All-electron Quasiparticle Self-consistent GW for Molecules and Periodic Systems within the Numerical Atomic Orbital Framework

Abstract

We report an all-electron implementation of the quasiparticle self-consistent GW (QSGW) method for molecules and periodic systems within the framework of numerical atomic orbitals (NAOs), as implemented in the LibRPA software package. We present systematic benchmark calculations on molecular systems, as well as a diverse set of periodic systems including typical semiconductors and wide-gap insulators. Our results demonstrate that the present NAO-based QSGW workflow yields molecular ionization potentials and quasiparticle band gaps for periodic solids that are consistent with established reference benchmarks, supporting the correctness of the implementation.

Figures (6)

• Figure 1: Workflow of the QS$GW$ calculation, starting from a KS-DFT calculation and converging to self-consistency using the LibRPA software. Indices $i,j$ label NAO basis functions (composite index $i=\{I,\kappa,l,m\}$ with magnetic quantum number $m$ in $S_{lm}$); $p,p'$ label single-particle eigenstates $\psi_{p\mathbf{k}}$; $\mathbf{k}$ is the Bloch wave vector; $\mathbf{q}$ is the momentum transfer used in $W_{\mu\nu}^0(\mathbf{q},\mathrm{i}\omega)$; $I,J$ represent atom indices; and $n$ is the iteration number.
• Figure 2: Basis representation dependence tests of analytic continuation in QS$GW$ for "Mode A" and "Mode B" for Si (top) and MgO (bottom). The Si calculations employed a $6\times6\times6$ k-point mesh, while MgO employed an $8\times8\times8$ mesh. Both used a frequency grid of 16 points and the intermediate_gw basis sets. In the figure, "initial" denotes analytic continuation performed in the initial KS-DFT eigenbasis, while "updated" denotes analytic continuation performed in the updated QS$GW$ eigenbasis from the previous iteration.
• Figure 3: k-point convergence of the Si QS$GW$ band gap with respect to uniform $n\times n\times n$${\bf k}$-point meshes ($n=6$--10) using the intermediate_gw basis sets. The $10\times10\times10$ mesh is taken as the reference.
• Figure 4: Frequency grid convergence of the Si QS$GW$ band gap using an $8\times8\times8$${\bf k}$-point mesh and the intermediate_gw basis sets (see Sec. \ref{['sec:basis_set_convergence']}). The calculation with 16 frequency points (highlighted with a red star) is adopted for the periodic systems in this work. The band gap varies by only 2.6 meV between 16 and 28 frequency points, confirming adequate convergence.
• Figure 5: Band gaps of semiconductors and insulators computed with PBE, $G^0W^0$@PBE, and QS$GW$ under the numerical atomic orbital (NAO) framework in LibRPA.