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Manuscript/MRGW.tex
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@ -1,5 +1,5 @@
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -64,16 +64,13 @@
\newcommand{\EHF}{E^\text{HF}} \newcommand{\EHF}{E^\text{HF}}
% orbital energies % orbital energies
\newcommand{\eps}[2]{\epsilon_{#1}^{#2}} \newcommand{\eps}{\epsilon}
\newcommand{\reps}[2]{\Tilde{\epsilon}_{#1}^{#2}} \newcommand{\reps}{\Tilde{\epsilon}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}} \newcommand{\Om}{\Omega}
% Matrix elements % Matrix elements
\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} \newcommand{\SigC}{\Sigma^\text{c}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}}
\newcommand{\rSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
\newcommand{\MO}[1]{\phi_{#1}} \newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\SO}[1]{\psi_{#1}} \newcommand{\SO}[1]{\psi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)} \newcommand{\ERI}[2]{(#1|#2)}
@ -85,21 +82,20 @@
\newcommand{\bO}{\boldsymbol{0}} \newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}} \newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bH}{\boldsymbol{H}} \newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bvc}{\boldsymbol{v}} \newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}_{#1}^{\text{c}}}
\newcommand{\be}{\boldsymbol{\epsilon}} \newcommand{\be}{\boldsymbol{\epsilon}}
\newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}} \newcommand{\bOm}{\boldsymbol{\Omega}}
\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}} \newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}} \newcommand{\bB}{\boldsymbol{B}}
\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}} \newcommand{\bC}{\boldsymbol{C}}
\newcommand{\bD}[2]{\boldsymbol{D}_{#1}^{#2}} \newcommand{\bD}{\boldsymbol{D}}
\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}} \newcommand{\bU}{\boldsymbol{U}}
\newcommand{\bW}[2]{\boldsymbol{W}_{#1}^{#2}} \newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}} \newcommand{\bW}{\boldsymbol{W}}
\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}} \newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}} \newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bc}[2]{\boldsymbol{c}_{#1}^{#2}} \newcommand{\bZ}{\boldsymbol{Z}}
\newcommand{\bc}{\boldsymbol{c}}
% orbitals, gaps, etc % orbitals, gaps, etc
\newcommand{\IP}{I} \newcommand{\IP}{I}
@ -111,6 +107,11 @@
\newcommand{\EgOpt}{\Eg^\text{opt}} \newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B} \newcommand{\EB}{E_B}
% shortcuts for greek letters
\newcommand{\si}{\sigma}
\newcommand{\la}{\lambda}
\newcommand{\RHH}{R_{\ce{H-H}}} \newcommand{\RHH}{R_{\ce{H-H}}}
\newcommand{\ii}{\mathrm{i}} \newcommand{\ii}{\mathrm{i}}
@ -119,7 +120,7 @@
\begin{document} \begin{document}
\title{Undressing $GW$ one determinant at a time} \title{Stupid ideas}
\author{Pierre-Fran\c{c}ois \surname{Loos}} \author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
@ -143,73 +144,73 @@ Here comes the abstract.
Here comes the introduction. Here comes the introduction.
