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arxiv: 2605.17083 · v1 · pith:GAZ2F2NLnew · submitted 2026-05-16 · ⚛️ physics.comp-ph · cond-mat.dis-nn· cond-mat.str-el· cond-mat.supr-con· physics.chem-ph

Basis-free neural-network geminal and Jastrow factors for variational Monte Carlo

Jan Kessler , Thomas D. K\"uhne This is my paper

Pith reviewed 2026-05-20 15:06 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.dis-nncond-mat.str-elcond-mat.supr-conphysics.chem-ph
keywords variational Monte Carloneural network quantum statesantisymmetrized geminal powerJastrow factorhydrogen systemsnodal surfacedynamical correlationbasis-free ansatz
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The pith

Neural-network replacements for basis sets in geminal and Jastrow factors separate nodal definition from dynamical correlation in variational Monte Carlo.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a basis-free ansatz that pairs an antisymmetrized geminal power determinant with neural-network Jastrow factors for use in variational Monte Carlo. The AGP component sets the nodal surface while the neural networks handle dynamical correlation without altering the nodes. This design allows the authors to isolate errors from static correlation versus dynamical effects. Tests on the hydrogen molecule and a rectangular hydrogen tetramer reach sub-millihartree accuracy when the nodes are suitable, but reveal remaining problems when the geometry challenges the nodal surface.

Core claim

The authors show that replacing conventional basis-set expansions with feed-forward neural networks in both the geminal and Jastrow constructions yields a compact wave function that achieves sub-millihartree accuracy for the hydrogen molecule and the rectangular hydrogen tetramer whenever the AGP nodes are adequate, while highlighting the residual nodal limitation near the large-radius square geometry of the hydrogen tetramer.

What carries the argument

The basis-free Jastrow-AGP ansatz in which an antisymmetrized geminal power determinant defines the nodal surface and a neural-network Jastrow factor recovers dynamical correlation at fixed nodes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending this separation to larger systems could help identify when additional nodal optimization is needed beyond neural Jastrow factors.
  • Similar neural replacements might apply to other antisymmetric wave function forms to reduce basis dependence.
  • Testing on systems with known exact nodes would confirm the isolation of dynamical correlation improvements.

Load-bearing premise

The AGP determinant supplies an adequate nodal surface so the neural-network Jastrow factor can recover dynamical correlation at fixed nodes without adjusting the nodes.

What would settle it

A variational Monte Carlo calculation on the square hydrogen tetramer at large separation that fails to reach sub-millihartree accuracy even after full optimization of the neural-network parameters would indicate that the nodal limitation cannot be overcome by the Jastrow factor alone.

Figures

Figures reproduced from arXiv: 2605.17083 by Jan Kessler, Thomas D. K\"uhne.

Figure 1
Figure 1. Figure 1: FIG. 1. Optimization protocol used for the neural-network [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Geometry and parametrizations of the rectangular [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: b resolves the remaining deviations. The rela￾tive error increases below 2 Bohr, i.e., around and below the equilibrium distance of about 1.4 Bohr, reaching ap￾proximately 0.0025% or 0.025 mHa. At larger separations the error remains close to 0.001%. This behavior dif￾fers from many conventional single-reference electronic￾structure methods, which are often accurate near equi￾librium but deteriorate for st… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Energy differences to reference values [5, 44] for NNAGP + all-body-NNJF wave functions with different numbers of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

Neural-network quantum states offer a flexible route to compact many-electron wave functions, but their practical accuracy depends strongly on how fermionic antisymmetry, electron correlation, and optimization noise are treated. Here we combine an antisymmetrized geminal power (AGP) determinant with feed-forward neural networks that replace conventional basis-set expansions in the geminal and in two Jastrow-factor constructions. The resulting basis-free Jastrow--AGP ansatz is optimized by variational Monte Carlo and is designed to separate two tasks: the AGP part defines the nodal surface, while the neural-network Jastrow factor recovers dynamical correlation at fixed nodes. This separation makes it possible to distinguish errors associated with dynamical correlation from those caused by static, multireference correlation. Applications to the hydrogen molecule and the rectangular hydrogen tetramer show sub-millihartree accuracy when the AGP nodes are adequate, and expose the residual nodal limitation near the large-radius square geometry of the hydrogen tetramer. These results clarify where neural-network building blocks can improve a compact geminal ansatz and where additional nodal flexibility is required.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a basis-free variational Monte Carlo ansatz that replaces conventional basis expansions with feed-forward neural networks for both the antisymmetrized geminal power (AGP) geminal and two Jastrow-factor constructions. The central design choice fixes the nodal surface via the AGP determinant while using the neural-network Jastrow solely to recover dynamical correlation at those fixed nodes. Numerical results on H2 and the rectangular H4 geometry reach sub-millihartree accuracy relative to reference values when the AGP nodes are adequate; the same framework is used to identify a residual error at the large-radius square H4 geometry that is attributed to nodal limitations.

