2022 lines
115 KiB
TeX
2022 lines
115 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[
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colorlinks=true,
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breaklinks=true
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]{hyperref}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\T}[1]{#1^{\intercal}}
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% coordinates
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bx}{\boldsymbol{x}}
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% methods
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\newcommand{\NO}[1]{\{#1\}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\GW}{GW}
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\newcommand{\GOWO}{G_0W_0}
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\newcommand{\dRPA}{\text{dRPA}}
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\newcommand{\RPAx}{\text{RPAx}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\CC}{\text{CC}}
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%
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\newcommand{\Ne}{N}
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\newcommand{\Norb}{K}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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% operators
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\newcommand{\hHN}{\Hat{H}_{\text{N}}}
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\newcommand{\hVN}{\Hat{V}_{\text{N}}}
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\newcommand{\hh}{\Hat{h}}
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\newcommand{\bH}{\Bar{H}}
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\newcommand{\bHN}{\Bar{H}_{\text{N}}}
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\newcommand{\tHN}{\Tilde{H}_{\text{N}}}
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\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
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\newcommand{\ca}[2]{\Hat{a}^{#1}_{#2}}
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\newcommand{\cF}{\mathcal{F}}
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\newcommand{\cW}{\mathcal{W}}
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% energies
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\newcommand{\Enuc}{E^\text{nuc}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\ECC}{E_\text{CC}}
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% orbital energies
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\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
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\newcommand{\eHF}[1]{\epsilon_{#1}^\text{HF}}
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\newcommand{\eKS}[1]{\epsilon_{#1}^\text{KS}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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\newcommand{\om}[2]{\omega_{#1}^{#2}}
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% Matrix elements
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\newcommand{\MO}[2]{\phi_{#1}^{#2}}
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\newcommand{\SO}[2]{\psi_{#1}^{#2}}
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\newcommand{\ERI}[2]{\braket{#1}{#2}}
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\newcommand{\sERI}[2]{(#1|#2)}
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\newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
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\newcommand{\wERI}[2]{\widetilde{W}_{#1 #2}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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% Matrices
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bbH}{\Bar{\boldsymbol{H}}}
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\newcommand{\bOm}[2]{\boldsymbol{\Omega}_{#1}^{#2}}
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\newcommand{\bom}[2]{\boldsymbol{\omega}_{#1}^{#2}}
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\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}}
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\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}}
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\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
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\newcommand{\bD}[2]{\boldsymbol{D}_{#1}^{#2}}
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\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
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\newcommand{\bR}[2]{\boldsymbol{R}_{#1}^{#2}}
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\newcommand{\bT}[2]{\boldsymbol{T}_{#1}^{#2}}
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\newcommand{\bU}[2]{\boldsymbol{U}_{#1}^{#2}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
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\newcommand{\be}[2]{\boldsymbol{\epsilon}_{#1}^{#2}}
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\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
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\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}^\text{c}_{#1}}
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% orbitals, gaps, etc
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\ii}{\mathrm{i}}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LPT}{Laboratoire de Physique Th\'eorique, Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\ETSF}{European Theoretical Spectroscopy Facility (ETSF)}
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\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut N\'EEL, F-38042 Grenoble, France}
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\begin{document}
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\title{Connections between many-body perturbation and coupled-cluster theories}
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\author{Ra\'ul \surname{Quintero-Monsebaiz}}
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\affiliation{\LCPQ}
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\author{Enzo \surname{Monino}}
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\affiliation{\LCPQ}
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\author{Antoine \surname{Marie}}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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Here, we build on the works of Scuseria \textit{et al.} [\href{http://dx.doi.org/10.1063/1.3043729}{J.~Chem.~Phys.~\textbf{129}, 231101 (2008)}] and Berkelbach [\href{https://doi.org/10.1063/1.5032314}{J.~Chem.~Phys.~\textbf{149}, 041103 (2018)}] to show connections between the Bethe-Salpeter equation (BSE) formalism combined with the $GW$ approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels.
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In particular, we show how to recast the $GW$ and Bethe-Salpeter equations as non-linear CC-like equations.
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Similitudes between BSE@$GW$ and the similarity-transformed equation-of-motion CC method introduced by Nooijen are also put forward.
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The present work allows to easily transfer key developments and general knowledge gathered in CC theory to many-body perturbation theory.
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In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ and BSE frameworks.
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{RPA Physics and Beyond}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The random-phase approximation (RPA), introduced by Bohm and Pines \cite{Bohm_1951,Pines_1952,Bohm_1953} in the context of the uniform electron gas, \cite{Loos_2016} is a quasibosonic approximation where one treats fermion products as bosons.
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In the particle-hole (ph) channel, which is quite popular in the electronic structure community, \cite{Ren_2012,Chen_2017} particle-hole fermionic excitations and deexcitations are assumed to be bosons.
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Because ph-RPA takes into account dynamical screening by summing up to infinity the (time-independent) ring diagrams, it is adequate in the high-density (or weakly correlated) regime and captures effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958}
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Another important feature of ph-RPA compared to finite-order perturbation theory is that it does not exhibit divergences for small-gap or metallic systems. \cite{Gell-Mann_1957}
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Roughly speaking, the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} of many-body perturbation theory \cite{Martin_2016} can be seen as a cheap and efficient way of introducing correlation in order to go \textit{beyond} RPA physics.
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In the ph channel, BSE is commonly performed on top of a $GW$ calculation \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} from which one extracts the quasiparticle energies as well as the dynamically-screened Coulomb potential $W$.
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Practically, $GW$ produces accurate \textit{``charged''} excitations providing a faithful description of the fundamental gap via the computation of the RPA polarizability obtained by a resummation of all time-dependent ring diagrams.
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The remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing overall accurate \textit{``neutral''} excitations.
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BSE@$GW$ has been shown to be highly successful to compute low-lying excited states of various natures (charge transfer, Rydberg, valence, etc) in molecular systems with a very attractive accuracy/cost ratio.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Connection between RPA and CC}
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\label{sec:RPAx}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Interestingly, RPA has strong connections with coupled-cluster (CC) theory, \cite{Freeman_1977,Scuseria_2008,Jansen_2010,Scuseria_2013,Peng_2013,Berkelbach_2018,Rishi_2020} the workhorse of molecular electronic structure when one is looking for high accuracy. \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009}
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In a landmark paper, Scuseria \textit{et al.} \cite{Scuseria_2008} have proven that ring CC with doubles (rCCD) is equivalent to RPA with exchange (RPAx) for the computation of the correlation energy, solidifying in the process the numerical evidences provided by Freeman many years before. \cite{Freeman_1977}
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Assuming the existence of $\bX{}{-1}$ (which can be proven as long as the RPAx problem is stable \cite{Scuseria_2008}), this proof can be quickly summarized starting from the RPAx linear eigensystem
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\begin{equation}
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\label{eq:RPA}
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\begin{pmatrix}
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\bA{}{} & \bB{}{} \\
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-\bB{}{} & -\bA{}{} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{}{} \\
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\bY{}{} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\bX{}{} \\
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\bY{}{} \\
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\end{pmatrix}
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\cdot
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\bOm{}{}
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\end{equation}
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from which one gets, by introducing $\bT{}{} = \bY{}{} \cdot \bX{}{-1}$,
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\begin{equation}
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\begin{pmatrix}
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\bA{}{} & \bB{}{} \\
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-\bB{}{} & -\bA{}{} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bI \\
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\bT{}{} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\bI \\
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\bT{}{} \\
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\end{pmatrix}
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\cdot
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\bR{}{}
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\end{equation}
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where $\bR{}{} = \bX{}{} \cdot \bOm{}{} \cdot \bX{}{-1}$, or equivalently, the two following equations
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\begin{subequations}
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\begin{align}
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\label{eq:RPA_1}
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\bA{}{} + \bB{}{} \cdot \bT{}{} & = \bR{}{}
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\\
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\label{eq:RPA_2}
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-\bB{}{} - \bA{}{} \cdot \bT{}{} & = \bT{}{} \cdot \bR{}{}
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\end{align}
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\end{subequations}
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Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following Riccati equation
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\begin{equation}
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\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
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\end{equation}
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that matches the rCCD amplitude (or residual) equations
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\begin{multline}
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\label{eq:rCCD}
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r_{ij}^{ab}
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= \dbERI{ij}{ab}
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+ \Delta_{ijab} t_{ij}^{ab}
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+ \sum_{kc} \dbERI{ic}{ak} t_{kj}^{cb}
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\\
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+ \sum_{kc} \dbERI{kb}{cj} t_{ik}^{ac}
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+ \sum_{klcd} \dbERI{kl}{cd} t_{ik}^{ac} t_{lj}^{db} = 0
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\end{multline}
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knowing that
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\begin{subequations}
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\begin{align}
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\label{eq:A_RPAx}
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A_{ia,jb} & = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj}
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\\
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\label{eq:B_RPAx}
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B_{ia,jb} & = \dbERI{ij}{ab}
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\end{align}
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\end{subequations}
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where $\Delta_{ijab} = \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{}$.
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We assume real quantities throughout this paper, $\e{p}{}$ is the one-electron energy associated with the Hartree-Fock (HF) spinorbital $\SO{p}{}(\bx)$ and
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\begin{equation}
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\label{eq:ERI}
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\braket{pq}{rs} = \iint \SO{p}{}(\bx_1) \SO{q}{}(\bx_2) \frac{1}{\abs{\br_1 - \br_2}} \SO{r}{}(\bx_1) \SO{s}{}(\bx_2) d\bx_1 d\bx_2
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\end{equation}
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are two-electron repulsion integrals, while
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\begin{equation}
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\dbERI{pq}{rs} = \braket{pq}{rs} - \braket{pq}{sr}
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\end{equation}
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are their anti-symmetrized versions.
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The composite variable $\bx$ gathers spin and spatial ($\br$) variables.
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The indices $i$, $j$, $k$, and $l$ are occupied (hole) orbitals; $a$, $b$, $c$, and $d$ are unoccupied (particle) orbitals; $p$, $q$, $r$, and $s$ indicate arbitrary orbitals; and $m$ labels single excitations or deexcitations.
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In the following, $O$ and $V$ are the number of occupied and virtual spinorbitals, respectively, and $N = O + V$ is the total number.
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There are various ways of computing the RPAx correlation energy, \cite{Furche_2008,Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula \cite{Sawada_1957a,Rowe_1968,Schuck_book} yields\footnote{The factor $1/4$ in Eq.~\eqref{eq:EcRPAx} is sometimes replaced by a factor $1/2$, which corresponds to a different choice for the interaction kernel. See Ref.~\onlinecite{Angyan_2011} for more details}
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\begin{equation}
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\label{eq:EcRPAx}
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\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
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\end{equation}
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and matches the rCCD correlation energy
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\begin{equation}
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\Ec^\text{rCCD} = \frac{1}{4} \sum_{ijab} \dbERI{ij}{ab} t_{ij}^{ab} = \frac{1}{4} \Tr(\bB{}{} \cdot \bT{}{})
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\end{equation}
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because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}.
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Note that, in the case of RPAx, the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem \cite{Furche_2005} (ACFDT) when exchange is included in the interaction kernel. \cite{Angyan_2011}
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This simple and elegant proof was subsequently extended to excitation energies by Berkelbach, \cite{Berkelbach_2018} who showed that similitudes between equation-of-motion (EOM) rCCD (EOM-rCCD) \cite{Stanton_1993} and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation (see also Ref.~\onlinecite{Rishi_2020}).
