clean up last section of theory (more to come)

BSE-PES.tex

@@ -352,7 +352,7 @@ In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral (direct) RPA excitation energie
 \subsection{Ground- and excited-state BSE energy}
 \label{sec:BSE_energy}
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-\titou{The key quantity to define here is the total BSE energy as we are going to compute PES following this definition.}
+The key quantity to define in the present context is the total BSE energy.
 Although not unique, we propose to define the BSE total energy of the $m$th state as
 \begin{equation}
 \label{eq:EtotBSE}
@@ -364,11 +364,15 @@ where $\Enuc$ and $\EHF$ are the state-independent nuclear repulsion energy and
 	\EcBSE = \frac{1}{2} \qty[ {\sum_m}' \OmBSE{m} - \Tr(\bA) ]
 \end{equation}
 is the ground-state BSE correlation energy computed with the so-called trace formula, \cite{Schuck_Book, Rowe_1968, Sawada_1957b} and $\OmBSE{m}$ is the $m$th BSE excitation energy with the convention that, for the ground state ($m=0$), $\OmBSE{0} = 0$. 
+An elegant derivation of Eq.~\eqref{eq:EcBSE} has been recently proposed within the BSE formalism by Olevano and coworkers. \cite{Li_2020} 
+Note that, at the RPA level, an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
+These two RPA formulations have been found to be equivalent in practice for both the uniform electron gas \cite{Sawada_1957b, Fukuta_1964, Furche_2008} and in molecules. \cite{Li_2020} 
+However, in the case of BSE, there is no guarantee that the two formalisms (trace \textit{vs} adiabatic connection) yields the same values.
+
 Equation \eqref{eq:EtotBSE} defines unambiguously the total BSE energy of the system for both ground and (singlet and triplet) excited states.
-Note that the prime in the summation of Eq.~\eqref{eq:EcBSE} means that we only consider the \textit{positive} excitation energies from the singlet and triplet manifolds.
+It has also the indisputable advantage of treating on equal footing (\ie, at the same level of theory) the ground state and the excited states.
+Note that the prime in the sum of Eq.~\eqref{eq:EcBSE} means that we only consider in the summation the \textit{positive} excitation energies from the singlet and triplet manifolds.
 Indeed, in case of triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered. 
-Note also that an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
-These two formulation have been found to be equivalent in practice for both the uniform electron gas \cite{Sawada_1957b, Fukuta_1964, Furche_2008} and in molecules. \cite{Li_2020} \xavier{WELL, THIS IS UNCLEAR TO ME : this is true for direct RPA, but is this true for BSE ... I am really not sure ...} 
 
 From a practical point of view, it is also convenient to split $\EcBSE = \EcsBSE + \EctBSE$ in its singlet ($\sigma \neq \sigma^{\prime}$) and triplet ($\sigma = \sigma^{\prime}$) contributions, \ie,
 \begin{equation}
@@ -376,7 +380,7 @@ From a practical point of view, it is also convenient to split $\EcBSE = \EcsBSE
 	= \underbrace{\frac{1}{2} \qty[ {\sum_m} \OmsBSE{m} - \Tr(\bAs) ]}_{\EcsBSE}
 	+ \underbrace{\frac{1}{2} \qty[ {\sum_m}'  \OmtBSE{m} - \Tr(\bAt) ]}_{\EctBSE}.
 \end{equation}
-As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, RPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the different level of theory.
+As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, RPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the these levels of theory.
 
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 \section{Computational details}