savinf work in intro

Manuscript/Ec.tex

@@ -42,6 +42,8 @@
 \affiliation{\LCPQ}
 \author{Anthony Scemama}
 \affiliation{\LCPQ}
+\author{Michel Caffarel}
+\affiliation{\LCPQ}
 \author{Denis Jacquemin}
 \affiliation{\CEISAM}
 \author{Pierre-Fran\c{c}ois Loos}
@@ -62,16 +64,62 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
 % Title
 \maketitle
 
+%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+Electronic structure theory relies heavily on approximations.
+Loosely speaking, to make any theory useful, three main approximations must be enforced.
+The first fundamental approximation, known as the Born-Oppenheimer approximation, usually consists in assuming that the motion of nuclei and electrons are decoupled.
+The second central approximation which makes calculations feasable by a computer is the basis set approximation where one introduces a set of pre-defined basis functions to represent the many-electron wave function of the system.
+In most molecular calculations, a set of one-electron, atom-centered gaussian basis functions are introduced to expand the so-called one-electron molecular orbitals which are then used to build the many-electron Slater determinants.
+The third and most relevant approximation in the present context is the ansatz (or form) of the electronic wave function $\Psi$.
+For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator
+\begin{equation}
+	\Hat{T} = \sum_{k=1}^N \Hat{T}_k
+\end{equation}
+where $N$ is the number of electrons.
+The truncation of $\Hat{T}$ defines as well-defined hierarchy of non-variational and size-extensive methods with improved accuracy.
+CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} 
+CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} 
+CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} and 
+with corresponding computational scalings of $\order*{N^{6}}$, $\order*{N^{8}}$, and $\order*{N^{10}}$, respectively (where $N$ denotes the number of orbitals).
+CC2, \cite{Christiansen_1995a} 
+CC3, \cite{Christiansen_1995b,Koch_1997} and 
+CC4 \cite{Kallay_2005} series of models which have been introduced by the Aarhus group in the context of CC response theory. \cite{Christiansen_1998}
+These iterative methods scale as $\order*{N^{5}}$, $\order*{N^{7}}$, and $\order*{N^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ, by skipping the most expensive terms and avoiding the storage of the higher-excitation amplitudes.
 
+A similar systematic truncation strategy can be applied to CI methods leading to alternative family of methods known as CISD, CISDT, CISDTQ, \ldots~which consists in increasing the maximum excitation degree of the determinants tkaen into account. 
+Except for full CI for which all determinants with excitation degree up to $N$ are taken into account, truncated CI methods are variational but lack size-consistency.
+The non-variationnality of CC methods being less of an issue than the size-inconsistency of the CI methods, it is fair to say that truncated CC methods have naturally overshadowed truncated CI methods in the electronic structure landscape.
+
+
+
+M{\o}ller-Plesset pertrubation theory \cite{Moller_1934}
+MP3 \cite{Pople_1976}
+MP4 \cite{Krishnan_1980}
+MP5 \cite{Kucharski_1989}
+MP6 \cite{He_1996a,He_1996b}
+CCSD(T) \cite{Raghavachari_1989} is the gold-standard
+
+
+
+Reviews. \cite{Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009}
+
+Coupled-cluster methods have been particularly successful for small- and medium-sized molecules  properties
+\cite{Kallay_2003,Kallay_2004a,Gauss_2006,Kallay_2006,Gauss_2009}  
+
+%%% FIG 1 %%%
 \begin{figure*}
 	\includegraphics[width=\linewidth]{mol}
 	\caption{
 	Five-membered rings (top) and six-membered rings (bottom) considered in this study.
 	\label{fig:mol}}
 \end{figure*}
+%%% FIG 1 %%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
+%%%%%%%%%%%%%%%%%%%%%%%%%
 The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of geometry and have been extracted from a previous study. \cite{Loos_2020a}
 The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations have been performed with Cfour, \cite{cfour} while the CCSD(T) and MP5 calculations have been performed in Gaussian 09. \cite{g09}
 For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ) which consists of Hilbert space sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
@@ -161,7 +209,7 @@ We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for
 
 \begin{acknowledgements}
 This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005.
-PFL and AS have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
+PFL, AS, and MC have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
 \end{acknowledgements}
 
 \section*{Data availability statement}