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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
\usepackage[version=4]{mhchem}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage[
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\SupMat}{\textcolor{blue}{supplementary material}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}
% coordinates
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bx}{\boldsymbol{x}}
% methods
\newcommand{\NO}[1]{\{#1\}}
\newcommand{\HF}{\text{HF}}
\newcommand{\CC}{\text{CC}}
%
\newcommand{\Ne}{N}
\newcommand{\Norb}{K}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}
% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hHN}{\Hat{H}_{\text{N}}}
\newcommand{\hFN}{\Hat{F}_{\text{N}}}
\newcommand{\hVN}{\Hat{V}_{\text{N}}}
\newcommand{\hh}{\Hat{h}}
\newcommand{\hf}{\Hat{f}}
\newcommand{\bH}{\Bar{H}}
\newcommand{\bHN}{\Bar{H}_{\text{N}}}
\newcommand{\hT}{\Hat{T}}
\newcommand{\hS}{\Hat{S}}
\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
\newcommand{\ani}[1]{\Hat{a}_{#1}}
\newcommand{\ca}[2]{\Hat{a}^{#1}_{#2}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cW}{\mathcal{W}}
% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\ECC}{E_\text{CC}}
% orbital energies
\newcommand{\eps}{\epsilon}
% Matrix elements
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\SO}[1]{\psi_{#1}}
\newcommand{\ERI}[2]{\mel{#1}{}{#2}}
% Matrices
\newcommand{\bdO}{\boldsymbol{0}}
\newcommand{\bdI}{\boldsymbol{1}}
\newcommand{\bdH}{\Bar{\boldsymbol{H}}}
% orbitals, gaps, etc
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}{E_\text{g}}
\newcommand{\EgFun}{\Eg^\text{fund}}
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
\newcommand{\ii}{\mathrm{i}}
% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LPT}{Laboratoire de Physique Th\'eorique, Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\ETSF}{European Theoretical Spectroscopy Facility (ETSF)}
\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut N\'EEL, F-38042 Grenoble, France}
\begin{document}
\title{Bethe-Salpeter ground- and excited-state energies from approximate coupled-cluster theory}
\author{Ra\'ul \surname{Quintero-Monsebaiz}}
\affiliation{\LCPQ}
\author{Enzo \surname{Monino}}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
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 intimate connections between the Bethe-Salpeter equation (BSE) formalism from many-body perturbation theory and coupled-cluster (CC) theory at both the ground- and excited-state levels.
Similitudes between BSE@$GW$ and the similarity-transformed equation-of-motion CC method introduced by Nooijen are put forward.
The present work allows to easily transfer key developments and general knowledge gathered in CC theory to many-body perturbation theory.
In particular, it provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
When one is looking for high accuracy, the workhorse of molecular electronic structure is certainly coupled cluster theory.
At low orders, \eg, coupled-cluster with singles and doubles, CC can treat weakly correlated systems, while when one can afford to ramp up the excitation degree to triplets or quadruples, even strongly correlated systems can be treated, hence eschewing to resort to multireference methods.
Link between BSE and STEOM-CC.
A route towards the obtention of BSE gradients for ground and excited states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Many-body perturbation theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Coupled-cluster theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The Schr\"odinger equation is
%\begin{equation}
% \hH \Psi = E \Psi
%\end{equation}
%where the bare Hamiltonian reads
%\begin{equation}
% \hH = \sum_{pq} h_{p}^{q} \cre{p} \ani{q} + \frac{1}{4} \sum_{pqrs} v_{pq}^{rs} \cre{p} \cre{q} \ani{s} \ani{r}
%\end{equation}
%where $h_{p}^{q} = \mel{p}{\hh}{q}$ are the core Hamiltonian elements and $v_{pq}^{rs} = \ERI{pq}{rs}$ are antisymmetrized two-electron integrals.
