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arxiv: 2603.27292 · v2 · submitted 2026-03-28 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Low-scaling textit{GW} calculations of quasi-particle energies for extended systems within the numerical atomic orbital framework

Min-Ye Zhang , Peize Lin , Rong Shi , Xinguo Ren
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:53 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords GW approximationlow-scaling algorithmnumerical atomic orbitalsquasi-particle energieslocalized resolution of identityspace-time methodpolarization functionself-energy
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The pith

Space-time algorithm in numerical atomic orbitals reduces GW scaling to O(N^2) for quasi-particle energies.

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

The paper develops a low-scaling GW method for computing quasi-particle energies in extended periodic systems. It performs the polarization function and self-energy evaluations in real space within the numerical atomic orbital basis, using the localized resolution of identity to achieve formal O(N^2) scaling or better instead of the usual O(N^4). Benchmarks on crystalline solids show that the resulting energies agree closely with those from the conventional high-scaling k-space approach. The implementation already delivers lower overall cost for systems containing fewer than 100 atoms.

Core claim

By computing the polarization function under the random phase approximation and the GW self-energy in real space with the localized resolution of identity technique inside the numerical atomic orbital framework, the scaling of the rate-limiting steps is formally reduced to O(N^2) with respect to system size, while the quasi-particle energies remain in close agreement with those obtained from the conventional O(N^4) k-space formalism for the crystalline solids examined.

What carries the argument

The localized resolution of identity technique applied to real-space evaluation of the polarization function and self-energy inside the numerical atomic orbital basis.

If this is right

  • GW quasi-particle energies become accessible for systems with hundreds of atoms at substantially lower cost than before.
  • The observed scaling reduction makes the approach faster than the canonical method already below 100 atoms.
  • Periodic solids can be treated with accuracy comparable to the established high-scaling implementation.
  • Larger-scale simulations of material electronic structures become practical without changing the underlying GW approximation.

Where Pith is reading between the lines

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

  • The real-space formulation may extend naturally to supercell models containing defects or interfaces.
  • Further gains could arise by combining the approach with existing low-scaling techniques for the underlying density-functional calculation.
  • Time-dependent generalizations might enable efficient spectra calculations for similarly sized systems.

Load-bearing premise

The localized resolution of identity technique must preserve enough accuracy for the polarization function and self-energy when used on extended periodic systems in the numerical atomic orbital basis.

What would settle it

A benchmark on a crystalline solid with roughly 50 atoms that shows quasi-particle energies deviating substantially from the conventional O(N^4) k-space results would disprove the accuracy preservation.

Figures

Figures reproduced from arXiv: 2603.27292 by Min-Ye Zhang, Peize Lin, Rong Shi, Xinguo Ren.

