saving work in intro

Manuscript/Ec.tex

@@ -70,31 +70,27 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
 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 nuclei coordinates can then be treated as parameters in the electronic Hamiltonian.
 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.
+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 $\Hat{T} = \sum_{k=1}^n \Hat{T}_k$ (where $n$ is the number of electrons).
 
-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.
+The truncation of $\Hat{T}$ allows to define a 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} with corresponding computational scalings of $\order*{N^{6}}$, $\order*{N^{8}}$, and $\order*{N^{10}}$, respectively (where $N$ denotes the number of orbitals).
+Parallel to the ``complete'' CC series presented above, an alternative series of approximate iterative CC models have been developed by the Aarhus group in the context of CC response theory \cite{Christiansen_1998} where one skips the most expensive terms and avoids the storage of the higher-excitation amplitudes:  CC2, \cite{Christiansen_1995a} CC3, \cite{Christiansen_1995b,Koch_1997} and CC4 \cite{Kallay_2005} 
+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.
+
+A similar systematic truncation strategy can be applied to CI methods leading to the well-established family of methods known as CISD, CISDT, CISDTQ, \ldots~where one systematically increases the maximum excitation degree of the determinants taken into account. 
+Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $N$) are considered, truncated CI methods are variational but lack size-consistency.
+The non-variationality of truncated CC methods being less of an issue than the size-inconsistency of the truncated CI methods, the formers have naturally overshadowed the latters in the electronic structure landscape.
+However, a different strategy has recently made a come back in the context of CI methods.
+Indeed, selected CI (SCI) methods where one iteratively selects the energetically relevant determinants from the FCI space has been recently shown to be highly successful to produce reference energies for ground and excited states in small- and medium-size molecules.
 
 
-
-M{\o}ller-Plesset pertrubation theory \cite{Moller_1934}
+A rather different strategy in order to reach the holy grail FCI limit is to resort to M{\o}ller-Plesset perturbation theory. \cite{Moller_1934}
+Again, at least in theory, one can obtain the exact energy of the system by ramping up the degree of the pertrubative series.
+The second-order  M{\o}ller-Plesset (MP2) method is very well known but its higher-order 
 MP3 \cite{Pople_1976}
 MP4 \cite{Krishnan_1980}
 MP5 \cite{Kucharski_1989}