Done with IIIA

Manuscript/EPAWTFT.tex

@@ -694,22 +694,9 @@ slowly convergent, or catastrophically divergent results.\cite{Gill_1986,Gill_19
 Furthermore, the convergence properties of the MP series can depend strongly on the choice of restricted or
 unrestricted reference orbitals.
 
-% HGAB: I don't think this parapgrah tells us anything we haven't discussed before
-%A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable. 
-%By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$, 
-%and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions).
-%As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series. 
-%These singularities of the energy function are exactly the EPs connecting the electronic states as mentioned in Sec.~\ref{sec:intro}. 
-%The direct computation of the terms of the series is quite manageable up to fourth order in perturbation, while the fifth and sixth order in perturbation can still be obtained but at a rather high cost. \cite{JensenBook}
-%In order to better understand the behaviour of the MP series and how it is connected to the singularity structure, we have to access high-order terms. 
-%For small systems, one can access the whole terms of the series using full configuration interaction (FCI). 
-%If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI space, one gets the exact energies (in this finite Hilbert space) and the Taylor expansion with respect to $\lambda$ allows to access the MP perturbation series at any order.
-
-
 Although practically convenient for electronic structure calculations, the MP partitioning is not 
-the only possibility and alternative partitionings have been considered including: %proposed in the literature:
+the only possibility and alternative partitionings have been considered including: 
 i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926} 
-%Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
 ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and 
 iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
 While an in-depth comparison of these different approaches can offer insight into