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CCvsMBPT.bib
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@ -1,13 +1,65 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-04-27 14:39:59 +0200 %% Created for Pierre-Francois Loos at 2022-10-04 16:49:04 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Bintrim_2022,
author = {Bintrim,Sylvia J. and Berkelbach,Timothy C.},
date-added = {2022-10-04 16:48:32 +0200},
date-modified = {2022-10-04 16:48:48 +0200},
doi = {10.1063/5.0074434},
journal = {J. Chem. Phys.},
number = {4},
pages = {044114},
title = {Full-frequency dynamical Bethe--Salpeter equation without frequency and a study of double excitations},
volume = {156},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0074434}}
@article{Monino_2022,
author = {Monino,Enzo and Loos,Pierre-Fran{\c c}ois},
date-added = {2022-10-04 14:11:52 +0200},
date-modified = {2022-10-04 14:12:06 +0200},
doi = {10.1063/5.0089317},
journal = {J. Chem. Phys.},
number = {23},
pages = {231101},
title = {Unphysical discontinuities, intruder states and regularization in GW methods},
volume = {156},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0089317}}
@article{Berkelbach_2018,
author = {Berkelbach,Timothy C.},
date-added = {2022-10-04 14:10:44 +0200},
date-modified = {2022-10-04 14:11:05 +0200},
doi = {10.1063/1.5032314},
journal = {J. Chem. Phys.},
number = {4},
pages = {041103},
title = {Communication: Random-phase approximation excitation energies from approximate equation-of-motion coupled-cluster doubles},
volume = {149},
year = {2018},
bdsk-url-1 = {https://doi.org/10.1063/1.5032314}}
@article{Scuseria_2008,
author = {Scuseria,Gustavo E. and Henderson,Thomas M. and Sorensen,Danny C.},
date-added = {2022-10-04 14:09:59 +0200},
date-modified = {2022-10-04 14:10:15 +0200},
doi = {10.1063/1.3043729},
journal = {J. Chem. Phys.},
number = {23},
pages = {231101},
title = {The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach},
volume = {129},
year = {2008},
bdsk-url-1 = {https://doi.org/10.1063/1.3043729}}
@article{Loos_2022, @article{Loos_2022,
author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina}, author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina},
date-added = {2022-04-27 14:39:38 +0200}, date-added = {2022-04-27 14:39:38 +0200},
@ -454,10 +506,10 @@
year = {2014}, year = {2014},
bdsk-url-1 = {https://doi.org/10.1063/1.4865816}} bdsk-url-1 = {https://doi.org/10.1063/1.4865816}}
@article{Bintrim_2021a, @article{Bintrim_2021,
author = {Bintrim,Sylvia J. and Berkelbach,Timothy C.}, author = {Bintrim,Sylvia J. and Berkelbach,Timothy C.},
date-added = {2021-11-03 22:50:51 +0100}, date-added = {2021-11-03 22:50:51 +0100},
date-modified = {2021-11-03 22:51:09 +0100}, date-modified = {2022-10-04 16:48:56 +0200},
doi = {10.1063/5.0035141}, doi = {10.1063/5.0035141},
journal = {J. Chem. Phys.}, journal = {J. Chem. Phys.},
number = {4}, number = {4},
@ -467,16 +519,6 @@
year = {2021}, year = {2021},
bdsk-url-1 = {https://doi.org/10.1063/5.0035141}} bdsk-url-1 = {https://doi.org/10.1063/5.0035141}}
@misc{Bintrim_2021b,
archiveprefix = {arXiv},
author = {Sylvia J. Bintrim and Timothy C. Berkelbach},
date-added = {2021-11-03 22:49:47 +0100},
date-modified = {2021-11-03 22:50:18 +0100},
eprint = {2110.03850},
primaryclass = {cond-mat.mtrl-sci},
title = {Full-frequency dynamical Bethe-Salpeter equation without frequency and a study of double excitations},
year = {2021}}
@article{Loos_2021, @article{Loos_2021,
author = {Loos, Pierre-Fran{\c c}ois and Comin, Massimiliano and Blase, Xavier and Jacquemin, Denis}, author = {Loos, Pierre-Fran{\c c}ois and Comin, Massimiliano and Blase, Xavier and Jacquemin, Denis},
date-added = {2021-11-03 16:57:23 +0100}, date-added = {2021-11-03 16:57:23 +0100},

552
CCvsMBPT.tex
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@ -37,6 +37,10 @@
