VQZ

BSE-PES.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-01-28 17:57:57 +0100 
+%% Created for Pierre-Francois Loos at 2020-02-03 16:35:45 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -13050,22 +13050,15 @@
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113},
 	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
 
-%%%% Xavier
-
-@misc{complexw,
- note = {In the case of complex molecular orbitals, see Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$. }
-}
-
 @article{Holzer_2019,
-author = {Holzer,Christof  and Teale,Andrew M.  and Hampe,Florian  and Stopkowicz,Stella  and Helgaker,Trygve  and Klopper,Wim },
+	Author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim},
-title = {GW quasiparticle energies of atoms in strong magnetic fields},
+	Doi = {10.1063/1.5093396},
-journal = { J. Chem. Phys. },
+	Eprint = {https://doi.org/10.1063/1.5093396},
-volume = {150},
+	Journal = {J. Chem. Phys.},
-number = {21},
+	Number = {21},
-pages = {214112},
+	Pages = {214112},
-year = {2019},
+	Title = {GW quasiparticle energies of atoms in strong magnetic fields},
-doi = {10.1063/1.5093396},
+	Url = {https://doi.org/10.1063/1.5093396},
-URL = {   https://doi.org/10.1063/1.5093396},
+	Volume = {150},
-eprint = {  https://doi.org/10.1063/1.5093396}
+	Year = {2019},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.5093396}}
-

BSE-PES.tex

@@ -307,9 +307,9 @@ With the Mulliken notation for 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 $\lambda$ 
+ 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^{\lambda}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}')  \dbr{} \dbr{}',
+	\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:
 \begin{subequations}
@@ -318,20 +318,23 @@ the $(\bA{\IS},\bB{\IS})$  BSE matrix elements read:
 	\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb}   -  \W{ij,ab}{\IS} ],
 	\\ 
 	\label{eq:LR_BSE-B}
-	\BBSE{ia,jb}{\IS} & =  \lambda \qty[ 2 \ERI{ia}{bj} -   \W{ib,aj}{\IS} ],
+	\BBSE{ia,jb}{\IS} & =  \IS \qty[ 2 \ERI{ia}{bj} -   \W{ib,aj}{\IS} ],
 \end{align}
 \end{subequations}
 where $\eGW{p}$ are the $GW$ quasiparticle energies. 
-%In the standard BSE implementation,  the screened Coulomb potential $W^{\lambda}$ is taken to be static $(\omega \rightarrow 0)$. 
+%In the standard BSE implementation,  the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$. 
-In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is built within the direct RPA scheme:  
+In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is built within the direct RPA scheme:  
 \begin{subequations}
 \label{eq:wrpa}
 \begin{align}
-	W^{\lambda}({\bf r},{\bf r}') &= \int d{\bf r}_1 \; \frac{\epsilon_{\lambda}^{-1}({\bf r},{\bf r}_1; \omega=0) } {   |{\bf r}_1-{\bf r}' | }, \\ 
+	\W{}{\IS}(\br{},\br{}') 
-	\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) &= \delta({\bf r}-{\bf r}') - \lambda  \int d{\bf r}_1 \; \frac{   \chi_{0}({\bf r},{\bf r}_1; \omega) }{ |{\bf r}_1 - {\bf r}'| },
+	& = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}}, 
+	\\ 
+	\epsilon_{\IS}(\br{},\br{}'; \omega) 
+	& = \delta(\br{}-\br{}') - \IS  \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}},
 \end{align}
 \end{subequations}
-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}
+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 $W$.}
 \begin{multline}
 \label{eq:W}
 	\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} +  2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
@@ -339,7 +342,7 @@ with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$
 	\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 $\lambda$ read
+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}.
 \end{equation}
@@ -357,8 +360,8 @@ In Eq.~\eqref{eq:W},  $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
 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  
-%namely setting  $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ 
+%namely setting  $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ 
-so that   $W^{\lambda}$ 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. 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: 
 \begin{subequations}
 \begin{align}
 	\label{eq:LR_RPAx-A}
@@ -368,7 +371,7 @@ so that   $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit,  t
 	\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.  
+%This allows to understand that the strength parameter $\IS$ enters twice in the $\IS W^{\IS}$ contribution, one time to renormalize the screening efficiency, and a second  time to renormalize the direct electron-hole interaction.  
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\subsection{Ground-state BSE energy}
@@ -582,9 +585,9 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown
 \begin{figure*}
 	\includegraphics[height=0.26\linewidth]{N2_GS_VQZ}
 	\includegraphics[height=0.26\linewidth]{CO_GS_VQZ}
-	\includegraphics[height=0.26\linewidth]{BF_GS_VTZ}
+	\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 \titou{cc-pVQZ} basis set. 
+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}.
 \label{fig:PES-N2-CO-BF}
 }