Low-scaling \textit{GW} calculation of quasi-particle energies within numerical atomic orbital framework

Abstract

The many-body perturbation theory within the $GW$ approximation is a widely used method for describing the electronic band structures in real materials. Its application to large-scale systems is, however, impeded by its high computational cost. The rate-limiting steps in a typical $GW$ implementation are the evaluation of the polarization function under the random phase approximation (RPA) and the evaluation of the $GW$ self-energy, both of which have a canonical $O(N^4)$ scaling with $N$ being the system size. The conventional space-time algorithm within the plane-wave basis sets reduces the scaling from $O(N^4)$ to $O(N^3)$, albeit with a large prefactor and increased memory cost. Here, we present a space-time algorithm within the numerical atomic orbital (NAO) basis-set framework, for which the evaluation of the polarization function and self-energy is formally reduced to $O(N^2)$ or better with respect to system size. This is achieved by computing these quantities in real space, where low-scaling algorithms can be formulated by leveraging the localized resolution of identity (LRI) technique. The resulting NAO-based, LRI-enhanced space-time $GW$ algorithm has been implemented in the LibRPA library interfaced with the FHI-aims code package. Benchmark calculations for crystalline solids show that the low-scaling implementation yields quasi-particle energies in close agreement with the conventional $O(N^4)$ k-space formalism previously implemented in FHI-aims. For the systems studied here, the observed overall scaling is substantially reduced relative to the canonical approach, and the low-scaling implementation becomes advantageous already for systems containing fewer than 100 atoms.

Figures (11)

• Figure 1: Schematic diagrams of the contributions from an atom quartet $\expval{IJKL}$ to the atom-pair block of the real-space imaginary-time response function $\chi^0_{\mathcal{U}=I,\mathcal{V}=J}(\mathrm{i}\tau)$ in Eq. \ref{['eq:chi0-mn-ap-block']}. Here, $\mathbf{G}^{\pm} = \mathbf{G}(\pm\mathrm{i}\tau)$. From left to right, the diagrams correspond to the four terms in Eq. \ref{['eq:chi0-mn-ap-block']}, respectively. The atom indices denote atoms in the BvK cell. The dashed lines indicate the intrinsic binding of atom pairs in the RI-coefficient tensor $\mathbf{C}^{\mathcal{U}}_{IK}$ under the LRI approximation. The solid straight lines represent one-electron NAO basis indices that are contracted during the computation. The auxiliary-basis indices are denoted by wavy lines and remain uncontracted for the response function.
• Figure 2: Schematic diagrams of the contributions from an atom quartet $\expval{IJKL}$ to the atom-pair block of the correlation self-energy $\boldsymbol{\Sigma}^{\mathrm{c}}_{IJ}$ in Eq. \ref{['eq:sigmac-ij-Rt-label-sum']}. From left to right, the diagrams correspond to Eqs. \ref{['eq:sigmac-ij-Rt-label-IJ']}, \ref{['eq:sigmac-ij-Rt-label-KJ']}, \ref{['eq:sigmac-ij-Rt-label-IL']}, and \ref{['eq:sigmac-ij-Rt-label-KL']}, respectively. The line conventions are the same as those in Fig. \ref{['fig:chi0-tensor-contract-pattern']}. The imaginary-time arguments of $\mathbf{G}$ and $\mathbf{W}^{\mathrm{c}}$ are omitted for brevity.
• Figure 3: Schematic diagrams of the rearranged terms in Eq. \ref{['eq:sigmac-rearrange-labels']}, illustrating how a fixed screened-interaction block $\mathbf{W}^{\mathrm{c}}_{\mathcal{U}=I,\mathcal{V}=J}$ contributes to multiple atom-pair blocks of the correlation self-energy from an atom quartet $\expval{IJKL}$. The line conventions are the same as those in Fig. \ref{['fig:sigmac-IJ-pattern']}, and the imaginary-time arguments of $\mathbf{G}$ and $\mathbf{W}^{\mathrm{c}}$ are omitted for brevity.
• Figure 4: Flowchart of the low-scaling $G^0W^0$ implementation in LibRPA. The region enclosed by the red dashed lines indicates the internal workflow in LibRPA for computing the quasi-particle energies $\epsilon^{\mathrm{QP}}_{n\mathbf{k}}$. The quantities shown in blue blocks are input data provided by an external first-principles code, which is FHI-aimsFHIaims25arxiv in the present work. "AC" denotes the analytic continuation discussed in Sec. \ref{['sssec:ac']}.
• Figure 5: Convergence of the fundamental band gaps of eight selected semiconductors and insulators with respect to the number of minimax grid points in low-scaling $G^0W^0$@PBE calculations. The results from the canonical implementation in FHI-aims are taken as reference. The shaded region indicates an error range of $\pm 10$ meV. The mean absolute error (MAE) over all eight materials is shown by the black solid line. The intermediate_gw basis set and an $8\times 8 \times 8$$\mathbf{k}$ grid are used. In the canonical calculations, 60 modified Gauss--Legendre grid points are employed.