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To be more specific, restricting ourselves to CCD, \ie, $\hT = \hT_2$, the elements of the 1h1p block of the EOM Hamiltonian read \cite{Stanton_1993}
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\begin{equation}
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\mel*{ \Psi_{i}^{a} }{ \bHN}{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
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\end{equation}
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where $\bHN = e^{-\hT} \hH_{N} e^{\hT} - \ECC $ is the (shifted) similarity-transformed normal-ordered Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
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\begin{subequations}
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\begin{align}
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\label{eq:cFab}
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\cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \dbERI{kl}{bc} t_{kl}^{ac}
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\\
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\label{eq:cFij}
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\cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \dbERI{ik}{cd} t_{jk}^{cd}
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\end{align}
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\end{subequations}
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and the two-body term is
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\begin{equation}
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\label{eq:cWibaj}
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\cW_{ibaj} = \dbERI{ib}{aj} + \sum_{kc} \dbERI{ik}{ac} t_{kj}^{cb}
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\end{equation}
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Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab} and \eqref{eq:cFij}] and relying on the rCCD amplitudes in the two-body terms, Eq.~\eqref{eq:cWibaj}, yields
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\begin{equation}
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\label{eq:EOM-rCCD}
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\begin{split}
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\mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} }
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& = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj} + \sum_{kc} \dbERI{ik}{ac} t_{kj}^{cb}
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\\
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& = (\bA{}{} + \bB{}{} \cdot \bT{}{})_{ia,jb}
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\end{split}
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\end{equation}
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which exactly matches Eq.~\eqref{eq:RPA_1}.
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Although the excitation energies of this approximate EOM-rCCD scheme are equal to the RPAx ones, it has been shown that the transition amplitudes (or residues) are distinct and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018}
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Equation \eqref{eq:EOM-rCCD} can be more systematically derived through the formulation of $\Lambda$ equations based on a rCCD effective Hamiltonian, as proposed by Rishi \textit{et al.} \cite{Rishi_2020}
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As we shall see below, the connection between a ph eigensystem with the structure of Eq.~\eqref{eq:RPA} and a set of CC-like amplitude equations does not hold only for RPAx as it is actually quite general and can be applied to most ph problems, such as time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995} BSE, and many others.
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This analysis has also been extended to the pp and hh sectors independently by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013}
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(See also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Connection between BSE and CC}
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\label{sec:BSE}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Within the usual static approximation of BSE, one must solve a very similar linear eigenvalue problem
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\begin{equation}
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\label{eq:BSE}
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\begin{pmatrix}
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\bA{}{\BSE} & \bB{}{\BSE} \\
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-\bB{}{\BSE} & -\bA{}{\BSE} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{}{\BSE} \\
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|
\bY{}{\BSE} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bX{}{\BSE} \\
|
|
\bY{}{\BSE} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\bOm{}{\BSE}
|
|
\end{equation}
|
|
where the matrix elements read
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:A_BSE}
|
|
A_{ia,jb}^{\BSE} & = \delta_{ij} \delta_{ab} (\e{a}{\GW} - \e{i}{\GW}) + \dbERI{ib}{aj} - W_{ij,ba}^\text{stat}
|
|
\\
|
|
\label{eq:B_BSE}
|
|
B_{ia,jb}^{\BSE} & = \dbERI{ij}{ab} - W_{ib,ja}^\text{stat}
|
|
\end{align}
|
|
\end{subequations}
|
|
The quasiparticle energies $\e{p}{\GW}$ are computed at the $GW$ level (see below) and
|
|
\begin{multline}
|
|
\label{eq:W}
|
|
W_{pq,rs}^\text{c}(\omega) = \sum_{m} \sERI{pq}{m} \sERI{rs}{m}
|
|
\\
|
|
\times \qty[ \frac{1}{\omega - \Om{m}{\dRPA} + \ii \eta} - \frac{1}{\omega + \Om{m}{\dRPA} - \ii \eta} ]
|
|
\end{multline}
|
|
are the elements of the correlation part of the dynamically-screened Coulomb potential which is set to its static limit \ie, $W_{pq,rs}^\text{stat} = W_{pq,rs}^\text{c}(\omega = 0)$.
|
|
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, the screened two-electron integrals are
|
|
\begin{equation}
|
|
\label{eq:sERI}
|
|
\sERI{pq}{m} = \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia}
|
|
\end{equation}
|
|
and $\Om{m}{\dRPA}$ is the $m$th (positive) eigenvalue and $\bX{m}{\dRPA} + \bY{m}{\dRPA}$ is constructed from the corresponding eigenvectors of the direct (\ie, without exchange) RPA (dRPA) problem defined as
|
|
\begin{equation}
|
|
\label{eq:dRPA}
|
|
\begin{pmatrix}
|
|
\bA{}{\dRPA} & \bB{}{\dRPA} \\
|
|
-\bB{}{\dRPA} & -\bA{}{\dRPA} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX{}{\dRPA} \\
|
|
\bY{}{\dRPA} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bX{}{\dRPA} \\
|
|
\bY{}{\dRPA} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\bOm{}{\dRPA}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:A_dRPA}
|
|
A_{ia,jb}^{\dRPA} & = \delta_{ij} \delta_{ab} (\e{a}{\GW} - \e{i}{\GW}) + \ERI{ib}{aj}
|
|
\\
|
|
\label{eq:B_dRPA}
|
|
B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
As readily seen in Eqs.~\eqref{eq:A_RPAx}, \eqref{eq:B_RPAx}, \eqref{eq:A_BSE} and \eqref{eq:B_BSE}, the only difference between RPAx and BSE lies in the definition of the matrix elements, where one includes, via the presence of the $GW$ quasiparticle energies in the one-body terms and the screening of the electron-electron interaction [see Eq.~\eqref{eq:W}] in the two-body terms, correlation effects at the BSE level.
|
|
Therefore, following the derivation detailed in Sec.~\ref{sec:RPAx}, one can show that the BSE correlation energy obtained using the trace formula
|
|
\begin{equation}
|
|
\Ec^\text{BSE} = \frac{1}{4} \Tr(\bOm{}{\BSE} - \bA{}{\BSE}) = \frac{1}{4} \sum_{ijab} \wERI{ij}{ab} \tilde{t}_{ij}^{ab}
|
|
\end{equation}
|
|
can be equivalently obtained via a set of rCCD-like amplitude equations, where one substitutes in Eq.~\eqref{eq:rCCD} the HF orbital energies by the $GW$ quasiparticle energies and all the antisymmetrized two-electron integrals $\dbERI{pq}{rs}$ by $\wERI{pq}{rs} = \dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$, \ie,
|
|
\begin{multline}
|
|
\label{eq:rCCD-BSE}
|
|
\Tilde{r}_{ij}^{ab}
|
|
= \wERI{ij}{ab}
|
|
+ \Delta_{ijab}^{\GW} \tilde{t}_{ij}^{ab}
|
|
+ \sum_{kc} \wERI{ic}{ak} \tilde{t}_{kj}^{cb}
|
|
\\
|
|
+ \sum_{kc} \wERI{kb}{cj} \tilde{t}_{ik}^{ac}
|
|
+ \sum_{klcd} \wERI{kl}{cd} \tilde{t}_{ik}^{ac} \tilde{t}_{lj}^{db} = 0
|
|
\end{multline}
|
|
with $\Delta_{ijab}^{\GW} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{j}{\GW}$.
|
|
Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these approximate CCD amplitude equations can be solved with $\order*{N^6}$ cost via the definition of appropriate intermediates.
|
|
As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, \footnote{\alert{To the best of our knowledge, the trace (or plasmon) formula has been first introduced by Sawada \cite{Sawada_1957a} to calculate the correlation energy of the uniform electron gas as an alternative to the Gell-Mann-Brueckner formulation \cite{Gell-Mann_1957} where one integrates along the adiabatic connection path.
|
|
More precisely, the trace formula can be justified via the introduction of a quadratic Hamiltonian made of boson transition operators (quasiboson approximation). See Ref.~\onlinecite{Li_2020} for more details}.} either based on the plasmon formula \cite{Li_2020,Li_2021,DiSabatino_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
|
|
|
|
|
|
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states.
|
|
Indeed, one can obtain an analog of the 1h1p block of the approximate EOM-rCCD Hamiltonian [see Eq.\eqref{eq:EOM-rCCD}] using the amplitudes resulting from Eq.~\eqref{eq:rCCD-BSE} as well as replacing $\bA{}{}$ and $\bB{}{}$ by their BSE counterparts, \ie,
|
|
\begin{equation}
|
|
\label{eq:EOM-rCCD-BSE}
|
|
\mel*{ \Psi_{i}^{a} }{ \tHN }{ \Psi_{j}^{b} }
|
|
= (\e{a}{\GW} - \e{i}{\GW}) \delta_{ij} \delta_{ab} + \wERI{ib}{aj} + \sum_{kc} \wERI{ik}{ac} \tilde{t}_{kj}^{cb}
|
|
\end{equation}
|
|
This equation provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}, and the corresponding $\Lambda$ equations based on the BSE effective Hamiltonian $\tHN$ can be derived following Ref.~\onlinecite{Rishi_2020}.
|
|
|
|
However, there is a significant difference with RPAx as the BSE involves $GW$ quasiparticle energies, where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}.
|
|
In other words, in the spirit of the Brueckner version of CCD, \cite{Handy_1989} \alert{the $GW$ pre-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic \cite{Scuseria_2008,Scuseria_2013} as well as additional diagrams, \cite{Lange_2018} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}.}
|
|
|
|
This observation evidences clear similitudes between BSE@$GW$ and the similarity-transformed EOM-CC (STEOM-CC) method introduced by Nooijen, \cite{Nooijen_1997c,Nooijen_1997b,Nooijen_1997a} where one performs a second similarity transformation to partially decouple the 1h determinants from the 2h1p ones in the ionization potential (IP) sector and the 1p determinants from the 1h2p ones in the electron affinity (EA) sector.
|
|
At the CC with singles and doubles (CCSD) level, for example, this is achieved by performing IP-EOM-CCSD \cite{Stanton_1994,Musial_2003a} (up to 2h1p) and EA-EOM-CCSD \cite{Nooijen_1995,Musial_2003b} (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling.
|
|
(An extended version of STEOM-CC has been proposed where the EOM treatment is pushed up to 2h2p. \cite{Nooijen_2000})
|
|
Following the same philosophy, in BSE@$GW$, one performs first a $GW$ calculation (which corresponds to an approximate and simultaneous treatment of the IP and EA sectors up to 2h1p and 2p1h \cite{Lange_2018,Monino_2022}) in order to renormalize the one-electron energies (see Sec.~\ref{sec:GW} for more details).
|
|
Then, a static BSE calculation is performed in the 1h1p sector with a two-body term dressed with correlation stemming from $GW$.
|
|
The dynamical version of BSE [where the BSE kernel is explicitly treated as frequency-dependent in Eq.~\eqref{eq:BSE}] takes partially into account the 2h2p configurations. \cite{Strinati_1980,Strinati_1982,Strinati_1984,Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Monino_2021,Bintrim_2022}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Connection between $GW$ and CC}
|
|
\label{sec:GW}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Because $GW$ is able to capture key correlation effects as illustrated above, it is therefore interesting to investigate if it is also possible to recast the $GW$ equations as a set of CC-like equations that can be solved iteratively using the CC machinery.
|
|
Connections between approximate IP/EA-EOM-CC schemes and the $GW$ approximation have been already studied in details by Lange and Berkelbach, \cite{Lange_2018} but we believe that the present work proposes a different perspective on this particular subject as we derive genuine CC equations that do not decouple the 2h1p and 2p1h sectors.
|
|
Note also that the procedure described below can be applied to other approximate self-energies such as second-order Green's function (or second Born) \cite{Stefanucci_2013,Ortiz_2013,Phillips_2014,Rusakov_2014,Hirata_2015,Hirata_2017} or $T$-matrix.\cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022}
|
|
|
|
Quite unfortunately, there are several ways of computing $GW$ quasiparticle energies. \cite{Loos_2018b}
|
|
Within the perturbative $GW$ scheme (commonly known as $G_0W_0$), the quasiparticle energies are obtained via a one-shot procedure (with or without linearization).
|
|
\cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007}
|
|
Partial self-consistency can be attained via the \textit{``eigenvalue''} self-consistent $GW$ (ev$GW$) \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018} or the quasiparticle self-consistent $GW$ (qs$GW$) \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} schemes.
|
|
|
|
In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation
|
|
\begin{equation}
|
|
\label{eq:GW}
|
|
\qty[ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
|
|
\end{equation}
|
|
\alert{which gives also access to the satellite solutions}.