% methods % methods
\newcommand{\NO}[1]{\{#1\}} \newcommand{\NO}[1]{\{#1\}}
\newcommand{\HF}{\text{HF}} \newcommand{\HF}{\text{HF}}
\newcommand{\GW}{GW}
\newcommand{\dRPA}{\text{dRPA}}
\newcommand{\RPAx}{\text{RPAx}}
\newcommand{\BSE}{\text{BSE}}
\newcommand{\CC}{\text{CC}} \newcommand{\CC}{\text{CC}}
% %
@ -70,17 +74,40 @@
\newcommand{\ECC}{E_\text{CC}} \newcommand{\ECC}{E_\text{CC}}
% orbital energies % orbital energies
\newcommand{\eps}{\epsilon} \newcommand{\e}[2]{\epsilon_{#1}^{#2}}
\newcommand{\eHF}[1]{\epsilon_{#1}^\text{HF}}
\newcommand{\eKS}[1]{\epsilon_{#1}^\text{KS}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
\newcommand{\om}[2]{\omega_{#1}^{#2}}
% Matrix elements % Matrix elements
\newcommand{\MO}[1]{\phi_{#1}} \newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\SO}[1]{\psi_{#1}} \newcommand{\SO}[1]{\psi_{#1}}
\newcommand{\ERI}[2]{\mel{#1}{}{#2}} \newcommand{\ERI}[2]{\braket{#1}{#2}}
\newcommand{\sERI}[2]{(#1|#2)}
\newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
% Matrices % Matrices
\newcommand{\bdO}{\boldsymbol{0}} \newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bdI}{\boldsymbol{1}} \newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bdH}{\Bar{\boldsymbol{H}}} \newcommand{\bbH}{\Bar{\boldsymbol{H}}}
\newcommand{\bOm}[2]{\boldsymbol{\Omega}_{#1}^{#2}}
\newcommand{\bom}[2]{\boldsymbol{\omega}_{#1}^{#2}}
\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}}
\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}}
\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
\newcommand{\bD}[2]{\boldsymbol{D}_{#1}^{#2}}
\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
\newcommand{\bR}[2]{\boldsymbol{R}_{#1}^{#2}}
\newcommand{\bT}[2]{\boldsymbol{T}_{#1}^{#2}}
\newcommand{\bU}[2]{\boldsymbol{U}_{#1}^{#2}}
\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
\newcommand{\be}[2]{\boldsymbol{\epsilon}_{#1}^{#2}}
\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
% orbitals, gaps, etc % orbitals, gaps, etc
\newcommand{\IP}{I} \newcommand{\IP}{I}
@ -102,7 +129,7 @@
\begin{document} \begin{document}
\title{Bethe-Salpeter ground- and excited-state energies from approximate coupled-cluster theory} \title{Connections between many-body perturbation theory and coupled-cluster theory}
\author{Ra\'ul \surname{Quintero-Monsebaiz}} \author{Ra\'ul \surname{Quintero-Monsebaiz}}
\affiliation{\LCPQ} \affiliation{\LCPQ}
@ -113,7 +140,7 @@
\affiliation{\LCPQ} \affiliation{\LCPQ}
\begin{abstract} \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. 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 and $GW$ approximation from many-body perturbation theory and coupled-cluster (CC) theory at 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. 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. 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. In particular, it provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.
@ -127,20 +154,383 @@ In particular, it provides a clear path for the computation of ground- and excit
\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} %\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
When one is looking for high accuracy, the workhorse of molecular electronic structure is certainly coupled cluster theory. %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. %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. %Link between BSE and STEOM-CC.
A route towards the obtention of BSE gradients for ground and excited states. %A route towards the obtention of BSE gradients for ground and excited states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Many-body perturbation theory} %\section{Bethe-Salpeter equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In a landmark paper, Scuseria \textit{et al.} \cite{Scuseria_2008} have proven that rCCD is equivalent to RPAx for the computation of the correlation energy.
