new values of Req

BSE-PES.tex

@@ -108,6 +108,7 @@
 %% bold in Table
 \newcommand{\bb}[1]{\textbf{#1}}
 \newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
+\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
 
 % excitation energies
 \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
@@ -303,15 +304,15 @@ where the excitation amplitudes are
 	\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
 \end{align}
 \end{subequations}
-With the Mulliken notation for the bare two-electron integrals
+With Mulliken's notation of the bare two-electron integrals
 \begin{equation}
 	\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
 \end{equation}
- and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$ 
+and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$ 
 \begin{equation}
 	\W{pq,rs}{\IS} = \iint  \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}')  \dbr{} \dbr{}',
 \end{equation}
-the $(\bA{\IS},\bB{\IS})$  BSE matrix elements read:
+the BSE matrix elements read
 \begin{subequations}
 \begin{align}
 	\label{eq:LR_BSE-A}
@@ -328,20 +329,19 @@ In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is buil
 \label{eq:wrpa}
 \begin{align}
 	\W{}{\IS}(\br{},\br{}') 
-	& = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}}, 
+	& = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , 
 	\\ 
 	\epsilon_{\IS}(\br{},\br{}'; \omega) 
-	& = \delta(\br{}-\br{}') - \IS  \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}},
+	& = \delta(\br{}-\br{}') - \IS  \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
 \end{align}
 \end{subequations}
-with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability.  In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real molecular orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
+with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability.  In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
 \begin{multline}
 \label{eq:W}
 	\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} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}),
 \end{multline}
-
 where the spectral weights at coupling strength $\IS$ read
 \begin{equation}
 	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
@@ -359,9 +359,10 @@ In Eq.~\eqref{eq:W},  $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
 \end{subequations}
 where $\eHF{p}$ are the HF orbital energies.
 
-The relation between the BSE formalism and the well-known RPAx approach can be obtained by switching off the screening  
+The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening  
 %namely setting  $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ 
-so that   $\W{}{\IS}$ reduces to the bare Coulomb potential. In that limit,  the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: 
+so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$. 
+In this limit,  the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: 
 \begin{subequations}
 \begin{align}
 	\label{eq:LR_RPAx-A}
@@ -412,10 +413,10 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
 	\end{pmatrix}
 \end{equation} 
 is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
-Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
+Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been named ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
 Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies.
 Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added.
-However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. 
+However, as commonly done within RPA and RPAx, \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. 
 
 Equation \eqref{eq:EcBSE} can also be straightforwardly applied to RPA and RPAx, the only difference being the expressions of $\bA{\IS}$ and $\bB{\IS}$ used to obtain the eigenvectors $\bX{\IS}$ and $\bY{\IS}$ entering the definition of $\bP{\IS}$ [see Eq.~\eqref{eq:2DM}].
 For RPA, these expressions have been provided in Eqs.~\eqref{eq:LR_RPA-A} and \eqref{eq:LR_RPA-B}, and their RPAx analogs in Eqs.~\eqref{eq:LR_RPAx-A} and \eqref{eq:LR_RPAx-B}.
@@ -468,6 +469,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
 \caption{
 Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. 
 The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
+The values in parenthesis have been obtained via a Morse fit of the PES.
 }
 \label{tab:Req}
 	
