Fully Analytic Nuclear Gradients for the Bethe--Salpeter Equation

Abstract

The Bethe-Salpeter equation (BSE) formalism, combined with the $GW$ approximation for ionization energies and electron affinities, is emerging as an efficient and accurate method for predicting optical excitations in molecules. In this letter, we present the first derivation and implementation of fully analytic nuclear gradients for the BSE@$G_0W_0$ method. Building on recent developments for $G_0W_0$ nuclear gradients, we derive analytic nuclear gradients for several BSE@$G_0W_0$ variants. We validate our implementation against numerical gradients and compare excited-state geometries and adiabatic excitation energies obtained from different BSE@$G_0W_0$ variants with those from state-of-the-art wavefunction methods.

Figures (3)

• Figure 1: Molecular structures and geometry parameters considered in this work [gray: C, red: O, white: H, yellow: S, purple: Se]: a) $n\to\pi^*$ transitions; b) $\pi\to\pi^*$ transitions.
• Figure 2: Deviation in absorption (abs.) and fluorescence (fluo.) transition energies ($S_0 \to S_1$), calculated using BSE@$GW$ with and without the TDA and different mean-field starting points (HF and BHLYP), relative to CC3 reference excitation energies of Ref. loos2019chemically. All excitation energies are computed in the aug-cc-pVTZ basis set. BSE@$GW$ calculations rely on the optimized geometries of this work.
• Figure 3: Deviation in adiabatic transition energies ($S_0 \to S_1$), calculated using BSE@$GW$ with and without the TDA and different mean-field starting points (HF and BHLYP), relative to CC3 reference excitation energies of Refs. loos2019chemically and budzak2017accurate (ketene and thioketene). All excitation energies are computed in the aug-cc-pVTZ basis set. BSE@$GW$ calculations rely on the optimized geometries of this work.