%The creation and annihilation operators fulfils the following anticommutation relationships:
%\begin{align}
% \cre{p} \cre{q} + \cre{q} \cre{p} & = 0
% &
% \ani{p} \ani{q} + \ani{q} \ani{p} & = 0
% &
% \cre{p} \ani{q} + \ani{p} \cre{q} & = \delta_{pq}
%\end{align}
%In terms of normal-ordered (with respect to the reference wave function $\Psi_0$) quantities (which are indicated with curly braces), the Schr\"odinger equation reads
%\begin{equation}
% \hHN \Psi = \Delta E \Psi
%\end{equation}
%where $\Delta E = E - E_0 = E - \mel{\Psi_0}{\hH}{\Psi_0}$ is the correlation energy and the normal-ordered Hamiltonian $\hHN = \hH - E_0$ is
%\begin{equation}
% \hHN = \hFN + \hVN = \sum_{pq} f_{p}^{q} \NO{\cre{p}\ani{q}} + \frac{1}{4} \sum_{pqrs} v_{pq}^{rs} \NO{\cre{p}\cre{q}\ani{s}\ani{r}}
%\end{equation}
%with $f_{p}^{q} = \mel{p}{\hf}{q}$ the elements of the Fock operator.
% and where the creation and annihilation operators are written compactly as
%\begin{equation}
% \ca{pq\cdots}{rs\cdots} = \cre{p} \cre{q} \cdots \ani{s} \ani{r}
%\end{equation}
%Within coupled-cluster theory, thanks to the following exponential ansatz of the wave function
%\begin{equation}
% \Psi = e^{\hT} \Psi_0
%\end{equation}
%and the introducing the following similarity-transformed Hamiltonian
%\begin{equation}
% \bH = e^{-\hT} \hHN e^{\hT}
%\end{equation}
%one recasts the Schr\"odinger equation as
%\begin{equation}
% \bH \Psi_0 = \Delta E \Psi_0
%\end{equation}
%where $e^{\hT}$ is the wave operator and $\hT$ is the excitation operator.
%The Schr\"odinger equation is recasted as
%
%Matrix elements involving $\bH$ can be efficiently and elegantly evaluated thanks to Wick's theorem and the so-called Campbell-Baker-Hausdorff (BCH) formula which leads to a linear combination of nested communtators
%\begin{equation}
%\begin{split}
% \bH
% & = \hHN + \comm{\hHN}{\hT{}{}}
% + \frac{1}{2!} \comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}}
% \\
% & + \frac{1}{3!} \comm{\comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}}}{\hT{}{}}
% + \frac{1}{4!} \comm{\comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}}}{\hT{}{}}
%\end{split}
%\end{equation}
%which naturally truncate at the fourth term due to the two-body nature of the bare Hamiltonian.
%
%In coupled-cluster theory, it is usual to truncate the excitation operator at the $n$th order, \ie,
%\begin{equation}
% \hT = \sum_{k=1}^{n} \hT_k
%\end{equation}
%where the normal-ordered $k$-fold ``neutral'' excitation operator
%\begin{equation}
% \hT_{k} = \frac{1}{(k!)^2} \sum_{ij\cdots} \sum_{ab\cdots} t_{ij\cdots}^{ab\cdots}
% \NO{\cre{a}\cre{b}\cdots}{\ani{i}\ani{j}\cdots}
%\end{equation}
%creates, when applied to $\ket*{\Psi_0}$, $k$-fold excited $\Ne$-electron determinants of the form $\ket*{\Psi_{ij\cdots}^{ab\cdots}}$.
%It is worth mentioning here that the excitation operators do commute with each other, \ie, $\comm{\hT_{k_1}}{\hT_{k_2}} = 0$ but do not commute with $\hHN$, \ie, $\comm{\hT_k}{\hHN} \neq 0$.
%======================
\subsection{EE-EOM-CCD}
%======================
Up to 2h2p, the EE-EOM-CC matrix has the simple form
\begin{equation}
\bdH_\text{2h2p}^\text{EE} =
\begin{pmatrix}
\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{k}^{c} } & \mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{kl}^{cd} }
\\
\mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} } & \mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} }
\\
\end{pmatrix}
\end{equation}
Restricting ourselves to CCD, the elements of the 1h1p block read
\begin{equation}
\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
\end{equation}
where the one-body terms are
\begin{subequations}
\begin{align}
\cF_{ab} & = \eps_{a} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac}
\\
\cF_{ij} & = \eps_{i} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd}
\end{align}
\end{subequations}
and the two-body term is
\begin{equation}
\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
\end{equation}
It is interesting to note that, in the case where $t_{ij}^{ab} = 0$, the 1h1p block reduces to the well-known random-phase approximation (with exchange).
