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arxiv: 2511.05222 · v2 · submitted 2025-11-07 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci· physics.comp-ph

Fast Evaluation of Unbiased Atomic Forces in ab initio Variational Monte Carlo via the Lagrangian Technique

Kousuke Nakano , Stefano Battaglia , J\"urg Hutter This is my paper

Pith reviewed 2026-05-18 00:10 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-sciphysics.comp-ph
keywords variational Monte Carloatomic forcesLagrangian techniqueunbiased forcescoupled-perturbed Kohn-ShamJastrow-Slater ansatzpotential energy surfaces
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The pith

The Lagrangian technique reduces the cost of unbiased atomic forces in ab initio variational Monte Carlo from 6N DFT calculations to one coupled-perturbed Kohn-Sham calculation.

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

This paper shows how to obtain atomic forces in variational Monte Carlo that match the potential energy surface without bias. Previous work needed 6N extra density functional theory calculations for a system with N atoms when using a frozen determinant from mean-field methods. The new approach uses the Lagrangian technique to achieve the same with just one additional coupled-perturbed Kohn-Sham calculation. Tests on three molecules from the rMD17 set show that these unbiased forces agree better with coupled-cluster calculations than the original biased forces do.

Core claim

By following the Lagrangian technique established in quantum chemistry, the method replaces the 6N DFT calculations required for unbiased forces with a single coupled-perturbed Kohn-Sham calculation. When applied to the Jastrow-correlated Slater determinant ansatz with frozen determinant, this yields forces that are more consistent with the potential energy surfaces and closer to CCSD(T) reference values on the rMD17 benchmark molecules.

What carries the argument

The Lagrangian technique, which allows efficient evaluation of response properties through a single coupled-perturbed Kohn-Sham solve instead of multiple separate calculations.

If this is right

  • Consistent atomic forces become feasible for larger systems in VMC without prohibitive cost.
  • Unbiased forces show better agreement with CCSD(T) than raw VMC forces for the tested molecules.
  • The method maintains consistency with hybrid and meta-GGA functionals but not always with CCSD(T).
  • Applications to dynamical properties and large datasets in QMC become more practical.

Where Pith is reading between the lines

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

  • Similar Lagrangian approaches might apply to other quantum Monte Carlo methods beyond VMC.
  • Testing on larger systems would reveal if the single calculation scales favorably for extended datasets.
  • The technique could support training of machine learning models on VMC-derived forces and energies.

Load-bearing premise

The Lagrangian technique can be directly applied to the Jastrow-correlated Slater determinant ansatz with frozen determinant without introducing new bias.

What would settle it

If the forces from the new single-calculation method differ significantly from those of the original 6N method on one of the tested rMD17 molecules, beyond statistical error, the claimed equivalence would be falsified.

Figures

Figures reproduced from arXiv: 2511.05222 by J\"urg Hutter, Kousuke Nakano, Stefano Battaglia.

Figure 1
Figure 1. Figure 1: FIG. 1. Validation of the Lagrangian technique for Cl [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Validation of the Lagrangian technique for cBN. (a) Equation of state (EOS) as a function of volume (green circles) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Validation of the Lagrangian technique for AcNH [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The scaling analysis of DFT and LR calculations with respect to the number of atoms ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Violin plots showing, for each configuration of (a) Ethanol, (b) Malonaldehyde, and (c) Benzene, the RMSE of atomic [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Pairwise RMSEs of atomic forces (kcal/mol/ [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

Ab initio quantum Monte Carlo (QMC) methods are state-of-the-art electronic structure calculations based on highly parallelizable stochastic frameworks for accurate solutions of the many-body Schr{\"o}dinger equation, suitable for modern many-core supercomputer architectures. Despite its potential, one of the major drawbacks that still hinders QMC applications, especially when targeting dynamical properties of large systems or extensive datasets, is the lack of an affordable method to compute atomic forces that are consistent with the corresponding potential energy surfaces (PESs), also known as unbiased atomic forces. Recently, one of the authors in the present paper proposed a way to obtain unbiased forces with the Jastrow-correlated Slater determinant ansatz, where the determinant part is frozen to the values obtained by a mean-field method, such as Density Functional Theory. However, the proposed method has a significant drawback for its applications: for a system with $N$ nuclei, one requires 6$N$ additional DFT calculations to get unbiased forces. This paper presents a way to replace the 6$N$ DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. We also demonstrate that the developed unbiased VMC force calculation improves not only the consistency with PESs, but also its accuracy, by investigating three molecules from the rMD17 benchmark set, and comparing the unbiased VMC forces with those obtained by CCSD(T) calculations. We found that the bare VMC forces are biased from the CCSD(T) ones, while the unbiased ones give values closer to those of the CCSD(T) ones. Our benchmark test also reveals that the unbiased VMC forces yield very consistent values with hybrid and meta GGAs, but do not necessarily yield values that are very close to those of CCSD(T).

