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REVIEW 3 major objections 6 minor 170 references

Periodic implementation of the random phase approximation with numerical atomic orbitals and dual reciprocal space grids

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A dual reciprocal-space grid makes periodic RPA correlation energies converge rapidly for two-dimensional materials.

desk verdict Solid NAO-based periodic RPA implementation with fast k-grid convergence; the headline CO/MgO error bar omits documented sensitivities and the Bajdich agreement rides on error cancellation. read the letter →

arxiv 2505.06021 v2 pith:G4REJGTX submitted 2025-05-09 cond-mat.mtrl-sci physics.chem-ph

Edoardo Spadetto , Pier Herman Theodoor Philipsen , Arno Förster , Lucas Visscher This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.chem-ph
keywords randomphaseapproximationperiodicRPAnumericalatomicorbitalspair-atomicdensityfittingdualk-gridschemeCoulombdampingadsorptionenergyMgO(001)
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 reports a practical implementation of the random phase approximation (RPA) for periodic systems using a localized numerical atomic orbital basis with pair-atomic density fitting. Its central claim is that a dual reciprocal-space grid—a regular momentum-transfer grid augmented near the $\Gamma$-point—combined with a k-grid-dependent damping of the Coulomb potential, makes RPA correlation energies converge quickly and reliably to the thermodynamic limit for 2D systems. For CO on MgO(001), the implementation yields an RPA@PBE adsorption energy of $-1.67 \pm 0.31$ kcal/mol after basis-set and coverage extrapolations, close to a previous periodic RPA value, while being less negative than experimental and CCSD(T) results, as expected for RPA@PBE. If the claim holds, it gives researchers a route to affordable canonical RPA calculations for molecule–surface interactions in a localized basis, with controlled convergence and good parallel scaling.

What carries the argument

The load-bearing object is the dual grid pair $(K_G, Q_G)$: the momentum-transfer grid $Q_G$ is a regular sampling with the $\Gamma$ point removed and extra points inserted at small displacements around it, while the k-grid $K_G$ is built so that all differences $k_j - k_l$ lie in $Q_G$. This keeps the computational cost of the polarizability proportional to the product of the two grid sizes while making memory growth linear in the number of added points. The second mechanism is a Fermi-Dirac damped Coulomb potential, with radius $r_0$ chosen automatically from the real-space supercell imposed by the k-sampling, which regularizes the $\Gamma$-point divergence without introducing an artificial short-range truncation. The third piece is a periodic projector that removes basis combinations whose overlap eigenvalues fall below a threshold, keeping the pair-atomic density fitting expansion stable.

What would settle it

Repeat the CO/MgO(001) calculation at 100% coverage with the projector threshold lowered from $10^{-3}$ to $5\times 10^{-4}$ while increasing the auxiliary basis until absolute RPA correlation energies stop shifting; if the RPA contribution to the adsorption energy moves by substantially more than the quoted 0.31 kcal/mol uncertainty, the error cancellation underlying the projector result breaks.

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Extended reading notes

Core claim

The paper's central claim is that the slow k-grid convergence of RPA correlation energies in periodic systems, caused by the Coulomb singularity at $q = \Gamma$, can be overcome by a dual-grid sampling scheme. A regular momentum-transfer grid has the $\Gamma$ point removed and a few extra points inserted close to it, and the k-grid is constructed so that differences of k-points cover this enhanced grid. Combined with an automatic Fermi-Dirac damping of the Coulomb potential whose range grows with the k-grid, this yields RPA correlation energies that converge with small meshes, independent of basis set. For CO on MgO(001) at the RPA@PBE level, the final low-coverage adsorption energy is $-1.67 \pm 0.31$ kcal/mol, matching a previous periodic RPA result while underestimating both experiment and recent embedded CCSD(T) values, as RPA@PBE is known to do.

Load-bearing premise

The calculation rests on the assumption that discarding basis combinations whose overlap eigenvalues fall below a fixed threshold removes only redundant directions, so that the large shifts in absolute correlation energies caused by the threshold cancel between the molecule and the surface.

Editorial extensions

If this is right

  • RPA correlation energies for 2D slabs and monolayers can be converged to the thermodynamic limit with modest k-meshes; the paper reports $9\times 9$ grids giving adsorption energies converged within 0.02–0.05 kcal/mol.
  • k-point convergence is nearly independent of the one-particle basis, so practitioners can converge the k-grid with a small basis and then extrapolate the basis at a fixed coarse k-grid.
  • The parallelization over q and frequency pairs achieves near-perfect strong scaling to thousands of cores, removing the wall-clock bottleneck for canonical RPA on small-unit-cell surface calculations.
  • At RPA@PBE, CO on MgO(001) binds by $-1.67 \pm 0.31$ kcal/mol in the low-coverage limit, confirming the known tendency of RPA@PBE to underestimate this adsorption energy relative to experiment and CCSD(T).
  • The same machinery—dual grids, automatic damping, and pair-atomic density fitting—is a direct base for more advanced ACFDT correlation kernels such as $\sigma$-functionals or renormalized adiabatic kernels.

