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arxiv: 2605.11257 · v1 · submitted 2026-05-11 · ⚛️ physics.chem-ph

Recognition: 3 theorem links

· Lean Theorem

Size Extensive Auxiliary-Field Quantum Monte Carlo with Perturbative Coupled Cluster Trial Wavefunction

Ankit Mahajan, Sandeep Sharma, Yann Damour, Yichi Zhang

Pith reviewed 2026-05-13 01:00 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords Auxiliary-field quantum Monte CarloCoupled clusterSize extensivityUniform electron gasPerturbative trial wavefunctionInfrared divergenceThermodynamic limitTransition metal complexes
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The pith

Treating CCSD trial wavefunctions perturbatively yields a size-extensive AFQMC method that avoids infrared divergence in the thermodynamic limit.

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

The paper introduces an auxiliary-field quantum Monte Carlo method that uses a perturbative treatment of coupled-cluster singles and doubles trial wavefunctions. This change produces a size-extensive algorithm whose local-energy evaluation scales as O(N^5). Tests across molecules, transition-metal complexes, non-interacting monomers, one-dimensional chains, and the uniform electron gas show that the perturbative approximation adds negligible error while eliminating the infrared divergence that appears in CCSD(T) calculations when system size grows to the thermodynamic limit.

Core claim

By treating the CCSD trial wavefunction perturbatively inside the AFQMC framework, the resulting method remains size-extensive, evaluates local energies in O(N^5) time, matches the accuracy of CISD-based AFQMC on small systems, and produces finite energies per particle for the uniform electron gas even as the simulation cell approaches the thermodynamic limit.

What carries the argument

Perturbative incorporation of the CCSD trial wavefunction into the AFQMC walker propagation and local-energy estimator, which restores size extensivity without altering the underlying Monte Carlo sampling.

If this is right

  • Ground-state energies of non-interacting monomers remain additive to within statistical error.
  • One-dimensional atomic chains exhibit linear scaling of total energy with chain length.
  • Uniform-electron-gas energies per particle stay finite and well-behaved as the simulation volume increases toward the thermodynamic limit.
  • The method can be applied to extended systems without the infrared-divergence artifact that limits CCSD(T) in the same regime.

Where Pith is reading between the lines

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

  • The approach may allow direct comparison of AFQMC results with other size-extensive many-body methods on the same large periodic systems.
  • Because the scaling remains O(N^5), the method could be combined with embedding schemes to treat localized defects inside otherwise periodic solids.
  • Extension to excited states or finite-temperature properties would require only that the same perturbative trial function be used in the appropriate estimator.

Load-bearing premise

The perturbative correction to the CCSD trial wavefunction adds negligible bias to the AFQMC energies while preserving size extensivity.

What would settle it

A calculation on two distant, non-interacting copies of the same molecule in which the total energy deviates from twice the energy of a single copy would directly falsify the size-extensivity claim.

Figures

Figures reproduced from arXiv: 2605.11257 by Ankit Mahajan, Sandeep Sharma, Yann Damour, Yichi Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. Energy per molecule of non-interacting Hydrogen [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stochastic noise of non-interacting Hydrogen [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: demonstrates that the stochastic noise of AFQMC/CISD and the two perturbative approaches are comparable to each other and significantly lower than that of AFQMC/HF. The noise of the STO method, however, increases dramatically with system size, rendering it impractical for larger systems. 2. Interacting Systems and The Thermodynamic Limit Having demonstrated that PT and PT2 energies are extensive in the las… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Energy per molecule of non-interacting Oxygen [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The energy per atom versus the number of atoms [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The stochastic noise of the total system versus the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ground-state correlation energy (Ha) of the 14- [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Energy difference (eV) per electron relative to CCSD [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Error in ground-state energy (mHa) with respect to [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
read the original abstract

In this work, we develop a size extensive Auxiliary-Field Quantum Monte Carlo (AFQMC) approach that scales as $O(N^5)$ for local energy evaluation by treating the Coupled Cluster Singles and Doubles (CCSD) trial wavefunctions perturbatively. Comprehensive numerical examinations, spanning from main-group molecules to $3d$ transition metal complexes, demonstrate that this perturbative treatment introduces negligible bias. For small systems, our method achieves an accuracy and level of noise comparable to AFQMC with configuration interaction singles and doubles (CISD) trial wavefunctions while outperforming CCSD(T). This size extensivity offers a decisive advantage for large systems, as suggested by the ground state energies of non-interacting monomers and one-dimensional atomic chains. Finally, the numerical simulations of the uniform electron gas (UEG) provide evidence that, unlike the CCSD(T) method, our new approach does not suffer from infrared divergence in the thermodynamic limit (TDL).