Significance. If the separation of nodal and dynamical-correlation errors can be rigorously validated, the work supplies a practical diagnostic for deciding when additional nodal flexibility (e.g., beyond a single AGP) is required in neural-network quantum states. The concrete accuracy numbers on two small systems and the explicit identification of a nodal bottleneck constitute a useful benchmark contribution for the VMC community.

major comments (1)
  1. [§4] §4 (H4 results) and the abstract: the claim that the residual error near the large-radius square geometry is caused by nodal limitations presupposes that the neural-network Jastrow has already saturated the dynamical correlation recoverable at the fixed AGP nodes. No scaling study with Jastrow network depth/width, no comparison against a converged conventional Jastrow at the same nodes, and no independent error decomposition are reported; without such evidence the attribution remains inconclusive.
minor comments (2)
  1. [§2] The precise functional form of the neural-network geminal (how the AGP coefficients are generated from the network output) should be written explicitly, preferably with an equation.
  2. [Figures] Figure captions for the H4 energy curves should state the reference method and basis used for the comparison values.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. The major comment raises a valid point about the strength of evidence supporting our attribution of the residual error to nodal limitations. We address this concern directly below and outline revisions that will be made to the manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (H4 results) and the abstract: the claim that the residual error near the large-radius square geometry is caused by nodal limitations presupposes that the neural-network Jastrow has already saturated the dynamical correlation recoverable at the fixed AGP nodes. No scaling study with Jastrow network depth/width, no comparison against a converged conventional Jastrow at the same nodes, and no independent error decomposition are reported; without such evidence the attribution remains inconclusive.

    Authors: We agree with the referee that a more rigorous demonstration that the neural-network Jastrow has saturated the dynamical correlation at the fixed AGP nodes would make the attribution of the residual error to nodal limitations more conclusive. The current results show that two distinct neural-network Jastrow constructions yield essentially the same residual error at the large-radius square H4 geometry (while both recover sub-millihartree accuracy at the rectangular geometry), which we interpret as evidence that further dynamical-correlation recovery is not possible at those nodes. However, we acknowledge that this interpretation would be strengthened by the additional checks the referee suggests. We will therefore add a scaling study with respect to Jastrow network depth and width, include a comparison against a converged conventional Jastrow factor at the same AGP nodes, and provide a clearer error decomposition in the revised manuscript. The language in the abstract and §4 will be adjusted to reflect that the nodal limitation is inferred from the saturation behavior observed with the present flexible Jastrow forms. revision: yes

Circularity Check

0 steps flagged

No circularity: variational optimization and external benchmarks keep results independent of input definitions

full rationale

The paper defines a variational Monte Carlo procedure in which neural-network parameters for the AGP geminal and Jastrow factors are optimized to minimize the energy expectation value. Reported sub-millihartree accuracies are obtained by direct comparison of these variational energies against known external reference values for the H2 and rectangular H4 systems. The separation of tasks (AGP supplying fixed nodes, Jastrow recovering dynamical correlation) is presented explicitly as a design choice that enables diagnostic attribution of residual errors, not as a definitional identity that forces the numerical outcomes. No equation reduces a reported energy or accuracy figure to a quantity defined by the same fitted parameters, and the provided text contains no load-bearing self-citations or imported uniqueness theorems. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the standard antisymmetry requirement for fermionic wave functions and on the assumption that a fixed nodal surface from the AGP can be used while the Jastrow recovers correlation; no new particles or forces are introduced.

axioms (2)
  • standard math Fermionic wave functions must be antisymmetric under particle exchange.
    Invoked when the AGP determinant is introduced to enforce antisymmetry.
  • domain assumption The nodal surface can be held fixed while dynamical correlation is added by the Jastrow factor.
    Central design choice stated in the abstract that enables separation of error sources.

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Reference graph

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