., no cc-pCVTZ basis used for the isolated H atom) or include the value.","section":"Table II"}],"recommendation":"major_revision","confidential_remarks":"The paper presents a genuinely novel and useful optimization scheme with a clean analytic gradient derivation and reproducible implementation. The main risk is the unvalidated fidelity of the finite-basis/xTC variance proxy outside Li; this is fixable with additional calibration studies and should be addressed before publication. The abstract slightly overstates the energy comparison with energy-minimized VMC; Table IV shows differences of tens of mEh for heavier atoms. I recommend major revision but am optimistic about the outcome if the proxy validation is added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a genuinely new and useful way to remove stochastic noise from Jastrow optimization. The authors minimize a finite-basis proxy for the variance of the transcorrelated reference energy, with analytic gradients, and show it produces Jastrow parameters that yield VMC energies close to energy minimization and variances close to variance minimization. That is a nice trade-off and a step forward for reproducibility in transcorrelated and VMC workflows.\n\nThe gradient derivation is the core of the paper and it is careful. They work through the derivative integrals for both K and L terms and the xTC intermediates, and the implementation is benchmarked against VMC. The transferability result—optimize in a core-valence basis, then use a smaller basis for post-HF—is practically useful and backed by the IP data.\n\nThe soft spot is the proxy itself. Fig. 1 shows the finite-basis variance is hyper-sensitive to whether the basis has core-valence functions; cc-pVTZ underestimates it and shifts the minimum. That calibration was done for Li only, with one parameter. For the heavier atoms and molecules, there is no direct check that the deterministic variance curve tracks the true VMC variance. And for hydrides they use cc-pVTZ on H, which is exactly the regime where their own Fig. 1 says the proxy fails. They assert it doesn't introduce significant error, but don't demonstrate it. The xTC approximation adds another layer of approximation to the cost function that is not separately validated. These are real limitations, but not fatal: the final VMC energies and variances in Table IV are direct measurements of the actual wavefunction, and they confirm the method works for Li-Ne. The proxy could be wrong for H without invalidating the central claim for atoms.\n\nThe bigger practical worry is that the integral code is \"to be released; available from the authors upon reasonable request.\" That should be resolved before publication. I'd also ask for a few more proxy-vs-VMC calibration curves, at least for one heavier atom and one molecule, to show the Li calibration is not a one-off.\n\nThis deserves a serious referee. I'd send it out, with a request for those calibrations along with the code release. The paper is a solid, honest methodological contribution that the TC/VMC community will use.","headline":"A genuinely new deterministic Jastrow optimization with a solid derivation and benchmarks; the main caveat is a basis-set calibration that is thinner than the authors admit.","tokens_in":17947,"tokens_out":3138,"would_cite":true,"duration_ms":35940,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A deterministic alternative to VMC Jastrow optimisation, minimising the finite-basis variance of the transcorrelated reference energy, yields reproducible Slater-Jastrow wavefunctions with low variance and VMC energies close to…","keywords":["Jastrow factors","transcorrelated methods","variational Monte Carlo","variance optimisation","analytical gradients","deterministic optimisation","core-valence basis sets","Slater-Jastrow wavefunctions"],"falsifier":"Compute the deterministic variance curve of Eq. (26) for the water molecule in cc-pVDZ, cc-pVTZ, cc-pCVTZ, and cc-pCVQZ, and compare the minimising Jastrow parameters with those from VMC variance optimisation; if the minimum moves significantly between cc-pCVTZ and cc-pCVQZ, or disagrees with the stochastic optimum, the finite-basis approximation is not faithful.","tokens_in":16956,"feed_emoji":"⚛️","tokens_out":11211,"duration_ms":115866,"temperature":0.7,"pith_summary":"This paper proposes a way to optimise Jastrow factors without the stochastic noise of variational Monte Carlo: minimise, in a finite one-particle basis, the variance of the transcorrelated reference energy, i.e. the energy spread of the Jastrow-transformed Hamiltonian on the Hartree-Fock determinant. The target is a sum of squared off-diagonal matrix elements of the transcorrelated Hamiltonian, so it can be evaluated from TC integrals, and the paper derives analytic derivatives that make gradient-based minimisation possible. On first-row atoms and small molecules, the resulting Slater-Jastrow wavefunctions have variances almost as low as VMC variance-optimised ones and energies closer to those from VMC energy optimisation, while being reproducible from run to run. If this holds, Jastrow factors for transcorrelated or plain VMC calculations can be fixed deterministically, removing a noise source that can hide weak intermolecular interactions.","feed_headline":"Jastrow factors get a noise-free optimisation route","feed_subtitle":"A finite-basis transcorrelated variance fixes Jastrow parameters deterministically with better energy-variance trade-off.","key_machinery":"The load-bearing object is the finite-basis variance of the TC reference energy, Eq. (26): it converts the stochastic quantity $\\langle\\Phi_0|(\\hat{H}_{\\mathrm{TC}}-E_{\\mathrm{ref}})^2|\\Phi_0\\rangle$ into a sum of squared Hamiltonian matrix elements to all excited determinants in the one-particle basis, computable from TC integrals. The method is completed by analytic derivatives of the TC Hamiltonian matrix