GWΓ+Bethe-Salpeter equation approach for photoabsorption spectra: Importance of self-consistentGWΓcalculations in small atomic systems

Abstract

The self-consistent GWΓ method satisfies the Ward-Takahashi identity (i.e., the gauge invariance or the local charge continuity) for arbitrary energy (ω) and momentum (q) transfers. Its self-consistent firstprinciples treatment of the vertex Γ = Γ v or Γ W is possible to first order in the bare (v) or dynamicallyscreened (W) Coulomb interaction. It is developed within a linearized scheme and combined with the Bethe-Salpeter equation (BSE) to accurately calculate photoabsorption spectra (PAS) and photoemission (or inverse photoemission) spectra (PES) simultaneously. The method greatly improves the PAS of Na, Na 3 , B 2 , and C 2 H 2 calculated using the standard one-shot G 0 W 0 + BSE method that results in significantly redshifted PAS by 0.8-3.1 eV, although the PES are well reproduced already in G 0 W 0 . 1The quasiparticle (QP) equation method in many-body perturbation theory [1] is powerful for simultaneously determining the photoemission (or inverse photoemission) spectra (PES), i.e., QP energy spectra, and QP wave functions of target materials from first-principles. In this method, we expand the skeleton diagrams, i.e., the diagrams drawn with the full Green's function lines, for the self-energy in terms of the electron-electron Coulomb interaction v, and solve the QP equation, which is equivalent to the Dyson equation, as a self-consistent (SC) eigenvalue problem. The Hartree-Fock (HF) approach provides the first-order approximation. In Hedin's set of equations[1] known as the GWΓ approach, the exchange-correlation part of the self-energy is expressed aswhere G σ and Γ σ are the one-particle Green's function and the vertex function (σ is the spin index), respectively, and W = (1 − vP) −1 v represents the dynamically screened Coulomb interaction (P = −i σ G σ G σ Γ σ is the polarization function). The simplest approximation is to assume Γ σ = 1, which is called the GW approximation.It is well known that the SC GW method usually overestimates the energy gap [2, 3], while the one-shot GW approach (G 0 W 0 ) using the Kohn-Sham (KS) wave functions and eigenvalues[4] results in a better energy gap. However, quite recently, it has been pointed out that the photoabsorption spectra (PAS) for small molecules obtained by solving the Bethe-Salpeter equation (BSE) [5, 6] using G 0 W 0 are often significantly redshifted by about 1 eV [7,8]. The use of the Heyd-Scuseria-Ernzerhof (HSE) functional or the SC GW calculation (hereafter referred to as GW) improves the results, but they are not perfect [8,9]. For a spin-polarized sodium atom (Na) and trimer (Na 3 ), G 0 W 0 + BSE is extremely bad, although the G 0 W 0 QP energies are reasonably good [10]. The calculated and experimental [11] optical gaps for Na are 1.32 eV and 2.10 eV, respectively, and the calculated and experimental [12] PAS for Na 3 are shown in Fig. 1. These calculated results are far off from the experimental data [13].Here, we develop a GWΓ method, which involves a SC treatment of the vertex Γ = Γ v or Γ W and satisfies the Ward-Takahashi identity [14][...

Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies

“…44 Furthermore, they do not outperform simple G 0 W 0 . 40,43,45 Non-local vertex functions seem to improve the description of the QP energies; 44,[46][47][48][49] however, they are costly and suffer from steep scaling (N 6 e ). 49 Alternatively, the vertex term has been approximated up to the second order, 36,50,51 but this is associated with only mild cost reduction (N 5 e scaling).…”

“…51,52 Consequently, the beyond GW calculations have been applied only to model or few-electron systems. 46,48,49,53 Here, numerical and theoretical developments are combined to overcome this limitation. A self-consistent expression for the self-energy with non-local Γ is constructed using derivatives of the inverse Green's function.…”

GW Vertex Corrected Calculations for Molecular Systems

“…To the best of our knowledge, vertex corrections have been routinely considered only for extended systems [15,22,[40][41][42][43][44]. For finite systems, calculations have been performed for atoms [45], for very simple molecules (within the Tamm-Dancoff approximation) [46], or with the SOSEX [26,47], which approximates the vertex only in second order. A local vertex Λ LDA [48] has also been used for a set of aromatic molecules, however, this simple two-point vertex behaves quite differently than the four-point many-body vertex used here.…”

GW(Γ) method without the Bethe-Salpeter equation for photoabsorption energies of spin-polarized systems

“…Indeed, it has been recently reported by several authors that the GW + BSE method significantly underestimates the experimental PAEs of atoms and small molecules. [14][15][16] The use of the Heyd-Scuseria-Ernzerhof (HSE) functional or the self-consistent GW calculation improves the results, but they are not perfect. [15,17] This problem is difficult to solve in MBPT unless one uses a more sophisticated approach such as the self-consistent LGWΓ + BSE approach.…”

“…[15,17] This problem is difficult to solve in MBPT unless one uses a more sophisticated approach such as the self-consistent LGWΓ + BSE approach. [16] In any approach solving the BSE, the resulting two-particle (electronhole) wave functions are the complicated linear combinations of the products of the electron and hole QP wave functions. Therefore, there is no one-to-one correspondence between each peak of the PA specturm (corresponding a N-particle EES) and the QP energy level (corresponding to a N ± 1particle EES).…”