minor

BSE-PES.tex

@@ -63,7 +63,7 @@
 \newcommand{\Ec}{E_\text{c}}
 \newcommand{\EHF}{E^\text{HF}}
 \newcommand{\EBSE}[1]{E_{#1}^\text{BSE}}
-\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
+\newcommand{\EcRPA}{E_\text{c}^\text{dRPA}}
 \newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
 \newcommand{\EcsBSE}{{}^1\EcBSE}
 \newcommand{\EctBSE}{{}^3\EcBSE}
@@ -236,7 +236,7 @@ The paper is organized as follows.
 In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
 In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}.
 Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
-In Sec.~\ref{sec:comp_details}, computational details are reported.
+Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
 Section \ref{sec:PES} reports PES of the ground- and excited-states for various diatomic molecules. 
 Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
 
@@ -346,7 +346,7 @@ are the bare two-electron integrals, $\delta_{pq}$ is the Kronecker delta, and
 		1,	&	\sigma = \sigma^{\prime} \text{ (triplet manifold)}.
 	\end{cases}
 \end{equation}
-In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral (direct) RPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal.
+In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral direct (\ie, without exchange) dRPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Ground- and excited-state BSE energy}
@@ -365,8 +365,8 @@ where $\Enuc$ and $\EHF$ are the state-independent nuclear repulsion energy and
 \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} 
+Note that, at the dRPA level, an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
+These two dRPA 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.
@@ -380,7 +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 these levels of theory.
+As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, dRPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the these levels of theory.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}