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Efficient $GW$ band structure calculations using Gaussian basis functions and application to atomically thin transition-metal dichalcogenides

Rémi Pasquier, María Camarasa-Gómez, Anna-Sophia Hehn, Daniel Hernangómez-Pérez, Jan Wilhelm

TL;DR

This work develops a $GW$ space-time algorithm based on Gaussian basis functions with spin-orbit coupling for periodic systems, incorporating explicit lattice summations to treat density response and self-energy alongside $k$-point sampling for the screened interaction. The approach is benchmarked on atomically thin TMDCs (MoS$_2$, MoSe$_2$, WS$_2$, WSe$_2$), showing $G_0W_0$ band gaps within roughly 50 meV of plane-wave references, and it demonstrates practical Computational scalability, achieving full band-structure calculations on a laptop in about a day and on 1024 cores in tens of minutes. The method leverages RI approximations and minimax time-frequency grids to manage computational cost, while offering SOC through HGH pseudopotentials and a perturbative correction. Collectively, these results establish a scalable, accurate framework for GW calculations in low-dimensional materials and pave the way for broader applications beyond small unit cells, including real-space grid integrations as a future enhancement.

Abstract

We present a $GW$ space-time algorithm for periodic systems in a Gaussian basis including spin-orbit coupling. We employ lattice summation to compute the irreducible density response and the self-energy, while we employ $k$-point sampling for computing the screened Coulomb interaction. Our algorithm enables accurate and computationally efficient quasiparticle band structure calculations for atomically thin transition-metal dichalcogenides. For monolayer MoS$_\text{2}$, MoSe$_\text{2}$, WS$_\text{2}$, and WSe$_\text{2}$, computed $GW$ band gaps agree on average within 50 meV with plane-wave-based reference calculations. $G_0W_0$ band structures are obtained in less than two days on a laptop (Intel i5, 192 GB RAM) or in less than 30 minutes using 1024 cores. Overall, our work provides an efficient and scalable framework for $GW$ calculations on atomically thin materials.

Efficient $GW$ band structure calculations using Gaussian basis functions and application to atomically thin transition-metal dichalcogenides

TL;DR

This work develops a space-time algorithm based on Gaussian basis functions with spin-orbit coupling for periodic systems, incorporating explicit lattice summations to treat density response and self-energy alongside -point sampling for the screened interaction. The approach is benchmarked on atomically thin TMDCs (MoS, MoSe, WS, WSe), showing band gaps within roughly 50 meV of plane-wave references, and it demonstrates practical Computational scalability, achieving full band-structure calculations on a laptop in about a day and on 1024 cores in tens of minutes. The method leverages RI approximations and minimax time-frequency grids to manage computational cost, while offering SOC through HGH pseudopotentials and a perturbative correction. Collectively, these results establish a scalable, accurate framework for GW calculations in low-dimensional materials and pave the way for broader applications beyond small unit cells, including real-space grid integrations as a future enhancement.

Abstract

We present a space-time algorithm for periodic systems in a Gaussian basis including spin-orbit coupling. We employ lattice summation to compute the irreducible density response and the self-energy, while we employ -point sampling for computing the screened Coulomb interaction. Our algorithm enables accurate and computationally efficient quasiparticle band structure calculations for atomically thin transition-metal dichalcogenides. For monolayer MoS, MoSe, WS, and WSe, computed band gaps agree on average within 50 meV with plane-wave-based reference calculations. band structures are obtained in less than two days on a laptop (Intel i5, 192 GB RAM) or in less than 30 minutes using 1024 cores. Overall, our work provides an efficient and scalable framework for calculations on atomically thin materials.

