From eb7206ca5c6ad92c60810f79e494eae640086905 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 2 May 2022 14:42:45 +0200 Subject: [PATCH] wiring notes --- Manuscript/MRGW.tex | 126 +++++++++++++++++++++++++++----------------- 1 file changed, 79 insertions(+), 47 deletions(-) diff --git a/Manuscript/MRGW.tex b/Manuscript/MRGW.tex index 6af845a..c25f50d 100644 --- a/Manuscript/MRGW.tex +++ b/Manuscript/MRGW.tex @@ -86,7 +86,8 @@ \newcommand{\bI}{\boldsymbol{1}} \newcommand{\bH}{\boldsymbol{H}} \newcommand{\bvc}{\boldsymbol{v}} -\newcommand{\bSig}[1]{\boldsymbol{\Sigma}^{#1}} +\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}} +\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}_{#1}^{\text{c}}} \newcommand{\be}{\boldsymbol{\epsilon}} \newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}} \newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}} @@ -125,6 +126,7 @@ \affiliation{\LCPQ} \begin{abstract} +Here comes the abstract. %\bigskip %\begin{center} % \boxed{\includegraphics[width=0.5\linewidth]{TOC}} @@ -134,44 +136,61 @@ \maketitle + +%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%% +Here comes the introduction. + +%%%%%%%%%%%%%%%%% +\section{Theory} +%%%%%%%%%%%%%%%%% +In the case of {\GOWO}, the quasiparticle equation reads \begin{equation} \label{eq:qp_eq} \eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0 \end{equation} +where $\eps{p}{\HF}$ is the HF one-electron energy of the spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is \begin{equation} \label{eq:SigC} \SigC{p}(\omega) = \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}} + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}} \end{equation} +where \begin{equation} \ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA \end{equation} +are the screened two-electron repulsion integrals where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system: \begin{equation} \bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA} \end{equation} +with \begin{equation} A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj} \end{equation} +and \begin{equation} \ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2 \end{equation} +The spectral weight of the solution $\eps{p,s}{\GW}$ is given by \begin{equation} \label{eq:Z} 0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1 \end{equation} +with the following sum rules: \begin{align} \sum_{s} Z_{p,s} & = 1 & \sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF} \end{align} - - +As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution: \begin{equation} \bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)} \end{equation} +with \begin{equation} \label{eq:Hp} \bH^{(p)} = @@ -183,83 +202,96 @@ \T{(\bV{p}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \end{pmatrix} \end{equation} +and where the expressions of the 2h1p and 2p1h blocks reads \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{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} + 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} +with the following expressions for the coupling blocks: \begin{align} - V^\text{2h1p}_{p,klc} & = \sqrt{2} \ERI{pk}{cl} + V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL} & - V^\text{2p1h}_{p,kcd} & = \sqrt{2} \ERI{pd}{kc} + V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC} \end{align} - +By solving the secular equation \begin{equation} \det[ \bH^{(p)} - \omega \bI ] = 0 \end{equation} +we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, \begin{equation} \SigC{p}(\omega) = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} \end{equation} +with \begin{equation} \label{eq:Z_proj} Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2} \end{equation} +In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space. +Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$) +\begin{equation} +\label{eq:Hp_qia} + \bH^{(p,qia)} = + \begin{pmatrix} + \eps{p}{\HF} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} + \\ + V_{qia,p} & C_{qia,qia} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}} + \\ + \T{(\bV{p}{\text{2h1p}})} & \T{(\bC{qia}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO + \\ + \T{(\bV{p}{\text{2p1h}})} & \T{(\bC{qia}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} + \end{pmatrix} +\end{equation} +with new blocks defined as +\begin{gather} + V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia} + \\ + C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ] + \\ + C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK} + \\ + C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD} +\end{gather} + +Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy \begin{equation} \label{eq:Hp} - \bH^{(P,QIA)} = + \bSigC{p,qia}(\omega) = \begin{pmatrix} - \eps{P}{\HF} & W_{P,QIA} & \bW{P}{\text{2h1p}} & \bW{P}{\text{2p1h}} + \eps{p}{\HF} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega) \\ - W_{QIA,P} & D_{QIA,QIA} & \bD{QIA}{\text{2h1p}} & \bD{QIA}{\text{2p1h}} + V_{qia,p} + \SigC{qia,p}(\omega) & C_{qia,qia} + \SigC{qia}(\omega) \\ - \T{(\bW{P}{\text{2h1p}})} & \T{(\bD{QIA}{\text{2h1p}})} & \bD{}{\text{2h1p}} & \bO - \\ - \T{(\bW{P}{\text{2p1h}})} & \T{(\bD{QIA}{\text{2p1h}})} & \bO & \bD{}{\text{2p1h}} \end{pmatrix} \end{equation} with \begin{gather} - W_{P,QIA} = \sqrt{2} \ERI{PQ}{IA} + \SigC{p}(\omega) + = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} \\ - D_{QIA,QIA} = \text{sgn}(\eps{Q}{\HF} - \mu) \qty[ \qty(\eps{Q}{\HF} + \eps{A}{\HF} - \eps{I}{\HF} ) + 2 \ERI{IA}{IA} ] + \SigC{qia}(\omega) + = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} + + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} \\ - D_{QIA,klc}^\text{2h1p} = - 2 \ERI{IA}{cl} \delta_{Qk} + \SigC{p,qia}(\omega) + = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} + + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} \\ - D_{QIA,klc}^\text{2p1h} = + 2 \ERI{IA}{kc} \delta_{Qd} + \SigC{qia,p}(\omega) + = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} \end{gather} +Of course, the present procedure can be generalized to any number of states. -\begin{equation} -\label{eq:Hp} - \bH^{(P,QIA)} = - \begin{pmatrix} - \eps{P}{\HF} + \SigC{P}(\omega) & W_{P,QIA} + \SigC{P,QIA}(\omega) - \\ - W_{QIA,P} + \SigC{QIA,P}(\omega) & D_{QIA,QIA} + \SigC{QIA}(\omega) - \\ - \end{pmatrix} -\end{equation} -with -\begin{align} - \SigC{P}(\omega) - & = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})} - + \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})} - \\ - \SigC{QIA}(\omega) - & = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})} - + \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})} - \\ - \SigC{P,QIA}(\omega) - & = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})} - + \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})} - \\ - \SigC{QIA,P}(\omega) - & = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})} - + \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})} -\end{align} - +%%%%%%%%%%%%%%%%%%%%%% +\section{Conclusion} +%%%%%%%%%%%%%%%%%%%%%% +Here comes the conclusion. +d %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}