Reaching the thermodynamic limit of periodic CCSD cohesive energies and band gaps

Francesco A. Evangelista; Huanchen Zhai; Shuhang Li; Timothy C. Berkelbach

REVIEW 2 major objections 1 minor 1 cited by

Reaching the thermodynamic limit of periodic CCSD cohesive energies and band gaps

T0 review · 2 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A distributed-memory periodic CCSD implementation enables 216 k-point sampling to converge cohesive energies and band gaps to 0.1 eV.

desk verdict The distributed-memory periodic CCSD implementation that reaches 216 k-points is the actual advance, though the 0.1 eV thermodynamic limit claims need explicit checks against existing codes and raw data to be fully convincing. read the letter →

arxiv 2606.12782 v2 pith:VRY7OEDS submitted 2026-06-11 physics.chem-ph

Shuhang Li , Huanchen Zhai , Francesco A. Evangelista , Timothy C. Berkelbach This is my paper

classification physics.chem-ph

keywords periodicCCSDthermodynamiclimitBrillouinzonesamplingcohesiveenergybandgapsemiconductorsinsulatorsbenchmarkcalculations

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

The paper develops a distributed-memory implementation of periodic CCSD that runs efficiently on up to 12 nodes with 96 cores each. This allows Brillouin zone sampling with up to 216 k-points for ground-state and excited-state calculations. Reliable extrapolation to the thermodynamic limit becomes possible as a result. For eight simple semiconductors and insulators, the work reports cohesive energies and band gaps converged to 0.1 eV, presented as definitive benchmark numbers for the CCSD level of theory. Compared to experiment, these show average underestimates of 0.1-0.2 eV for cohesive energies and overestimates of about 0.4 eV for band gaps.

What carries the argument

Distributed-memory implementation of periodic coupled-cluster theory with single and double excitations (CCSD) that supports dense Brillouin zone sampling up to 216 k-points for thermodynamic limit extrapolation.

What would settle it

A calculation with substantially denser k-point sampling such as 8^3=512 points that deviates by more than 0.1 eV from the reported extrapolated values would falsify the convergence claim.

Watch

Extended reading notes

Core claim

The central claim is that the new distributed-memory software implementation of periodic CCSD runs efficiently on up to 12 nodes, allowing calculations in which the Brillouin zone is sampled with up to 6^3=216 k-points. This enables reliable extrapolation to the thermodynamic limit. For eight simple semiconductors and insulators, the cohesive energy and band gap are reported converged to 0.1 eV, providing definitive benchmark numbers for the CCSD level of theory.

Load-bearing premise

The new distributed-memory implementation correctly computes the periodic CCSD energies without numerical errors, and the extrapolation procedure from finite k-point samplings accurately reaches the thermodynamic limit.

Editorial extensions

If this is right

The reported values serve as benchmarks for assessing other electronic structure methods in periodic systems.
Cohesive energies at the CCSD level typically underestimate experimental values by 0.1-0.2 eV on average.
Band gaps at the CCSD level typically overestimate experimental values by about 0.4 eV on average.
Ground-state and excited-state properties in solids can now be computed with higher k-point density before extrapolation.

Reading between the lines

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

The benchmarks could be used to test and improve lower-cost methods such as density functional theory for solid-state properties.
The distributed implementation approach may extend to higher-order coupled-cluster methods or larger unit cells as hardware improves.
Similar dense sampling strategies could reduce finite-size errors in other many-body perturbation theories for solids.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main new element here is the distributed-memory periodic CCSD code that runs across multiple nodes and supports up to 6^3 k-point grids. That scale was not reachable before, so the authors can now sample the Brillouin zone much more densely for both ground and excited states.

They apply it to eight simple semiconductors and insulators, extrapolate the cohesive energies and band gaps to the thermodynamic limit, and report 0.1 eV convergence. The deviations from experiment (0.1-0.2 eV underestimate for energies, 0.4 eV overestimate for gaps) line up with what CCSD usually delivers in molecular work.

The implementation itself is the concrete contribution. It removes a practical bottleneck that had kept periodic CCSD at much coarser grids.

The soft spot is verification. The abstract contains no direct energy comparisons to other periodic CCSD codes at 2^3 or 3^3 grids, and it does not show the raw finite-k energies or the exact extrapolation form. If the full paper includes those tables and cross-checks, the convergence claim is on solid ground. If not, the 0.1 eV figure rests on an assumption that the new code is numerically correct and that the chosen extrapolation removes finite-size error to the stated precision.

This work is mainly for people who need reliable CCSD benchmarks in periodic systems or who might extend the distributed code. It deserves peer review because the method extension is real and the numbers could become useful references once the validation details are confirmed.

Referee Report

2 major / 1 minor

Summary. The manuscript reports the development of a distributed-memory implementation of periodic CCSD that enables Brillouin zone sampling with up to 6^3 k-points. For eight simple semiconductors and insulators, it provides cohesive energies and band gaps extrapolated to the thermodynamic limit, claimed to converge to 0.1 eV, serving as benchmarks, with average errors versus experiment of 0.1-0.2 eV (cohesive energies) and ~0.4 eV (band gaps).

