SRGGW/Manuscript/SRGGW.tex

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% coordinates
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bx}{\boldsymbol{x}}
\newcommand{\dbr}{d\br}
\newcommand{\dbx}{d\bx}

% methods
\newcommand{\GW}{\text{$GW$}}	
\newcommand{\evGW}{ev$GW$}	
\newcommand{\qsGW}{qs$GW$}	
\newcommand{\GOWO}{$G_0W_0$}	
\newcommand{\Hxc}{\text{Hxc}}
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% 
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% energies
\newcommand{\Enuc}{E^\text{nuc}}
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% orbital energies
\newcommand{\eps}{\epsilon}
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% Matrix elements
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\newcommand{\ERI}[2]{(#1|#2)}
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% Matrices
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bH}{\boldsymbol{H}}
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% orbitals, gaps, etc
\newcommand{\IP}{I}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\RHH}{R_{\ce{H-H}}}
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% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}

\begin{document}	

\title{A Similarity Renormalization Group Approach to $GW$}

\author{Antoine \surname{Marie}}
	\email{amarie@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\begin{abstract}
Here comes the abstract.
%\bigskip
%\begin{center}
%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
Here comes the introduction.

%%%%%%%%%%%%%%%%%
\section{Undressing $GW$ one determinant at a time}
%%%%%%%%%%%%%%%%%
In the case of {\GOWO}, the quasiparticle equation reads
\begin{equation}
\label{eq:qp_eq}
	\eps_p + \SigC_p(\omega) - \omega = 0
\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
\begin{equation}
\label{eq:SigC}
	\SigC_p(\omega) 
	= \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{}}
\end{equation}
where 
\begin{equation}
	\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}
\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:
\begin{equation}
	\bA^{\RPA} \cdot \bX_m = \Om_m \bX_m
\end{equation}
with
\begin{equation}
	A_{ia,jb}^{} = (\eps_a - \eps_i) \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 a solution $\eps_{p,\si}^{\GW}$ (where $\si$ numbers the solution for a given orbital $p$) is given by
\begin{equation}
\label{eq:Z}
	0 \le Z_{p,\si} = \qty[ 1 - \eval{\pdv{\SigC_p(\omega)}{\omega}}_{\omega = \eps_{p,\si}^{\GW}} ]^{-1} \le 1
\end{equation} 
with the following sum rules:
\begin{align}
	\sum_{\si} Z_{p,\si} & = 1
	&
	\sum_{\si} Z_{p,\si} \eps_{p,\si}^{\GW} & = \eps_{p}
\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.
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,\si} = \eps_{p,\si}^{\GW} \bc_{p,\si}
\end{equation}
with
\begin{equation}
\label{eq:Hp}
	\bH_p = 
	\begin{pmatrix}
		\eps_{p}		&	\bV_p^{\text{2h1p}}	&	\bV_p^{\text{2p1h}}
		\\
		\T{(\bV_p^{\text{2h1p}})}	&	\bC^{\text{2h1p}}			&	\bO
		\\
		\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{subequations}
\begin{align}
	\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} 
	\\
	\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} 
\end{align}
\end{subequations}
with the following expressions for the coupling blocks:
\begin{subequations}
\begin{align}
	\label{eq:V2h1p}
	V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
	\\
	\label{eq:V2p1h}
	V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
\end{align}
\end{subequations}
Here, we use lower case letters for the electronic configurations belonging to the reference model state and upper case letters for the external determinants (\ie, the perturbers).

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{multline}
	\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{multline}
with
\begin{equation}
\label{eq:Z_proj}
	Z_{p,\si} = \qty[ c_{p,\si,1} ]^{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 ($qia = iab$) that one wants to consider explicitly in the model space.
Equation \label{eq:Hp} can then be written exactly as
\begin{equation}
\label{eq:Hp_qia}
	\bH_{p,qia} = 
	\begin{pmatrix}
		\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}}
		\\
		\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{subequations}
\begin{gather}
	\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,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
	\\
	V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
\end{gather}
\end{subequations}
where $\text{sgn}$ is the sign function and $\mu$ is the chemical potential.

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.
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
\begin{equation}
\label{eq:Hp}
	\bSigC_{p,qia}(\omega)  = 
	\begin{pmatrix}
		\eps_p + \SigC_p(\omega)				&	V_{p,qia} + \SigC_{p,qia}(\omega)		
		\\
		V_{qia,p} + \SigC_{qia,p}(\omega)		&	\eps_{qia}	+ \SigC_{qia}(\omega)				
		\\
	\end{pmatrix}
\end{equation}
with the dynamical self-energies
\begin{subequations}
\begin{gather}
	\begin{split}
		\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{split}
	\\
	\begin{split}
	\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}})} 	
	\end{split}
	\\
	\begin{split}
	\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}})} 
	\end{split}
	\\
	\begin{split}
	\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{split}
\end{gather}
\end{subequations}
Of course, the present procedure can be generalized to any number of states.

Solving
\begin{equation}
	\bH_{p,qia} \cdot \bc_{p,qia,\si} = \eps_{p,qia,\si}^{\GW} \bc_{p,qia,\si}
\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:
\begin{equation}
\label{eq:Z_proj}
	Z_{p,qia,\si} = \qty[ c_{p,qia,\si,1} ]^{2} + \qty[ c_{p,qia,\si,2} ]^{2}
\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.
\\

%%%%%%%%%%%%%%%%%%%%%%
\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}

%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion.

%%%%%%%%%%%%%%%%%%%%%%%%
\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).}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article.% and its supplementary material.

%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{MRGW}
%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}