elements, built from derivatives of the four basic operators $K_1=\\nabla^2 u$, $K_2=(\\nabla u)^2$, $K_3=\\nabla u\\cdot\\nabla$, and $L_1=(\\nabla_1 u)\\cdot(\\nabla_1 u')$; in the xTC approximation the two-body correction derivatives follow the same intermediate chain as the integrals themselves. The derivatives only require gradients and Laplacians of the individual Jastrow basis functions, so L-BFGS can minimise the variance.","core_discovery":"The central claim is that the parameters of a Jastrow factor can be determined by minimising $\\sigma^2_{\\mathrm{ref}} = \\sum_{I\\neq 0} \\langle\\Phi_I|\\hat{H}_{\\mathrm{TC}}|\\Phi_0\\rangle^2$ in a finite basis, where $\\Phi_0$ is the Hartree-Fock determinant and the sum runs over all other determinants reachable in that basis. With core-valence basis sets such as cc-pCVTZ this finite-basis variance reproduces the location and magnitude of the true variance minimum, so the same quality of Jastrow is obtained without Monte Carlo sampling. Used alone or as a refinement of VMC guesses, the deterministic Jastrows give transcorrelated and variational energies of the same accuracy as VMC-optimised ones, with standard deviations across independent runs reduced by more than an order of magnitude in the reference energy and lower variational energies than VMC variance minimisation.","pith_inferences":["Because Eq. (25) is written for a general reference $|\\Phi\\rangle$, the same deterministic target could in principle be applied to multi-configurational or orbital-optimised references; the paper does not implement this, but nothing in the derivation forces a single-determinant reference.","Removing stochastic noise makes the Jastrow-optimisation pipeline differentiable, which could enable direct gradients of the variance with respect to nuclear coordinates or basis-set parameters, opening a route to geometry optimisation within the transcorrelated framework that the paper does not pursue.","The observed sensitivity to core-valence completeness suggests immediate tests with effective-core-potential or frozen-core transcorrelated computations: if the core-like virtual orbitals can be omitted without shifting the minimum, the cost of deterministic optimisation drops substantially."],"forward_implications":["A Jastrow factor can be produced from scratch with no stochastic sampling, and repeated optimisations converge to essentially the same parameters; the paper demonstrates this for the oxygen atom starting from zero and from a VMC guess.","Reference-energy standard deviations across independent runs drop by more than an order of magnitude when deterministic optimisation refines VMC Jastrows, making small energy differences such as ionisation potentials easier to resolve.","Post-Hartree-Fock correlation can be carried out in a smaller basis than the one used for Jastrow optimisation, because deterministically optimised Jastrows transfer without losing accuracy, reducing the cost of xTC-CCSD(T) calculations.","In plain VMC, the deterministic protocol gives energies between variance minimisation and energy minimisation while keeping variances close to the variance-minimised ones, a more favourable energy-variance trade-off."],"supporting_citations":[{"why":"Defines the variance of the TC reference energy and establishes it as the preferred VMC optimisation target that this work makes deterministic.","marker":"[53]"},{"why":"Supplies the xTC approximation and its intermediate-chain formulas, which the analytic derivatives extend to Jastrow parameters.","marker":"[54]"},{"why":"Introduces the transcorrelated Hamiltonian formalism and one of the Jastrow parametrisations used in the paper.","marker":"[31]"},{"why":"Supplies the flexible Jastrow parametrisation used in all benchmark optimisations.","marker":"[55]"},{"why":"Provides the variational Monte Carlo engine used to compare stochastic and deterministic optimisation.","marker":"[9]"},{"why":"Supplies the cc-pVTZ basis used for post-Hartree-Fock calculations and for hydrogen in hydrides.","marker":"[70]"},{"why":"Supplies the cc-pCVTZ core-valence basis set whose completeness is required for the finite-basis variance to be faithful.","marker":"[71]"}],"fun_headline_variants":["Jastrow fitting goes deterministic via TC variance","Noise-free Jastrow optimization using TC reference variance","Deterministic Jastrow parameters beat VMC energy-variance","TC variance makes Jastrow optimization reproducible","Optimize Jastrow factors without Monte Carlo noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The truncated finite-basis sum in Eq. (26), evaluated in cc-pCVTZ, has its minimum at essentially the same Jastrow parameters as the true variance; if the basis is too small, the computed minimum shifts and the optimised Jastrow is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Jastrow fitting goes deterministic via TC variance","Noise-free Jastrow optimization using TC reference variance","Deterministic Jastrow parameters beat VMC energy-variance","TC variance makes Jastrow optimization reproducible","Optimize Jastrow factors without Monte Carlo noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00016,"raw_usage":{"total_tokens":1255,"prompt_tokens":993,"completion_tokens":262,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":185}},"tokens_in":609,"tokens_out":262,"duration_ms":3435,"temperature":1.0,"reasoning_tokens":185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:30:44.419944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the deterministic variance curve of Eq. (26) for the water molecule in cc-pVDZ, cc-pVTZ, cc-pCVTZ, and cc-pCVQZ, and compare the minimising Jastrow parameters with those from VMC variance optimisation; if the minimum moves significantly between cc-pCVTZ and cc-pCVQZ, or disagrees with the stochastic optimum, the finite-basis approximation is not faithful.","supporting_citations":[],"review_version":1}