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CCvsMBPT.tex
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@ -20,6 +20,7 @@
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}} \newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\ant}[1]{\textcolor{purple}{#1}}
\newcommand{\SupMat}{\textcolor{blue}{supplementary material}} \newcommand{\SupMat}{\textcolor{blue}{supplementary material}}
\newcommand{\mc}{\multicolumn} \newcommand{\mc}{\multicolumn}
@ -167,12 +168,13 @@ In particular, it may provide a path for the computation of ground- and excited-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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. 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.
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. 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.
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} Because \ant{ph-}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} \ant{Maybe explain more this last sentence, I feel like it's a bit fast.}
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. 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.
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 $W$. 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 $W$.
Practically, $GW$ produces accurate ``charged'' excitations and provides a faithful description of the fundamental gap, while 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 accurate ``neutral'' excitations. Practically, $GW$ produces accurate ``charged'' excitations and provides a faithful description of the fundamental gap, while 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 accurate ``neutral'' excitations.
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} 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}
\ant{In this paragraph we say that BSE goes beyond RPA and then we say that BSE is done on top of GW without expliciting the link between GW and RPA. I guess it's obvious for GW people but maybe just saying that the GW polarizability is obtained by summing the time dependent ring diagrams would make the connection a bit clearer.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Connection between RPA and CC} \section{Connection between RPA and CC}
@ -368,7 +370,7 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, the screened two-electr
\label{eq:sERI} \label{eq:sERI}
\sERI{pq}{m} = \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \sERI{pq}{m} = \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia}
\end{equation} \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 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} \begin{equation}
\label{eq:dRPA} \label{eq:dRPA}
\begin{pmatrix} \begin{pmatrix}
@ -398,6 +400,7 @@ with
B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab} B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
\end{align} \end{align}
\end{subequations} \end{subequations}
\ant{We maybe we can remove the above equations and just say that it's Eq.~\eqref{eq:RPA}\eqref{eq:A_RPAx}\eqref{eq:B_RPAx} without double bars. If we keep the equations, I think the GW superscript on the energies is wrong.}
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. 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 Therefore, following the derivation detailed in Sec.~\ref{sec:RPAx}, one can show that the BSE correlation energy obtained using the trace formula
@ -419,7 +422,7 @@ 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. 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, either based on the plasmon formula \cite{,Li_2020,Li_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e} As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, either based on the plasmon formula \cite{,Li_2020,Li_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e}
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states: the EOM treatment (restricted to 1h1p) of the approximate similarity-transformed rCCD Hamiltonian [see Eq.~\eqref{eq:rCCD-BSE}] provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}. Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states: the EOM treatment (restricted to 1h1p) of the approximate similarity-transformed rCCD Hamiltonian [see Eq.~\eqref{eq:rCCD-BSE}] provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}. \ant{I'm not sure that we can call it an EOM treatment of an approximate similarity transformed rCCD Hamiltonian. Maybe we can say that we can obtain an analog of the approximate-EOM equation Eq.~(\ref{eq:EOM-rCCD}) by using the amplitudes defined by [see Eq.~\eqref{eq:rCCD-BSE}] as well as replacing $A$ and $B$ by their BSE counterparts and this equation gives the same excitations as the linear response equations}.
However, there is a significant difference with RPAx as the BSE 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) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}. However, there is a significant difference with RPAx as the BSE 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) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}.
In other words, in the spirit of the Brueckner version of CCD, the $GW$ post-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic diagrams, \cite{Scuseria_2008,Scuseria_2013} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}. In other words, in the spirit of the Brueckner version of CCD, the $GW$ post-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic diagrams, \cite{Scuseria_2008,Scuseria_2013} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}.