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Deterministic Optimisation of Jastrow Factors

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.04895 v1 pith:I6AFBU6S submitted 2025-06-05 physics.chem-ph

Maria-Andreea Filip , Evelin M.C. Christlmaier , J. Philip Haupt , Daniel Kats , Pablo López Ríos , Ali Alavi This is my paper
classification physics.chem-ph
keywords JastrowfactorstranscorrelatedmethodsvariationalMonteCarlovarianceoptimisationanalyticalgradientsdeterministiccore-valencebasissetsSlater-Jastrowwavefunctions
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [Sec. IV A and IV D; Eq. (26)] 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.
  2. [Sec. II C and IV; Eq. (26)] 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.
  3. [Abstract and Sec. V; Table IV] 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'.
minor comments (5)
  1. [Eq. (33)] 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.
  2. [Eq. (35)] 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.
  3. [Fig. 1 caption] 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.
  4. [Sec. IV D] 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.
  5. [Table II] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the deterministic Jastrow optimization minimizes a well-defined variance cost function, and the reported energies are assessed against independent external benchmarks, not against the fitted inputs.

full rationale

The claimed derivation chain is self-contained and not circular. The cost function is the finite-basis variance of the transcorrelated reference energy, Eq. (26), which is a direct algebraic quantity built from the TC Hamiltonian matrix elements; its analytic gradients are derived in Eqs. (27)-(49) directly from the Hamiltonian integrals. Minimizing this variance is a variational parameter optimization, not a fit to the energies, ionization potentials, or atomization energies that are later reported. The final VMC and xTC-CCSD(T) energies are compared with HEAT reference values and with VMC-optimized Jastrows, which are external benchmarks rather than outputs of the optimization. The calibration of the cc-pCVTZ basis on the Li atom is a methodological basis-set choice informed by one external VMC comparison, not a fitted parameter that is later relabeled as a prediction; the B-Ne atomic and molecular results are genuine out-of-sample predictions. Self-citations to Refs. 53 and 54 supply the variance-of-reference-energy target and the xTC approximation of three-body terms, but both are prior published approximations whose validity is not the paper's central claim, and the deterministic optimization algorithm with analytic gradients is new work that does not reduce to those citations. The paper's own caveat that hydrogen lacks core-valence basis functions and that valence-only basis sets underestimate the variance (Fig. 1) is a robustness limitation, not a circular reduction: it concerns the fidelity of the proxy, not whether the method fits its own output. No equation in the paper is equivalent to its input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method relies on the xTC approximation for Hamiltonian integrals, a finite-basis variance proxy, and core-valence basis sets; these are the main assumptions on which the deterministic optimization rests. No new physical entities are introduced.

free parameters (5)
  • Basis set for deterministic optimisation = cc-pCVTZ
    Chosen because cc-pVTZ underestimates the variance and shifts the minimum (Sec. IV A, Fig. 1); cc-pCVTZ gives variance minima matching VMC. This choice is load-bearing for the method.
  • Quadrature grid levels = 1 for optimisation, 2 for final energies
    Grid level 1 reduces integral cost during optimization; grid level 2 is used for final energy estimates (Sec. IV). The exact grids are those used in PySCF.
  • Variance convergence threshold = 1e-6 Hartree^2
    L-BFGS optimization is run until the variance converges to this value; looser thresholds would give noisier parameters.
  • VMC target standard deviation = 0.1 mHartree
    When VMC is used to generate an initial guess, the number of configurations targets a 0.1 mHartree standard deviation in Eref (Sec. IV C).
  • Jastrow polynomial orders = Nu=Nchi=4, Nf=2
    The DTN Jastrow parametrisation for all benchmarks; fewer terms than the one used in Ref. 54, noted in Sec. IV D.
assumptions (5)
  • standard math The BCH expansion of e^{-tau} H e^{tau} terminates at the second nested commutator for a two-electron Jastrow tau.
    Used in Eq. (3). Relies on tau being a sum of pairwise functions; for three-body Jastrows this would not hold.
  • domain assumption The xTC approximation, which neglects fully three-body L integrals with all six indices distinct, has little effect on energies and variances.
    Inherited from Ref. 54 and used in the cost function and energy evaluations (Eq. 21). Systematic error is not quantified in this paper.
  • ad hoc to paper The finite-basis sum in Eq. (26) is a good approximation to the true variance of the TC reference energy when cc-pCVTZ is used.
    The paper shows cc-pVTZ fails, and chooses cc-pCVTZ based on Li benchmarks; no general proof that this holds for all systems, especially those without core-like basis functions.
  • domain assumption The Hartree-Fock determinant |Phi0> is an adequate reference for variance minimization.
    All optimizations use a single-determinant HF reference. For multireference systems this may be insufficient (the authors note future work on multireference TC in Ref. 57).
  • domain assumption The density matrix of the reference and the normal-ordering corrections are computed with the same reference used in post-HF.
    Standard in TC methods.

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Pith. "Pith review of Deterministic Optimisation of Jastrow Factors." pith.science (2026). https://pith.science/paper/I6AFBU6S

@misc{pith2026250604895,
  author       = {Pith},
  title        = {Pith review of: Deterministic Optimisation of Jastrow Factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6AFBU6S}},
  note         = {Machine review of arXiv:2506.04895}
}
read the original abstract

Highly flexible Jastrow factors have found significant use in stochastic electronic structure methods such as variational Monte Carlo (VMC) and diffusion Monte Carlo, as well as in quantum chemical transcorrelated (TC) approaches, which have recently seen great success in generating highly accurate electronic energies using moderately sized basis sets. In particular for the latter, the intrinsic noise in the Jastrow factor due to its optimisation by VMC can pose a problem, especially when targeting weak (non-covalent) interactions. In this paper, we propose a deterministic alternative to VMC Jastrow optimisation, based on minimising the "variance of the TC reference energy" in a standard basis set. Analytic expressions for the derivatives of the TC Hamiltonian matrix elements are derived and implemented. This approach can be used to optimise the parameters in the Jastrow functions, either from scratch or to refine an initial VMC-based guess, to produce noise-free Jastrows in a reproducible manner. Applied to the first row atoms and molecules, the results show that the method yields Slater-Jastrow wavefunctions whose variances are almost as low as those obtained from standard VMC variance optimisation, but whose energies are lower, and comparable to those obtained from energy-minimisation VMC. We propose that the method can be used both in the context of the transcorrelated method or in standard VMC as a new way to optimise Jastrow functions.

Figures

Figures reproduced from arXiv: 2506.04895 by the authors.

Figure 1
Figure 1. FIG. 1. The variance of the reference energy (top) and the reference energy itself (bottom) for the Li atom, as a function of the parameter [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Variation of the variance of the reference energy, the reference energy and the various parameters in the Jastrow over the course of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The error in the IPs of 1st row elements as computed using VMC and reference energies (top left), the corresponding standard [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The error in the IPs of 1st row elements as computed using reference energies (top left), the corresponding standard deviations (top [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. VMC energies and variances obtained by variational en [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.