Stochastic GW Calculations for Molecules

Abstract

Quasiparticle (QP) excitations are extremely important for understanding and predicting charge transfer and transport in molecules, nanostructures, and extended systems. Since density functional theory (DFT) within the Kohn-Sham (KS) formulation does not provide reliable QP energies, many-body perturbation techniques such as the GW approximation are essential. The main practical drawback of GW implementations is the high computational scaling with system size, prohibiting its use in extended, open boundary systems with many dozens of electrons or more. Recently, a stochastic formulation of GW (sGW) was presented (Phys. Rev. Lett. 2014, 113, 076402) with a near-linear-scaling complexity, illustrated for a series of silicon nanocrystals reaching systems of more than 3000 electrons. This advance provides a route for many-body calculations on very large systems that were impossible with previous approaches. While earlier we have shown the gentle scaling of sGW, its accuracy was not extensively demonstrated. Therefore, we show that this new sGW approach is very accurate by calculating the ionization energies of a group of sufficiently small molecules where a comparison to other GW codes is still possible. Using a set of 10 such molecules, we demonstrate that sGW provides reliable vertical ionization energies in close agreement with benchmark deterministic GW results (J. Chem. Theory Comput, 2015, 11, 5665), with mean (absolute) deviation of 0.05 and 0.09 eV. For completeness, we also provide a detailed review of the sGW theory and numerical implementation.

Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies

“…This method was applied to the G 0 W 0 approximation as described in earlier publications. 56,60,61 In contrast to the previous work, the starting point is no longer constrained to DFT with local functionals. The stochastic formulation of Σ GW Γ X c is a new development.…”

“…Note the χ is a time-ordered quantity, but it is trivially related to the standard retarded response function. 60,61 Consequently, the polarization self-energy has the following form:…”

“…This is consistent with previous observations. 56,[59][60][61]76 In practice, stochastic computation of the induced charge density requires only a few random states η (typically between 4 and 20), i.e., the number of states that are propagated by Eq. 42 is much smaller than the number of occupied states.…”

“…79 In all cases, N ζ < 5, 000, which is slightly lower compared to the fully stochastic G 0 W 0 based on a KS DFT starting point. 60 Inclusion of vertex corrections shifts QPs to higher energies. G 0 W tc 0 usually provides higher estimates than G 0 W tc 0 Γ x .…”

Stochastic many-body perturbation theory for Moiré states in twisted bilayer phosphorene

“…low-dimensional structures, however, require modified boundary conditions. Furthermore, strongly localized states (such as Moiré impurity states) are expected to worsen the statistical sampling as seen, for example, in calculations involving localized molecular orbitals [34]. Localized states in periodic systems were not studied by stochastic methods up to now.…”