pith. sign in

arxiv: 2605.19746 · v1 · pith:S5YHAR7Hnew · submitted 2026-05-19 · ❄️ cond-mat.mtrl-sci

G₀W₀@HF and BSE methods in periodic systems from Hartree-Fock theory: gaussian orbital and density fitting approach

Charles H. Patterson This is my paper

Pith reviewed 2026-05-20 03:43 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords G0W0Hartree-FockGaussian orbitalsdensity fittingBethe-Salpeter equationRPA screeningquasi-particle energiesband gaps
0
0 comments X

The pith

G0W0 calculations begun from Hartree-Fock in Gaussian orbitals correct HF overestimations of band gaps and valence band widths in solids.

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

The paper develops G0W0 and BSE methods that start from a Hartree-Fock Hamiltonian rather than a DFT one and use a basis of Gaussian orbitals together with density fitting. Screening of the Coulomb interaction is obtained directly from the RPA polarizability expressed as W = v + vΠv without a plasmon-pole model, and the polarizability itself is obtained by solving Bethe-Salpeter equations at each Q point. A practical convergence scheme treats the majority of the self-energy with full G0W0 and the high-energy virtual-state tail with second-order perturbation theory. The resulting quasi-particle energies are tested on diamond, silicon, MgO, and both phases of TiO2. The calculations remove the typical HF errors of overestimated gaps and valence-band widths while the RPA part alone tends to overestimate gaps.

Core claim

G0W0 calculations initiated from the Hartree-Fock Hamiltonian in a Gaussian orbital basis, with the screened interaction obtained via W = v + vΠv in the RPA without plasmon pole approximation, and using a mixed treatment of virtual states for convergence, produce quasi-particle band structures for elemental semiconductors and transition metal oxides that correct the overestimation of band gaps and valence band widths typical of HF theory.

What carries the argument

The G0W0 self-energy evaluated from an HF starting point, with RPA polarizability Π obtained from Bethe-Salpeter equations for each Q and completed by second-order perturbation theory for the high-virtual tail, all performed with density fitting in the Coulomb metric.

If this is right

  • The RPA screened interaction alone overestimates band gaps while the full G0W0 self-energy supplies the band-width renormalization needed for agreement with experiment in diamond and silicon.
  • The same framework yields corrected band structures for MgO and both rutile and anatase TiO2.
  • Gaussian-orbital plus density-fitting implementations supply an alternative route to periodic GW calculations that avoids plane-wave expansions.
  • HF initial Hamiltonians combined with this RPA screening remove the common overestimation of both gaps and widths seen in pure HF calculations.

Where Pith is reading between the lines

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

  • The Gaussian-orbital route may scale more favorably than plane-wave methods when unit cells become large or when defects are introduced.
  • Because the starting point is HF rather than a density-functional approximation, the residual starting-point dependence of the final gaps may be smaller than in conventional G0W0@DFT workflows.
  • Extending the same density-fitting machinery to include vertex corrections beyond RPA could be tested on the same set of oxides to quantify further improvements.

Load-bearing premise

The split between full G0W0 treatment for lower virtual states and second-order perturbation theory for the high-energy tail accurately captures the remaining self-energy contribution without introducing significant uncontrolled errors.

What would settle it

A G0W0 calculation on diamond or silicon that produces a valence-band width differing substantially from the experimental value after the high-energy tail is treated at second order would show the convergence strategy fails.

Figures

Figures reproduced from arXiv: 2605.19746 by Charles H. Patterson.

Figure 1
Figure 1. Figure 1: FIG. 1. Band energy versus wave vector diagram for indirect [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Feynman diagrams which contribute to the direct part [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Feynman diagrams which contribute to the exchange pa [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Charge density at point, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagrams for matrix elements of the second order self- [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Band structures of Si and C in the diamond structure fr [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Band structures of rock salt MgO from HF and [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Band structures of anatase TiO [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Dielectric functions of diamond (left panel) and Si [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Dielectric functions of MgO from BSE-TDA calculati [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Dielectric functions of anatase (upper panel) and r [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
read the original abstract

The $GW$ method for calculating quasi-particle energies of solids commonly begin from a DFT Hamiltonian and Kohn-Sham orbitals in a plane wave basis. Screening of the coulomb interaction is implemented using the inverse dielectric function in the random phase approximation (RPA). We present $G_0W_0$ calculations which begin from the Hartree-Fock method in a basis of gaussian orbitals. The screened coulomb interaction, $W$, is obtained using a $W$ = $v$ + $v\Pi v$ approach without invoking a plasmon pole approximation. The polarizability, $\Pi$, in $W$ is treated at the RPA level. RPA polarizabilities require solution of Bethe-Salpeter equations (BSE) for each unique $\textbf{Q}$ point. A strategy for obtaining self-energies which are converged with respect to number of virtual states is employed in which $G_0W_0$ yields the majority of the self-energy and the remaining part from high energy virtual levels is evaluated at second-order. The methods are evaluated by applying them to elemental semiconductors (C, Si) and oxides (MgO and anatase and rutile TiO$_2$). Common errors of HF theory applied to materials include overestimation of both the band gap and valence band widths. These are corrected in the approach employed here. Typically, the RPA screened interaction results in overestimation of band gaps while the $G_0W_0$ self-energy band width renormalization yields band widths for diamond and Si which are in good agreement with experiment. HF calculations are performed in gaussian orbital basis sets and $G_0W_0$ and BSE calculations are performed using density fitting with a coulomb metric.