|
|
In Eq.~\eqref{eq:GW}, $\be{}{}$ is a diagonal matrix gathering the HF orbital energies and the elements of the correlation part of the dynamical (and non-hermitian) $GW$ self-energy are
|
|
\begin{equation}
|
|
\begin{split}
|
|
\Sig{pq}{\GW}(\omega)
|
|
& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{\GW} + \Om{m}{\dRPA} - \ii \eta}
|
|
\\
|
|
& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{\GW} - \Om{m}{\dRPA} + \ii \eta}
|
|
\end{split}
|
|
\end{equation}
|
|
Because both the left- and right-hand sides of Eq.~\eqref{eq:GW} depend on $\e{p}{\GW}$, this equation has to be solved iteratively via a self-consistent procedure.
|
|
|
|
As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the quasiparticle equation \eqref{eq:GW} can be recast as a larger set of linear and frequency-independent equations (that still needs to be solved self-consistently), which reads in the Tamm-Dancoff approximation
|
|
\begin{equation}
|
|
\label{eq:GWlin}
|
|
\begin{pmatrix}
|
|
\be{}{} & \bV{}{\text{2h1p}} & \bV{}{\text{2p1h}} \\
|
|
\T{(\bV{}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\
|
|
\T{(\bV{}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX{}{} \\
|
|
\bY{}{\text{2h1p}} \\
|
|
\bY{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bX{}{} \\
|
|
\bY{}{\text{2h1p}} \\
|
|
\bY{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\be{}{\GW}
|
|
\end{equation}
|
|
where $\be{}{\GW}$ is a diagonal matrix collecting the quasiparticle energies, the 2h1p and 2p1h matrix elements are
|
|
\begin{subequations}
|
|
\begin{align}
|
|
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \e{i}{\GW} + \e{j}{\GW} - \e{a}{\GW}) \delta_{jl} \delta_{ac} - \ERI{jc}{al} ] \delta_{ik}
|
|
\\
|
|
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW}) \delta_{ik} \delta_{ac} + \ERI{ak}{ic} ] \delta_{bd}
|
|
\end{align}
|
|
\end{subequations}
|
|
and the corresponding coupling blocks read
|
|
\begin{align}
|
|
V^\text{2h1p}_{p,klc} & = \ERI{pc}{kl}
|
|
&
|
|
V^\text{2p1h}_{p,kcd} & = \ERI{pk}{dc}
|
|
\end{align}
|
|
Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}
|
|
Note that, contrary to the IP/EA-EOM-CC equations, $GW$ does couple the IP and EA sectors due to the lack of exponential parametrization of the wave function. \cite{Nooijen_1995,Rishi_2020}
|
|
However, it allows to generate higher-order diagrams. \cite{Lange_2018,Schirmer_2018}
|
|
|
|
Let us suppose that we are looking for the $N$ \textit{``principal''} (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}.
|
|
Therefore, $\bX{}{}$ and $\be{}{\GW}$ are square matrices of size $N \times N$.
|
|
Assuming the existence of $\bX{}{-1}$ and introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have
|
|
\begin{equation}
|
|
\begin{pmatrix}
|
|
\be{}{} & \bV{}{\text{2h1p}} & \bV{}{\text{2p1h}} \\
|
|
\T{(\bV{}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\
|
|
\T{(\bV{}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bI \\
|
|
\bT{}{\text{2h1p}} \\
|
|
\bT{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bI \\
|
|
\bT{}{\text{2h1p}} \\
|
|
\bT{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\bR{}{}
|
|
\end{equation}
|
|
with $\bR{}{} = \bX{}{} \cdot \be{}{\GW} \cdot \bX{}{-1}$, which yields the three following equations
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\be{}{} + \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} & = \bR{}{}
|
|
\label{eq:R}
|
|
\\
|
|
\T{(\bV{}{\text{2h1p}})} + \bC{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} & = \bT{}{\text{2h1p}} \cdot \bR{}{}
|
|
\label{eq:T1R}
|
|
\\
|
|
\T{(\bV{}{\text{2p1h}})} + \bC{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} & = \bT{}{\text{2p1h}} \cdot \bR{}{}
|
|
\label{eq:T2R}
|
|
\end{align}
|
|
\end{subequations}
|
|
Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one gets two coupled Riccati equations
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\begin{split}
|
|
\T{(\bV{}{\text{2h1p}})}
|
|
+ \bC{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}}
|
|
- \bT{}{\text{2h1p}} \cdot \be{}{}
|
|
- \bT{}{\text{2h1p}} \cdot \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}}
|
|
\\
|
|
- \bT{}{\text{2h1p}} \cdot \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
|
|
= \bO
|
|
\end{split}
|
|
\\
|
|
\begin{split}
|
|
\T{(\bV{}{\text{2p1h}})}
|
|
+ \bC{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
|
|
- \bT{}{\text{2p1h}} \cdot \be{}{}
|
|
- \bT{}{\text{2p1h}} \cdot \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}}
|
|
\\
|
|
- \bT{}{\text{2p1h}} \cdot \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
|
|
= \bO
|
|
\end{split}
|
|
\end{align}
|
|
\end{subequations}
|
|
that can be converted to the following CC-like residual equations
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:r_2h1p}
|
|
\begin{split}
|
|
r_{ija,p}^{\text{2h1p}}
|
|
& = \ERI{pa}{ij}
|
|
+ \Delta_{ija,p}^{\text{2h1p}} t_{ija,p}^{\text{2h1p}}
|
|
- \sum_{kc} \ERI{jc}{ak} t_{ikc,p}^{\text{2h1p}}
|
|
\\
|
|
& - \sum_{klcq} \ERI{qc}{kl} t_{ija,q}^{\text{2h1p}} t_{klc,p}^{\text{2h1p}}
|
|
- \sum_{kcdq} \ERI{qk}{dc} t_{ija,q}^{\text{2h1p}} t_{kcd,p}^{\text{2p1h}}
|
|
= 0
|
|
\end{split}
|
|
\\
|
|
\label{eq:r_2p1h}
|
|
\begin{split}
|
|
r_{iab,p}^{\text{2p1h}}
|
|
& = \ERI{pi}{ba}
|
|
+ \Delta_{iab,p}^{\text{2p1h}} t_{iab,p}^{\text{2p1h}}
|
|
+ \sum_{kc} \ERI{ak}{ic} t_{kcb,p}^{\text{2p1h}}
|
|
\\
|
|
& - \sum_{klcq} \ERI{qc}{kl} t_{iab,q}^{\text{2p1h}} t_{klc,p}^{\text{2h1p}}
|
|
- \sum_{kcdq} \ERI{qk}{dc} t_{iab,q}^{\text{2p1h}} t_{kcd,p}^{\text{2p1h}}
|
|
= 0
|
|
\end{split}
|
|
\end{align}
|
|
\end{subequations}
|
|
with $\Delta_{ija,p}^{\text{2h1p}} = \e{i}{\GW} + \e{j}{\GW} - \e{a}{\GW} - \e{p}{}$ and $\Delta_{iab,p}^{\text{2p1h}} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{p}{}$.
|
|
To determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one can then rely on the usual quasi-Newton iterative procedure to solve these quadratic equations by updating the amplitudes via
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:t_2h1p_update}
|
|
t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \qty( \Delta_{ija,p}^{\text{2h1p}} )^{-1} r_{ija,p}^{\text{2h1p}}
|
|
\\
|
|
\label{eq:t_2p1h_update}
|
|
t_{iab,p}^{\text{2p1h}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \qty( \Delta_{iab,p}^{\text{2p1h}} )^{-1} r_{iab,p}^{\text{2p1h}}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $\be{}{} + \bSig{}{\GW}$, where
|
|
\begin{equation}
|
|
\bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
|
|
\end{equation}
|
|
\alert{Due to the non-linear nature of these equations, the iterative procedure proposed in Eqs.~\eqref{eq:t_2h1p_update} and \eqref{eq:t_2p1h_update} can potentially converge to satellite solutions.
|
|
This is also the case at the CC level when one relies on more elaborated algorithms to converge the amplitude equations to higher-energy solutions. \cite{Piecuch_2000,Mayhall_2010,Lee_2019,Kossoski_2021,Marie_2021b}}
|
|
|
|
Again, similarly to the dynamical equations defined in Eq.~\eqref{eq:GW} which requires the diagonalization of the dRPA eigenproblem [see Eq.~\eqref{eq:dRPA}], the CC equations reported in Eqs.~\eqref{eq:r_2h1p} and \eqref{eq:r_2p1h} can be solved with $\order*{N^6}$ cost by defining judicious intermediates.
|
|
Cholesky decomposition, density fitting, and other related techniques may be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
|
|
The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equations for each value of $p$ separately.
|
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%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%
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Here, we have unveiled exact similarities between CC and many-body perturbation theory at the ground- and excited-state levels.
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More specifically, we have shown how to recast $GW$ and BSE as non-linear CC-like equations that can be solved with the usual CC machinery at the same computational cost.
|
|
The conventional and CC-based versions of the BSE and $GW$ schemes that we have described in the present work have been implemented in the electronic structure package QuAcK \cite{QuAcK} (available at \url{https://github.com/pfloos/QuAcK}) with which we have numerically checked these exact equivalences.
|
|
Similitudes between BSE@$GW$ and STEOM-CC have been also highlighted, and may explain the reliability of BSE@$GW$ for the computation of optical excitations in molecular systems.
|
|
|
|
We hope that the present work may provide a consistent approach for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} and BSE \cite{IsmailBeigi_2003,Caylak_2021,Knysh_2022} frameworks, hence broadening the applicability of these formalisms in computational photochemistry.
|
|
However, several challenges lie ahead as one must derive, for example, the $\Lambda$ equations associated with $GW$\cite{Bartlett_1986,Rishi_2020} and the response of the static screening with respect to the external perturbation at the BSE level.
|
|
The present connections between CC and $GW$ could also provide new directions for the development of multireference $GW$ methods \cite{Brouder_2009,Linner_2019} in order to treat strongly correlated systems. \cite{Lyakh_2012,Evangelista_2018}
|
|
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section*{Supplementary Material}
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%\label{sec:supmat}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%
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\acknowledgments{
|
|
PFL thanks Xavier Blase, Pina Romaniello, and Francesco Evangelista for useful discussions.
|
|
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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%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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%%%%%%%%%
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%\appendix
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%%%%%%%%%
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%merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
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\bibitem [{\citenamefont {Bohm}\ and\ \citenamefont {Pines}(1951)}]{Bohm_1951}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Bohm}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Pines}},\
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}\href {\doibase 10.1103/PhysRev.82.625} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages}
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{625} (\bibinfo {year} {1951})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Pines}\ and\ \citenamefont
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{Bohm}(1952)}]{Pines_1952}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Pines}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Bohm}},\
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}\href {\doibase 10.1103/PhysRev.85.338} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {85}},\ \bibinfo {pages}
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{338} (\bibinfo {year} {1952})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Bohm}\ and\ \citenamefont {Pines}(1953)}]{Bohm_1953}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Bohm}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Pines}},\
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}\href {\doibase 10.1103/PhysRev.92.609} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {92}},\ \bibinfo {pages}
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{609} (\bibinfo {year} {1953})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont {Gill}(2016)}]{Loos_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1002/wcms.1257} {\bibfield {journal} {\bibinfo
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{journal} {Wiley Interdiscip. Rev. Comput. Mol. Sci.}\ }\textbf {\bibinfo
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{volume} {6}},\ \bibinfo {pages} {410} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Ren}\ \emph {et~al.}(2012)\citenamefont {Ren},
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\citenamefont {Rinke}, \citenamefont {Joas},\ and\ \citenamefont
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{Scheffler}}]{Ren_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Ren}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Rinke}}, \bibinfo
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{author} {\bibfnamefont {C.}~\bibnamefont {Joas}}, \ and\ \bibinfo {author}
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{\bibfnamefont {M.}~\bibnamefont {Scheffler}},\ }\href {\doibase
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10.1007/s10853-012-6570-4} {\bibfield {journal} {\bibinfo {journal} {J.