%================================
\subsection{IP- and EA-EOM-CCD}
%================================
Up to 2h1p and 2p1h, the IP-EOM-CC and EA-EOM-CC matrices, respectively, have the simple form
\begin{subequations}
\begin{align}
\bdH_\text{2h1p}^\text{IP} & =
\begin{pmatrix}
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & \mel{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} }
\\
\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & \mel{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} }
\\
\end{pmatrix}
\\
\bdH_\text{2p1h}^\text{EA} & =
\begin{pmatrix}
\mel*{ \Phi^{a} }{ \bHN }{ \Phi^{c} } & \mel*{ \Phi^{a} }{ \bHN }{ \Phi_{k}^{cd} }
\\
\mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} } & \mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} }
\\
\end{pmatrix}
\end{align}
\end{subequations}
\begin{align}
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & = \titou{??}
\\
\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} } & = \titou{??}
\\
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } & = \cW_{ickl}
\\
\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & = \ERI{ij}{ka}
\end{align}
and
\begin{multline}
\cW_{iakl} = \ERI{ia}{kl} + \sum_{me} \ERI{im}{ke} t_{lm}^{ae}
\\
- \sum_{me} \ERI{im}{le} t_{km}^{ae}
+ \frac{1}{2} \sum_{ef} \ERI{ia}{ef} t_{kl}^{ef}
\end{multline}
%================================
%\subsection{DIP- and DEA-EOM-CCD}
%================================
%Up to 3h1p and 3p1h, the DIP-EOM-CC and DEA-EOM-CC matrices, respectively, have the simple form
%\begin{subequations}
%\begin{align}
% \bdH_\text{3h2p}^\text{DIP} & =
% \begin{pmatrix}
% \mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{lm} } & \mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{lmn}^{d} }
% \\
% \mel*{ \Phi_{ijk}^{a} }{ \bHN }{ \Phi_{lm} } & \mel*{ \Phi_{ijk}^{a} }{ \bHN }{ \Phi_{lmn}^{d} }
% \\
% \end{pmatrix}
% \\
% \bdH_\text{3p2h}^\text{DEA} & =
% \begin{pmatrix}
% \mel*{ \Phi_{i}^{abc} }{ \bHN }{ \Phi^{de} } & \mel*{ \Phi_{i}^{abc} }{ \bHN }{ \Phi_{l}^{def} }
% \\
% \mel*{ \Phi^{ab} }{ \bHN }{ \Phi^{de} } & \mel*{ \Phi^{ab} }{ \bHN }{ \Phi_{l}^{def} }
% \end{pmatrix}
%\end{align}
%\end{subequations}
%
%\begin{subequations}
%\begin{align}
% \mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{kl} } & = - \cF_{ij} \delta_{kl} - \delta_{ij} \cF_{kl}+ \cW_{ijkl}
% \\
% \mel*{ \Phi^{ab} }{ \bHN }{ \Phi^{cd} } & = + \cF_{ab} \delta_{cd} + \delta_{ab} \cF_{cd} + \cW_{abcd}
%\end{align}
%\end{subequations}
%with
%\begin{subequations}
%\begin{align}
% \cW_{ijkl} & = \ERI{ij}{kl} + \sum_{a<b} \ERI{ij}{ab} t_{ij}^{ab}
% \\
% \cW_{abcd} & = \ERI{ab}{cd} + \sum_{i<j} \ERI{ab}{ij} t_{ij}^{ab}
%\end{align}
%\end{subequations}
%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material}
%\label{sec:supmat}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
\acknowledgments{
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%
%\appendix
%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{CCvsMBPT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}