$G_0W_0$@HF and BSE methods in periodic systems from Hartree-Fock theory: gaussian orbital and density fitting approach

This paper presents a $G_0W_0$@HF and Bethe-Salpeter equation framework for periodic systems using Gaussian orbitals and density fitting, which corrects Hartree-Fock overestimations of band gaps and valence band widths in semiconductors and oxides by employing an exact RPA screening without plasmon pole approximations and a hybrid convergence strategy for virtual states.

The Gist

Imagine you are trying to understand how a solid material, like a diamond or a piece of silicon, behaves when light hits it or when electricity flows through it. To do this, scientists need to calculate the exact energy levels of the electrons inside the material. Think of these energy levels as the "floors" in a skyscraper where electrons live. If you know exactly where the floors are, you know how the building functions.

For decades, the standard way to map these floors has been using a method called Density Functional Theory (DFT). However, DFT is like using a slightly blurry map; it gets the general shape of the building right but often misses the precise height of the floors. To get a sharper picture, scientists use a more advanced technique called GW (named after the symbols G and W in the equations). This method is like switching from a blurry sketch to a high-definition 3D model, but it is extremely computationally expensive and usually requires a specific type of mathematical "grid" (called plane waves) that is hard to work with for certain types of materials.

The New Approach: A Different Lens
This paper, written by Charles H. Patterson, introduces a new way to build that high-definition 3D model. Instead of using the standard blurry map (DFT) as a starting point, the author starts with a different, very sharp but overly rigid map called Hartree-Fock (HF).

• The Problem with the Starting Point: The Hartree-Fock method is like a map drawn with a ruler that is too strict. It predicts the floors are too far apart (the "band gap" is too big) and the rooms are too wide (the "band width" is too large). If you just used this map, your predictions would be wrong.
• The Solution: The author uses a clever strategy. They start with this strict Hartree-Fock map and then apply a "correction lens" (the GW method) to fix the errors. The paper shows that this correction lens is actually very good at shrinking the overly wide rooms back to their real size, resulting in a final map that matches experimental reality very well.

The Tools: Gaussian Orbitals and Density Fitting
Most GW calculations use "plane waves" (like a grid of infinite flat sheets) to describe the electrons. This paper uses Gaussian Orbitals instead.

• The Analogy: Imagine describing a complex sculpture. A plane wave approach is like trying to describe it by stacking millions of flat, square tiles. A Gaussian approach is like using soft, round clay blobs that can mold perfectly around the curves of the sculpture. This is often more efficient for complex molecules and crystals.
• Density Fitting: To make the math work with these clay blobs without the computer crashing, the author uses a technique called Density Fitting. Think of this as a "compression algorithm." Instead of calculating the interaction between every single pair of clay blobs (which would take forever), the method groups them into clusters and calculates the interaction for the group. It's like estimating the weight of a crowd by weighing a few representative people and multiplying, rather than weighing every single person individually.

The "No-Approximation" Trick
A common shortcut in these calculations is the "plasmon pole approximation."

• The Analogy: Imagine you are trying to predict the sound of a drum. The shortcut method says, "Let's just assume the drum only makes one specific note and ignore the rest." It's fast, but it misses the nuance.
• The Paper's Claim: This paper avoids that shortcut. It calculates the full, complex sound of the drum (the full frequency dependence of the electron interactions) without assuming it's just one note. This is more accurate but requires solving a massive, complex puzzle (the Bethe-Salpeter equation) for every point in the material's structure.

What Did They Find?
The author tested this new method on four materials: Diamond, Silicon, Magnesium Oxide (MgO), and Titanium Dioxide (TiO2).

1. Diamond and Silicon: The standard Hartree-Fock method predicted the "rooms" (valence bands) were about 25% too wide. The new method corrected this, shrinking them down to match exactly what experiments measure.
2. Oxides (MgO and TiO2): The method successfully predicted the energy gaps (the distance between floors) and how the material absorbs light. While the predicted gaps were slightly larger than what is seen in experiments (a common issue in this field), the overall shape of the energy map was very accurate.
3. Light Absorption: When simulating how these materials absorb light (their "optical spectra"), the method reproduced the positions of the peaks (the colors absorbed) very well. However, for the oxides, the method predicted the light absorption was slightly too intense, similar to how a microphone might pick up a sound that is a bit too loud.

The Bottom Line
This paper demonstrates that you can build a highly accurate, high-definition map of electron energies in solids by starting with a strict "Hartree-Fock" model and applying a sophisticated "GW" correction, all while using a flexible "Gaussian" mathematical language and a smart "compression" technique (density fitting). It proves that you don't need the standard "plane wave" grid to get excellent results; in fact, this alternative approach can correct the specific errors of the starting method to produce results that agree with real-world experiments.

Technical Summary

Technical Summary: GoWo@HF and BSE Methods in Periodic Systems from Hartree-Fock Theory

Problem Statement
The ab-initio GW and Bethe-Salpeter equation (BSE) methods are standard tools for calculating quasi-particle (QP) energies and optical properties of crystalline materials. However, the vast majority of existing implementations rely on plane-wave (PW) basis sets and Density Functional Theory (DFT) Hamiltonians (typically LDA or PBE) as the starting point. While effective, these approaches often require large basis sets and specific approximations, such as the plasmon pole approximation (PPA), to handle the frequency dependence of the dielectric function efficiently. Furthermore, DFT starting points can introduce specific errors in band structures that are difficult to disentangle from many-body corrections. There is a need to validate and extend these methods to Gaussian orbital (GO) bases, which are standard in quantum chemistry and advantageous for systems with large unit cells or open volumes, while exploring the efficacy of starting from Hartree-Fock (HF) theory rather than DFT.