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Mater. Sci.}\ }\textbf {\bibinfo {volume} {47}},\ \bibinfo {pages} {7447}
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(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Chen}\ \emph {et~al.}(2017)\citenamefont {Chen},
|
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\citenamefont {Voora}, \citenamefont {Agee}, \citenamefont {Balasubramani},\
|
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and\ \citenamefont {Furche}}]{Chen_2017}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~P.}\ \bibnamefont
|
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{Chen}}, \bibinfo {author} {\bibfnamefont {V.~K.}\ \bibnamefont {Voora}},
|
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\bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont {Agee}}, \bibinfo
|
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{author} {\bibfnamefont {S.~G.}\ \bibnamefont {Balasubramani}}, \ and\
|
|
\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Furche}},\ }\href
|
|
{\doibase 10.1146/annurev-physchem-040215-112308} {\bibfield {journal}
|
|
{\bibinfo {journal} {Ann. Rev. Phys. Chem.}\ }\textbf {\bibinfo {volume}
|
|
{68}},\ \bibinfo {pages} {421} (\bibinfo {year} {2017})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Gell-Mann}\ and\ \citenamefont
|
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{Brueckner}(1957)}]{Gell-Mann_1957}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Gell-Mann}}\ and\ \bibinfo {author} {\bibfnamefont {K.~A.}\ \bibnamefont
|
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{Brueckner}},\ }\href {\doibase 10.1103/PhysRev.106.364} {\bibfield
|
|
{journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume}
|
|
{106}},\ \bibinfo {pages} {364} (\bibinfo {year} {1957})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Nozi\`eres}\ and\ \citenamefont
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{Pines}(1958)}]{Nozieres_1958}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Nozi\`eres}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
|
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{Pines}},\ }\href {\doibase 10.1103/PhysRev.111.442} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {111}},\
|
|
\bibinfo {pages} {442} (\bibinfo {year} {1958})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Salpeter}\ and\ \citenamefont
|
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{Bethe}(1951)}]{Salpeter_1951}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~E.}\ \bibnamefont
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{Salpeter}}\ and\ \bibinfo {author} {\bibfnamefont {H.~A.}\ \bibnamefont
|
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{Bethe}},\ }\href {\doibase 10.1103/PhysRev.84.1232} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {84}},\
|
|
\bibinfo {pages} {1232} (\bibinfo {year} {1951})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Strinati}(1988)}]{Strinati_1988}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
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{Strinati}},\ }\href {\doibase 10.1007/BF02725962} {\bibfield {journal}
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{\bibinfo {journal} {Riv. Nuovo Cimento}\ }\textbf {\bibinfo {volume}
|
|
{11}},\ \bibinfo {pages} {1} (\bibinfo {year} {1988})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Blase}, \citenamefont {Duchemin},\ and\ \citenamefont
|
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{Jacquemin}(2018)}]{Blase_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Jacquemin}},\ }\href
|
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{\doibase 10.1039/C7CS00049A} {\bibfield {journal} {\bibinfo {journal}
|
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{Chem. Soc. Rev.}\ }\textbf {\bibinfo {volume} {47}},\ \bibinfo {pages}
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{1022} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Blase}\ \emph {et~al.}(2020)\citenamefont {Blase},
|
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\citenamefont {Duchemin}, \citenamefont {Jacquemin},\ and\ \citenamefont
|
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{Loos}}]{Blase_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
|
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\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Jacquemin}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}},\ }\href
|
|
{\doibase 10.1021/acs.jpclett.0c01875} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Phys. Chem. Lett.}\ }\textbf {\bibinfo {volume} {11}},\
|
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\bibinfo {pages} {7371} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Martin}, \citenamefont {Reining},\ and\ \citenamefont
|
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{Ceperley}(2016)}]{Martin_2016}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~M.}\ \bibnamefont
|
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{Martin}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Reining}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont {Ceperley}},\
|
|
}\href@noop {} {\emph {\bibinfo {title} {Interacting Electrons: Theory and
|
|
Computational Approaches}}}\ (\bibinfo {publisher} {Cambridge University
|
|
Press},\ \bibinfo {year} {2016})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hedin}(1965)}]{Hedin_1965}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
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{Hedin}},\ }\href {\doibase 10.1103/PhysRev.139.A796} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {139}},\
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\bibinfo {pages} {A796} (\bibinfo {year} {1965})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Aryasetiawan}\ and\ \citenamefont
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{Gunnarsson}(1998)}]{Aryasetiawan_1998}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
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{Aryasetiawan}}\ and\ \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
|
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{Gunnarsson}},\ }\href {\doibase 10.1088/0034-4885/61/3/002} {\bibfield
|
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{journal} {\bibinfo {journal} {Rep. Prog. Phys.}\ }\textbf {\bibinfo
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{volume} {61}},\ \bibinfo {pages} {237} (\bibinfo {year} {1998})}\BibitemShut
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\bibitem [{\citenamefont {Onida}, \citenamefont {Reining},\ and\ \citenamefont
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{Rubio}(2002)}]{Onida_2002}%
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{Onida}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Reining}}, \
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and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Rubio}},\ }\href
|
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{\doibase 10.1103/RevModPhys.74.601} {\bibfield {journal} {\bibinfo
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{journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {74}},\ \bibinfo
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{pages} {601} (\bibinfo {year} {2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Reining}(2017)}]{Reining_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
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{Reining}},\ }\href {\doibase 10.1002/wcms.1344} {\bibfield {journal}
|
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{\bibinfo {journal} {Wiley Interdiscip. Rev. Comput. Mol. Sci.}\ }\textbf
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{\bibinfo {volume} {8}},\ \bibinfo {pages} {e1344} (\bibinfo {year}
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{2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Golze}, \citenamefont {Dvorak},\ and\ \citenamefont
|
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{Rinke}(2019)}]{Golze_2019}%
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{Golze}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Dvorak}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Rinke}},\ }\href
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{\doibase 10.3389/fchem.2019.00377} {\bibfield {journal} {\bibinfo
|
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{journal} {Front. Chem.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages}
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{377} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\citenamefont {van Setten}(2021)}]{Bruneval_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
|
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{Bruneval}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Dattani}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {M.~J.}\ \bibnamefont {van Setten}},\
|
|
}\href {\doibase 10.3389/fchem.2021.749779} {\bibfield {journal} {\bibinfo
|
|
{journal} {Front. Chem.}\ }\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages}
|
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{749779} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rohlfing}\ and\ \citenamefont
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{Louie}(1999)}]{Rohlfing_1999a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Rohlfing}}\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
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{Louie}},\ }\href {\doibase 10.1103/PhysRevLett.82.1959} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {82}},\ \bibinfo {pages} {1959} (\bibinfo {year}
|
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{1999})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {van~der Horst}\ \emph {et~al.}(1999)\citenamefont
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{van~der Horst}, \citenamefont {Bobbert}, \citenamefont {Michels},
|
|
\citenamefont {Brocks},\ and\ \citenamefont {Kelly}}]{Horst_1999}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-W.}\ \bibnamefont
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{van~der Horst}}, \bibinfo {author} {\bibfnamefont {P.~A.}\ \bibnamefont
|
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{Bobbert}}, \bibinfo {author} {\bibfnamefont {M.~A.~J.}\ \bibnamefont
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{Michels}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Brocks}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont {Kelly}},\ }\href
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{\doibase 10.1103/PhysRevLett.83.4413} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {83}},\ \bibinfo
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{pages} {4413} (\bibinfo {year} {1999})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Puschnig}\ and\ \citenamefont
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{Ambrosch-Draxl}(2002)}]{Puschnig_2002}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Puschnig}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
|
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{Ambrosch-Draxl}},\ }\href {\doibase 10.1103/PhysRevLett.89.056405}
|
|
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
|
|
{\bibinfo {volume} {89}},\ \bibinfo {pages} {056405} (\bibinfo {year}
|
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{2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Tiago}, \citenamefont {Northrup},\ and\ \citenamefont
|
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{Louie}(2003)}]{Tiago_2003}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~L.}\ \bibnamefont
|
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{Tiago}}, \bibinfo {author} {\bibfnamefont {J.~E.}\ \bibnamefont {Northrup}},
|
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\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont {Louie}},\
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}\href {\doibase 10.1103/PhysRevB.67.115212} {\bibfield {journal} {\bibinfo
|
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {67}},\ \bibinfo
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{pages} {115212} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rocca}, \citenamefont {Lu},\ and\ \citenamefont
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{Galli}(2010)}]{Rocca_2010}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Rocca}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Lu}}, \ and\
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\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Galli}},\ }\href
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{\doibase 10.1063/1.3494540} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {133}},\ \bibinfo {pages} {164109}
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(\bibinfo {year} {2010})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Boulanger}\ \emph {et~al.}(2014)\citenamefont
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{Boulanger}, \citenamefont {Jacquemin}, \citenamefont {Duchemin},\ and\
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\citenamefont {Blase}}]{Boulanger_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Boulanger}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Jacquemin}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href
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{\doibase 10.1021/ct401101u} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages}
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{1212} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Jacquemin}, \citenamefont {Duchemin},\ and\
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\citenamefont {Blase}(2015{\natexlab{a}})}]{Jacquemin_2015a}%
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\BibitemOpen
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{Jacquemin}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href
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{\doibase 10.1021/acs.jctc.5b00304} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {11}},\
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\bibinfo {pages} {3290} (\bibinfo {year} {2015}{\natexlab{a}})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Bruneval}, \citenamefont {Hamed},\ and\ \citenamefont
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{Neaton}(2015)}]{Bruneval_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
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{Bruneval}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Hamed}},
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\ and\ \bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont {Neaton}},\
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}\href {\doibase 10.1063/1.4922489} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {142}},\ \bibinfo
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{pages} {244101} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Jacquemin}, \citenamefont {Duchemin},\ and\
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\citenamefont {Blase}(2015{\natexlab{b}})}]{Jacquemin_2015b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Jacquemin}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href
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{\doibase 10.1021/acs.jctc.5b00619} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {11}},\
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\bibinfo {pages} {5340} (\bibinfo {year} {2015}{\natexlab{b}})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Hirose}, \citenamefont {Noguchi},\ and\ \citenamefont
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{Sugino}(2015)}]{Hirose_2015}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Hirose}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Noguchi}}, \
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and\ \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Sugino}},\ }\href
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{\doibase 10.1103/PhysRevB.91.205111} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo
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{pages} {205111} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Jacquemin}, \citenamefont {Duchemin},\ and\
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\citenamefont {Blase}(2017)}]{Jacquemin_2017a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Jacquemin}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href
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{\doibase 10.1021/acs.jpclett.7b00381} {\bibfield {journal} {\bibinfo
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{journal} {J. Phys. Chem. Lett.}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo
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{pages} {1524} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Jacquemin}\ \emph {et~al.}(2017)\citenamefont
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{Jacquemin}, \citenamefont {Duchemin}, \citenamefont {Blondel},\ and\
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\citenamefont {Blase}}]{Jacquemin_2017b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Jacquemin}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Blondel}}, \ and\
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\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href
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{\doibase 10.1021/acs.jctc.6b01169} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {13}},\
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\bibinfo {pages} {767} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rangel}\ \emph {et~al.}(2017)\citenamefont {Rangel},
|
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\citenamefont {Hamed}, \citenamefont {Bruneval},\ and\ \citenamefont
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{Neaton}}]{Rangel_2017}%
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\BibitemOpen
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{Rangel}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Hamed}},
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\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Bruneval}}, \ and\
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\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont {Neaton}},\ }\href
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{\doibase 10.1063/1.4983126} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {146}},\ \bibinfo {pages} {194108}
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(\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Krause}\ and\ \citenamefont
|
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{Klopper}(2017)}]{Krause_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Krause}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
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{Klopper}},\ }\href {\doibase 10.1002/jcc.24688} {\bibfield {journal}
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{\bibinfo {journal} {J. Comput. Chem.}\ }\textbf {\bibinfo {volume} {38}},\
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\bibinfo {pages} {383} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gui}, \citenamefont {Holzer},\ and\ \citenamefont
|
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{Klopper}(2018)}]{Gui_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Gui}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Holzer}}, \ and\
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\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},\ }\href
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{\doibase 10.1021/acs.jctc.8b00014} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {14}},\
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\bibinfo {pages} {2127} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Liu}\ \emph {et~al.}(2020)\citenamefont {Liu},
|
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\citenamefont {Kloppenburg}, \citenamefont {Yao}, \citenamefont {Ren},
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\citenamefont {Appel}, \citenamefont {Kanai},\ and\ \citenamefont
|
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{Blum}}]{Liu_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Liu}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Kloppenburg}},
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\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Yao}}, \bibinfo {author}
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{\bibfnamefont {X.}~\bibnamefont {Ren}}, \bibinfo {author} {\bibfnamefont
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{H.}~\bibnamefont {Appel}}, \bibinfo {author} {\bibfnamefont
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{Y.}~\bibnamefont {Kanai}}, \ and\ \bibinfo {author} {\bibfnamefont
|
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{V.}~\bibnamefont {Blum}},\ }\href {\doibase 10.1063/1.5123290} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
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{152}},\ \bibinfo {pages} {044105} (\bibinfo {year} {2020})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Holzer}\ and\ \citenamefont
|
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{Klopper}(2018)}]{Holzer_2018a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Holzer}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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{Klopper}},\ }\href {\doibase 10.1063/1.5051028} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibinfo {pages} {101101} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Holzer}\ \emph {et~al.}(2018)\citenamefont {Holzer},
|
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\citenamefont {Gui}, \citenamefont {Harding}, \citenamefont {Kresse},
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\citenamefont {Helgaker},\ and\ \citenamefont {Klopper}}]{Holzer_2018b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Holzer}}, \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Gui}},
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\bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont {Harding}}, \bibinfo
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{author} {\bibfnamefont {G.}~\bibnamefont {Kresse}}, \bibinfo {author}
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{\bibfnamefont {T.}~\bibnamefont {Helgaker}}, \ and\ \bibinfo {author}
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{\bibfnamefont {W.}~\bibnamefont {Klopper}},\ }\href {\doibase
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10.1063/1.5047030} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {144106}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ \emph {et~al.}(2020)\citenamefont {Loos},
|
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\citenamefont {Scemama}, \citenamefont {Duchemin}, \citenamefont
|
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{Jacquemin},\ and\ \citenamefont {Blase}}]{Loos_2020e}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}},
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\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}}, \bibinfo
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{author} {\bibfnamefont {D.}~\bibnamefont {Jacquemin}}, \ and\ \bibinfo
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{author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href {\doibase
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10.1021/acs.jpclett.0c00460} {\bibfield {journal} {\bibinfo {journal} {J.
