saving work

CCvsMBPT.bib

@@ -1,13 +1,30 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% http://bibdesk.sourceforge.net/
+%% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2022-10-04 16:49:04 +0200 
+%% Created for Pierre-Francois Loos at 2022-10-05 10:58:37 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Emrich_1981,
+	abstract = {The expS method (coupled cluster formalism) is extended to excited states of finite and infinite systems. We obtain equations which are formally similar to the known ground-state equations of the expS theory. The method is applicable to Fermi as well as Bose systems.},
+	author = {K. Emrich},
+	date-added = {2022-10-05 10:57:15 +0200},
+	date-modified = {2022-10-05 10:58:35 +0200},
+	doi = {https://doi.org/10.1016/0375-9474(81)90179-2},
+	issn = {0375-9474},
+	journal = {Nuc. Phys. A},
+	number = {3},
+	pages = {379-396},
+	title = {An extension of the coupled cluster formalism to excited states (I)},
+	url = {https://www.sciencedirect.com/science/article/pii/0375947481901792},
+	volume = {351},
+	year = {1981},
+	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0375947481901792},
+	bdsk-url-2 = {https://doi.org/10.1016/0375-9474(81)90179-2}}
+
 @article{Bintrim_2022,
 	author = {Bintrim,Sylvia J. and Berkelbach,Timothy C.},
 	date-added = {2022-10-04 16:48:32 +0200},

CCvsMBPT.tex

@@ -58,6 +58,7 @@
 \newcommand{\hf}{\Hat{f}}
 \newcommand{\bH}{\Bar{H}}
 \newcommand{\bHN}{\Bar{H}_{\text{N}}}
+\newcommand{\tHN}{\Tilde{H}_{\text{N}}}
 \newcommand{\hT}{\Hat{T}}
 \newcommand{\hS}{\Hat{S}}
 \newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
@@ -87,6 +88,7 @@
 \newcommand{\ERI}[2]{\braket{#1}{#2}}
 \newcommand{\sERI}[2]{(#1|#2)}
 \newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
+\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
 
 % Matrices
 \newcommand{\bO}{\boldsymbol{0}}
@@ -129,7 +131,7 @@
 
 \begin{document}	
 
-\title{Connections between many-body perturbation theory and coupled-cluster theory}
+\title{Connections between many-body perturbation and coupled-cluster theories}
 
 \author{Ra\'ul \surname{Quintero-Monsebaiz}}
 	\affiliation{\LCPQ}
@@ -154,8 +156,16 @@ In particular, it provides a clear path for the computation of ground- and excit
 \maketitle
 
 %%%%%%%%%%%%%%%%%%%%%%
-%\section{Introduction}
+%\section{RPA Physics and Beyond}
 %%%%%%%%%%%%%%%%%%%%%%
+The random-phase approximation (RPA), introduced by Bohm and Pines in the context of the uniform electron gas, \cite{Loos_2016} is a quasibosonic approximation where one treats fermion products as bosons.
+In the particle-hole (ph) channel, which is quite popular in the electronic structure community, particle-hole fermionic excitations and deexcitations are assumed to be bosons. 
+Because RPA corresponds to a resummation of all ring diagrams, it is adequate in the high-density (or weakly correlated) regime and catch effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958} 
+
+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 effective way of introducing correlation to go \textit{beyond} RPA physics.
+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 $W$. 
+This scheme 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}
+
 
 %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.
@@ -166,7 +176,7 @@ In particular, it provides a clear path for the computation of ground- and excit
 %\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
+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 ubiquitous RPAx eigensystem
 \begin{equation}
 	\label{eq:RPA}
 	\begin{pmatrix}
@@ -220,7 +230,7 @@ Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following
 \begin{equation}
 	\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
 \end{equation}
-that matches the rCCD equations
+that matches the well-known rCCD equations
 \begin{multline}
 \label{eq:rCCD}
 	\dbERI{ij}{ab} 
@@ -240,8 +250,7 @@ knowing that
 	B_{ia,jb} & = \dbERI{ij}{ab}
 \end{align}
 \end{subequations}
-We assume real quantities throughout this paper. 
+We assume real quantities throughout this paper, $\e{p}{}$ is the one-electron energy associated with the Hartree-Fock spinorbital $\SO{p}(\bx)$ and 
-$\e{p}{}$ 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
@@ -253,22 +262,24 @@ are two-electron repulsion integrals, while
 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 $N = O + V$.
+In the following, $O$ and $V$ are the number of occupied and virtual spinorbitals, respectively, and $N = O + V$ is the total number.
 
 
-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
+There are various ways of computing the RPAx correlation energy, \cite{Angyan_2011} but the usual plasmon (or trace) formula yields
 \begin{equation}
+\label{eq:EcRPAx}
 	\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
 \end{equation}
-which matches the rCCD correlation energy
+and 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}.
+Note that the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem.\cite{Angyan_2011}
 
-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.
+This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EE-EOM-rCCD and RPAx exist when the EOM space is restricted to 1h1p configurations and only the two-body terms are dressed by 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
+To be more specific, 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}
@@ -284,15 +295,21 @@ 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
+Neglecting the effect of $\hT_2$ on the one-body terms and relying on the rCCD ampltiudes in the two-body terms yields 
 \begin{equation}
-	\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
+\begin{split}
+	\mel*{ \Psi_{i}^{a} }{ \tHN }{ \Psi_{j}^{b} } 
+	& = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj} + \sum_{kc} \dbERI{ik}{ac} t_{kj}^{cb}
+	\\
+	& = (\bA{}{} + \bB{}{} \cdot \bT{}{})_{ia,jb}
+\end{split}
 \end{equation}
+which exactly matches Eq.~\eqref{eq:RPA_1}.
+Although the excitation energies of this approximate EE-EOM-rCCD scheme are equal to the RPAx ones, it has been shown that the transition amplitudes (or residues) are distincts and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018}
 
 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}
@@ -367,6 +384,28 @@ with
 \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.
+
+Note also that, within BSE, the orbital energies in Eqs.~\eqref{eq:A_BSE} and \eqref{eq:B_BSE} are usually computed at the $GW$ level to be consistent with the introduction of $W$ in the BSE kernel.
+For example, at the one-shot $GW$ level, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} we have
+\begin{equation}
+	\label{eq:G0W0}
+	\e{p}{\GW} = \e{p}{\HF} + Z_p \SigC{p}(\omega = \e{p}{\HF})
+\end{equation}
+where the elements of the correlation part of the dynamical self-energy are
+\begin{equation}
+\begin{split}
+	\SigC{pq}(\omega) 
+	& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{\HF} + \Om{m}{\dRPA} - \ii \eta}
+	\\
+	& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{\HF} - \Om{m}{\dRPA} + \ii \eta}
+\end{split}
+\end{equation}
+and the renormalization factor is
+\begin{equation}
+	Z_p = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1}
+\end{equation}
+
+
 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})