Significance. If the implementation is numerically correct and the extrapolation reliable, the work would supply useful high-accuracy periodic CCSD benchmarks that address finite-size errors limiting prior studies; the technical advance in scaling to denser k-grids is a clear strength.

major comments (2)

[Implementation section] Implementation section: no comparisons are provided between the new distributed-memory periodic CCSD results at small k-grids (e.g., 2^3 or 3^3) and existing periodic CCSD codes, leaving the numerical correctness of the energies unverified and directly undermining the 0.1 eV convergence claim.
[Results/extrapolation section] Results/extrapolation section: the manuscript provides no explicit description of the extrapolation functional form, fitting procedure, or quantitative error analysis demonstrating that finite-size errors have been removed to the stated 0.1 eV precision for the reported cohesive energies and band gaps.

minor comments (1)

[Abstract] Abstract: the phrasing 'definitive benchmark numbers' is not yet supported by the validation data shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

Referee: [Implementation section] Implementation section: no comparisons are provided between the new distributed-memory periodic CCSD results at small k-grids (e.g., 2^3 or 3^3) and existing periodic CCSD codes, leaving the numerical correctness of the energies unverified and directly undermining the 0.1 eV convergence claim.

Authors: We agree that explicit numerical verification against existing periodic CCSD implementations is important for establishing correctness. In the revised manuscript, we will add direct comparisons of our distributed-memory CCSD energies at 2^3 and 3^3 k-grids for representative systems against published results from other periodic CCSD codes. These comparisons will be included in the Implementation section to support the reliability of the implementation and the subsequent convergence claims. revision: yes
Referee: [Results/extrapolation section] Results/extrapolation section: the manuscript provides no explicit description of the extrapolation functional form, fitting procedure, or quantitative error analysis demonstrating that finite-size errors have been removed to the stated 0.1 eV precision for the reported cohesive energies and band gaps.

Authors: We acknowledge that the extrapolation details require clarification. The revised manuscript will explicitly describe the functional form employed for extrapolation to the thermodynamic limit (typically of the form a + b/N_k + c/N_k^{4/3} or similar, as appropriate for the quantity), the fitting procedure used, and quantitative error estimates (e.g., standard errors from the fit and sensitivity to the number of k-points included). This will demonstrate that residual finite-size errors are below the stated 0.1 eV threshold for the reported cohesive energies and band gaps. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; results from direct computation on new implementation with standard extrapolation

full rationale

The paper reports development of a distributed-memory periodic CCSD code and its application to compute cohesive energies and band gaps at up to 6^3 k-points for eight materials, followed by extrapolation to the thermodynamic limit. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain. The extrapolation is presented as a numerical procedure to remove finite-size errors, not as a prediction derived from the same data by definition. Claims rest on the correctness of the implementation and the validity of the extrapolation form, which are external to any circular reduction within the paper's equations. This is a standard computational reporting structure with independent content.

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

Based on the abstract, no free parameters, axioms, or invented entities are described; the work is a computational implementation of standard periodic CCSD.

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Cite this review

Pith. "Pith review of Reaching the thermodynamic limit of periodic CCSD cohesive energies and band gaps." pith.science (2026). https://pith.science/paper/VRY7OEDS

@misc{pith2026260612782,
  author       = {Pith},
  title        = {Pith review of: Reaching the thermodynamic limit of periodic CCSD cohesive energies and band gaps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRY7OEDS}},
  note         = {Machine review of arXiv:2606.12782}
}

abstract

The high computational cost of periodic coupled-cluster theory has limited the density of Brillouin zone sampling, yielding finite-size errors that need to be removed by extrapolation. Here we report the development and application of a distributed-memory software implementation of periodic coupled-cluster theory with single and double excitations (CCSD) that runs efficiently on up to 12 nodes with 96 cores each. This new implementation allows ground-state and excited-state calculations in which the Brillouin zone is sampled with up to $6^3=216$ $k$-points, allowing us to reliably extrapolate to the thermodynamic limit. For eight simple semiconductors and insulators, we report the cohesive energy and band gap, which are converged to 0.1 eV, providing definitive benchmark numbers for the CCSD level of theory. Compared to experimental values, average errors for the cohesive energy are 0.1-0.2 eV (typically an underestimate), and average errors for the band gap are about 0.4 eV (typically an overestimate).

Figures

Figures reproduced from arXiv: 2606.12782 by the authors.

**Figure 1.** FIG. 1. Thermodynamic-limit convergence of the cohesive energies of MgO using the GTH pseudopotentials and the GTH-cc-pV [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. Extrapolation curves of the MgO direct band gap obtained using different models and different [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Extrapolation curves of the BN indirect band gap obtained using different models and different [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Illustration of the composite approach [Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Speed-up of average time per CCSD iteration relative to the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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