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Phys. Chem. Lett.}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo {pages}
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{3536} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ \emph {et~al.}(2021)\citenamefont {Loos},
|
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\citenamefont {Comin}, \citenamefont {Blase},\ and\ \citenamefont
|
|
{Jacquemin}}]{Loos_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Comin}},
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\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}}, \ and\ \bibinfo
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{author} {\bibfnamefont {D.}~\bibnamefont {Jacquemin}},\ }\href {\doibase
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10.1021/acs.jctc.1c00226} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages}
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{3666} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {McKeon}\ \emph {et~al.}(2022)\citenamefont {McKeon},
|
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\citenamefont {Hamed}, \citenamefont {Bruneval},\ and\ \citenamefont
|
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{Neaton}}]{McKeon_2022}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
|
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{McKeon}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Hamed}},
|
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\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Bruneval}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont {Neaton}},\ }\href
|
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{\doibase 10.1063/5.0097582} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {157}},\ \bibinfo {pages} {074103}
|
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(\bibinfo {year} {2022})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Freeman}(1977)}]{Freeman_1977}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~L.}\ \bibnamefont
|
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{Freeman}},\ }\href {\doibase 10.1103/PhysRevB.15.5512} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {15}},\
|
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\bibinfo {pages} {5512} (\bibinfo {year} {1977})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Scuseria}, \citenamefont {Henderson},\ and\
|
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\citenamefont {Sorensen}(2008)}]{Scuseria_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}}, \bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont
|
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{Henderson}}, \ and\ \bibinfo {author} {\bibfnamefont {D.~C.}\ \bibnamefont
|
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{Sorensen}},\ }\href {\doibase 10.1063/1.3043729} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {129}},\
|
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\bibinfo {pages} {231101} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Jansen}, \citenamefont {Liu},\ and\ \citenamefont
|
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{{\'A}ngy{\'a}n}(2010)}]{Jansen_2010}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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{Jansen}}, \bibinfo {author} {\bibfnamefont {R.-F.}\ \bibnamefont {Liu}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {J.~G.}\ \bibnamefont
|
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{{\'A}ngy{\'a}n}},\ }\href {\doibase 10.1063/1.3481575} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {133}},\
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\bibinfo {pages} {154106} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Scuseria}, \citenamefont {Henderson},\ and\
|
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\citenamefont {Bulik}(2013)}]{Scuseria_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}}, \bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont
|
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{Henderson}}, \ and\ \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont
|
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{Bulik}},\ }\href {\doibase 10.1063/1.4820557} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {139}},\
|
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\bibinfo {pages} {104113} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Peng}\ \emph {et~al.}(2013)\citenamefont {Peng},
|
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\citenamefont {Steinmann}, \citenamefont {{van Aggelen}},\ and\ \citenamefont
|
|
{Yang}}]{Peng_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Peng}}, \bibinfo {author} {\bibfnamefont {S.~N.}\ \bibnamefont {Steinmann}},
|
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\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {{van Aggelen}}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}},\ }\href {\doibase
|
|
10.1063/1.4820556} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
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Phys.}\ }\textbf {\bibinfo {volume} {139}},\ \bibinfo {pages} {104112}
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(\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Berkelbach}(2018)}]{Berkelbach_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.~C.}\ \bibnamefont
|
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{Berkelbach}},\ }\href {\doibase 10.1063/1.5032314} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibinfo {pages} {041103} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rishi}, \citenamefont {Perera},\ and\ \citenamefont
|
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{Bartlett}(2020)}]{Rishi_2020}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
|
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{Rishi}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Perera}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont {Bartlett}},\
|
|
}\href {\doibase 10.1063/5.0023862} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {153}},\ \bibinfo
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{pages} {234101} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{\v C}{\'\i}{\v z}ek}(1966)}]{Cizek_1966}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {{\v
|
|
C}{\'\i}{\v z}ek}},\ }\href {\doibase 10.1063/1.1727484} {\bibfield
|
|
{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
|
|
{45}},\ \bibinfo {pages} {4256} (\bibinfo {year} {1966})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Paldus}, \citenamefont {\ifmmode \check{C}\else
|
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\v{C}\fi{}\'{\i}\ifmmode~\check{z}\else \v{z}\fi{}ek},\ and\ \citenamefont
|
|
{Shavitt}(1972)}]{Paldus_1972}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
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{Paldus}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {\ifmmode
|
|
\check{C}\else \v{C}\fi{}\'{\i}\ifmmode~\check{z}\else \v{z}\fi{}ek}}, \ and\
|
|
\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Shavitt}},\ }\href
|
|
{\doibase 10.1103/PhysRevA.5.50} {\bibfield {journal} {\bibinfo {journal}
|
|
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {5}},\ \bibinfo {pages} {50}
|
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(\bibinfo {year} {1972})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Crawford}\ and\ \citenamefont
|
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{Schaefer}(2000)}]{Crawford_2000}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.~D.}\ \bibnamefont
|
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{Crawford}}\ and\ \bibinfo {author} {\bibfnamefont {H.~F.}\ \bibnamefont
|
|
{Schaefer}},\ }in\ \href {\doibase 10.1002/9780470125915.ch2} {\emph
|
|
{\bibinfo {booktitle} {Reviews in {{Computational Chemistry}}}}}\ (\bibinfo
|
|
{publisher} {{John Wiley \& Sons, Ltd}},\ \bibinfo {year} {2000})\ pp.\
|
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\bibinfo {pages} {33--136}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Piecuch}\ \emph {et~al.}(2002)\citenamefont
|
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{Piecuch}, \citenamefont {Kowalski}, \citenamefont {Pimienta},\ and\
|
|
\citenamefont {Mcguire}}]{Piecuch_2002}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Piecuch}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Kowalski}},
|
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\bibinfo {author} {\bibfnamefont {I.~S.~O.}\ \bibnamefont {Pimienta}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {M.~J.}\ \bibnamefont {Mcguire}},\ }\href
|
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{\doibase 10.1080/0144235021000053811} {\bibfield {journal} {\bibinfo
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{journal} {Int. Rev. Phys. Chem.}\ }\textbf {\bibinfo {volume} {21}},\
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\bibinfo {pages} {527} (\bibinfo {year} {2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Bartlett}\ and\ \citenamefont
|
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{Musia{\l}}(2007)}]{Bartlett_2007}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
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{Bartlett}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Musia{\l}}},\ }\href {\doibase 10.1103/RevModPhys.79.291} {\bibfield
|
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{journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume}
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{79}},\ \bibinfo {pages} {291} (\bibinfo {year} {2007})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Shavitt}\ and\ \citenamefont
|
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{Bartlett}(2009)}]{Shavitt_2009}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Shavitt}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
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{Bartlett}},\ }\href {\doibase 10.1017/CBO9780511596834} {\emph {\bibinfo
|
|
{title} {Many-{{Body Methods}} in {{Chemistry}} and {{Physics}}: {{MBPT}} and
|
|
{{Coupled}}-{{Cluster Theory}}}}},\ Cambridge {{Molecular Science}}\
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(\bibinfo {publisher} {{Cambridge University Press}},\ \bibinfo {address}
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{{Cambridge}},\ \bibinfo {year} {2009})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Furche}(2008)}]{Furche_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
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{Furche}},\ }\href {\doibase 10.1063/1.2977789} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {129}},\
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\bibinfo {pages} {114105} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Angyan}\ \emph {et~al.}(2011)\citenamefont {Angyan},
|
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\citenamefont {Liu}, \citenamefont {Toulouse},\ and\ \citenamefont
|
|
{Jansen}}]{Angyan_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~G.}\ \bibnamefont
|
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{Angyan}}, \bibinfo {author} {\bibfnamefont {R.-F.}\ \bibnamefont {Liu}},
|
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\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}}, \ and\
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\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Jansen}},\ }\href
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{\doibase 10.1021/ct200501r} {\bibfield {journal} {\bibinfo {journal} {J.