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

1 major / 0 minor

Summary. The manuscript presents G0W0@HF calculations for periodic systems in a Gaussian orbital basis with density fitting. Screening is obtained from RPA polarizabilities via BSE solution for each Q-point without plasmon-pole approximation. A hybrid virtual-state convergence strategy is introduced in which full G0W0 is applied to lower virtual states while the high-energy tail is treated at bare second-order perturbation theory. The method is applied to C, Si, MgO, and TiO2 (anatase/rutile), with the central claim that HF overestimations of band gaps and valence-band widths are corrected and that G0W0 self-energy renormalization produces band widths for diamond and Si in good agreement with experiment.

Significance. If the hybrid convergence strategy is shown to be numerically controlled, the work supplies a practical Gaussian-basis route to G0W0 starting from Hartree-Fock rather than DFT, which may be advantageous for systems where the HF reference is preferred or where localized orbitals simplify the calculation. The direct comparison of computed band widths to experiment for diamond and Si is a concrete strength that supports the utility of the approach.

major comments (1)
  1. Virtual-state convergence strategy (described in the abstract and methods): the hybrid split assumes that RPA screening and higher-order diagrams become negligible above the chosen cutoff so that bare second-order perturbation theory suffices for the tail. This assumption is load-bearing for the reported self-energies and for the claim of experimental agreement on band widths; no independent validation (e.g., a fully converged reference calculation, cutoff-sensitivity table, or comparison against a larger virtual-space run) is indicated, leaving open the possibility that residual screening in the Gaussian basis still affects the tail and renders the quoted widths uncontrolled.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the potential utility of a Gaussian-orbital G0W0@HF route. We address the single major comment below and will incorporate the requested validation into the revised manuscript.

read point-by-point responses

  1. Referee: Virtual-state convergence strategy (described in the abstract and methods): the hybrid split assumes that RPA screening and higher-order diagrams become negligible above the chosen cutoff so that bare second-order perturbation theory suffices for the tail. This assumption is load-bearing for the reported self-energies and for the claim of experimental agreement on band widths; no independent validation (e.g., a fully converged reference calculation, cutoff-sensitivity table, or comparison against a larger virtual-space run) is indicated, leaving open the possibility that residual screening in the Gaussian basis still affects the tail and renders the quoted widths uncontrolled.

    Authors: We agree that explicit validation of the hybrid cutoff is necessary to substantiate the reported band widths. The strategy rests on the observation that the RPA screening contribution to the self-energy decays rapidly with virtual-state energy, so that the high-energy tail can be safely approximated by bare second-order perturbation theory. In the original submission we performed internal convergence tests to select the cutoff but did not present them. In the revision we will add a dedicated subsection (or supplementary table) showing the dependence of the quasiparticle energies and valence-band widths on the virtual-state cutoff for diamond and silicon. The table will demonstrate that the self-energy contributions from states above the chosen threshold change by less than 0.05 eV when the cutoff is increased by 50 %, thereby confirming that residual screening in the tail is negligible at the level of accuracy claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents a G0W0@HF computational scheme in Gaussian orbitals with density fitting, RPA screening via BSE without plasmon-pole approximation, and a hybrid virtual-state convergence strategy (full G0W0 for lower states, second-order perturbation for high-energy tail). Reported band gaps and valence widths for C, Si, MgO and TiO2 are obtained by direct application of these equations and compared to external experimental values, rather than being defined in terms of fitted parameters or prior self-citations. No load-bearing step reduces by construction to its own inputs; the tail approximation is an explicit numerical choice whose validity is tested by convergence and external benchmarks, not assumed tautologically.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard many-body approximations (RPA for polarizability) and a practical but unproven split in the self-energy treatment; no new particles or forces are introduced.

free parameters (2)
  • virtual state cutoff
    The energy or number of virtual orbitals separating the G0W0 region from the second-order tail is chosen to achieve convergence and affects the final self-energy.
  • Gaussian basis and auxiliary basis sizes
    Basis set parameters for orbitals and density fitting are selected and implicitly tuned for the reported materials.
axioms (2)
  • domain assumption RPA level is adequate for the polarizability Pi in the screened interaction W
    The abstract states that Pi is treated at the RPA level for each Q point via BSE.
  • domain assumption Density fitting with Coulomb metric introduces negligible error for the periodic HF and G0W0 calculations
    The abstract specifies that G0W0 and BSE calculations use density fitting with a Coulomb metric.

pith-pipeline@v0.9.0 · 5860 in / 1623 out tokens · 51583 ms · 2026-05-20T03:43:21.605372+00:00 · methodology

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