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% methods
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% energies
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% orbital energies
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\newcommand { \A } [1]{ A_ { #1} }
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\newcommand { \ISCD } { Institut des Sciences du Calcul et des Donn\' ees, Sorbonne Universit\' e, Paris, France}
\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\' e de Toulouse, CNRS, UPS, France}
\newcommand { \LCT } { Laboratoire de Chimie Th\' eorique (UMR 7616), Sorbonne Universit\' e, CNRS, Paris, France}
\begin { document}
\title { A Density-Based Basis Set Correction for GW Methods}
\author { Bath\' elemy Pradines}
\affiliation { \LCT }
\affiliation { \ISCD }
\author { Emmanuel Giner}
\affiliation { \LCT }
\author { Anthony Scemama}
\affiliation { \LCPQ }
\author { Julien Toulouse}
\affiliation { \LCT }
\author { Pierre-Fran\c { c} ois Loos}
\email [Corresponding author: ] { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
% \centering
% \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure}
\end { abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%
\section { Introduction}
\label { sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section { Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here, we provide self-contained summary of the main equations and quantities behind { \GOWO } and { \evGW } .
More details can be found, for example, in Refs.~\citenum { vanSetten_ 2013, Kaplan_ 2016, Bruneval_ 2016} .
For a given (occupied or virtual) orbital $ p $ , the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
\begin { equation}
\label { eq:SigC}
\SigC { p} (\omega ) = \SigCp { p} (\omega ) + \SigCh { p} (\omega ),
\end { equation}
which, within the GW approximation, read
\begin { subequations}
\begin { align}
\label { eq:SigCh}
\SigCh { p} (\omega )
& = 2 \sum _ { i} ^ \text { occ} \sum _ { x} \frac { [pi|x]^ 2} { \omega - \e { i} + \Om { x} - i \eta } ,
\\
\label { eq:SigCp}
\SigCp { p} (\omega )
& = 2 \sum _ { a} ^ \text { virt} \sum _ { x} \frac { [pa|x]^ 2} { \omega - \e { a} - \Om { x} + i \eta } ,
\end { align}
\end { subequations}
where $ \eta $ is a positive infinitesimal.
The screened two-electron integrals
\begin { equation}
[pq|x] = \sum _ { ia} (pq|ia) (\bX +\bY )_ { ia} ^ { x}
\end { equation}
are obtained via the contraction of the bare two-electron integrals \cite { Gill_ 1994} $ ( pq|rs ) $ and the transition densities $ ( \bX + \bY ) _ { ia } ^ { x } $ originating from a random phase approximation (RPA) calculation \cite { Casida_ 1995, Dreuw_ 2005}
\begin { equation}
\label { eq:LR}
\begin { pmatrix}
\bA & \bB \\
\bB & \bA \\
\end { pmatrix}
\begin { pmatrix}
\bX \\
\bY \\
\end { pmatrix}
=
\bOm
\begin { pmatrix}
\boldsymbol { 1} & \boldsymbol { 0} \\
\boldsymbol { 0} & \boldsymbol { -1} \\
\end { pmatrix}
\begin { pmatrix}
\bX \\
\bY \\
\end { pmatrix} ,
\end { equation}
with
\begin { align}
\label { eq:RPA}
A_ { ia,jb} & = \delta _ { ij} \delta _ { ab} (\epsilon _ a - \epsilon _ i) + 2 (ia|jb),
&
B_ { ia,jb} & = 2 (ia|bj),
\end { align}
and $ \delta _ { pq } $ is the Kronecker delta. \cite { NISTbook}
The one-electron energies $ \epsilon _ p $ in \eqref { eq:SigCh} , \eqref { eq:SigCp} and \eqref { eq:RPA} are either the HF or the GW quasiparticle energies.
Equation \eqref { eq:LR} also provides the neutral excitation energies $ \Om { x } $ .
In practice, there exist two ways of determining the { \GOWO } QP energies. \cite { Hybertsen_ 1985a, vanSetten_ 2013}
In its ``graphical'' version, they are provided by one of the many solutions of the (non-linear) QP equation
\begin { equation}
\label { eq:QP-G0W0}
\omega = \eHF { p} + \Re [\SigC{p}(\omega)] .
\end { equation}
In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution.
In particular, it is usually worth calculating its renormalization weight (or factor), $ \Z { p } ( \eHF { p } ) $ , where
\begin { equation}
\label { eq:Z}
\Z { p} (\omega ) = \qty [ 1 - \pdv{\Re[\SigC{p}(\omega)] } { \omega } ]^ { -1} .
\end { equation}
Because of sum rules, \cite { Martin_ 1959, Baym_ 1961, Baym_ 1962, vonBarth_ 1996} the other solutions, known as satellites, share the remaining weight.
In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $ 0 . 7 $ -$ 0 . 9 $ .
Within the linearized version of { \GOWO } , one assumes that
\begin { equation}
\label { eq:SigC-lin}
\SigC { p} (\omega ) \approx \SigC { p} (\eHF { p} ) + (\omega - \eHF { p} ) \left . \pdv { \SigC { p} (\omega )} { \omega } \right |_ { \omega = \eHF { p} } ,
\end { equation}
that is, the self-energy behaves linearly in the vicinity of $ \omega = \eHF { p } $ .
Substituting \eqref { eq:SigC-lin} into \eqref { eq:QP-G0W0} yields
\begin { equation}
\label { eq:QP-G0W0-lin}
\eGOWO { p} = \eHF { p} + \Z { p} (\eHF { p} ) \Re [\SigC{p}(\eHF{p})] .
\end { equation}
Unless otherwise stated, in the remaining of this paper, the { \GOWO } QP energies are determined via the linearized method.
In the case of { \evGW } , the QP energy, $ \eGW { p } $ , are obtained via Eq.~\eqref { eq:QP-G0W0} , which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref { eq:SigC} ]. \cite { Hybertsen_ 1986, Shishkin_ 2007, Blase_ 2011, Faber_ 2011}
At least in the weakly correlated regime where a clear QP solution exists, we believe that, within { \evGW } , the self-consistent algorithm should select the solution of the QP equation \eqref { eq:QP-G0W0} with the largest renormalization weight $ \Z { p } ( \eGW { p } ) $ .
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%%%%%%%%%%%%%%%%%%%%%%%%
\section { Computational details}
\label { sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section { Results and Discussion}
\label { sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section { Conclusion}
\label { sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin { acknowledgements}
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), CALMIP (Toulouse) under allocation 2019-18005 and the Jarvis-Alpha cluster from the \textit { Institut Parisien de Chimie Physique et Th\' eorique} .
\end { acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography { GW,GW-srDFT,GW-srDFT-control}
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\end { document}