Reading between the lines

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

  • The reported sensitivity of absolute correlation energies to the projector threshold suggests that the final adsorption-energy accuracy leans on error cancellation between subsystems; for systems where this cancellation is less complete, total energies may carry a larger unquantified error than the quoted $\pm 0.31$ kcal/mol.
  • The dual-grid construction is not tied to RPA: the same $\Gamma$-point treatment could accelerate periodic MP2 correlation energies or GW self-energies in localized-basis codes, where analogous k-grid convergence problems appear.
  • Because damping is tied to the k-grid rather than to a fixed physical cutoff, the method's converged absolute energies are range-separated intermediates; direct comparisons of absolute RPA correlation energies between codes with different k-grids may be misleading even when energy differences converge.
  • If RPA@PBE0 (hybrid input orbitals) indeed makes adsorption energies more negative, the implementation's natural next step—combining it with self-consistent periodic exact exchange—would directly test whether the gap to experiment and CCSD(T) is an input-orbital issue or an RPA kernel issue.
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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

3 major / 6 minor

Summary. The manuscript reports a periodic implementation of the RPA correlation energy in the BAND module of AMS, using numerical atomic orbitals and pair-atomic density fitting (PADF). The core methodological contributions are a Fermi-Dirac damping of the Coulomb potential whose range is tied to the k-grid ('AUTO damping'), a dual reciprocal-space grid scheme that adds points near the Gamma point to accelerate convergence to the thermodynamic limit, and a parallelization strategy based on splitting the (q, omega) integration over separate ScaLapack contexts. The implementation is tested on h-BN monolayers and bilayers, on fluoro-polyacetylene, and on CO adsorbed on MgO(001). For the latter, the authors report a counterpoise-corrected RPA@PBE adsorption energy of -1.67 +/- 0.31 kcal/mol in the low-coverage limit, which they describe as in excellent agreement with the earlier periodic RPA@PBE value of Bajdich et al. (-1.65 kcal/mol) and as less negative than experiment and embedded CCSD(T) results.

Significance. If the claims hold, the paper provides a useful and efficient canonical RPA implementation for two-dimensional and slab systems in a localized basis, with three notable strengths: (i) the k-grid convergence is rapid and shown to be nearly independent of the primary basis set, which is practically valuable; (ii) the AUTO damping criterion removes a user-tuned parameter for the Coulomb truncation, and the dual-grid scheme demonstrably reduces the Gamma-point sampling error; and (iii) the parallelization benchmark shows near-perfect strong scaling to thousands of cores, which is a practical advantage for production calculations. The convergence tests are systematic, spanning k-grids, damping radii, primary and auxiliary basis sets, projector thresholds, and coverage. However, the final application to CO/MgO carries a headline uncertainty that omits several documented sources of error, and the CBS extrapolation rests on a three-point linear fit; these issues affect the reliability of the stated final result but do not invalidate the methodological contribution.

