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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{Unphysical Discontinuities in $GW$ Methods and the Role of Intruder States}

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

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

Here,  for the sake of simplicity, we consider a restricted Hartree-Fock (HF) starting point but the present analysis can be straightforwardly extended to a Kohn-Sham (KS) starting point.
Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
\begin{equation}
\label{eq:qp_eq}
	\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
\end{equation}
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{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}
	\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are respectively the $m$th eigenvalue and eigenvector of the RPA problem in the Tamm-Dancoff approximation, \ie, 
\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}
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,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) $\eps{p}{} \equiv \eps{p,s=0}{}$ has a large weight $Z_{} \equiv Z_{p,=0}$
Note that we have the following important conservation rules
\begin{align}
	\sum_{s} Z_{p,s} & = 1
	&
	\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_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
are upfolded from the 1h and 1p sectors.
For each orbital $p$, this yields a linear eigenvalue problem of the form
\begin{equation}
	\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)}
\end{equation}
with
\begin{equation}
\label{eq:Hp}
	\bH^{(p)} = 
	\begin{pmatrix}
		\eps{p}{\HF}		&	\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}
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 $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$, and this eigenvalue problem has to be solved for each orbital that one wants to correct.
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}.
This can be further illustrate by expanding the secular equation associated with Eq.~\eqref{eq:Hp}
\begin{equation}
	\det[ \bH^{(p)} - \omega \bI ] = 0 
\end{equation}
and comparing it with Eq.~\eqref{eq:qp_eq} by setting
\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}
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.

One can see this downfolding process as the construction of an frequency-dependent effective Hamiltonian where the reference (zeroth-order) space is composed by a single determinant of the 1h or 1p sector and the external (first-order) space by all the singly-excited states built from to this reference electronic configuration.  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An illustrative example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Multiple solution issues in $GW$ appears all the time, especially for orbitals that are far in energy from the Fermi level.
Therefore, such issues are ubiquitous when one wants to compute core ionized states for example.

In order to illustrate the appearance and the origin of these multiple solutions, we consider the hydrogen molecule in the 6-31G basis set which corresponds to a system with 2 electrons and 4 spatial orbitals (one occupied and three virtual).
This example was already considered in our previous work but here we provide further insights on the origin of the appearances of these multiple solutions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	FIGURE 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
%	\includegraphics[width=\linewidth]{fig1a}
%	\includegraphics[width=\linewidth]{fig1b}
	\caption{
	\label{fig:H2}
	Quasiparticle energies (left), correlation part of the self-energy (center) and renormalization factor (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) for various orbitals of \ce{H2} at the {\GOWO}@HF/6-31G level.
}
\end{figure*}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Figure \ref{fig:H2} shows the evolution of the quasiparticle energies at the {\GOWO}@HF/6-31G level as a function on the internuclear distance $\RHH$.
As one can see there are two problematic regions showing obvious discontinuities around $\RHH = \SI{1}{\AA}$ and $\RHH = \SI{1}{\AA}$

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

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\bibliography{ufGW}
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\end{document}