|
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages}
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{3116} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Sawada}(1957)}]{Sawada_1957a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
|
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{Sawada}},\ }\href {\doibase 10.1103/PhysRev.106.372} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {106}},\
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\bibinfo {pages} {372} (\bibinfo {year} {1957})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rowe}(1968)}]{Rowe_1968}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
|
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{Rowe}},\ }\href {\doibase 10.1103/PhysRev.175.1283} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {175}},\
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\bibinfo {pages} {1283} (\bibinfo {year} {1968})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ring}\ and\ \citenamefont
|
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{Schuck}(2004)}]{Schuck_book}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Ring}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Schuck}},\
|
|
}\href@noop {} {\emph {\bibinfo {title} {The Nuclear Many-Body Problem}}}\
|
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(\bibinfo {publisher} {Springer},\ \bibinfo {year} {2004})\BibitemShut
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{NoStop}%
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\bibitem [{Note1()}]{Note1}%
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\BibitemOpen
|
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\bibinfo {note} {The factor $1/4$ in Eq.~\protect \textup {\hbox
|
|
{\mathsurround \z@ \protect \normalfont (\ignorespaces \ref
|
|
{eq:EcRPAx}\unskip \@@italiccorr )}} is sometimes replaced by a factor $1/2$,
|
|
which corresponds to a different choice for the interaction kernel. See
|
|
Ref.~\protect \rev@citealpnum {Angyan_2011} for more details}\BibitemShut
|
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{NoStop}%
|
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\bibitem [{\citenamefont {Furche}\ and\ \citenamefont {{Van
|
|
Voorhis}}(2005)}]{Furche_2005}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
|
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{Furche}}\ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {{Van
|
|
Voorhis}}},\ }\href {\doibase 10.1063/1.1884112} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {122}},\
|
|
\bibinfo {pages} {164106} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Stanton}\ and\ \citenamefont
|
|
{Bartlett}(1993)}]{Stanton_1993}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~F.}\ \bibnamefont
|
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{Stanton}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.464746} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {98}},\
|
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\bibinfo {pages} {7029} (\bibinfo {year} {1993})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Emrich}(1981)}]{Emrich_1981}%
|
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\BibitemOpen
|
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
|
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{Emrich}},\ }\href {\doibase https://doi.org/10.1016/0375-9474(81)90179-2}
|
|
{\bibfield {journal} {\bibinfo {journal} {Nuc. Phys. A}\ }\textbf {\bibinfo
|
|
{volume} {351}},\ \bibinfo {pages} {379} (\bibinfo {year}
|
|
{1981})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Runge}\ and\ \citenamefont
|
|
{Gross}(1984)}]{Runge_1984}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Runge}}\ and\ \bibinfo {author} {\bibfnamefont {E.~K.~U.}\ \bibnamefont
|
|
{Gross}},\ }\href {\doibase 10.1103/PhysRevLett.52.997} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {52}},\
|
|
\bibinfo {pages} {997} (\bibinfo {year} {1984})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Casida}(1995)}]{Casida_1995}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont
|
|
{Casida}},\ }\enquote {\bibinfo {title} {Time-dependent density functional
|
|
response theory for molecules},}\ \ (\bibinfo {publisher} {World Scientific,
|
|
Singapore},\ \bibinfo {year} {1995})\ pp.\ \bibinfo {pages}
|
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{155--192}\BibitemShut {NoStop}%
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\bibitem [{Note2()}]{Note2}%
|
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\BibitemOpen
|
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\bibinfo {note} {\protect \leavevmode {\color {black}To the best of our
|
|
knowledge, the trace (or plasmon) formula has been first introduced by Sawada
|
|
\cite {Sawada_1957a} to calculate the correlation energy of the uniform
|
|
electron gas as an alternative to the Gell-Mann-Brueckner formulation \cite
|
|
{Gell-Mann_1957} where one integrates along the adiabatic connection path.
|
|
More precisely, the trace formula can be justified via the introduction of a
|
|
quadratic Hamiltonian made of boson transition operators (quasiboson
|
|
approximation). See Ref.~\protect \rev@citealpnum {Li_2020} for more
|
|
details}.}\BibitemShut {Stop}%
|
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\bibitem [{\citenamefont {Li}\ \emph {et~al.}(2020)\citenamefont {Li},
|
|
\citenamefont {Duchemin}, \citenamefont {Blase},\ and\ \citenamefont
|
|
{Olevano}}]{Li_2020}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
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{Li}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
|
|
\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}}, \ and\ \bibinfo
|
|
{author} {\bibfnamefont {V.}~\bibnamefont {Olevano}},\ }\href {\doibase
|
|
10.21468/SciPostPhys.8.2.020} {\bibfield {journal} {\bibinfo {journal}
|
|
{SciPost Phys.}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {20}
|
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(\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Li}\ and\ \citenamefont {Olevano}(2021)}]{Li_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
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{Li}}\ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Olevano}},\
|
|
}\href {\doibase 10.1103/PhysRevA.103.012809} {\bibfield {journal} {\bibinfo
|
|
{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo
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{pages} {012809} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Di~Sabatino}, \citenamefont {Loos},\ and\
|
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\citenamefont {Romaniello}(2021)}]{DiSabatino_2021}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
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{Di~Sabatino}}, \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
|
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{Loos}}, \ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
|
{Romaniello}},\ }\href {\doibase 10.3389/fchem.2021.751054} {\bibfield
|
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{journal} {\bibinfo {journal} {Front. Chem.}\ }\textbf {\bibinfo {volume}
|
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{9}},\ \bibinfo {pages} {751054} (\bibinfo {year} {2021})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Maggio}\ and\ \citenamefont
|
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{Kresse}(2016)}]{Maggio_2016}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Maggio}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
|
{Kresse}},\ }\href {\doibase 10.1103/PhysRevB.93.235113} {\bibfield
|
|
{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
|
|
{93}},\ \bibinfo {pages} {235113} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
|
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\bibitem [{\citenamefont {Berger}, \citenamefont {Loos},\ and\ \citenamefont
|
|
{Romaniello}(2020)}]{Berger_2021}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
|
|
{Berger}}, \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}}, \
|
|
and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Romaniello}},\
|
|
}\href {\doibase 10.1021/acs.jctc.0c00896} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {17}},\
|
|
\bibinfo {pages} {191} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Handy}\ \emph {et~al.}(1989)\citenamefont {Handy},
|
|
\citenamefont {Pople}, \citenamefont {Head-Gordon}, \citenamefont
|
|
{Raghavachari},\ and\ \citenamefont {Trucks}}]{Handy_1989}%
|
|
\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
|
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{Handy}}, \bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont {Pople}},
|
|
\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Head-Gordon}}, \bibinfo
|
|
{author} {\bibfnamefont {K.}~\bibnamefont {Raghavachari}}, \ and\ \bibinfo
|
|
{author} {\bibfnamefont {G.~W.}\ \bibnamefont {Trucks}},\ }\href {\doibase
|
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https://doi.org/10.1016/0009-2614(89)85013-4} {\bibfield {journal} {\bibinfo
|
|
{journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {164}},\ \bibinfo
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{pages} {185} (\bibinfo {year} {1989})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Lange}\ and\ \citenamefont
|
|
{Berkelbach}(2018)}]{Lange_2018}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~F.}\ \bibnamefont
|
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{Lange}}\ and\ \bibinfo {author} {\bibfnamefont {T.~C.}\ \bibnamefont
|
|
{Berkelbach}},\ }\href {\doibase 10.1021/acs.jctc.8b00455} {\bibfield
|
|
{journal} {\bibinfo {journal} {J. Chem. Theory. Comput.}\ }\textbf {\bibinfo
|
|
{volume} {14}},\ \bibinfo {pages} {4224} (\bibinfo {year}
|
|
{2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Nooijen}\ and\ \citenamefont
|
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{Bartlett}(1997{\natexlab{a}})}]{Nooijen_1997c}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Nooijen}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.473635} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {106}},\
|
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\bibinfo {pages} {6449} (\bibinfo {year} {1997}{\natexlab{a}})}\BibitemShut
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{NoStop}%
|
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\bibitem [{\citenamefont {Nooijen}\ and\ \citenamefont
|
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{Bartlett}(1997{\natexlab{b}})}]{Nooijen_1997b}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Nooijen}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.474922} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {107}},\
|
|
\bibinfo {pages} {6812} (\bibinfo {year} {1997}{\natexlab{b}})}\BibitemShut
|
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{NoStop}%
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\bibitem [{\citenamefont {Nooijen}\ and\ \citenamefont
|
|
{Bartlett}(1997{\natexlab{c}})}]{Nooijen_1997a}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
|
{Nooijen}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.474000} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {106}},\
|
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\bibinfo {pages} {6441} (\bibinfo {year} {1997}{\natexlab{c}})}\BibitemShut
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{NoStop}%
|
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\bibitem [{\citenamefont {Stanton}\ and\ \citenamefont
|
|
{Gauss}(1994)}]{Stanton_1994}%
|
|
\BibitemOpen
|
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~F.}\ \bibnamefont
|
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{Stanton}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
|
{Gauss}},\ }\href {\doibase 10.1063/1.468022} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo
|
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{pages} {8938} (\bibinfo {year} {1994})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Musia{\l}}, \citenamefont {Kucharski},\ and\
|
|
\citenamefont {Bartlett}(2003)}]{Musial_2003a}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Musia{\l}}}, \bibinfo {author} {\bibfnamefont {S.~A.}\ \bibnamefont
|
|
{Kucharski}}, \ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.1527013} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {118}},\
|
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\bibinfo {pages} {1128} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Nooijen}\ and\ \citenamefont
|
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{Bartlett}(1995)}]{Nooijen_1995}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Nooijen}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.468592} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {102}},\
|
|
\bibinfo {pages} {3629} (\bibinfo {year} {1995})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Musia{\l}}\ and\ \citenamefont
|
|
{Bartlett}(2003)}]{Musial_2003b}%
|
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\BibitemOpen
|
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
|
{Musia{\l}}}\ and\ \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
|
{Bartlett}},\ }\href {\doibase 10.1063/1.1584657} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {119}},\
|
|
\bibinfo {pages} {1901} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Nooijen}\ and\ \citenamefont
|
|
{Lotrich}(2000)}]{Nooijen_2000}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Nooijen}}\ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
|
|
{Lotrich}},\ }\href {\doibase 10.1063/1.481828} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {113}},\
|
|
\bibinfo {pages} {494} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Monino}\ and\ \citenamefont
|
|
{Loos}(2022)}]{Monino_2022}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
|
{Monino}}\ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
|
|
{Loos}},\ }\href {\doibase 10.1063/5.0089317} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {156}},\ \bibinfo
|
|
{pages} {231101} (\bibinfo {year} {2022})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Strinati}, \citenamefont {Mattausch},\ and\
|
|
\citenamefont {Hanke}(1980)}]{Strinati_1980}%
|
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\BibitemOpen
|
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
|
{Strinati}}, \bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
|
|
{Mattausch}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
|
{Hanke}},\ }\href {\doibase 10.1103/PhysRevLett.45.290} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {45}},\
|
|
\bibinfo {pages} {290} (\bibinfo {year} {1980})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Strinati}(1982)}]{Strinati_1982}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
|
{Strinati}},\ }\href {\doibase 10.1103/PhysRevLett.49.1519} {\bibfield
|
|
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {49}},\ \bibinfo {pages} {1519} (\bibinfo {year}
|
|
{1982})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Strinati}(1984)}]{Strinati_1984}%
|
|
\BibitemOpen
|
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
|
{Strinati}},\ }\href {\doibase 10.1103/PhysRevB.29.5718} {\bibfield
|
|
{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
|
|
{29}},\ \bibinfo {pages} {5718} (\bibinfo {year} {1984})}\BibitemShut
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{NoStop}%
|
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\bibitem [{\citenamefont {Rohlfing}\ and\ \citenamefont
|
|
{Louie}(2000)}]{Rohlfing_2000}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
|
{Rohlfing}}\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
|
{Louie}},\ }\href {\doibase 10.1103/PhysRevB.62.4927} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {62}},\
|
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\bibinfo {pages} {4927} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Romaniello}\ \emph {et~al.}(2009)\citenamefont
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{Romaniello}, \citenamefont {Sangalli}, \citenamefont {Berger}, \citenamefont
|
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{Sottile}, \citenamefont {Molinari}, \citenamefont {Reining},\ and\
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\citenamefont {Onida}}]{Romaniello_2009b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Romaniello}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Sangalli}}, \bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
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{Berger}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Sottile}},
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\bibinfo {author} {\bibfnamefont {L.~G.}\ \bibnamefont {Molinari}}, \bibinfo