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
\section{Theory} \section{Undressing $GW$ one determinant at a time}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
In the case of {\GOWO}, the quasiparticle equation reads In the case of {\GOWO}, the quasiparticle equation reads
\begin{equation} \begin{equation}
\label{eq:qp_eq} \label{eq:qp_eq}
\eps{p}{} + \SigC{p}(\omega) - \omega = 0 \eps_p + \SigC_p(\omega) - \omega = 0
\end{equation} \end{equation}
where $\eps{p}{}$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is where $\eps_p$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
\begin{equation} \begin{equation}
\label{eq:SigC} \label{eq:SigC}
\SigC{p}(\omega) \SigC_p(\omega)
= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{} + \Om{m}{}} = \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps_i + \Om_m{}}
+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{} - \Om{m}{}} + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps_a - \Om_m{}}
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m} \ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}
\end{equation} \end{equation}
are the screened two-electron repulsion integrals where $\Om{m}{}$ and $\bX{m}{}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation: are the screened two-electron repulsion integrals where $\Om_m$ and $\bX_m$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation:
\begin{equation} \begin{equation}
\bA{}{\RPA} \cdot \bX{m}{} = \Om{m}{\RPA} \bX{m}{} \bA^{\RPA} \cdot \bX_m = \Om_m \bX_m
\end{equation} \end{equation}
with with
\begin{equation} \begin{equation}
A_{ia,jb}^{} = (\eps{a}{} - \eps{i}{}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj} A_{ia,jb}^{} = (\eps_a - \eps_i) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
\end{equation} \end{equation}
and and
\begin{equation} \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 \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} \end{equation}
The spectral weight of the solution $\eps{p,s}{\GW}$ is given by The spectral weight of a solution $\eps_{p,\si}^{\GW}$ (where $\si$ numbers the solution for a given orbital $p$) is given by
\begin{equation} \begin{equation}
\label{eq:Z} \label{eq:Z}
0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1 0 \le Z_{p,\si} = \qty[ 1 - \eval{\pdv{\SigC_p(\omega)}{\omega}}_{\omega = \eps_{p,\si}^{\GW}} ]^{-1} \le 1
\end{equation} \end{equation}
with the following sum rules: with the following sum rules:
\begin{align} \begin{align}
\sum_{s} Z_{p,s} & = 1 \sum_{\si} Z_{p,\si} & = 1
& &
\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{} \sum_{\si} Z_{p,\si} \eps_{p,\si}^{\GW} & = \eps_{p}
\end{align} \end{align}
Here, we $p,q,r$ indicate arbitrary (\ie, occupied or unoccupied) orbitals, $i,j,k,l$ are occupied orbitals, while $a,b,c,d$ are unoccupied (virtual) orbitals. Here, we $p,q,r$ indicate arbitrary (\ie, occupied or unoccupied) orbitals, $i,j,k,l$ are occupied orbitals, while $a,b,c,d$ are unoccupied (virtual) orbitals.
As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution: As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
\begin{equation} \begin{equation}
\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)} \bH_p \cdot \bc_{p,\si} = \eps_{p,\si}^{\GW} \bc_{p,\si}
\end{equation} \end{equation}
with with
\begin{equation} \begin{equation}
\label{eq:Hp} \label{eq:Hp}
\bH^{(p)} = \bH_p =
\begin{pmatrix} \begin{pmatrix}
\eps{p}{} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} \eps_{p} & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\ \\
\T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \T{(\bV_p^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO
\\ \\
\T{(\bV{p}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \T{(\bV_p^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}}
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
and where the expressions of the 2h1p and 2p1h blocks reads and where the expressions of the 2h1p and 2p1h blocks reads
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:C2h1p} \label{eq:C2h1p}
C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{} + \eps{J}{} - \eps{A}{}) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK} C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps_I + \eps_J - \eps_A) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK}
\\ \\
\label{eq:C2p1h} \label{eq:C2p1h}
C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{} + \eps{B}{} - \eps{I}{}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD} C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps_A + \eps_B - \eps_I) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD}
\end{align} \end{align}
\end{subequations} \end{subequations}
with the following expressions for the coupling blocks: with the following expressions for the coupling blocks:
@ -226,19 +227,19 @@ Here, we use lower case letters for the electronic configurations belonging to t
By solving the secular equation By solving the secular equation
\begin{equation} \begin{equation}
\det[ \bH^{(p)} - \omega \bI ] = 0 \det[ \bH_p - \omega \bI ] = 0
\end{equation} \end{equation}
we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
\begin{multline} \begin{multline}
\SigC{p}(\omega) \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{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}})} + \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{multline} \end{multline}
with with
\begin{equation} \begin{equation}
\label{eq:Z_proj} \label{eq:Z_proj}
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2} Z_{p,\si} = \qty[ c_{p,\si,1} ]^{2}
\end{equation} \end{equation}
In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space. In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
@ -246,21 +247,21 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $
Equation \label{eq:Hp} can then be written exactly as Equation \label{eq:Hp} can then be written exactly as
\begin{equation} \begin{equation}
\label{eq:Hp_qia} \label{eq:Hp_qia}
\bH^{(p,qia)} = \bH_{p,qia} =
\begin{pmatrix} \begin{pmatrix}
\eps{p}{} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} \eps_p & V_{p,qia} & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\ \\
V_{qia,p} & \eps{qia}{} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}} V_{qia,p} & \eps_{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{2h1p}})} & \T{(\bC_{qia}^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO
\\ \\
\T{(\bV{p}{\text{2p1h}})} & \T{(\bC{qia}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \T{(\bV_p^{\text{2p1h}})} & \T{(\bC_{qia}^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}}
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
with new blocks defined as with new blocks defined as
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\eps{qia}{} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ] \eps_{qia} = \text{sgn}(\eps_q - \mu) \qty[ \qty(\eps_q + \eps_a - \eps_i ) + 2 \ERI{ia}{ia} ]
\\ \\
C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK} C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
\\ \\
@ -271,18 +272,18 @@ with new blocks defined as
\end{subequations} \end{subequations}
where $\text{sgn}$ is the sign function and $\mu$ is the chemical potential. where $\text{sgn}$ is the sign function and $\mu$ is the chemical potential.
