Efficient $G_0W_0$ and BSE calculations of heterostructures within an all-electron framework

Claudia Draxl; Ignacio Gonzalez Oliva; Maximilian Schebek

Efficient $G_0W_0$ and BSE calculations of heterostructures within an all-electron framework

T0 review · 0 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The expansion-and-addition screening method, ported to the mixed-product basis of (L)APW all-electron codes, reproduces exact $G_0W_0$ and BSE results for weakly bound heterostructures at a fraction of the polarizability cost.

desk verdict A careful, well-validated extension of EAS to all-electron (L)APW with clear derivation; the boundary assumptions are honestly stated, so the paper deserves peer review. read the letter →

arxiv 2507.17960 v1 pith:VLAO7XC3 submitted 2025-07-23 cond-mat.mtrl-sci

Maximilian Schebek , Ignacio Gonzalez Oliva , Claudia Draxl This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords G0W0Bethe-Salpeterequationexpansionandadditionscreening(EAS)mixed-productbasis(L)APWall-electronmethodpolarizabilityvanderWaalsheterostructurestwo-dimensionalmaterials

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

All-electron $G_0W_0$ and Bethe-Salpeter calculations of van der Waals heterostructures are normally very expensive because building the polarizability of the combined supercell grows steeply with atom number. This paper establishes that for weakly bound stacks the polarizability can be treated additively: each component is computed in its own small unit cell, folded into the heterostructure supercell, and then summed. The authors derive the folding and summation transformations for the mixed-product basis of the (L)APW all-electron method and implement them in both the $G_0W_0$ and BSE screening steps. Benchmarks on bilayer WSe$_2$ and pyridine@MoS$_2$ reproduce exact fundamental gaps within $0.04$ eV and optical gaps within $0.02$ eV while cutting the polarizability cost by roughly half to two-thirds. This makes reference-quality all-electron calculations of much larger interfaces feasible.

What carries the argument

The machinery is the EAS identity $P^{HS}=P^1+P^2$ together with two basis-aware steps. In the expansion step, a polarizability computed in a component's smallest unit cell is folded into the heterostructure supercell using the mapping of reciprocal vectors; in the addition step, the two expanded polarizabilities are superposed by transforming each into the heterostructure's basis via overlap matrices $C^\mu_{aA}$. The mixed-product basis makes this nontrivial: inside muffin-tin spheres it consists of radial functions times spherical harmonics, while in the interstitial region it consists of orthonormalized planewave combinations. The paper shows how each block of the polarizability matrix—muffin-tin, interstitial, and mixed—must be transformed separately, and how the disjointness of the two muffin-tin regions simplifies the addition to a small set of block expressions.

What would settle it

Take a heterostructure whose geometry is optimized after stacking and compute the exact polarizability matrix blocks; compare them with the EAS sum $P^1+P^2$ and with exact $G_0W_0$ gaps. If the difference in fundamental gaps exceeds the roughly $0.04$ eV seen here, the additivity premise fails.

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Extended reading notes

Core claim

The central claim is that the expansion-and-addition screening (EAS) ansatz—$P^{HS}=P^1+P^2$, with each component polarizability first folded from its primitive cell to the heterostructure supercell—can be implemented exactly in the mixed-product basis used by all-electron (L)APW codes. The paper derives the required transformations blockwise: muffin-tin matrix elements are folded through a real-to-reciprocal-space sum, interstitial elements through the planewave mapping $\mathbf{q}_{uc}+\mathbf{G}_{uc}=\mathbf{q}+\mathbf{G}$, and mixed blocks by applying the corresponding transformation to rows or columns. For the addition step, the component polarizabilities are transformed into the heterostructure basis with overlap matrices $C^\mu_{aA}=\langle\chi^\mu_a|\chi_A\rangle$, using the fact that muffin-tin spheres of different components are disjoint and their radial functions are assumed identical. The same additive screening is used for the electron-hole interaction in BSE as for $W$ in $G_0W_0$. On bilayer WSe$_2$ and pyridine@MoS$_2$, the method reproduces the exact quasi-particle and optical spectra: fundamental gaps of 2.39 vs 2.41 eV and 2.32 vs 2.36 eV, and optical gaps of 2.63 vs 2.64 eV and 2.00 vs 2.02 eV.

Load-bearing premise

The whole construction rests on the assumption that a heterostructure's polarizability is exactly the sum of its two components' polarizabilities, with atoms sitting at exact replica positions and identical atom-centered basis functions used in both calculations.

