Challenges with relativistic GW calculations in solids and molecules

TL;DR

The paper addresses the challenge of performing relativistic $GW$ calculations in solids and molecules containing heavy elements, where relativity and electronic correlation jointly shape electronic and structural properties. It employs an exact two-component relativistic Hamiltonian within the X2C framework ($X2C1e$ and sfX2C1e) combined with self-consistent $GW$ on the Matsubara axis and uses the Birch–Murnaghan equation of state to connect energy and volume. The authors identify three major bottlenecks—deficiencies in pseudopotentials for Green's-function methods, slow or incomplete all-electron basis-set convergence for heavy elements, and linear dependencies in large basis representations—and illustrate these with case studies spanning Si, Ge, $\alpha$-Sn, ZnX, AgBr, CdSe, and HgCl$_2$, highlighting the need for AE relativistic treatments, solids-optimized basis sets, and better experimental benchmarks. They discuss potential remedies, including development of correlated-consistent basis sets for solids, orthogonal orbital representations, and benchmark-guided improvements, to enable robust, predictive relativistic $GW$ calculations in challenging materials and molecules.

Abstract

For molecules and solids containing heavy elements, accurate electronic structure calculations require accounting not only for electronic correlations but also for relativistic effects. In molecules, relativity can lead to severe changes in the ground-state description. In solids, the interplay between both correlation and relativity can change the stability of phases or it can lead to an emergence of completely new phases. Traditionally, the simplest illustration of relativistic effects can be done either by including pseudopotentials in non-relativistic calculations or alternatively by employing large all electron basis sets in relativistic methods. By analyzing different electronic properties (band structure, equilibrium lattice constant and bulk modulus) in semiconductors and insulators, we show that capturing the interplay of relativity and electron correlation can be rather challenging in Green's function methods. For molecular problems with heavy elements, we also observe that similar problems persist. We trace these challenges to three major problems: deficiencies in pseudopotential treatment as applied to Green's function methods, the scarcity of accurate and compact all-electron basis-sets that can be converged with respect to the basis-set size, and linear dependencies arising in all-electron basis-sets particularly when employing Gaussian orbitals. Our analysis provides detailed insight into these problems and opens a discussion about potential approaches to mitigate them.

Figures (5)

• Figure 1: Band structure of germanium calculated with Top panels: PBE and Lower panels: sc$GW$ theories, using Left:GTH-PBE pseudopotential with GTH-DZVP-MOLOPT-SR basis set, Middle:x2c-SVPall all-electron basis set with no relativistic corrections, and Right:x2c-SVPall basis set with sfX2C1e Hamiltonian. For the GTH result, relativity is intrinsically accounted for in the pseudopotential. Diamond lattice with lattice constant $a = 5.657 \mathrm{\AA}$ was used, and all calculations were performed with a $6\times 6\times 6$$k$-mesh sampling in the BZ.
• Figure 2: Errors in Top: lattice constants $a$, and Bottom: bulk moduli $B_0$ for selected materials. All compounds are calculated with GTH-DZVP-MOLOPT-SR, x2c-SVPall and x2c-TZVPall basis sets. Both AE-PBE and AE-sc$GW$ results are reported using the sfX2C1e Hamiltonian, with a $4\times 4\times 4$$k$-mesh sampling. Experimental values for lattice constants and bulk moduli can be found in Refs. fukumori_measurements_1988nelmes_chapter_1998 and cline_volume_1965blakemore_semiconducting_1982pellicer-porres_high-pressure_2005, respectively.
• Figure 3: Graphical representation of convergence trends in \ref{['tab:mol_ip_convergence']}. All IPs are reported relative to the respective extrapolated values, which shows that sc$GW$ exhibits a much slower convergence than PBE0.
• Figure 4: Left panel: Lattice constants $a$ and bulk modulus $B_0$ for silicon, calculated using def2-TZVP basis with and without diffuse orbitals (Gaussian exponent $< 0.1$). Both PBE and sc$GW$ results, along with experimental values, are shown. Non-relativistic Hamiltonian was employed for these calculations. Right panel: The corresponding energy-volume curves.
• Figure 5: Comparison of experimental spectra with sc$GW$ results for HgCl$_2$, obtained with both scalar and two-component relativistic effects. For these calculations, we have used the x2c-QZVPall basis. Both experimental spectra, exp-1 by Bogges et al. bogges_photoelectron_1973 and exp-2 by Eland et al. eland_photoelectron_1970, are shifted by 0.35 eV to match the associated IP peaks.