From c3fc975ce091f0bd867d945d3cf5f91f50843437 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 10:41:07 +0100 Subject: [PATCH] ok with Sec II --- Manuscript/SRGGW.tex | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 22d4df8..3f894c1 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -205,13 +205,13 @@ with B_{ia,jb} & = \eri{ij}{ab}, \end{align} \end{subequations} -and +and where \begin{equation} \braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2) }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2 \end{equation} are bare two-electron integrals in the spin-orbital basis. -The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}. +The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problem defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}. In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$). As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. @@ -231,7 +231,6 @@ These solutions can be characterized by their spectral weight given by the renor \end{equation} The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions). However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined. -%These additional solutions with large weights are the previously mentioned intruder states. One obvious drawback of the one-shot scheme mentioned above is its starting point dependence. Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead. @@ -263,7 +262,7 @@ One of the main results of the present manuscript is the derivation, from first Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level. Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy. Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions. -If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021} +If it is not the case, the self-consistent qs$GW$ scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021} The satellites causing convergence problems are the above-mentioned intruder states. \cite{Monino_2022} One can deal with them by introducing \textit{ad hoc} regularizers.