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REVIEW 2 major objections 5 minor 31 references

Similarity transformation of the electronic Schr\"odinger equation via Jastrow factorisation

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.02882 v1 pith:VPAKXLGO submitted 2019-08-08 physics.chem-ph cond-mat.otherquant-ph

Aron J. Cohen , Hongjun Luo , Kai Guther , Werner Dobrautz , David P. Tew , Ali Alavi This is my paper · ORCID
classification physics.chem-phcond-mat.otherquant-ph
keywords JastrowfactorsimilaritytransformationfullconfigurationinteractionquantumMonteCarlotranscorrelatedHamiltonianthree-bodyintegralsnon-Hermitianexplicitlycorrelatedmethodsfirst-rowatoms
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

By factorising the electronic wavefunction as $\Psi = e^{\tau}\Phi$, with $\tau = \sum_{i

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 5 minor

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.

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 (2)
  1. [Supplementary, Eqs. (S5)–(S8); main text, p.3] 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.
  2. [Tables I–III; p.4] 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.
minor comments (5)
  1. [p.2; Supplementary after Eq. (S5); final paragraph] 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).
  2. [p.2, after Eq. (2)] 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.
  3. [Eq. (2) and following definitions of K̂ and L̂] 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.
  4. [Table I; main text p.3] 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.
  5. [Table I caption; p.4 text] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the similarity-transformed Hamiltonian is derived in-paper, and the Jastrow parameters are external literature inputs, not fitted to the reported energies.

full rationale

The core derivation is self-contained: Eq. (2) follows from the commutator expansion of e^{-tau} H e^{tau}, and the explicit K and L operators are written out in the paper. The matrix-element evaluation is a numerical quadrature prescription, independent of the final energies. The Jastrow parameters are taken from the published Schmidt-Moskowitz VMC study (ref. 26), not optimized in this paper, and the ionization potentials are computed with the same Jastrow without re-optimization, providing a genuine transferability test. The FCIQMC implementation is cited from prior work by the same group, but that citation supplies a solver, not the target energies, so the accuracy claim is not forced by a self-citation chain. One benchmark caveat exists: the total-energy comparisons use a Jastrow that was previously variance-optimized for the same first-row atoms, so those total energies are not parameter-free predictions; however, the ST-FCIQMC energies are not equal to the VMC inputs by construction, and the IPs use fixed transferred parameters. The quadrature-convergence concern raised by the reviewer is a numerical correctness risk, not a circularity, because no equation in the paper defines the reported energies in terms of the quadrature tolerance.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on two external inputs: the Jastrow parameters fitted in a prior VMC study of the same atoms, and the numerical quadrature accuracy which is asserted but not demonstrated. The FCIQMC initiator approximation is an additional domain assumption inherited from prior work.

free parameters (2)
  • SM7 Jastrow coefficients (7 parameters) = From Schmidt-Moskowitz VMC (1990), not listed in paper
    Fitted via variance minimization on the same first-row atoms; used as fixed inputs, not re-optimized.
  • SM17 Jastrow coefficients (17 parameters) = From Schmidt-Moskowitz VMC (1990), not listed in paper
    Fitted via variance minimization on the same first-row atoms; used as fixed inputs, not re-optimized. The presence of these fitted parameters weakens the fully ab initio nature of the benchmark.
assumptions (3)
  • domain assumption The chosen Boys-Handy Jastrow form is sufficiently flexible to capture the electron-electron cusp and short-range correlation.
    The accuracy of the method depends on the correlation factor; the s-wave cusp is fulfilled but the p-wave is not, which the authors note limits convergence.
  • domain assumption The numerical quadrature over atom-centered grids converges rapidly to accurate matrix elements.
    The paper states this but provides no grid-convergence data.
  • domain assumption The FCIQMC initiator approximation yields an accurate approximation to the ground state of the non-Hermitian transformed Hamiltonian.
    Inherited from prior FCIQMC studies; no convergence with initiator threshold is reported.

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Pith. "Pith review of Similarity transformation of the electronic Schr\"odinger equation via Jastrow factorisation." pith.science (2026). https://pith.science/paper/VPAKXLGO

@misc{pith2026190802882,
  author       = {Pith},
  title        = {Pith review of: Similarity transformation of the electronic Schr\"odinger equation via Jastrow factorisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPAKXLGO}},
  note         = {Machine review of arXiv:1908.02882}
}
read the original abstract

By expressing the electronic wavefunction in an explicitly-correlated (Jastrow-factorised) form, a similarity-transformed effective Hamiltonian can be derived. The effective Hamiltonian is non-Hermitian and contains three-body interactions. The resulting ground-state eigenvalue problem can be solved projectively using a stochastic configuration-interaction formalism. Our approach permits use of highly flexible Jastrow functions, which we show to be effective in achieving extremely high accuracy, even with small basis sets. Results are presented for the total energies and ionisation potentials of the first-row atoms, achieving accuracy within a mH of the basis-set limit, using modest basis sets and computational effort.

Figures

Figures reproduced from arXiv: 1908.02882 by the authors.

Figure 1
Figure 1. FIG. 1. Errors in the total energies of the atoms, in H, for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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