Figure 1
Figure 1. Figure 1: Schematic diagrams of the contributions from an atom quartet [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagrams of the contributions from an atom quartet [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagrams of the rearranged terms in Eq. (47), illustrating how a [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Flowchart of the low-scaling G0W0 implementation in LibRPA. The region enclosed by the red dashed lines indicates the internal workflow in LibRPA for computing the quasi￾particle energies ϵ QP nk . The quantities shown in blue blocks are input data provided by an external first-principles code, which is FHI-aims 57 in the present work. “AC” denotes the analytic continuation discussed in Sec. 2.4.4. plement… view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the fundamental band gaps of eight selected semiconductors and [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: G0W0@PBE quasi-particle band structures of (a) Si and (b) MgO computed by the canonical implementation in FHI-aims (black) and the low-scaling implementation in LibRPA (red). Dots indicate the quasi-particle energies evaluated on the uniform k-grid. The FHI-aims calculations use 60 modified Gauss-Legendre grid points, while the LibRPA calculations use 32 minimax time/frequency points. For both materials, t… view at source ↗
Figure 7
Figure 7. Figure 7: Error in the G0W0@PBE fundamental gap Eg of diamond, computed with the low-scaling algorithm using different filtering thresholds (red triangles, in meV): (a) RI co￾efficients ηC and (b) Green’s function ηG for the response function χ 0 ; (c) RI coefficients ηC, (d) screened Coulomb matrix ηW , and (e) Green’s function ηG for the correlation self-energy Σ c . The result obtained with the smallest tested th… view at source ↗
Figure 8
Figure 8. Figure 8: Wall-time scaling (in core hours) of G0W0@PBE calculations for silicon with respect to the number of k points in the first Brillouin zone, comparing the canonical imple￾mentation in FHI-aims (black) and the low-scaling implementation in LibRPA (red). The reported wall time includes the computation of χ 0 , the construction of Wc from χ 0 , and the evaluation of Σc . The intermediate gw species default from… view at source ↗
Figure 9
Figure 9. Figure 9: Wall-time scaling (in core hours) of G0W0@PBE calculations for diamond with respect to the number of atoms in the unit cell, comparing the canonical implementation in FHI-aims (black circles) and the low-scaling implementation in LibRPA (red triangles). The reported wall times correspond to (a) χ 0 , (b) the construction of Wc from χ 0 , and (c) Σc . The light species default and a single Γ point are used.… view at source ↗
Figure 10
Figure 10. Figure 10: Same as Fig. 9, but for the total computer time summed over the three compo [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the strong scaling of LibRPA for diamond supercells of carbon containing 256 and 512 atoms. For the 256-atom supercell, the calculation with 24 threads per MPI task scales up to 81 nodes (7776 cores), yielding a speedup of 16.7 relative to the single-node run and a parallel efficiency of 20.6%. With 48 threads per MPI task, the same system scales up to 128 nodes (12288 cores), with a speedup of 12.8… view at source ↗
read the original abstract

The many-body perturbation theory within the $GW$ approximation is a widely used method for describing the electronic band structures in real materials. Its application to large-scale systems is, however, impeded by its high computational cost. The rate-limiting steps in a typical $GW$ implementation are the evaluation of the polarization function under the random phase approximation (RPA) and the evaluation of the $GW$ self-energy, both of which have a canonical $O(N^4)$ scaling with $N$ being the system size. The conventional space-time algorithm within the plane-wave basis sets reduces the scaling from $O(N^4)$ to $O(N^3)$, albeit with a large prefactor and increased memory cost. Here, we present a space-time algorithm within the numerical atomic orbital (NAO) basis-set framework, for which the evaluation of the polarization function and self-energy is formally reduced to $O(N^2)$ or better with respect to system size. This is achieved by computing these quantities in real space, where low-scaling algorithms can be formulated by leveraging the localized resolution of identity (LRI) technique. The resulting NAO-based, LRI-enhanced space-time $GW$ algorithm has been implemented in the LibRPA library interfaced with the FHI-aims code package. Benchmark calculations for crystalline solids show that the low-scaling implementation yields quasi-particle energies in close agreement with the conventional $O(N^4)$ k-space formalism previously implemented in FHI-aims. For the systems studied here, the observed overall scaling is substantially reduced relative to the canonical approach, and the low-scaling implementation becomes advantageous already for systems containing fewer than 100 atoms.

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 / 0 minor

Summary. The manuscript presents a space-time GW algorithm implemented in the numerical atomic orbital (NAO) basis within the FHI-aims code, using the localized resolution of identity (LRI) to evaluate the RPA polarization function and self-energy in real space. This yields a formal reduction to O(N^2) scaling or better, in contrast to the canonical O(N^4) k-space approach. Benchmarks on crystalline solids are reported to produce quasi-particle energies in close agreement with the existing O(N^4) implementation, with the low-scaling version becoming advantageous for systems containing fewer than 100 atoms.