Assuming the existence of $\bX{}{-1}$ (which can be proven as long as the RPAx problem is stable), this proof can be quickly summarised starting from the RPAx eigensystem
\begin{equation}
\label{eq:RPA}
\begin{pmatrix}
\bA{}{} & \bB{}{} \\
-\bB{}{} & -\bA{}{} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{}{} \\
\bY{}{} \\
\end{pmatrix}
=
\begin{pmatrix}
\bX{}{} \\
\bY{}{} \\
\end{pmatrix}
\cdot
\bOm{}{}
\end{equation}
from which one gets, by introducing $\bT{}{} = \bY{}{} \cdot \bX{}{-1}$,
\begin{equation}
\begin{pmatrix}
\bA{}{} & \bB{}{} \\
-\bB{}{} & -\bA{}{} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bI \\
\bT{}{} \\
\end{pmatrix}
=
\begin{pmatrix}
\bI \\
\bT{}{} \\
\end{pmatrix}
\cdot
\bR{}{}
\end{equation}
where $\bR{}{} = \bX{}{} \cdot \bOm{}{} \cdot \bX{}{-1}$, or equivalently, the two following equations
\begin{subequations}
\begin{align}
\label{eq:RPA_1}
\bA{}{} + \bB{}{} \cdot \bT{}{} & = \bR{}{}
\\
\label{eq:RPA_2}
-\bB{}{} - \bA{}{} \cdot \bT{}{} & = \bT{}{} \cdot \bR{}{}
\end{align}
\end{subequations}
Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following Riccati equation
\begin{equation}
\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
\end{equation}
that matches the rCCD equations
\begin{multline}
\label{eq:rCCD}
\dbERI{ij}{ab}
+ \qty( \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{} ) t_{ij}^{ab}
+ \sum_{kc} \dbERI{ic}{ak} t_{kj}^{cb}
\\
+ \sum_{kc} \dbERI{kb}{cj} t_{ik}^{ac}
+ \sum_{klcd} \dbERI{kl}{cd} t_{ik}^{ac} t_{lj}^{db} = 0
\end{multline}
knowing that
\begin{subequations}
\begin{align}
\label{eq:A_RPAx}
A_{ia,jb} & = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj}
\\
\label{eq:B_RPAx}
B_{ia,jb} & = \dbERI{ij}{ab}
\end{align}
\end{subequations}
We assume real quantities throughout this paper.
$\e{p}{}$ is the one-electron energy associated with the Hartree-Fock (spin)orbital $\SO{p}(\bx)$ and
\begin{equation}
\label{eq:ERI}
\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
\end{equation}
are two-electron repulsion integrals, while
\begin{equation}
\dbERI{pq}{rs} = \braket{pq}{rs} - \braket{pq}{sr}
\end{equation}
are their anti-symmetrized versions.
The composite variable $\bx$ gathers spin and spatial ($\br$) variables.
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.
In the following, $O$ and $V$ are the number of occupied and virtual spinorbitals, respectively, and $N = O + V$.
There are various ways of computing the RPAx correlation energy, \cite{Angyan_2011} but the usual plasmon (or trace) formula (or, equivalently, the adiabatic connection fluctuation dissipation theorem) yields
\begin{equation}
\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
\end{equation}
which matches the rCCD correlation energy
\begin{equation}
\Ec^\text{rCCD} = \frac{1}{4} \sum_{ijab} \dbERI{ij}{ab} t_{ij}^{ab} = \frac{1}{4} \Tr(\bB{}{} \cdot \bT{}{})
\end{equation}
because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}.
This proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EOM-rCCD and RPAx exist when the EOM space is restricted to 1h1p configurations and only the two-body terms are dressed by the rCCD correlation.
To be more specific, by restricting ourselves to CCD, \ie, $\hT = \hT_2$, the elements of the 1h1p block of the EOM Hamiltonian read
\begin{equation}
\mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
\end{equation}
where $\bHN = e^{-\hT} \hH e^{\hT} - E_\text{CCD}$ is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
\begin{subequations}
\begin{align}
\cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac}
\\
\cF_{ij} & = \e{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}
Remembering that $\bT{}{} = \bY{}{} \cdot \bX{}{-1}$ in the case of rCCD, we have
\begin{equation}
\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
\end{equation}
This does not hold only for RPAx as it is actually quite general and can be applied to any ph problem with the structure of Eq.~\eqref{eq:RPA}, such as TD-DFT, BSE, and many others.