The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has to remove the contribution from the 2h1p or 2p1h configuration $qia$. The expressions of $\bC^{\text{2h1p}}$, $\bC^{\text{2p1h}}$, $\bV_p^{\text{2h1p}}$, and $\bV_p^{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has to remove the contribution from the 2h1p or 2p1h configuration $qia$.
While $\eps{p}{}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration with respect to the $N$-electron HF reference determinant. While $\eps_p$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps_{qia} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration with respect to the $N$-electron HF reference determinant.
Therefore, when $\eps{p}{}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems. Therefore, when $\eps_p$ and $\eps_{qia}$ becomes of similar magnitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix
\begin{equation} \begin{equation}
\label{eq:Hp} \label{eq:Hp}
\bSigC{p,qia}(\omega) = \bSigC_{p,qia}(\omega) =
\begin{pmatrix} \begin{pmatrix}
\eps{p}{} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega) \eps_p + \SigC_p(\omega) & V_{p,qia} + \SigC_{p,qia}(\omega)
\\ \\
V_{qia,p} + \SigC{qia,p}(\omega) & \eps{qia}{} + \SigC{qia}(\omega) V_{qia,p} + \SigC_{qia,p}(\omega) & \eps_{qia} + \SigC_{qia}(\omega)
\\ \\
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
@ -290,31 +291,31 @@ with the dynamical self-energies
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\begin{split} \begin{split}
\SigC{p}(\omega) \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{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}})} & + \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{split} \end{split}
\\ \\
\begin{split} \begin{split}
\SigC{qia}(\omega) \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{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}})} & + \bC_{qia}^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2p1h}})}
\end{split} \end{split}
\\ \\
\begin{split} \begin{split}
\SigC{p,qia}(\omega) \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{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}})} & + \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2p1h}})}
\end{split} \end{split}
\\ \\
\begin{split} \begin{split}
\SigC{qia,p}(\omega) \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{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}})} & + \bC_{qia}^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{split} \end{split}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
@ -322,15 +323,104 @@ Of course, the present procedure can be generalized to any number of states.
Solving Solving
\begin{equation} \begin{equation}
\bH^{(p,qia)} \cdot \bc{}{(p,qia,s)} = \eps{p,s}{\GW} \bc{}{(p,qia,s)} \bH_{p,qia} \cdot \bc_{p,qia,\si} = \eps_{p,qia,\si}^{\GW} \bc_{p,qia,\si}
\end{equation} \end{equation}
Because both the 1h or 1p configuration $p$ and the 2h1p or 2p1h configuration $qia$ are in the internal space, we have a new definition of the spectral weight: Because both the 1h or 1p configuration $p$ and the 2h1p or 2p1h configuration $qia$ are in the internal space, we have a new definition of the spectral weight:
\begin{equation} \begin{equation}
\label{eq:Z_proj} \label{eq:Z_proj}
Z_{p,qia,s} = \qty[ c_{1}^{(p,qia,s)} ]^{2} + \qty[ c_{2}^{(p,qia,s)} ]^{2} Z_{p,qia,\si} = \qty[ c_{p,qia,\si,1} ]^{2} + \qty[ c_{p,qia,\si,2} ]^{2}
\end{equation} \end{equation}
Without doubt, the present procedure has similarities with the dressed time-dependent density-functional theory method developed by Maitra and coworkers, \cite{Cave_2004,Maitra_2004} where one doubly-excited configuration is included in the space of single excitations, hence resulting in a dynamical kernel. Without doubt, the present procedure has similarities with the dressed time-dependent density-functional theory method developed by Maitra and coworkers, \cite{Cave_2004,Maitra_2004} where one doubly-excited configuration is included in the space of single excitations, hence resulting in a dynamical kernel.