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{author} {\bibfnamefont {L.}~\bibnamefont {Reining}}, \ and\ \bibinfo
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{author} {\bibfnamefont {G.}~\bibnamefont {Onida}},\ }\href {\doibase
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10.1063/1.3065669} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
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Phys.}\ }\textbf {\bibinfo {volume} {130}},\ \bibinfo {pages} {044108}
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(\bibinfo {year} {2009})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
|
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{Blase}(2020)}]{Loos_2020h}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}},\
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}\href {\doibase 10.1063/5.0023168} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {153}},\ \bibinfo
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{pages} {114120} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Authier}\ and\ \citenamefont
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{Loos}(2020)}]{Authier_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Authier}}\ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1063/5.0028040} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {153}},\ \bibinfo
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{pages} {184105} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Monino}\ and\ \citenamefont
|
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{Loos}(2021)}]{Monino_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Monino}}\ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1021/acs.jctc.1c00074} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{17}},\ \bibinfo {pages} {2852} (\bibinfo {year} {2021})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Bintrim}\ and\ \citenamefont
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{Berkelbach}(2022)}]{Bintrim_2022}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~J.}\ \bibnamefont
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{Bintrim}}\ and\ \bibinfo {author} {\bibfnamefont {T.~C.}\ \bibnamefont
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{Berkelbach}},\ }\href {\doibase 10.1063/5.0074434} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {156}},\
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\bibinfo {pages} {044114} (\bibinfo {year} {2022})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Stefanucci}\ and\ \citenamefont {van
|
|
Leeuwen}(2013)}]{Stefanucci_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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{Stefanucci}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {van
|
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Leeuwen}},\ }\href@noop {} {\emph {\bibinfo {title} {Nonequilibrium Many-Body
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Theory of Quantum Systems: A Modern Introduction}}}\ (\bibinfo {publisher}
|
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{{Cambridge University Press}},\ \bibinfo {address} {Cambridge},\ \bibinfo
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{year} {2013})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ortiz}(2013)}]{Ortiz_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~V.}\ \bibnamefont
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{Ortiz}},\ }\href {\doibase 10.1002/wcms.1116} {\bibfield {journal}
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{\bibinfo {journal} {Wiley Interdiscip. Rev. Comput. Mol. Sci.}\ }\textbf
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{\bibinfo {volume} {3}},\ \bibinfo {pages} {123} (\bibinfo {year}
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{2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Phillips}\ and\ \citenamefont
|
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{Zgid}(2014)}]{Phillips_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont
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{Phillips}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Zgid}},\ }\href {\doibase 10.1063/1.4884951} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo
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{pages} {241101} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rusakov}, \citenamefont {Phillips},\ and\
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\citenamefont {Zgid}(2014)}]{Rusakov_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
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{Rusakov}}, \bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont
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{Phillips}}, \ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Zgid}},\ }\href {\doibase 10.1063/1.4901432} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {141}},\ \bibinfo
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{pages} {194105} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hirata}\ \emph {et~al.}(2015)\citenamefont {Hirata},
|
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\citenamefont {Hermes}, \citenamefont {Simons},\ and\ \citenamefont
|
|
{Ortiz}}]{Hirata_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Hirata}}, \bibinfo {author} {\bibfnamefont {M.~R.}\ \bibnamefont {Hermes}},
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\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Simons}}, \ and\ \bibinfo
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{author} {\bibfnamefont {J.~V.}\ \bibnamefont {Ortiz}},\ }\href {\doibase
|
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10.1021/acs.jctc.5b00005} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo {pages}
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{1595} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hirata}\ \emph {et~al.}(2017)\citenamefont {Hirata},
|
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\citenamefont {Doran}, \citenamefont {Knowles},\ and\ \citenamefont
|
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{Ortiz}}]{Hirata_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Hirata}}, \bibinfo {author} {\bibfnamefont {A.~E.}\ \bibnamefont {Doran}},
|
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\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont {Knowles}}, \ and\
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\bibinfo {author} {\bibfnamefont {J.~V.}\ \bibnamefont {Ortiz}},\ }\href
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{\doibase 10.1063/1.4994837} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {147}},\ \bibinfo {pages} {044108}
|
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(\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Romaniello}, \citenamefont {Bechstedt},\ and\
|
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\citenamefont {Reining}(2012)}]{Romaniello_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Romaniello}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
|
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{Bechstedt}}, \ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
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{Reining}},\ }\href {\doibase 10.1103/PhysRevB.85.155131} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
|
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{85}},\ \bibinfo {pages} {155131} (\bibinfo {year} {2012})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Zhang}, \citenamefont {Su},\ and\ \citenamefont
|
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{Yang}(2017)}]{Zhang_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Zhang}}, \bibinfo {author} {\bibfnamefont {N.~Q.}\ \bibnamefont {Su}}, \
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and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}},\ }\href
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{\doibase 10.1021/acs.jpclett.7b01275} {\bibfield {journal} {\bibinfo
|
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{journal} {J. Phys. Chem. Lett.}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo
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{pages} {3223} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Li}, \citenamefont {Chen},\ and\ \citenamefont
|
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{Yang}(2021)}]{Li_2021b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Li}}, \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Chen}}, \ and\
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\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}},\ }\href {\doibase
|
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10.1021/acs.jpclett.1c01723} {\bibfield {journal} {\bibinfo {journal} {J.
|
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Phys. Chem. Lett.}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo {pages}
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{6203} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
|
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{Romaniello}(2022)}]{Loos_2022}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Romaniello}},\ }\href {\doibase 10.1063/5.0088364} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {156}},\
|
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\bibinfo {pages} {164101} (\bibinfo {year} {2022})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}, \citenamefont {Romaniello},\ and\
|
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\citenamefont {Berger}(2018)}]{Loos_2018b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~F.}\ \bibnamefont
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{Loos}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Romaniello}}, \
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and\ \bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont {Berger}},\
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}\href {\doibase 10.1021/acs.jctc.8b00260} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {14}},\
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\bibinfo {pages} {3071} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hybertsen}\ and\ \citenamefont
|
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{Louie}(1985)}]{Hybertsen_1985a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont
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{Hybertsen}}\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
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{Louie}},\ }\href {\doibase 10.1103/PhysRevLett.55.1418} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
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{volume} {55}},\ \bibinfo {pages} {1418} (\bibinfo {year}
|
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{1985})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hybertsen}\ and\ \citenamefont
|
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{Louie}(1986)}]{Hybertsen_1986}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont
|
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{Hybertsen}}\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
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{Louie}},\ }\href {\doibase 10.1103/PhysRevB.34.5390} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {34}},\
|
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\bibinfo {pages} {5390} (\bibinfo {year} {1986})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Godby}, \citenamefont {Schl\"uter},\ and\
|
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\citenamefont {Sham}(1988)}]{Godby_1988}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~W.}\ \bibnamefont
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{Godby}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Schl\"uter}}, \
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and\ \bibinfo {author} {\bibfnamefont {L.~J.}\ \bibnamefont {Sham}},\ }\href
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{\doibase 10.1103/PhysRevB.37.10159} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {37}},\ \bibinfo
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{pages} {10159} (\bibinfo {year} {1988})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {von~der Linden}\ and\ \citenamefont
|
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{Horsch}(1988)}]{Linden_1988}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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{von~der Linden}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Horsch}},\ }\href {\doibase 10.1103/PhysRevB.37.8351} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {37}},\
|
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\bibinfo {pages} {8351} (\bibinfo {year} {1988})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Northrup}, \citenamefont {Hybertsen},\ and\
|
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\citenamefont {Louie}(1991)}]{Northrup_1991}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~E.}\ \bibnamefont
|
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{Northrup}}, \bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont
|
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{Hybertsen}}, \ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
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{Louie}},\ }\href {\doibase 10.1103/PhysRevLett.66.500} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {66}},\
|
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\bibinfo {pages} {500} (\bibinfo {year} {1991})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Blase}, \citenamefont {Zhu},\ and\ \citenamefont
|
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{Louie}(1994)}]{Blase_1994}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
|
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{Blase}}, \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Zhu}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont {Louie}},\ }\href
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{\doibase 10.1103/PhysRevB.49.4973} {\bibfield {journal} {\bibinfo
|
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {49}},\ \bibinfo
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{pages} {4973} (\bibinfo {year} {1994})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rohlfing}, \citenamefont {Kr{\"u}ger},\ and\
|
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\citenamefont {Pollmann}(1995)}]{Rohlfing_1995}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Rohlfing}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Kr{\"u}ger}}, \ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
|
{Pollmann}},\ }\href {\doibase 10.1103/PhysRevB.52.1905} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
|
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{52}},\ \bibinfo {pages} {1905} (\bibinfo {year} {1995})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Shishkin}\ and\ \citenamefont
|
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{Kresse}(2007)}]{Shishkin_2007}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Shishkin}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
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{Kresse}},\ }\href {\doibase 10.1103/PhysRevB.75.235102} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
|
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{75}},\ \bibinfo {pages} {235102} (\bibinfo {year} {2007})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Blase}\ and\ \citenamefont
|
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{Attaccalite}(2011)}]{Blase_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Attaccalite}},\ }\href {\doibase 10.1063/1.3655352} {\bibfield {journal}
|
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{\bibinfo {journal} {Appl. Phys. Lett.}\ }\textbf {\bibinfo {volume} {99}},\
|
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\bibinfo {pages} {171909} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Faber}\ \emph
|
|
{et~al.}(2011{\natexlab{a}})\citenamefont {Faber}, \citenamefont
|
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{Attaccalite}, \citenamefont {Olevano}, \citenamefont {Runge},\ and\
|
|
\citenamefont {Blase}}]{Faber_2011}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Faber}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Attaccalite}},
|
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\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Olevano}}, \bibinfo
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{author} {\bibfnamefont {E.}~\bibnamefont {Runge}}, \ and\ \bibinfo {author}
|
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{\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href {\doibase
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10.1103/PhysRevB.83.115123} {\bibfield {journal} {\bibinfo {journal} {Phys.
