modif xav

BSE-PES.bib

@@ -13049,3 +13049,23 @@
 	Year = {2016},
 	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 },
+title = {GW quasiparticle energies of atoms in strong magnetic fields},
+journal = { J. Chem. Phys. },
+volume = {150},
+number = {21},
+pages = {214112},
+year = {2019},
+doi = {10.1063/1.5093396},
+URL = {   https://doi.org/10.1063/1.5093396},
+eprint = {  https://doi.org/10.1063/1.5093396}
+}
+

BSE-PES.tex

@@ -303,42 +303,71 @@ 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}
-In the case of BSE, the specific expression of the matrix elements are
+With the Mulliken notation for the bare two-electron integrals
-\begin{subequations}
-\begin{align}
-	\label{eq:LR_BSE}
-	\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \W{ia,bj}{\IS}(\omega = 0)  - \lambda \ERI{ia}{jb},
-	\\ 
-	\BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ia}{bj} ,
-\end{align}
-\end{subequations}
-where $\eGW{p}$ are the $GW$ quasiparticle energies, 
-\begin{multline}
-\label{eq:W}
-	\W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb} 
-	\\
-	+ \sum_m^{\Nocc \Nvir} \sERI{ia}{m} \sERI{jb}{m} \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} + \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta})
-\end{multline}
-are the elements of the screened Coulomb operator, 
-\begin{equation}
-	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
-\end{equation}
-are the screened 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}
-are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
+ and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\lambda$ 
+\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{}',
+\end{equation}
+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],
+	\\ 
+	\BBSE{ia,jb}{\IS} & =  \lambda \left[ \ERI{ia}{bj} -   \W{ib,aj}{\IS} \right],
+\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 approach, the screened Coulomb potential $W^{\lambda}$ 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}' | }, \\ 
+	\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}'| },
+\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}
+\begin{multline}
+\label{eq:W}
+	\W{ij,ab}{\IS}(\omega) = \textcolor{red}{\sout{2}} \ERI{ij}{ab} +  \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})
+\end{multline}
+where the \xavier{ \sout{screened two-electron integrals}  spectral weights} $\sERI{pq}{m}$  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}
+\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}
 \begin{align}
 	\label{eq:LR_RPA}
-	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj},
+	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) +   \IS \ERI{ia}{jb},
 	\\ 
-	\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb},
+	\BRPA{ia,jb}{\IS} & =   \IS \ERI{ia}{bj},
 \end{align}
 \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  
+%namely setting  $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ 
+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.~\ref{eq:LR_BSE} to the RPAx equations: 
+\begin{subequations}
+\begin{align}
+	\label{eq:LR_RPAx}
+	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \left[ \ERI{ia}{jb} -    \ERI{ij}{ab} \right],
+	\\ 
+	\BRPAx{ia,jb}{\IS} & =  \IS \left[ \ERI{ia}{bj} -   \ERI{ib}{aj} \right].
+\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}
@@ -378,21 +407,13 @@ 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 be 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 labeled ``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. 
 
 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 Eq.~\eqref{eq:LR_RPA}, and their RPAx analogs read
+For RPA, these expressions have been provided in Eq.~\eqref{eq:LR_RPA}, and their RPAx analogs in Eq.~\eqref{eq:LR_RPAx}.
-\begin{subequations}
-\begin{align}
-	\label{eq:LR_RPAx}
-	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj} -  \IS \ERI{ia}{jb},
-	\\ 
-	\BRPAx{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb} - \IS \ERI{ia}{bj}.
-\end{align}
-\end{subequations}
 In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively.
 Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA} by the $GW$ quasiparticles energies.