Editorial extensions

If this is right

The polarizability step for pyridine@MoS$_2$ drops from 3511 to 1545 CPU hours in $G_0W_0$ and from 9462 to 2962 CPU hours in BSE, so weakly bound interfaces with large supercells become tractable in all-electron reference calculations.
Optical absorption spectra inherit the screening from the same additive construction, so BSE exciton physics, not just quasi-particle gaps, is reproduced to within tens of meV.
Because the expansion step is exact only for exact replicas, one primitive-cell polarizability can be reused for many heterostructures containing the same unrelaxed component, amortizing the cost across a family of interfaces.
All-electron benchmarks can now target heterostructures too large for direct $G_0W_0$ or BSE reference runs, giving pseudopotential and planewave results a more demanding comparison point.

Reading between the lines

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

A natural stress test would be a heterostructure that is relaxed after stacking or binds more strongly; there the exact-replica and additivity assumptions weaken, so the tens-of-meV agreement should degrade, and quantifying this would map the method's validity boundary.
Since only the polarizability is accelerated, the self-energy and BSE Hamiltonian still dominate; combining EAS with low-rank factorization of those steps could bring total wall-time reductions far beyond the 25–46% reported here.
The block structure of the addition step should extend recursively to three or more components, so multilayer stacks and multi-molecule hybrid interfaces could reuse the same component polarizabilities without additional screening calculations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The genuinely new thing here is the translation of the EAS polarizability trick from planewave codes to the mixed-product basis of the (L)APW method. That is not trivial because the basis splits into muffin-tin and interstitial pieces, and the expansion and addition steps need separate treatments. The authors derive the transformations in detail (Section III and Appendix B) and implement them in the exciting code. They also apply the same additive idea to BSE, which is a new application. The validation is solid: WSe2 bilayer fundamental gap 2.39 vs 2.41 eV exact, pyridine@MoS2 gap 2.32 vs 2.36 eV, optical gaps within 0.02 eV. Those numbers support the claim that EAS works for weakly bound heterostructures.

What the paper does well: the derivation is self-contained, the limitations are stated explicitly (expansion exact only for exact replicas, additive ansatz assumes weak coupling), and the CPU-hour table gives a realistic picture—the polarizability part is cut by more than half, but total time by only a quarter to a half. That is honest.

The soft spots are real but proportionate. The additive ansatz and the assumption that radial muffin-tin functions are the same in the components and the heterostructure are untested for relaxed geometries or stronger interlayer hybridization. The pyridine@MoS2 test uses MoS2 fixed at ideal lattice sites, which satisfies the exact-replica condition but is not the typical practical use case. The paper does not estimate how errors grow when the structure is relaxed. That is a boundary-of-applicability gap, not an internal inconsistency. I would ask the authors to add a paragraph on this and maybe a test with a slightly relaxed geometry, but it does not undermine the demonstrated results.

A minor point: no input files or commit hash are provided, which makes reproduction harder. That is common for methods papers in this field, so not a blocker.

The central claim holds within the demonstrated regime. The citation pattern looks appropriate, and there is no circularity: EAS is an approximation, not a target quantity, and the validation uses independent exact references. The paper is a useful contribution for anyone doing all-electron GW/BSE on heterostructures, especially for benchmarking.

This deserves a serious referee. I would send it to peer review and suggest minor revision: add a discussion of error behavior outside the tested regime and provide reproducibility artifacts if possible. Read it if you work on all-electron many-body methods; otherwise it is still a clean methods paper worth skimming.

Referee Report

0 major / 4 minor

Summary. The paper extends the expansion-and-addition screening (EAS) method, previously developed in a planewave representation, to the mixed-product basis used in the all-electron (L)APW method. It derives the basis transformations for both the expansion step (folding a component's polarizability from its unit cell to a supercell) and the addition step (superposing the polarizabilities of two components), and implements them in the G0W0 and BSE modules of the exciting code. The method is validated on bilayer WSe2 and pyridine@MoS2 against exact reference calculations, showing quasi-particle gap differences of at most 0.04 eV and optical-gap differences of at most 0.02 eV, with reductions in total runtime of about 25% for G0W0 and 46% for BSE.