|
|
Rev. B}\ }\textbf {\bibinfo {volume} {83}},\ \bibinfo {pages} {115123}
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(\bibinfo {year} {2011}{\natexlab{a}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rangel}\ \emph {et~al.}(2016)\citenamefont {Rangel},
|
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\citenamefont {Hamed}, \citenamefont {Bruneval},\ and\ \citenamefont
|
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{Neaton}}]{Rangel_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Rangel}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Hamed}},
|
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\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Bruneval}}, \ and\
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\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont {Neaton}},\ }\href
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{\doibase 10.1021/acs.jctc.6b00163} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {12}},\
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\bibinfo {pages} {2834} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Faleev}, \citenamefont {{van Schilfgaarde}},\ and\
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\citenamefont {Kotani}(2004)}]{Faleev_2004}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~V.}\ \bibnamefont
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{Faleev}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{van
|
|
Schilfgaarde}}}, \ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
|
{Kotani}},\ }\href {\doibase 10.1103/PhysRevLett.93.126406} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {93}},\ \bibinfo {pages} {126406} (\bibinfo {year}
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{2004})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{van Schilfgaarde}}, \citenamefont {Kotani},\ and\
|
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\citenamefont {Faleev}(2006)}]{vanSchilfgaarde_2006}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{van
|
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Schilfgaarde}}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
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{Kotani}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
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{Faleev}},\ }\href {\doibase 10.1103/PhysRevLett.96.226402} {\bibfield
|
|
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {96}},\ \bibinfo {pages} {226402} (\bibinfo {year}
|
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{2006})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kotani}, \citenamefont {{van Schilfgaarde}},\ and\
|
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\citenamefont {Faleev}(2007)}]{Kotani_2007}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Kotani}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{van
|
|
Schilfgaarde}}}, \ and\ \bibinfo {author} {\bibfnamefont {S.~V.}\
|
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\bibnamefont {Faleev}},\ }\href {\doibase 10.1103/PhysRevB.76.165106}
|
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo
|
|
{volume} {76}},\ \bibinfo {pages} {165106} (\bibinfo {year}
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{2007})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ke}(2011)}]{Ke_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.-H.}\ \bibnamefont
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{Ke}},\ }\href {\doibase 10.1103/PhysRevB.84.205415} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {84}},\
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\bibinfo {pages} {205415} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kaplan}\ \emph {et~al.}(2016)\citenamefont {Kaplan},
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\citenamefont {Harding}, \citenamefont {Seiler}, \citenamefont {Weigend},
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\citenamefont {Evers},\ and\ \citenamefont {{van Setten}}}]{Kaplan_2016}%
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\BibitemOpen
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{Kaplan}}, \bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont {Harding}},
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\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Seiler}}, \bibinfo
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{author} {\bibfnamefont {F.}~\bibnamefont {Weigend}}, \bibinfo {author}
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{\bibfnamefont {F.}~\bibnamefont {Evers}}, \ and\ \bibinfo {author}
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{\bibfnamefont {M.~J.}\ \bibnamefont {{van Setten}}},\ }\href {\doibase
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10.1021/acs.jctc.5b01238} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo {pages}
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{2528} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Bintrim}\ and\ \citenamefont
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{Berkelbach}(2021)}]{Bintrim_2021}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~J.}\ \bibnamefont
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{Bintrim}}\ and\ \bibinfo {author} {\bibfnamefont {T.~C.}\ \bibnamefont
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{Berkelbach}},\ }\href {\doibase 10.1063/5.0035141} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {154}},\
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\bibinfo {pages} {041101} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Schirmer}(2018)}]{Schirmer_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Schirmer}},\ }\href@noop {} {\emph {\bibinfo {title} {Many-Body Methods for
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Atoms, Molecules and Clusters}}}\ (\bibinfo {publisher} {Springer},\
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\bibinfo {year} {2018})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Piecuch}\ and\ \citenamefont
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{Kowalski}(2000)}]{Piecuch_2000}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Piecuch}}\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Kowalski}},\ }in\ \href {\doibase 10.1142/9789812792501_0001} {\emph
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{\bibinfo {booktitle} {Computational {{Chemistry}}: {{Reviews}} of {{Current
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Trends}}}}},\ \bibinfo {series} {Computational {{Chemistry}}: {{Reviews}} of
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{{Current Trends}}}, Vol.\ \bibinfo {volume} {Volume 5}\ (\bibinfo
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{publisher} {{WORLD SCIENTIFIC}},\ \bibinfo {year} {2000})\ pp.\ \bibinfo
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{pages} {1--104}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Mayhall}\ and\ \citenamefont
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{Raghavachari}(2010)}]{Mayhall_2010}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~J.}\ \bibnamefont
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{Mayhall}}\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Raghavachari}},\ }\href {\doibase 10.1021/ct100321k} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{6}},\ \bibinfo {pages} {2714} (\bibinfo {year} {2010})}\BibitemShut
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\bibitem [{\citenamefont {Lee}, \citenamefont {Small},\ and\ \citenamefont
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{{Head-Gordon}}(2019)}]{Lee_2019}%
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{Lee}}, \bibinfo {author} {\bibfnamefont {D.~W.}\ \bibnamefont {Small}}, \
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and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{Head-Gordon}}},\
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}\href {\doibase 10.1063/1.5128795} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {151}},\ \bibinfo
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{pages} {214103} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kossoski}\ \emph {et~al.}(2021)\citenamefont
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{Kossoski}, \citenamefont {Marie}, \citenamefont {Scemama}, \citenamefont
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{Caffarel},\ and\ \citenamefont {Loos}}]{Kossoski_2021}%
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\BibitemOpen
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{Kossoski}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Marie}},
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \bibinfo
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{author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}}, \ and\ \bibinfo
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{author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}},\ }\href {\doibase
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10.1021/acs.jctc.1c00348} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages}
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{4756} (\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Marie}, \citenamefont {Kossoski},\ and\ \citenamefont
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{Loos}(2021)}]{Marie_2021b}%
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\BibitemOpen
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{Marie}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Kossoski}}, \
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and\ \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}},\ }\href
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{\doibase 10.1063/5.0060698} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {155}},\ \bibinfo {pages} {104105}
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(\bibinfo {year} {2021})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {F{\"o}rster}\ and\ \citenamefont
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{Visscher}(2020)}]{Forster_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{F{\"o}rster}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Visscher}},\ }\href {\doibase 10.1021/acs.jctc.0c00693} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {16}},\ \bibinfo {pages} {7381} (\bibinfo {year}
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{2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {F{\"o}rster}\ and\ \citenamefont
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{Visscher}(2021)}]{Forster_2021}%
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\BibitemOpen
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{F{\"o}rster}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Visscher}},\ }\href {\doibase 10.3389/fchem.2021.736591} {\bibfield
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{journal} {\bibinfo {journal} {Front. Chem.}\ }\textbf {\bibinfo {volume}
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{9}},\ \bibinfo {pages} {736591} (\bibinfo {year} {2021})}\BibitemShut
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\bibitem [{\citenamefont {Duchemin}\ and\ \citenamefont
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{Blase}(2019)}]{Duchemin_2019}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Duchemin}}\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}},\ }\href {\doibase 10.1063/1.5090605} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {150}},\
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\bibinfo {pages} {174120} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Duchemin}\ and\ \citenamefont
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{Blase}(2020)}]{Duchemin_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Duchemin}}\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}},\ }\href {\doibase 10.1021/acs.jctc.9b01235} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{16}},\ \bibinfo {pages} {1742} (\bibinfo {year} {2020})}\BibitemShut
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\bibitem [{\citenamefont {Duchemin}\ and\ \citenamefont
|
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{Blase}(2021)}]{Duchemin_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Duchemin}}\ and\ \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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{Blase}},\ }\href {\doibase 10.1021/acs.jctc.1c00101} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{17}},\ \bibinfo {pages} {2383} (\bibinfo {year} {2021})}\BibitemShut
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\bibitem [{\citenamefont {Loos}(2019)}]{QuAcK}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.5281/zenodo.3745928} {\enquote {\bibinfo
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{title} {{{QuAcK: a software for emerging quantum electronic structure
|
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methods}}},}\ } (\bibinfo {year} {2019}),\ \bibinfo {note}
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{\url{https://github.com/pfloos/QuAcK}}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Lazzeri}\ \emph {et~al.}(2008)\citenamefont
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{Lazzeri}, \citenamefont {Attaccalite}, \citenamefont {Wirtz},\ and\
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\citenamefont {Mauri}}]{Lazzeri_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Lazzeri}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Attaccalite}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Wirtz}},
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\ and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Mauri}},\ }\href
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{\doibase 10.1103/PhysRevB.78.081406} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {78}},\ \bibinfo
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{pages} {081406} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Faber}\ \emph
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{et~al.}(2011{\natexlab{b}})\citenamefont {Faber}, \citenamefont {Janssen},
|
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\citenamefont {C\^ot\'e}, \citenamefont {Runge},\ and\ \citenamefont
|
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{Blase}}]{Faber_2011b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Faber}}, \bibinfo {author} {\bibfnamefont {J.~L.}\ \bibnamefont {Janssen}},
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\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {C\^ot\'e}}, \bibinfo
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{author} {\bibfnamefont {E.}~\bibnamefont {Runge}}, \ and\ \bibinfo {author}
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{\bibfnamefont {X.}~\bibnamefont {Blase}},\ }\href {\doibase
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10.1103/PhysRevB.84.155104} {\bibfield {journal} {\bibinfo {journal} {Phys.
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Rev. B}\ }\textbf {\bibinfo {volume} {84}},\ \bibinfo {pages} {155104}
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(\bibinfo {year} {2011}{\natexlab{b}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Yin}, \citenamefont {Kutepov},\ and\ \citenamefont
|
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{Kotliar}(2013)}]{Yin_2013}%
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{Yin}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Kutepov}}, \ and\
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\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Kotliar}},\ }\href
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{\doibase 10.1103/PhysRevX.3.021011} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. X}\ }\textbf {\bibinfo {volume} {3}},\ \bibinfo {pages}
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{021011} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Monserrat}(2016)}]{Montserrat_2016}%
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\BibitemOpen
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{Monserrat}},\ }\href {\doibase 10.1103/PhysRevB.93.100301} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
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{93}},\ \bibinfo {pages} {100301} (\bibinfo {year} {2016})}\BibitemShut
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\bibitem [{\citenamefont {Li}\ \emph {et~al.}(2019)\citenamefont {Li},
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\citenamefont {Antonius}, \citenamefont {Wu}, \citenamefont {da~Jornada},\
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and\ \citenamefont {Louie}}]{Zhenglu_2019}%
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\BibitemOpen
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{Li}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Antonius}},
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\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Wu}}, \bibinfo {author}
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{\bibfnamefont {F.~H.}\ \bibnamefont {da~Jornada}}, \ and\ \bibinfo {author}
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{\bibfnamefont {S.~G.}\ \bibnamefont {Louie}},\ }\href {\doibase
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10.1103/PhysRevLett.122.186402} {\bibfield {journal} {\bibinfo {journal}
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{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages}
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{186402} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ismail-Beigi}\ and\ \citenamefont
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{Louie}(2003)}]{IsmailBeigi_2003}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Ismail-Beigi}}\ and\ \bibinfo {author} {\bibfnamefont {S.~G.}\ \bibnamefont
|
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{Louie}},\ }\href {\doibase 10.1103/PhysRevLett.90.076401} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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{volume} {90}},\ \bibinfo {pages} {076401} (\bibinfo {year}
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{2003})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{\c C}aylak}\ and\ \citenamefont
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{Baumeier}(2021)}]{Caylak_2021}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {{\c
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C}aylak}}\ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
|
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{Baumeier}},\ }\href {\doibase 10.1021/acs.jctc.0c01099} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {17}},\ \bibinfo {pages} {879} (\bibinfo {year} {2021})}\BibitemShut
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\bibitem [{\citenamefont {Knysh}\ \emph {et~al.}(2022)\citenamefont {Knysh},
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\citenamefont {Duchemin}, \citenamefont {Blase},\ and\ \citenamefont
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{Jacquemin}}]{Knysh_2022}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Knysh}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Duchemin}},
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\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Blase}}, \ and\ \bibinfo
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{author} {\bibfnamefont {D.~M.}\ \bibnamefont {Jacquemin}},\ }\href {\doibase
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10.1063/5.0121121} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
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Phys.}\ }\textbf {\bibinfo {volume} {157}},\ \bibinfo {pages} {194102}
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(\bibinfo {year} {2022})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Bartlett}(1986)}]{Bartlett_1986}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
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{Bartlett}},\ }\enquote {\bibinfo {title} {Analytical evaluation of gradients
|
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in coupled-cluster and many-body perturbation theory},}\ in\ \href {\doibase
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10.1007/978-94-009-4584-5_4} {\emph {\bibinfo {booktitle} {Geometrical
|
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Derivatives of Energy Surfaces and Molecular Properties}}},\ \bibinfo
|
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{editor} {edited by\ \bibinfo {editor} {\bibfnamefont {P.}~\bibnamefont
|
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{J{\o}rgensen}}\ and\ \bibinfo {editor} {\bibfnamefont {J.}~\bibnamefont
|
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{Simons}}}\ (\bibinfo {publisher} {Springer Netherlands},\ \bibinfo
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{address} {Dordrecht},\ \bibinfo {year} {1986})\ pp.\ \bibinfo {pages}
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{35--61}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Brouder}, \citenamefont {Panati},\ and\ \citenamefont
|
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{Stoltz}(2009)}]{Brouder_2009}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Brouder}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Panati}}, \
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and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Stoltz}},\ }\href
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{\doibase 10.1103/PhysRevLett.103.230401} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo
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{pages} {230401} (\bibinfo {year} {2009})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Linn\'er}\ and\ \citenamefont
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{Aryasetiawan}(2019)}]{Linner_2019}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Linn\'er}}\ and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
|
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{Aryasetiawan}},\ }\href {\doibase 10.1103/PhysRevB.100.235106} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
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{100}},\ \bibinfo {pages} {235106} (\bibinfo {year} {2019})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Lyakh}\ \emph {et~al.}(2012)\citenamefont {Lyakh},
|
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\citenamefont {Musia{\l}}, \citenamefont {Lotrich},\ and\ \citenamefont
|
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{Bartlett}}]{Lyakh_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~I.}\ \bibnamefont
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{Lyakh}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Musia{\l}}},
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\bibinfo {author} {\bibfnamefont {V.~F.}\ \bibnamefont {Lotrich}}, \ and\
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\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont {Bartlett}},\ }\href
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{\doibase 10.1021/cr2001417} {\bibfield {journal} {\bibinfo {journal}
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{Chem. Rev.}\ }\textbf {\bibinfo {volume} {112}},\ \bibinfo {pages} {182}
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(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Evangelista}(2018)}]{Evangelista_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.~A.}\ \bibnamefont
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{Evangelista}},\ }\href {\doibase 10.1063/1.5039496} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibinfo {pages} {030901} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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