major comments (3)
  1. [Sec. 4.6.3, Eq. (34)] The uncertainty budget for the final adsorption energy is incomplete. The text states that 'the only source of error in this number stems from the basis set extrapolation' and combines only the CBS intercept error (0.29 kcal/mol) and the coverage correction error (0.12 kcal/mol). However, Table 4 shows that changing the projector threshold epsilon_d from 1e-3 to 5e-4 shifts the RPA contribution to the adsorption energy by 0.02 kcal/mol, while going to 1e-4 shifts it by 0.18 kcal/mol (from -7.53 to -7.35 kcal/mol). Table 3 shows the QZ4P auxiliary-basis result is not fully converged even at T4, with the relative correlation energy changing by 0.20 kcal/mol between T3 and T4. These documented, quantifiable sensitivities are not included in the reported +/- 0.31 kcal/mol error bar, so the stated uncertainty under-represents the known uncertainty of the final number.
  2. [Sec. 4.5.1 and Fig. 11] The complete-basis extrapolation is based on a linear fit of the RPA adsorption-energy contribution against the inverse number of basis functions using only three points (DZP, TZ2P, QZ4P). The extrapolated value (-9.25 +/- 0.29 kcal/mol) lies 0.89 kcal/mol below the largest-basis computed value (-8.36 kcal/mol), and the reported uncertainty is the intercept standard error of that three-point fit. This does not account for the model error of the assumed 1/N_bas linear form, nor for the residual auxiliary-basis incompleteness at the QZ4P/T4 point. The authors should either provide additional evidence for the extrapolation law (e.g., a four-point fit or a second functional form) or enlarge the error bar to reflect the sensitivity of the intercept to the choice of fitting points.
  3. [Sec. 4.5, text near Eqs. (29)-(30)] The paper correctly acknowledges that the projector method introduces an effect beyond conventional BSSE: the number of primary basis functions projected out in a subsystem depends on whether the other subsystem's basis is present. This means the counterpoise correction of Eq. (29) may not fully remove the basis-set superposition effect. The magnitude of this effect is, however, not quantified or bounded anywhere in the manuscript, and it is not included in the final uncertainty. Given that the final adsorption energy is a delicate balance between large absolute correlation energies (about 17 kcal/mol shifts in Table 4), the authors should estimate this contribution, for example by comparing projector thresholds in the dimer and monomer calculations, or at least justify why it is negligible at the CBS limit.
minor comments (6)
  1. [Abstract and Conclusions] The abstract and conclusions describe the agreement with Bajdich et al. as 'excellent', while the introduction says 'good agreement'. The quantitative basis for the word 'excellent' should be stated or the wording made consistent.
  2. [Sec. 4.3, Fig. 6 caption] In the figure caption, 'with respect to ts values calculated with a 17 x 17 k-mesh' appears to contain a typo; it should likely read 'its values calculated with a 17 x 17 k-mesh'.
  3. [Sec. 2.3, Eqs. (20)-(22)] The definition of the cumulative projector in Eq. (21) is unclear because the R(k) matrices are defined in the k-dependent eigenbases of S(k). Please specify the common basis in which the matrix product over k is taken, or clarify how the k-dependent matrices are combined.
  4. [Sec. 4.5.1, paragraph on TZ2P adjustment] The sentence 'the TZ2P results are adjusted for the resulting fit error of 0.14 kcal/mol' does not specify whether this adjustment is applied to the TZ2P entries in Table 3 or to the CBS extrapolation in Fig. 11. Please state where the adjustment enters.
  5. [Sec. 4.6.2, Table 5] The table does not report the k-grid used for the 50% and 25% coverage calculations. For reproducibility, the k-grid for each coverage entry should be stated.
  6. [Sec. 4.6.3, Eq. (33)] The meaning of the terms EQZ4P_HF, EQZ4P_PBE, and EQZ4P_XC@PBE in Eq. (33) should be defined explicitly, in particular whether these are total energies or adsorption-energy contributions and how the signs are chosen.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the implementation and benchmarks are self-contained, with heuristic parameters tested over ranges and the final adsorption energy compared against independent RPA@PBE, CCSD(T), and experimental values.

full rationale

The paper's central derivation starts from the standard ACFDT RPA expression in Eq. (4) and develops the periodic PADF-based matrix elements in Eqs. (14)-(17) without reintroducing the target adsorption energy as an input. The dual-grid and Fermi-Dirac damping parameters are presented as heuristic choices (Secs. 3.1-3.2), and their effect is assessed over parameter ranges: Fig. 6b shows convergence to the same thermodynamic limit for different r0 values, which is an internal consistency check rather than a fit to the final answer. The auxiliary basis and projector threshold sensitivities are documented explicitly in Tables 3 and 4, including the large absolute correlation energy shifts with epsilon_d, and the adsorption-energy stability is reported rather than hidden. The final CO/MgO adsorption energy in Eq. (34) combines a CBS-extrapolated RPA contribution with separately computed HF and PBE terms, and is then compared with the independent periodic RPA@PBE result of Bajdich et al., embedded CCSD(T), and experiment; no parameter is fitted to those benchmarks. The self-citations to the authors' prior PADF work (Refs. 123, 126) are not load-bearing circularity: the PADF approximation is re-derived in the paper's own equations, its auxiliary-basis convergence is tested here, and the method is independently benchmarked against external results. The documented projector-threshold and auxiliary-basis sensitivities are a legitimate uncertainty/correctness concern, but they do not make the derivation circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The method's central claims rest on heuristic choices (r0, beta, d, epsilon_d) and on error cancellation between subsystems. No new physical entities are introduced. The CBS extrapolation is a numerical fitting step that contributes directly to the final adsorption energy.