\\
%%%%%%%%%%%%%%%%%%%%%%
\section{Similarity renormalization group of the $GW$ equations}
%%%%%%%%%%%%%%%%%%%%%%
Following the similarity renormalization group (SRG) formalism, we perform a unitary transformation of the linear $GW$ equations
\begin{equation}
\bH_p(s) = \bU(s) \, \bH_p \, \bU^\dag(s)
\end{equation}
where the so-called flow parameter, $ 0 \le s < \infty$, is a time-like parameter that controls the extent of the transformation.
The purpose of this transformation is to partially decouple the internal and external spaces, or, more precisely in this case, the 1h or 1p sector from the 2h1p and 2p1h sectors, hence avoiding intruder state issues.
By construction, if $s=0$, then $\bU(s) = \bI$, \ie, $\bH_p(s=0) = \bH_p$, while, in the limit $s\to\infty$, $\bH_p(s)$ becomes diagonal.
The SRG flow equation is
\begin{equation}
\dv{\bH_p(s)}{s} = \comm{\boldsymbol{\eta}_p(s)}{\bH_p(s)}
\end{equation}
where the flow generator
\begin{equation*}
\boldsymbol{\eta}_p(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}_p^\dag(s)
\end{equation*}
is an anti-hermitian operator.
We consider Wegner's canonical generator
\begin{equation}
\boldsymbol{\eta}_p^\text{W}(s) = \comm{\bH_p^\text{d}(s)}{\bH_p(s)} = \comm{\bH_p^\text{d}(s)}{\bH_p^\text{od}(s)}
\end{equation}
where one partitions $\bH_p(s)$ [see Eq.~\eqref{eq:Hp}] into its diagonal $\bH_p^\text{d}(s)$ and off-diagonal $\bH_p^\text{od}(s)$ parts, \ie,
\begin{equation}
\bH_{p}(s) = \underbrace{\bH_p^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH_p^\text{od}(s)}_{\text{off-diagonal}}
\end{equation}
where we have $\lim_{s\to\infty} \bH_p^\text{od}(s) = \bO$.
Let us now perform a perturbative analysis of the SRG equation.
For $s=0$, the partition the initial problem as
\begin{equation}
\bH_p(0) = \bH_p^{(0)}(0) + \la \bH_p^{(1)}(0)
\end{equation}
with
\begin{gather}
\bH_p^{(0)}(0) =
\begin{pmatrix}
\eps_{p} & \bO & \bO
\\
\bO & \bC^{\text{2h1p}} & \bO
\\
\bO & \bO & \bC^{\text{2p1h}}
\end{pmatrix}
\\
\bH_p^{(1)}(0) =
\begin{pmatrix}
0 & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\
\T{(\bV_p^{\text{2h1p}})} & \bO & \bO
\\
\T{(\bV_p^{\text{2p1h}})} & \bO & \bO
\end{pmatrix}
\end{gather}
where $\la$ is the usual parameter that controls the magnitude of the perturbation.
This partitioning is reminiscent from Epstein-Nest perturbation theory.
We then expand both $\bH_p(s)$ and $\eps_{p,\si}(s)$ as power series in $\la$, such that
\begin{subequations}
\begin{align}
\bH_p(s) & = \bH_p^{(0)}(s) + \la \bH_p^{(1)}(s) + \la^2 \bH_p^{(2)}(s) + \cdots
\\
\eps_{p,\si}(s) & = \eps_{p,\si}^{(0)}(s) + \la \eps_{p,\si}^{(1)}(s) + \la^2 \eps_{p,\si}^{(2)}(s) + \cdots
\end{align}
\end{subequations}
\begin{gather}
\bH_p^{(0)}(s) =
\begin{pmatrix}
\eps_{p}(s) & \bO & \bO
\\
\bO & \bC^{\text{2h1p}}(s) & \bO
\\
\bO & \bO & \bC^{\text{2p1h}}(s)
\end{pmatrix}
\\
\bH_p^{(1)}(s) =
\begin{pmatrix}
0 & \bV_p^{\text{2h1p}}(s) & \bV_p^{\text{2p1h}}(s)
\\
\T{(\bV_p^{\text{2h1p}}(s))} & \bO & \bO
\\
\T{(\bV_p^{\text{2p1h}}(s))} & \bO & \bO
\end{pmatrix}
\end{gather}
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\section{Conclusion} \section{Conclusion}