This can be be also extended to the pp and hh sectors as shown by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} (see also Ref.~\onlinecite{Berkelbach_2018} for an extension to excitation energies).
At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem
\begin{equation}
\label{eq:BSE}
\begin{pmatrix}
\bA{}{\BSE} & \bB{}{\BSE} \\
-\bB{}{\BSE} & -\bA{}{\BSE} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{}{\BSE} \\
\bY{}{\BSE} \\
\end{pmatrix}
=
\begin{pmatrix}
\bX{}{\BSE} \\
\bY{}{\BSE} \\
\end{pmatrix}
\cdot
\bOm{}{\BSE}
\end{equation}
where
\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}
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) eigenvalues 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}{} - \e{i}{}) + \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 screening of the electron-electron interaction [see Eq.~\eqref{eq:W}], correlation effects at the BSE level.
Therefore, following a similar procedure, 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})
\end{equation}
can be obtained via a set of rCCD equations \eqref{eq:rCCD}, where one substitutes all the two-electron integrals $\dbERI{pq}{rs}$ by $\dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$.
Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost.
\titou{Comments on cost and extension to gradients.}
Following Berkelbach's analysis, one can extend the connection to excitation energies.
However, there is a significant difference as the BSE equations involves $GW$ quasiparticle energies [see Eq.~\eqref{eq:A_BSE}], where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) HF orbital energies
It is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations.
Connections between EOM-CC and the $GW$ approximation have been already studied by Lange and Berkelbach, \cite{Lange_2018} but we believe that the present work proposes a different perspective on this particular subject.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$GW$ approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the non-linear $GW$ equations can be recast as a set of linear equations, which reads in the Tamm-Dancoff approximation
\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}
\bX{}{} \\
\bY{}{\text{2h1p}} \\
\bY{}{\text{2p1h}} \\
\end{pmatrix}
=
\begin{pmatrix}
\bX{}{} \\
\bY{}{\text{2h1p}} \\
\bY{}{\text{2p1h}} \\
\end{pmatrix}
\cdot
\bOm{}{}
\end{equation}
where $\be{}{}$ is a diagonal matrix gathered the orbital energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \ERI{jc}{al} ] \delta_{ik}
\\
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \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}
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 $\bR{}{} = \bX{}{} \cdot \bOm{}{} \cdot \bX{}{-1}$.
The matrix $\bX{}{-1}$ can be proven to exist when the system is stable.
This 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} yields the two following 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}
In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ij}^{a}$ and $t_{i}^{ab}$, it means that one must solve the following amplitude equations
\begin{subequations}
\begin{align}
\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} t_{ija,q}^{\text{2h1p}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}}
- \sum_{kcdq} t_{ija,q}^{\text{2h1p}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}}
\end{split}
\\
\begin{split}
r_{iab,p}^{\text{2p1h}}
& = \ERI{pi}{ba}
+ \Delta_{iab,p}^{\text{2p1h}} t_{ija,p}^{\text{2p1h}}
+ \sum_{kc} \ERI{ak}{ic} t_{kcb,p}^{\text{2p1h}}
\\
& - \sum_{klcq} t_{iab,q}^{\text{2p1h}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}}
- \sum_{kcdq} t_{iab,q}^{\text{2p1h}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}}
\end{split}
\end{align}
\end{subequations}
with
\begin{subequations}
\begin{align}
\Delta_{ija,p}^{\text{2h1p}} & = \e{i}{} + \e{j}{} - \e{a}{} - \e{p}{}
\\
\Delta_{iab,p}^{\text{2p1h}} & = \e{a}{} + \e{b}{} - \e{i}{} - \e{p}{}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \frac{r_{ija,p}^{\text{2h1p}}}{\Delta_{ij}^{ap}}
\\
t_{iab,p}^{\text{2h1p}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \frac{r_{iab,p}^{\text{2p1h}}}{\Delta_{ab}^{ip}}
\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}
is the correlation part of the $GW$ self-energy.
