Self-Consistent GW via Conservation of Spectral Moments

Abstract

We expand on a recently introduced alternate framework for GW simulation of charged excitations [Scott, C. J. C. J. Chem. Phys. 2023, 158, 124102], based around the conservation of directly computed spectral moments of the GW self-energy. Featuring a number of desirable formal properties over other implementations, we also detail efficiency improvements and a parallelism strategy, resulting in an implementation with a demonstrably similar scaling to an established Hartree-Fock code, with only an order of magnitude increase in cost. We also detail the applicability of a range of self-consistent GW variants within this framework, including a scheme for full self-consistency of all dynamical variables, while avoiding the Matsubara axis or analytic continuation, allowing formal convergence at zero temperature. By investigating a range of self-consistency protocols over the GW100 molecular test set, we find a little-explored self-consistent variant based around a simpler coupled chemical potential and Fock matrix optimization to be the most accurate self-consistent GW approach. Additionally, we validate recently observed evidence that Tamm-Dancoff-based screening approximations within GW lead to higher accuracy than traditional random phase approximation screening over these molecular test cases. Finally, we consider the Chlorophyll A molecule, finding agreement with experiment within the experimental uncertainty, and a description of the full-frequency spectrum of charged excitations.

1

Performance benchmark for the moment-conserving G ₀ W ₀ implementation, indicating CPU time (top) and memory usage (bottom) for increasing numbers of atomic orbitals in linear alkane chains up to C ₃₂ H ₆₆ in a cc-pVDZ basis, with maximum moment order n _mom ^max = 7, obtaining the full G ₀ W ₀ spectrum for each calculation. No natural auxiliary orbital compression was used in this consideration of resource scaling. Lines are shown for both TDA and RPA screening, and for Hartree–Fock via PySCF for comparison.

2

Convergence of the IP of Borane (BH₃) with respect to the number of conserved moments (n _mom ^max) of the self-energy in a def2-TZVPP basis set for single-shot G ₀ W ₀, with RPA screening (left) and TDA (right). A range of mean-field starting points are considered, as well as reference values for RPA screening from PySCF, implementing an $\mathcal{O}[N^6]$ full-frequency algorithm to remove any grid approximations. The remaining discrepancy likely comes from the diagonal approximation to the self-energy enforced in the reference values.

3

Convergence of the IP of borane (BH₃) with respect to the number of conserved moments (n _mom ^max) for self-consistent implementations of moment-conserved GW across HF and PBE reference states. We consider RPA (left) and TDA (right) screening on a def2-TZVPP basis. Reference fully self-consistent scGW values are included from Caruso et al. (ref A) and Wen et al. (ref B), relying on self-consistency on the Matsubara axis, followed by analytic continuation.

4

Convergence of the MAE for IP (x-axis) and EA (y-axis) across the GW100 test set with respect to CCSD­(T) reference values for many of the self-consistent GW approaches considered. Numbers in the circles represent the maximum order of the conserved self-energy moments for each method, showing convergence to the full-frequency limit. The left plot shows the aggregated results for RPA screening, while the right plot is for TDA screening. The inset of each plot depicts the aggregated MAE for moment order 7 across the different moment-conserving GW variants.

5

Histograms of the signed errors for IP and EA with G ₀ W ₀ and fsGW relative to CCSD­(T) over the GW100 set. Results presented for maximum moment order of 9 with Gaussians fit to the error distribution shown as dashed lines. The left plot shows RPA screening, right shows TDA screening with IP results shown top and EA bottom. The mean signed error (μ) and standard deviation (σ) for each method are shown in the legend.

6

Convergence of the moment-conserving G ₀ W ₀ IP (left) and EA (middle) and their quasiparticle weights (right) with increasing conserved moment order for the Chlorophyll A molecule in an aug-cc-pVDZ basis set. Also included are the mean-field orbital energies corresponding to the IP and EA at the level of HF and DFT with a B3LYP functional. In addition, we include an experimental estimate of the IP energy from ref , with associated experimental uncertainty shown by the gray shaded region.

7

Dyson orbitals for the HOMO (left) and LUMO (right) for chlorophyll in an aug-cc-pVDZ basis set for the moment-conserving G ₀ W ₀ method, using RPA and TDA screening, with maximum moment order n _mom ^max = 11.

8

Spectral function for Chlorophyll A in an aug-cc-pVDZ basis set for the moment-conserving GW method, using RPA and TDA screening, with maximum moment order n _mom ^max = 11. Thinner lines show the Hartree–Fock and B3LYP spectral functions. IP and EA locations are shown for the moment-conserving G ₀ W ₀@TDA.

9

Convergence of the IP of carbon monoxide (CO) with respect to the number of conserved moments (n _mom ^max) of the self-energy in a def2-TZVPP basis set for single-shot G ₀ W ₀, with RPA screening (left) and TDA (right). A range of mean-field starting points are considered, as well as reference values for RPA screening from PySCF, implementing an $\mathcal{O}[N^6]$ full-frequency algorithm to remove any grid approximations. The remaining discrepancy likely comes from the diagonal approximation to the self-energy enforced in the reference values.

10

Convergence of the IP of carbon monoxide (CO) using the def2-TZVPP basis set with respect to the number of conserved moments (n _mom ^max) for various self-consistent implementations across HF and PBE starting points, with RPA screening (left) and TDA (right). Reference fully self-consistent GW values are included from Caruso et al. (ref A) and Wen et al. (ref B), where the convergence takes place on the Matsubara axis. The scGW results for RPA screening are limited to 15 moments due to convergence issues for higher moments.