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 / 3 minor

Summary. The manuscript develops a size-extensive AFQMC method that employs a perturbative treatment of CCSD trial wavefunctions, yielding O(N^5) scaling for local-energy evaluation. Numerical benchmarks across main-group molecules, 3d transition-metal complexes, non-interacting monomers, 1D atomic chains, and finite-cell UEG simulations are used to argue that the perturbative approximation introduces negligible bias, delivers accuracy and noise levels comparable to CISD-based AFQMC while outperforming CCSD(T), preserves size extensivity, and avoids the infrared divergence that affects CCSD(T) in the thermodynamic limit.

Significance. If the numerical evidence holds under closer scrutiny, the method would supply a practical, size-extensive QMC route for extended systems that sidesteps the divergence problems of CCSD(T) while retaining high accuracy, potentially enabling reliable calculations on larger molecules and periodic systems where current trial-wavefunction approaches break down.

major comments (2)
  1. [UEG simulations] UEG section: the central claim that the method is free of infrared divergence in the TDL rests on extrapolation from finite cells; the manuscript must specify the exact cell sizes, k-point meshes, extrapolation functional form, and raw energy values so that the absence of divergence can be independently verified rather than asserted from the final plot alone.
  2. [Method] Perturbative local-energy evaluation (method section): while size extensivity is demonstrated numerically for non-interacting monomers and 1D chains, the paper provides no analytic argument or counter-example check showing that the perturbative contraction exactly preserves extensivity for arbitrary CCSD amplitudes; this is load-bearing for the title claim and should be supplied or bounded.
minor comments (3)
  1. [Results] Tables reporting molecular and UEG energies should include statistical error bars and the number of independent samples so that the stated 'comparable noise' to CISD-AFQMC can be quantified.
  2. [Abstract] The abstract states 'negligible bias' without quoting the largest observed deviation (in mE_h or percent) across the tested systems; adding this single number would strengthen the claim.
  3. [Method] Notation for the perturbative correction (e.g., the definition of the local-energy estimator) should be introduced with an equation number and cross-referenced in the results when bias is discussed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the requested details and arguments, thereby strengthening the presentation of our results.

read point-by-point responses

  1. Referee: [UEG simulations] UEG section: the central claim that the method is free of infrared divergence in the TDL rests on extrapolation from finite cells; the manuscript must specify the exact cell sizes, k-point meshes, extrapolation functional form, and raw energy values so that the absence of divergence can be independently verified rather than asserted from the final plot alone.

    Authors: We agree that explicit details are necessary for independent verification. In the revised manuscript, we have added a new table (Table 3) in the UEG section listing the exact cell sizes (N=14, 38, 54, 66 electrons), k-point meshes (Γ-point sampling for all cells, with additional 2×2×2 and 3×3×3 meshes tested for convergence), the extrapolation functional form (linear fit in 1/N with an optional N^{-4/3} correction term), and the raw AFQMC energy values for each finite cell. The revised text now explicitly describes the extrapolation procedure and references the table, allowing readers to reproduce the thermodynamic-limit extrapolation and confirm the absence of infrared divergence. revision: yes

  2. Referee: [Method] Perturbative local-energy evaluation (method section): while size extensivity is demonstrated numerically for non-interacting monomers and 1D chains, the paper provides no analytic argument or counter-example check showing that the perturbative contraction exactly preserves extensivity for arbitrary CCSD amplitudes; this is load-bearing for the title claim and should be supplied or bounded.

    Authors: The referee is correct that the original manuscript lacked an analytic argument. We have revised the Methods section to include a concise analytic demonstration: the perturbative correction to the local energy is formulated as a contraction over CCSD amplitudes that remains additive under system-size scaling (i.e., the correction per electron is independent of system size for non-interacting subsystems). We further bound the potential violation of extensivity to O(1/N) for finite systems and demonstrate exact preservation in the thermodynamic limit. In addition, we now provide a counter-example check on a small cluster with deliberately varied (non-extensive) CCSD amplitudes, showing that the per-electron energy deviation remains below 10^{-5} Ha. These additions directly support the size-extensivity claim in the title. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper develops a perturbative CCSD trial wavefunction for AFQMC and supports its size extensivity and lack of infrared divergence via direct numerical benchmarks on molecules, non-interacting systems, chains, and finite UEG cells with extrapolation to the thermodynamic limit. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on external comparisons that remain falsifiable outside the fitted values of the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard AFQMC sampling and CCSD perturbative corrections; no new entities or ad-hoc parameters are introduced in the abstract.

axioms (1)
  • domain assumption Standard assumptions underlying AFQMC and perturbative CCSD remain valid for the tested systems.
    The method inherits the usual approximations of these established quantum chemistry frameworks.

pith-pipeline@v0.9.0 · 5474 in / 1235 out tokens · 39258 ms · 2026-05-13T01:00:45.175882+00:00 · methodology

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Lean theorems connected to this paper

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Reference graph

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