From 4aceccde1ddf6fb3dcc16cc0a659e0a6f18e361d Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 17 Jan 2022 08:49:37 +0100 Subject: [PATCH] saving wotk --- Manuscript/ufGW.tex | 74 +++++++++++++++++++++++++++++++-------------- 1 file changed, 51 insertions(+), 23 deletions(-) diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index 293baad..d7b6393 100644 --- a/Manuscript/ufGW.tex +++ b/Manuscript/ufGW.tex @@ -84,10 +84,6 @@ \newcommand{\rbra}[1]{(#1|} \newcommand{\rket}[1]{|#1)} -%% bold in Table -\newcommand{\bb}[1]{\textbf{#1}} -\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} -\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} % Matrices \newcommand{\bO}{\boldsymbol{0}} @@ -104,6 +100,7 @@ \newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}} \newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}} \newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}} +\newcommand{\bc}[2]{\boldsymbol{c}_{#1}^{#2}} % orbitals, gaps, etc \newcommand{\IP}{I} @@ -148,6 +145,11 @@ \end{abstract} \maketitle +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +We consider {\GOWO} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Downfold: The non-linear $GW$ problem} @@ -162,8 +164,8 @@ Within the {\GOWO} approximation, in order to obtain the quasiparticle energies where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads \begin{equation} \SigC{p}(\omega) - = \sum_{im} \frac{\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta} - + \sum_{am} \frac{\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta} + = \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta} + + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta} \end{equation} Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by \begin{equation} @@ -177,41 +179,67 @@ with \begin{equation} A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj} \end{equation} -As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,\ell}{\GW}$ and their corresponding weight is given by the value of the so-called renormalization factor +As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{}$ and their corresponding weight is given by the value of the so-called renormalization factor \begin{equation} - 0 \le Z_{p,\ell} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,\ell}{\GW}} ]^{-1} \le 1 + 0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{}} ]^{-1} \le 1 \end{equation} -In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,\ell=0}{\GW}$ has a large weight $Z_{\ell} \equiv Z_{p,\ell=0}$ +In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{} \equiv \eps{p,s=0}{}$ has a large weight $Z_{} \equiv Z_{p,=0}$ Note that we have the following important conservation rules \begin{align} - \sum_{\ell} Z_{p,\ell} & = 1 + \sum_{s} Z_{p,s} & = 1 & - \sum_{\ell} Z_{p,\ell} \eps{p,\ell}{\GW} & = \eps{p}{\HF} + \sum_{s} Z_{p,s} \eps{p,s}{} & = \eps{p}{\HF} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Upfolding: the linear $GW$ problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The non-linear quasiparticle equation \eqref{eq:qp_ep} can be transformed into a larger linear problem via an upfolding process: +The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors +is downfolded on the 1h and 1p sectors via their interaction with the 2h1p and 2p1h sectors: \begin{equation} - C^\text{1h1p}_{iajb,kcld} = \qty[ \qty( \eps{b}{\GW} + \eps{a}{\HF} - \eps{i}{\GW} - \eps{j}{\HF} ) \delta_{jl} \delta_{ac} + 2 \ERI{ja}{cl} ] \delta_{ik} \delta_{bd} + \bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)} \end{equation} -\begin{align} - V^\text{2h1p}_{p,kld} & = \ERI{pk}{ld} - & - V^\text{2h1p}_{p,cld} & = \ERI{pc}{ld} -\end{align} - +with \begin{equation} - \bH = +\label{eq:Hp} + \bH^{(p)} = \begin{pmatrix} - \Tilde{\bA{}{}} & \bV{}{(1)} & \bV{}{(2)} + \eps{p}{\HF} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} \\ - \T{(\bV{}{(1)})} & \bC{}{} & \bO + \T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\ - \T{(\bV{}{(2)})} & \bO & \bC{}{} + \T{(\bV{p}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \end{pmatrix} \end{equation} +where +\begin{align} + C^\text{2h1p}_{ija,kcl} & = \qty[ \qty( \eps{i}{\HF} + \eps{j}{\HF} - \eps{a}{\HF}) \delta_{jl} \delta_{ac} - 2 \ERI{ja}{cl} ] \delta_{ik} + \\ + C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \eps{a}{\HF} + \eps{b}{\HF} - \eps{i}{\HF}) \delta_{ik} \delta_{ac} + 2 \ERI{ai}{kc} ] \delta_{bd} +\end{align} +and the corresponding coupling blocks read +\begin{align} + V^\text{2h1p}_{p,kld} & = \sqrt{2} \ERI{pk}{cl} + & + V^\text{2p1h}_{p,cld} & = \sqrt{2} \ERI{pd}{kc} +\end{align} +The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$. +Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie, +\begin{equation} + Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2} +\end{equation} + +It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}. +The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introducing regularized $GW$ methods} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +One way to hamper such issues is to resort to regularization of the $GW$ self-energy. +Of course, the way of regularizing the self-energy is not unique but here we consider 3 different ways directly imported from MP2 theory. + +This helps greatly convergence for (partially) self-consistent $GW$ methods. %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{