Significance. If the reported accuracy is robust, the work would provide a practical route to GW calculations on larger periodic systems in materials science, where conventional scaling has been prohibitive. The implementation in an established all-electron code and the explicit comparison to a prior k-space method are strengths that facilitate adoption.

major comments (2)
  1. [Abstract] Abstract and benchmark section: the central claim of 'close agreement' with the O(N^4) k-space formalism and the assertion that the low-scaling method becomes advantageous below 100 atoms are not supported by quantitative error metrics, mean absolute deviations, or explicit system-size scaling plots. Without these, the scaling advantage and accuracy cannot be assessed rigorously.
  2. [Method / Results] The accuracy for extended periodic systems rests on the LRI expansion of the Coulomb operator and the real-space cutoff applied to the polarization function P(r,r',iω). The manuscript does not specify how the cutoff radius is chosen relative to supercell size or demonstrate that truncation errors remain bounded (rather than scaling with surface area) as N increases, which is required to ensure convergence to the same thermodynamic limit as the k-space method given the slow 1/r decay of dielectric screening.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. We address the major comments below and plan to revise the manuscript to incorporate the suggested improvements where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract and benchmark section: the central claim of 'close agreement' with the O(N^4) k-space formalism and the assertion that the low-scaling method becomes advantageous below 100 atoms are not supported by quantitative error metrics, mean absolute deviations, or explicit system-size scaling plots. Without these, the scaling advantage and accuracy cannot be assessed rigorously.

    Authors: We agree that providing quantitative error metrics and scaling plots would make the claims more rigorous. In the revised version, we will add mean absolute deviations (MAD) between the low-scaling and conventional GW quasi-particle energies for the benchmark systems. Additionally, we will include explicit plots of computational time versus system size (number of atoms) to demonstrate the scaling behavior and the point at which the low-scaling method becomes advantageous (around 100 atoms). These additions will be placed in the benchmark section and referenced in the abstract if necessary. revision: yes

  2. Referee: [Method / Results] The accuracy for extended periodic systems rests on the LRI expansion of the Coulomb operator and the real-space cutoff applied to the polarization function P(r,r',iω). The manuscript does not specify how the cutoff radius is chosen relative to supercell size or demonstrate that truncation errors remain bounded (rather than scaling with surface area) as N increases, which is required to ensure convergence to the same thermodynamic limit as the k-space method given the slow 1/r decay of dielectric screening.

    Authors: We acknowledge the importance of this point for ensuring the method's reliability in the thermodynamic limit. The manuscript will be revised to specify the cutoff radius selection criterion, which is based on the localization of the NAO and LRI functions and set to ensure that the polarization function is captured within the supercell with a convergence threshold. Regarding the boundedness of truncation errors, we will include additional numerical evidence from calculations on progressively larger supercells, demonstrating that the deviation from k-space results remains small and does not increase with system size. While a rigorous mathematical proof of bounded errors for all N is challenging due to the long-range nature of screening, the practical convergence observed supports the applicability to extended systems. revision: partial

Circularity Check

0 steps flagged

No circularity: algorithmic reformulation with independent benchmarks

full rationale

The paper derives a space-time GW algorithm in the NAO basis by moving polarization and self-energy evaluations to real space and applying the LRI technique for low-scaling matrix operations. This yields a formal O(N^2) scaling claim that is implemented and then benchmarked numerically against the authors' own prior O(N^4) k-space code on crystalline solids. No equation reduces a result to a fitted parameter or to a self-citation by construction; the agreement with the conventional implementation is an external numerical test rather than a definitional identity. The LRI step is presented as a standard technique leveraged for the new implementation, not as an ansatz smuggled in or a uniqueness theorem imported from the same authors' prior work. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard GW/RPA assumptions plus the domain-specific claim that LRI localization errors remain controllable for periodic solids.

axioms (1)
  • domain assumption The localized resolution of identity (LRI) approximation is sufficiently accurate for the polarization function and self-energy in NAO-based GW calculations of extended systems.
    Invoked to justify the formal O(N^2) reduction without accuracy loss.

pith-pipeline@v0.9.0 · 5627 in / 1143 out tokens · 39857 ms · 2026-05-14T21:53:16.416659+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $G^0W^0$ implementation based on the pseudopotential and numerical-atomic-orbital basis-set framework: Algorithms and benchmarks

    cond-mat.mtrl-sci 2026-05 unverdicted novelty 7.0

    An efficient G0W0 framework is implemented in the NAO-PP basis via ABACUS+LibRPA with a novel LRI compression scheme, showing agreement with established codes on band structures and gaps.

Reference graph

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