Methodology
The authors present a framework for performing $G_0W_0$ (denoted as $GoW_0$) and BSE calculations for periodic systems using a Gaussian orbital basis set and a density fitting (resolution of identity) approach. The implementation utilizes the Exciton code and is characterized by the following key technical features:

• Hartree-Fock Starting Point: Unlike most solid-state GW applications, the unperturbed Hamiltonian and single-particle orbitals are derived from HF theory. The authors acknowledge that HF typically overestimates band gaps and valence band widths but posit that the subsequent $GoW_0$ self-energy corrections can renormalize these quantities to match experimental data.
• Density Fitting: To manage the computational cost of four-center integrals in periodic systems, the method employs density fitting with a Coulomb metric. Wavefunction products are projected onto an auxiliary Gaussian basis, reducing the integral complexity. The Coulomb matrix is represented in this auxiliary basis, and its inverse is used to construct screened interactions.
• RPA without Plasmon Pole Approximation: The screened Coulomb interaction, $W$, is calculated using the relation $W = v + v\Pi v$, where $\Pi$ is the polarizability treated at the Random Phase Approximation (RPA) level. Crucially, the method solves the RPA equations (diagonalizing large RPA Hamiltonians) at unique finite wave vectors ($Q$) in the Brillouin zone. This avoids the need for a plasmon pole approximation, allowing for an exact treatment of the frequency dependence of $W$ within the RPA and basis set limitations.
• Hybrid Self-Energy Strategy: To address the convergence of the self-energy with respect to the number of virtual states, a hybrid strategy is employed. The majority of the self-energy is calculated using $GoW_0$ with a limited set of virtual states (sufficient for RPA screening). The remaining contribution from high-energy virtual levels is evaluated using second-order perturbation theory ($\Sigma^{(2)}$). This approach balances computational cost with convergence.
• BSE and TDA: The Bethe-Salpeter equation is formulated using a linear response equation of motion approach. The Tamm-Dancoff approximation (TDA) is employed, neglecting the coupling between resonant and anti-resonant terms, which is deemed sufficient for optical absorption properties.
• Small Q Limit: Special treatment is applied to the $Q \to 0$ limit to handle divergences in the self-energy and screened electron-hole interaction, utilizing methods analogous to those used for exchange energies in cubic solids.

Key Results
The methods were applied to elemental semiconductors (Diamond, Silicon) and wide-bandgap oxides (MgO, Anatase, and Rutile $TiO_2$).

• Band Structure Renormalization:
  • Diamond and Silicon: HF theory significantly overestimates valence band widths (by ~25% for Si and Diamond). The $GoW_0@HF$ corrections successfully renormalize these widths, bringing them into excellent agreement with experimental photoemission data (e.g., Si valence width reduced from 17.00 eV in HF to 13.03 eV in $GoW_0@HF$, matching the experimental 12.5 eV).
  • Band Gaps: While HF overestimates gaps, $GoW_0@HF$ yields gaps that are generally larger than $GoW_0@LDA$ results. For Diamond, the indirect gap is 5.75 eV (vs. 5.48 eV exp); for Si, it is 1.38 eV (vs. 1.17 eV exp). For MgO, the gap is 9.13 eV, which is higher than the experimental 7.77 eV (even after accounting for zero-point renormalization).
• Optical Properties (BSE):
  • The dielectric functions calculated via BSE-TDA show good agreement with experimental peak positions for Diamond and Silicon.
  • For oxides (MgO and $TiO_2$), the peak positions are reasonably accurate (within 0.1–0.4 eV of experiment), but the method systematically overestimates oscillator strengths (intensities). This overestimation is attributed to the use of HF wavefunctions, which are more localized than DFT wavefunctions, leading to stronger electron-hole interactions in the BSE kernel.
  • For $TiO_2$, a small energy shift (0.2–0.4 eV) toward lower energies in the calculated spectra compared to experiment is observed, which the authors attribute to approximations in the small $Q$ limit of the screened electron-hole attraction.

Significance and Claims
The paper establishes a robust density-fitting, Gaussian orbital method for $GoW_0$ and BSE calculations in periodic systems that does not rely on the plasmon pole approximation. The authors demonstrate that:

1. HF Starting Points are Viable: Despite HF's known deficiencies in band gaps and widths, the $GoW_0$ self-energy corrections are capable of renormalizing the band structures of Si and Diamond to yield results in good agreement with experiment, comparable to or better than $GoW_0@LDA$ in terms of bandwidths.
2. Density Fitting Efficiency: The approach successfully handles the computational demands of periodic GW/BSE calculations, reducing the number of integrals required and making the method applicable to systems with large unit cells (like $TiO_2$) where plane-wave approaches might be less efficient or require different convergence strategies.
3. No Plasmon Pole Approximation: By diagonalizing RPA Hamiltonians at finite $Q$ vectors, the method captures the full frequency dependence of the screened interaction, offering a more rigorous alternative to PPA-based approaches.

The authors conclude that while the $GoW_0@HF$ approach yields band gaps that are typically slightly larger than experimental values (and larger than $GoW_0@LDA$), it provides a consistent and accurate description of quasiparticle excitations and optical spectra in gapped materials, validating the use of Gaussian basis sets and HF starting points for many-body perturbation theory in solids.