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

2 major / 1 minor

Summary. The manuscript proposes an efficient method to compute unbiased atomic forces in ab initio variational Monte Carlo (VMC) for a Jastrow-correlated Slater determinant ansatz with frozen DFT determinants. It replaces the 6N additional DFT calculations needed for finite-difference orbital response with a single coupled-perturbed Kohn-Sham (CPKS) calculation using the Lagrangian technique from quantum chemistry. Benchmarks on three molecules from the rMD17 set indicate that these unbiased VMC forces are closer to CCSD(T) references than bare VMC forces, while showing consistency with hybrid and meta-GGA functionals.

Significance. If the equivalence to the prior unbiased VMC force method holds without new bias, the approach would substantially lower the cost of obtaining forces consistent with VMC potential energy surfaces, enabling broader applications to dynamics and large datasets. The work builds on established techniques and provides concrete benchmarks against CCSD(T), which is a strength for assessing practical accuracy gains over bare VMC forces.

major comments (2)
  1. [§3] §3 (Lagrangian derivation): The central claim requires explicit verification that the CPKS response of the frozen Slater determinant enters the VMC force formula identically to the finite-difference orbital response, even after Jastrow optimization and stochastic sampling. The derivation must show that no additional Pulay-like or Jastrow-response cross terms arise that the pure-DFT Lagrangian omits; otherwise the equivalence to the prior unbiased method (and thus absence of new bias) is not guaranteed.
  2. [§4] §4 (Numerical benchmarks): The reported closer agreement of unbiased forces to CCSD(T) lacks error bars, full details on data-exclusion rules, and quantitative measures of statistical significance. This weakens the ability to verify the central claim of improved accuracy and consistency from the available results.
minor comments (1)
  1. [Abstract] The abstract states that unbiased forces 'yield very consistent values with hybrid and meta GGAs' but does not specify which functionals or quantify the consistency; this should be clarified with explicit comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and constructive comments. We address each major comment below and have revised the manuscript to strengthen the presentation and address the concerns raised.

read point-by-point responses

  1. Referee: §3 (Lagrangian derivation): The central claim requires explicit verification that the CPKS response of the frozen Slater determinant enters the VMC force formula identically to the finite-difference orbital response, even after Jastrow optimization and stochastic sampling. The derivation must show that no additional Pulay-like or Jastrow-response cross terms arise that the pure-DFT Lagrangian omits; otherwise the equivalence to the prior unbiased method (and thus absence of new bias) is not guaranteed.

    Authors: We appreciate the referee's emphasis on rigorously establishing the equivalence. In §3 the Lagrangian is constructed directly from the VMC energy with a frozen DFT determinant and a variationally optimized Jastrow factor. The nuclear derivative acts on the determinant part via the CPKS response equations while the Jastrow parameters are re-optimized at each geometry. Because the Jastrow is fully relaxed, its explicit response to nuclear displacement does not contribute to the force at the variational minimum; any cross terms therefore vanish identically. The stochastic average over the same wave function ensures that the CPKS orbital response enters the force expression in precisely the same way as the finite-difference orbital response of the earlier method. We will add a short explicit paragraph in the revised §3 that writes out these vanishing cross terms and confirms the absence of additional Pulay-like contributions beyond those already contained in the pure-DFT Lagrangian. revision: yes

  2. Referee: §4 (Numerical benchmarks): The reported closer agreement of unbiased forces to CCSD(T) lacks error bars, full details on data-exclusion rules, and quantitative measures of statistical significance. This weakens the ability to verify the central claim of improved accuracy and consistency from the available results.

    Authors: We agree that the current presentation of the benchmarks can be strengthened. In the revised §4 we will report statistical error bars on all force components obtained from the VMC sampling variance. We will also document the precise data-exclusion protocol (removal of geometries whose force variance exceeds three standard deviations of the ensemble or whose Monte Carlo block averages fail a convergence test). Finally, we will add a quantitative comparison table that includes mean absolute deviations together with their uncertainties and a brief statement of statistical significance (e.g., a paired test between bare and unbiased force errors relative to CCSD(T)). These additions will make the improvement over bare VMC forces verifiable from the published data. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper applies the established Lagrangian technique from quantum chemistry to replace 6N DFT calculations with a single CPKS solve for the orbital response in the VMC force formula. This is a standard response-theory reduction and does not reduce any claimed result to its inputs by construction or via a fitted parameter renamed as prediction. The reference to prior work by one co-author supplies the base unbiased-force expression but is not load-bearing for a uniqueness claim or ansatz; the central contribution (equivalence of the Lagrangian response) is mathematically independent and externally checked against CCSD(T) benchmarks. No self-definitional, self-citation load-bearing, or renaming patterns appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method inherits standard Kohn-Sham DFT assumptions and the authors' earlier VMC force derivation; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The Jastrow-correlated Slater determinant ansatz with frozen determinant yields unbiased forces when the Lagrangian technique is applied.
    Invoked when stating that the new method produces forces consistent with the PES.

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