free parameters (5)
  • r0 (Coulomb damping radius) = 0.5 Rc default; 0.3-0.7 Rc tested
    Defines where the Fermi-Dirac damping attenuates the Coulomb interaction. Chosen relative to the largest inscribed circle in the real-space Nyquist grid (Sec. 3.1).
  • beta (Fermi-Dirac decay parameter) = set so damping reaches 0.1% at 1.4 r0
    Controls smoothness of the Coulomb truncation; used in production runs (Sec. 3.1).
  • d (dual-grid displacement) = 0.1 of the regular q-grid step
    Additional q-points near Gamma; chosen to resolve the integrable Coulomb singularity (Sec. 3.2).
  • epsilon_d (projector threshold) = 1e-3
    Removes linearly dependent basis combinations; absolute RPA correlation energies depend strongly on this value (Table 4).
  • CBS extrapolation intercept = -9.25 ± 0.29 kcal/mol
    Linear fit in inverse basis size over DZP, TZ2P, QZ4P; used for the final adsorption energy (Fig. 11).
assumptions (6)
  • domain assumption The PADF expansion of orbital pair products (Eq. 9) is accurate enough for RPA correlation energies with the chosen auxiliary sets.
    Central to Eqs. 14-17; Table 3 shows strong auxiliary-basis dependence for QZ4P, with T1 and T2 not converged.
  • domain assumption Born-von Karman boundary conditions and the Nyquist cell contain the significant range of the Coulomb interaction once damping is applied.
    The Fermi-Dirac damping restricts interactions to the finite real-space grid (Sec. 3.1); if this range is too small, long-range correlation is truncated.
  • domain assumption The system is spin-compensated with fully occupied valence bands and no partial occupations.
    Eq. (15) uses occupation factors of 2 or 0; metals and spin polarization are excluded.
  • standard math The ACFDT/RPA correlation energy expression (Eq. 4) and the completeness of the auxiliary basis for the Coulomb and polarizability operators are valid.
    Standard result (Refs. 1,11); the appendix notes that practical incompleteness introduces an error.
  • ad hoc to paper Linear extrapolation in the inverse number of basis functions gives the complete-basis limit.
    Assumed in Sec. 4.5.1 and Fig. 11; based on only three basis-set points with no formal justification.
  • domain assumption The projector in Eq. (22) removes the same linear combinations at all k-points without biasing correlation energies.
    Sec. 2.3; Table 4 shows the sensitivity of absolute energies to the threshold.

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Pith. "Pith review of Periodic implementation of the random phase approximation with numerical atomic orbitals and dual reciprocal space grids." pith.science (2026). https://pith.science/paper/G4REJGTX

@misc{pith2026250506021,
  author       = {Pith},
  title        = {Pith review of: Periodic implementation of the random phase approximation with numerical atomic orbitals and dual reciprocal space grids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4REJGTX}},
  note         = {Machine review of arXiv:2505.06021}
}
abstract

The random phase approximation (RPA) has emerged as a prominent first-principles method in material science, particularly to study the adsorption and chemisorption of small molecules on surfaces. However, its widespread application is hampered by its relatively high computational cost. Here, we present a well-parallelised implementation of the RPA with localised atomic orbitals and pair-atomic density fitting, which is especially suitable for studying two-dimensional systems. Through a dual $\textbf{k}$-grid scheme, we achieve fast and reliable convergence of RPA correlation energies to the thermodynamic limit. We demonstrate the efficacy of our implementation through an application to the adsorption of CO on MgO(001) using PBE input orbitals (RPA@PBE) Our calculated adsorption energy is in good agreement with previously published RPA@PBE studies, but, as expected, overestimates the experimentally available adsorption energies as well as recent CCSD(T) results.

Figures

Figures reproduced from arXiv: 2505.06021 by the authors.

Figure 1
Figure 1. Pictorial representation of the Rx grid. Each grey cell represents a primitive cell. The whole grid contains 15 × 15 unit cells, which can be represented by 15 × 15 points in the k grid. The red circle represents the maximum circle that can fit in this grid and the blue shadow is the damping function for the Coulomb potential. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. In (A) we show the QG grid. The additional sampling near Γ is represented in blue, the neglected Γ point with a red cross, and the other regular points in black. In (B) we show the KG grid. The regular set Kreg, which is sampled at first is represented in green, in black all the other points which are added accordingly to QG. Using (27) from green points in (B) with black points in (A) will transform to a green poin… view at source ↗
Figure 3
Figure 3. Schematic overview of the parallelization strategy adopted in our implementation. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Effect of Fermi–Dirac damping on RPA@PBE correlation energy convergence for [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: in (A) Fluoro-polyacetylene geometry with its RPA correlation energy integrand [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Absolute deviation of the RPA contribution to the adsorption energy with respect [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Differences between the RPA contributions to the adsorption energy for the h [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Percentage speed up of RPA correlation energy calculation for fluoro-poliacetylene. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Percentage speed up of RPA correlation energy calculation by reducing the [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Geometry of Carbon Oxide on MgO adsorption for 100% coverage case [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Inverse linear fit of the RPA contribution to the adsorption energy in kcal/mol [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: RPA@PBE contribution to adsorption energy in kcal mol [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.