Significance. If the reported accuracy holds, this is a valuable contribution: it transfers the EAS idea to an all-electron framework, where the mixed-product basis makes the addition and expansion steps nontrivial, and it provides a way to reduce the O(N^4) polarizability cost for weakly bound heterostructures. The paper is commendable for deriving the transformation matrices in Appendix B, for validating against independent exact calculations, and for explicitly stating the exact-replica condition for the expansion step. The main limitation is that the demonstrated accuracy covers the weakly bound, unrelaxed (or replica-exact) regime; the authors acknowledge this, and it does not undermine the stated central claim.

minor comments (4)

[IV B / Table I] The sentence 'This results in a total runtime of 2962 CPU hours, corresponding to a reduction of almost 70% of the computational time' refers only to the polarizability step; the total BSE runtime is reduced by 46% (7661 vs 14161 CPU hours). Please reword to avoid implying a 70% total reduction.
[V] The final sentence of the Conclusions is broader than the validated regime: 'heterostructures much larger than those considered so far' invites application to relaxed geometries, for which the expansion step is not exact. Please add a caveat or soften the sentence.
[III A] The derivation of the MT-MT expansion formula, Eq. (9), is very terse; the origin of the prefactor N_sc/N_uc and the phase factor in the second equality is not self-evident. Please expand the derivation or provide a reference.
[IV A] There are minor typos: 'the the exact calculation' should be 'the exact calculation' and 'the obtained result show' should be 'the obtained results show'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EAS extension is an approximation validated against independent exact reference calculations, with no fitted parameters or self-referential definitions.

full rationale

The paper's central claim is that the additive ansatz P_HS = P1 + P2 can be transported from planewave to (L)APW mixed-product basis through expansion and addition transformations. The derivation in Sections III A and III B is self-contained: Eq. 9 maps unit-cell polarizability to supercell by a real-to-reciprocal-space folding, Eq. 12 transforms between mixed-product and planewave representations, and Eq. 14 defines the addition via overlap matrices C. These are not defined in terms of the target G0W0/BSE results. The additive ansatz itself is an approximation, explicitly delimited in Section IV B by the statement that 'the expansion step is only exact if the supercell is an exact replica of the unit cell,' not a definition of the heterostructure polarizability. The validation uses exact reference calculations that are independent of the EAS workflow, and no fitted input is renamed as a prediction; the quantitative agreement (2.39 vs 2.41 eV and 2.32 vs 2.36 eV) is a genuine comparison. The only mild concern is the assumption that radial functions are identical between component and heterostructure (Section III B), but this is a modeling approximation, not a circular reduction. Self-citations, such as Refs. [9], [19], [26], [31], and [32], provide background or prior data and are not load-bearing for the derivation. Consequently, there is no circular step to report.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the additive screening approximation, which is a physical assumption about weak interlayer coupling rather than a fitted parameter. The expansion step additionally assumes the supercell is an exact replica of the unit cell. No new physical entities or free parameters are introduced.

assumptions (3)

domain assumption The polarizability of the heterostructure equals the sum of the individual component polarizabilities, P = P1 + P2.
Neglects interlayer screening and hybridization; valid only for weakly bound systems (Section II A, Eq. 4).
domain assumption The radial functions in the muffin-tin spheres of the individual components are the same as those in the heterostructure.
Used to set C^{MTmu/MTnu} = delta_{munu} in the addition step (Section III B, Eq. 18 and surrounding text).
domain assumption The supercell is an exact replica of the unit cell for each component (no geometry relaxation).
The expansion step is only exact under this condition, as noted for pyridine@MoS2 (Section IV B).

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Cite this review

Pith. "Pith review of Efficient $G_0W_0$ and BSE calculations of heterostructures within an all-electron framework." pith.science (2026). https://pith.science/paper/VLAO7XC3

@misc{pith2026250717960,
  author       = {Pith},
  title        = {Pith review of: Efficient $G_0W_0$ and BSE calculations of heterostructures within an all-electron framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLAO7XC3}},
  note         = {Machine review of arXiv:2507.17960}
}

abstract

The combination of two-dimensional materials into heterostructures offers new opportunities for the design of optoelectronic devices with tunable properties. However, computing electronic and optical properties of such systems using state-of-the-art methodology is challenging due to their large unit cells. This is in particular so for highly-precise all-electron calculations within the framework of many-body perturbation theory, which come with high computational costs. Here, we extend an approach that allows for the efficient calculation of the non-interacting polarizability, previously developed for planewave basis sets, to the (linearized) augmented planewave (L)APW method. This approach is based on an additive ansatz, which computes and superposes the polarizabilities of the individual components in their respective unit cells. We implement this formalism in the $G_0W_0$ module of the exciting code and implement an analogous approach for BSE calculations. This allows the calculation of highly-precise optical spectra at low cost. So-obtained results of the quasi-particle band structure and optical spectra are demonstrated for bilayer WSe$_2$ and pyridine@MoS$_2$ in comparison with exact reference calculations.

Figures

Figures reproduced from arXiv: 2507.17960 by the authors.

**Figure 2.** FIG. 2. Top left: schematic representation of how to compute the additive screening, for the example of the WSe [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Top left: schematic representation of how to compute the EAS for the example of pyridine@MoS [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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