Merge branch 'master' into overleaf-2020-02-03-1521

BSE-PES.tex

@@ -315,9 +315,9 @@ the $(\bA{\IS},\bB{\IS})$  BSE matrix elements read:
 \begin{subequations}
 \label{eq:LR_BSE}
 \begin{align}
-	\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \left[ \ERI{ia}{jb}   -  \W{ij,ab}{\IS} \right],
+	\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb}   -  \W{ij,ab}{\IS} ],
 	\\ 
-	\BBSE{ia,jb}{\IS} & =  \lambda \left[ \ERI{ia}{bj} -   \W{ib,aj}{\IS} \right],
+	\BBSE{ia,jb}{\IS} & =  \lambda \qty[ 2 \ERI{ia}{bj} -   \W{ib,aj}{\IS} ],
 \end{align}
 \end{subequations}
 where $\eGW{p}$ are the $GW$ quasiparticle energies. 
@@ -333,22 +333,24 @@ In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is bu
 with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$ and $\chi_{0}$ the non-interacting polarizability.  In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $W^{\lambda}$ can be written as follows in the case of real molecular orbitals: \cite{complexw}
 \begin{multline}
 \label{eq:W}
-	\W{ij,ab}{\IS}(\omega) =  \ERI{ij}{ab} +  \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
+	\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} +  2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
 	\\
-	\times  \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} \textcolor{red}{-} \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta})
+	\times  \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}),
 \end{multline}
-where the  spectral weights $\sERI{pq}{m}$  at coupling strength $\lambda$ read:
+
+where the spectral weights at coupling strength $\lambda$ read
 \begin{equation}
-	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
+	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
 \end{equation}
 
 In Eq.~\eqref{eq:W},  $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
 \begin{subequations}
 \label{eq:LR_RPA}
 \begin{align}
-	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) +   \IS \ERI{ia}{jb},
+	\label{eq:LR_RPA}
+	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) +   2 \IS \ERI{ia}{jb},
 	\\ 
-	\BRPA{ia,jb}{\IS} & =   \IS \ERI{ia}{bj},
+	\BRPA{ia,jb}{\IS} & =   2 \IS \ERI{ia}{bj},
 \end{align}
 \end{subequations}
 where $\eHF{p}$ are the HF orbital energies.
@@ -359,15 +361,14 @@ so that   $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit,  t
 \begin{subequations}
 \label{eq:LR_RPAx}
 \begin{align}
-	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \left[ \ERI{ia}{jb} -    \ERI{ij}{ab} \right],
+	\label{eq:LR_RPAx}
+	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} -    \ERI{ij}{ab} ],
 	\\ 
-	\BRPAx{ia,jb}{\IS} & =  \IS \left[ \ERI{ia}{bj} -   \ERI{ib}{aj} \right].
+	\BRPAx{ia,jb}{\IS} & =  \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ].
 \end{align}
 \end{subequations}
 %This allows to understand that the strength parameter $\lambda$ enters twice in the $\lambda W^{\lambda}$ contribution, one time to renormalize the screening efficiency, and a second  time to renormalize the direct electron-hole interaction.  
 
-
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\subsection{Ground-state BSE energy}
 %\label{sec:BSE_energy}
@@ -564,9 +565,9 @@ Note that these irregularities would be genuine discontinuities in the case of {
 %%% FIG 2 %%%
 \begin{figure*}
 	\includegraphics[height=0.35\linewidth]{LiF_GS_VQZ}
-	\includegraphics[height=0.35\linewidth]{HCl_GS_VTZ}
+	\includegraphics[height=0.35\linewidth]{HCl_GS_VQZ}
 \caption{
-Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set. 
+Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVQZ basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-LiF-HCl}
 }
@@ -589,15 +590,15 @@ Additional graphs for other basis sets and within the frozen-core approximation
 \end{figure*}
 %%% %%% %%%
 
-The \ce{F2} molecule is a notoriously difficult case to treat due to the relative weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence the relatively long equilibrium bond length observed for \ce{F2}.
+The \ce{F2} molecule is a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
 Similarly to what we have observed for \ce{LiF} and \ce{BF}, there is an irregularities near the minimum of the {\GOWO}-based curves.
 However, BSE@{\GOWO}@HF is the closest to the CC3 curve
 
 %%% FIG 4 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{F2_GS_VTZ}
+	\includegraphics[width=\linewidth]{F2_GS_VQZ}
 \caption{
-Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set. 
+Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVQZ basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-F2}
 }