saving work

CCvsMBPT.bib

@@ -1,13 +1,57 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% https://bibdesk.sourceforge.io/
+%% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-10-05 10:58:37 +0200 
+%% Created for Pierre-Francois Loos at 2022-10-05 14:52:00 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Rishi_2020,
+	author = {Rishi,Varun and Perera,Ajith and Bartlett,Rodney J.},
+	date-added = {2022-10-05 14:51:45 +0200},
+	date-modified = {2022-10-05 14:52:00 +0200},
+	doi = {10.1063/5.0023862},
+	journal = {J. Chem. Phys.},
+	number = {23},
+	pages = {234101},
+	title = {A route to improving RPA excitation energies through its connection to equation-of-motion coupled cluster theory},
+	volume = {153},
+	year = {2020},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0023862}}
+
+@article{Jansen_2010,
+	author = {Jansen,Georg and Liu,Ru-Fen and {\'A}ngy{\'a}n,J{\'a}nos G.},
+	date-added = {2022-10-05 14:50:41 +0200},
+	date-modified = {2022-10-05 14:50:56 +0200},
+	doi = {10.1063/1.3481575},
+	journal = {J. Chem. Phys.},
+	number = {15},
+	pages = {154106},
+	title = {On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions},
+	volume = {133},
+	year = {2010},
+	bdsk-url-1 = {https://doi.org/10.1063/1.3481575}}
+
+@article{Freeman_1977,
+	author = {Freeman, David L.},
+	date-added = {2022-10-05 14:48:42 +0200},
+	date-modified = {2022-10-05 14:49:07 +0200},
+	doi = {10.1103/PhysRevB.15.5512},
+	issue = {12},
+	journal = {Phys. Rev. B},
+	month = {Jun},
+	numpages = {0},
+	pages = {5512--5521},
+	publisher = {American Physical Society},
+	title = {Coupled-cluster expansion applied to the electron gas: Inclusion of ring and exchange effects},
+	url = {https://link.aps.org/doi/10.1103/PhysRevB.15.5512},
+	volume = {15},
+	year = {1977},
+	bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.15.5512},
+	bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.15.5512}}
+
 @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},

CCvsMBPT.tex

@@ -162,19 +162,16 @@ The random-phase approximation (RPA), introduced by Bohm and Pines in the contex
 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.
+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 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 $W$. 
 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.
+Interestingly, RPA has strong connections with coupled-cluster (CC) theory, the workhorse of molecular electronic structure is when one is looking for high accuracy. \cite{Freeman_1977,Scuseria_2008,Jansen_2010,Scuseria_2013,Peng_2013,Berkelbach_2018,Rishi_2020}
 %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{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 ubiquitous RPAx eigensystem
 \begin{equation}
@@ -265,7 +262,7 @@ The indices $i$, $j$, $k$, and $l$ are occupied (hole) orbitals; $a$, $b$, $c$,
 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 yields
+There are various ways of computing the RPAx correlation energy, \cite{Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula yields
 \begin{equation}
 \label{eq:EcRPAx}
 	\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
@@ -310,6 +307,11 @@ Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab}
 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 distinct and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018}
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\section{Bethe-Salpeter 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 has been also extended to the pp and hh sectors by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} (see also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels).
 
@@ -417,6 +419,7 @@ can be obtained via a set of rCCD equations, where one substitutes in Eq.~\eqref
 Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost, and provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.
 
 Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excitation energies.
+
 However, there is a significant difference 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
 It is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations.
 Connections between approximate IP- and 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 and we do not decouple the 2h1p and 2p1h sectors.
@@ -426,6 +429,7 @@ Connections between approximate IP- and EA-EOM-CC schemes and the $GW$ approxima
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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}
+\label{eq:GWlin}
 	\begin{pmatrix}
 		\be{}{}						&	\bV{}{\text{2h1p}}		&	\bV{}{\text{2p1h}}	\\
 		\T{(\bV{}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO					\\
@@ -444,7 +448,7 @@ As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the non-linear $GW$ equa
 		\bY{}{\text{2p1h}}	\\
 	\end{pmatrix}
 	\cdot
-	\bOm{}{}
+	\bom{}{}
 \end{equation}
 where $\be{}{}$ is a diagonal matrix gathered the orbital energies, the 2h1p and 2p1h matrix elements are
 \begin{subequations}
@@ -462,6 +466,8 @@ and the corresponding coupling blocks read
 \end{align}
 Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}
 
+Let us suppose that we are looking for the $N$ ``principal'' (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}. 
+Therefore, $\bX{}{}$ and $\bom{}{}$ are of size $N \times N$, and we can safely define $\bX{}{-1}$.
 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}
@@ -484,8 +490,7 @@ Introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}
 	\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.
+with $\bR{}{} = \bX{}{} \cdot \bom{}{} \cdot \bX{}{-1}$.
 
 This yields the three following equations
 \begin{subequations}
@@ -570,7 +575,8 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $
 	\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}.
+The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equation for each value of $p$ separately.
+
 
 %======================
 %\subsection{EE-EOM-CCD}