Paper Structure

This paper contains 25 sections, 118 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. GW space-time algorithm
  3. KS-DFT with periodic boundary conditions and Gaussian basis functions
  4. GW space-time method with Gaussian basis functions and periodic boundary conditions
  5. Computational Details
  6. Overview
  7. Atomic geometries of MoS$_\text{2}$, MoS$\text{e}_\text{2}$, WS$_\text{2}$, WS$\text{e}_\text{2}$
  8. GW space-time calculations (CP2K)
  9. Reference GW calculations with a plane-wave-based algorithm (BerkeleyGW)
  10. Reference GW calculations with the projector-augmented-wave scheme (VASP)
  11. DFT band structure of $\text{MoS}_\text{2}$, $\text{MoSe}_\text{2}$, WS$_\text{2}$, WS$\text{e}_\text{2}$
  12. GW band gap of WS$\text{e}_\text{2}$: Convergence study
  13. GW band structure of $\text{MoS}_\text{2}$, $\text{MoSe}_\text{2}$, WS$_\text{2}$, WS$\text{e}_\text{2}$
  14. Conclusion and outlook
  15. Periodic RI
  16. ...and 10 more sections

Figures (11)

  • Figure 1: PBE+SOC Bandstructures of monolayer MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$, computed from Eq. \ref{['e31']} using Gaussian basis sets (CP2K) and a plane-wave basis (QE). The computational details are given in Sec. \ref{['subsec:compparamcp2k']} and \ref{['subsec:paramQEBGW']}. The numerical values of the direct band gap at K are reported in Table \ref{['t1']}.
  • Figure 2: $G_0W_0$@PBE+SOC band gap of monolayer WSe$_2$ and execution time as a function of the number of time points $\tau$ (Sec. \ref{['sec:IV']}), the $k$-mesh (Eqs. \ref{['e21a']}, \ref{['e23']}, \ref{['e29']}), the filter threshold (for Eqs. \ref{['chiT']}, \ref{['SigmaR']} and \ref{['SigmaxR']}, see Eqs. \ref{['Frobcut']} and \ref{['Frobnorm']}), the simulation cell box height (Sec. \ref{['subsec:compparamcp2k']}), the size factor $\Delta$ for $V_{PQ}(\mathbf{k})$ (Eq. \ref{['lattsacle']}) and the number of basis functions (Sec. \ref{['subsec:compparamcp2k']}). Default parameters are reported on top. In (a), we also show $G_0W_0$@PBE+SOC with Hedin's shift hedin1965newHedin1999Pollehn1998Martin_Reining_Ceperley_2016Golze2019Golze2022 to avoid poles of the self-energy close to the quasiparticle solution Veril2018Schambeck2024.
  • Figure 3: Direct $G_0W_0$@PBE+SOC band gap of monolayer WSe$_2$ as function of the RI basis set size, the truncation radius $r_\text{c}$ of the truncated Coulomb metric \ref{['e5c']}/\ref{['e11a']} and the regularization parameter $\alpha$, Eq. \ref{['e12']}. Tight parameters and the aug-SZV-MOLOPT basis sets are used to expand the KS orbitals. Each point corresponds to an RI basis set with a given relative deviation of the RI-MP2 Weigend1998 correlation energy compared to MP2: the RI basis set with $N_\text{RI}\space{=}\space 88$ functions gives a relative RI-MP2 error of $10^{-2}$; the other RI basis set sizes and relative RI-MP2 errors are: $N_\text{RI}= 130$ and $10^{-3}$, $N_\text{RI}= 141$ and $10^{-4}$, $N_\text{RI}= 187$ and $10^{-5}$.
  • Figure 4: $G_0W_0$@PBE+SOC Bandstructures of monolayer MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$, computed from the algorithm presented in this work [CP2K code, Eq. \ref{['e58']}] and computed from BerkeleyGW. The computational details are given in Sec. \ref{['subsec:compparamcp2k']} and \ref{['subsec:paramQEBGW']}.
  • Figure 5: Average absolute deviation in meV across all monolayers TMDs of the $G_0W_0$@PBE+SOC band gap between the TZV2P-MOLOPT (T), aug-SZV-MOLOPT (light, aS), aug-DZVP-MOLOPT (aD), BerkeleyGW (BGW, B) and VASP (V) calculations. The $G_0W_0$@PBE+SOC band gaps for the individual monolayer TMDs are given in the four insets below the main figure and in Table \ref{['t1']}.
  • ...and 6 more figures