The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Coupled-cluster theory} %\section{Coupled-cluster theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The Schr\"odinger equation is %The Schr\"odinger equation is
%\begin{equation} %\begin{equation}
@ -214,77 +604,77 @@ A route towards the obtention of BSE gradients for ground and excited states.
%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, $\comm{\hT_k}{\hHN} \neq 0$. %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} %\subsection{EE-EOM-CCD}
%====================== %======================
Up to 2h2p, the EE-EOM-CC matrix has the simple form %Up to 2h2p, the EE-EOM-CC matrix has the simple form
\begin{equation} %\begin{equation}
\bdH_\text{2h2p}^\text{EE} = % \bdH_\text{2h2p}^\text{EE} =
\begin{pmatrix} % \begin{pmatrix}
\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{k}^{c} } & \mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{kl}^{cd} } % \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} } % \mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} } & \mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} }
\\ % \\
\end{pmatrix} % \end{pmatrix}
\end{equation} %\end{equation}
Restricting ourselves to CCD, the elements of the 1h1p block read %Restricting ourselves to CCD, the elements of the 1h1p block read
\begin{equation} %\begin{equation}
\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi} % \mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
\end{equation} %\end{equation}
where the one-body terms are %where the one-body terms are
\begin{subequations} %\begin{subequations}
\begin{align} %\begin{align}
\cF_{ab} & = \eps_{a} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac} % \cF_{ab} & = \e{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} % \cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd}
\end{align} %\end{align}
\end{subequations} %\end{subequations}
and the two-body term is %and the two-body term is
\begin{equation} %\begin{equation}
\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb} % \cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
\end{equation} %\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). %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} %\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 %Up to 2h1p and 2p1h, the IP-EOM-CC and EA-EOM-CC matrices, respectively, have the simple form
\begin{subequations} %\begin{subequations}
\begin{align} %\begin{align}
\bdH_\text{2h1p}^\text{IP} & = % \bdH_\text{2h1p}^\text{IP} & =
\begin{pmatrix} % \begin{pmatrix}
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & \mel{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } % \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} } % \mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & \mel{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} }
\\ % \\
\end{pmatrix} % \end{pmatrix}
\\ % \\
\bdH_\text{2p1h}^\text{EA} & = % \bdH_\text{2p1h}^\text{EA} & =
\begin{pmatrix} % \begin{pmatrix}
\mel*{ \Phi^{a} }{ \bHN }{ \Phi^{c} } & \mel*{ \Phi^{a} }{ \bHN }{ \Phi_{k}^{cd} } % \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} } % \mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} } & \mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} }
\\ % \\
\end{pmatrix} % \end{pmatrix}
\end{align} %\end{align}
\end{subequations} %\end{subequations}
%
\begin{align} %\begin{align}
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & = \titou{??} % \mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & = \titou{??}
\\ % \\
\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} } & = \titou{??} % \mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} } & = \titou{??}
\\ % \\
\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } & = \cW_{ickl} % \mel*{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } & = \cW_{ickl}
\\ % \\
\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & = \ERI{ij}{ka} % \mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & = \ERI{ij}{ka}
\end{align} %\end{align}
and %and
\begin{multline} %\begin{multline}
\cW_{iakl} = \ERI{ia}{kl} + \sum_{me} \ERI{im}{ke} t_{lm}^{ae} % \cW_{iakl} = \ERI{ia}{kl} + \sum_{me} \ERI{im}{ke} t_{lm}^{ae}
\\ % \\
- \sum_{me} \ERI{im}{le} t_{km}^{ae} % - \sum_{me} \ERI{im}{le} t_{km}^{ae}
+ \frac{1}{2} \sum_{ef} \ERI{ia}{ef} t_{kl}^{ef} % + \frac{1}{2} \sum_{ef} \ERI{ia}{ef} t_{kl}^{ef}
\end{multline} %\end{multline}
%================================ %================================
%\subsection{DIP- and DEA-EOM-CCD} %\subsection{DIP- and DEA-EOM-CCD}
%================================ %================================
@ -325,7 +715,7 @@ and
%\end{subequations} %\end{subequations}
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} %\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion. Here comes the conclusion.