@@ -490,17 +492,17 @@ The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold
 					&	cc-pVTZ		&	1.393	&	3.004	&	2.968	&	2.405	&	2.095	&	2.144	&	2.383	&	2.636	\\
 					&	cc-pVQZ		&	1.391	&	3.008	&	2.970	&	2.395	&	2.091	&	2.137	&	2.382	&	2.634	\\
 	BSE@{\GOWO}@HF	&	cc-pVDZ		&	1.437	&	3.042	&	3.000	&	2.454	&	2.107	&	2.153	&	2.407	&	2.700	\\
-					&	cc-pVTZ		&	1.404	& 	3.023	&			&	2.410	&	2.068	&	2.116	&			&			\\
-					&	cc-pVQZ		&\rb{1.399}	&\rb{3.017}	&\rb{}		&\rb{}		&\rb{}		&\rb{}		&\rb{}		&\rb{}		\\
+					&	cc-pVTZ		&	1.404	& 	3.023	&	(2.982)	&	2.410	&	2.068	&	2.116	&	(2.389)	&	(2.647)	\\
+					&	cc-pVQZ		&\rb{1.399}	&\rb{3.017}	&\rb{(2.974)}&\rb{(2.408)}&\rb{(2.070)}&\rb{(2.130)}&\rb{(2.383)}&\rb{(2.640)}	\\
 	RPA@{\GOWO}@HF	&	cc-pVDZ		&	1.426	&	3.019	&	2.994	&	2.436	&	2.083	&	2.144	&	2.403	&	2.691	\\
-					&	cc-pVTZ		&	1.388	&	3.013	&			&	2.408	&	2.065	&	2.114	&			&			\\
-					&	cc-pVQZ		&	1.382	&	3.013	&			&			&			&			&			&			\\
+					&	cc-pVTZ		&	1.388	&	3.013	&	(2.965)	&	2.408	&	2.065	&	2.114	&	(2.370)	&	(2.584)	\\
+					&	cc-pVQZ		&	1.382	&	3.013	&	(2.965)	&	(2.389)	&	(2.045)	&	(2.110)	&	(2.367)	&	(2.571)	\\
 	RPAx@HF			&	cc-pVDZ		&	1.428	&	3.040	&	2.998	&	2.424	&	2.077	&	2.130	&	2.417	&	2.611	\\
-					&	cc-pVTZ		&	1.395	&	3.003	&	2.971	&	2.400	&	2.046	&	2.110	&	2.368	&	2.568	\\
-					&	cc-pVQZ		&	1.394	&	3.011	&			&			&			&			&			&			\\
+					&	cc-pVTZ		&	1.395	&	3.003	&\gb{2.971}	&	2.400	&	2.046	&	2.110	&	2.368	&	2.568	\\
+					&	cc-pVQZ		&	1.394	&	3.011	&	(2.943)	&	(2.393)	&	(2.041)	&	(2.105)	&	(2.367)	&	(2.563)	\\
 	RPA@HF			&	cc-pVDZ		&	1.431	&	3.021	&	2.999	&	2.424	&	2.083	&	2.134	&	2.416	&	2.623	\\
 					&	cc-pVTZ		&	1.388	&	2.978	&	2.939	&	2.396	&	2.045	&	2.110	&	2.362	&	2.579	\\
-					&	cc-pVQZ		&	1.386	&	2.994	&			&			&			&			&			&			\\
+					&	cc-pVQZ		&	1.386	&	2.994	&	(2.946)	&	(2.385)	&	(2.042)	&	(2.104)	&	(2.365)	&	(2.571)	\\
 % FROZEN CORE VERSION
 %	Method			&	Basis		&	\ce{H2}	&	\ce{LiH}&	\ce{LiF}&	\ce{N2}	&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	&	\ce{HCl}\\
 %	\hline
@@ -551,7 +553,7 @@ Here again, the BSE@{\GOWO}@HF equilibrium bond length (obtained with cc-pVQZ) i
 	\includegraphics[width=0.49\linewidth]{LiH_GS_VQZ}
 \caption{
 Ground-state PES of \ce{H2} (left) and \ce{LiH} (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}.
+%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-H2-LiH}
 }
 \end{figure*}
@@ -572,7 +574,7 @@ Note that these irregularities would be genuine discontinuities in the case of {
 	\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 cc-pVQZ basis set. 
-Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
+%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-LiF-HCl}
 }
 \end{figure*}
@@ -588,7 +590,7 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown
 	\includegraphics[height=0.26\linewidth]{BF_GS_VQZ}
 \caption{
 Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (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}.
+%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-N2-CO-BF}
 }
 \end{figure*}
@@ -603,7 +605,7 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
 	\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 cc-pVQZ basis set. 
-Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
+%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-F2}
 }
 \end{figure}