loos/BSE-PES
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

labels

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-02-03 16:33:24 +01:00
parent fbea44ea73
commit c3a1ae1d7b
1 changed files with 9 additions and 6 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

15
BSE-PES.tex
View File

@ -313,10 +313,11 @@ With the Mulliken notation for the bare two-electron integrals
\end{equation} \end{equation}
the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read: the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read:
\begin{subequations} \begin{subequations}
\label{eq:LR_BSE}
\begin{align} \begin{align}
\label{eq:LR_BSE-A}
\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \W{ij,ab}{\IS} ], \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} & = \lambda \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ],
\end{align} \end{align}
\end{subequations} \end{subequations}
@ -345,10 +346,11 @@ where the spectral weights at coupling strength $\lambda$ read
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 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{subequations}
\label{eq:LR_RPA}
\begin{align} \begin{align}
\label{eq:LR_RPA-A}
\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb}, \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb},
\\ \\
\label{eq:LR_RPA-B}
\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj}, \BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj},
\end{align} \end{align}
\end{subequations} \end{subequations}
@ -356,12 +358,13 @@ 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 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 $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
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: 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:
\begin{subequations} \begin{subequations}
\label{eq:LR_RPAx}
\begin{align} \begin{align}
\label{eq:LR_RPAx-A}
\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \ERI{ij}{ab} ], \ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \ERI{ij}{ab} ],
\\ \\
\label{eq:LR_RPAx-B}
\BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ]. \BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ].
\end{align} \end{align}
\end{subequations} \end{subequations}
@ -412,9 +415,9 @@ Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from t
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 (\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}]. 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 in Eq.~\eqref{eq:LR_RPAx}. 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}.
In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively. 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. Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA-A} by the $GW$ quasiparticles energies.
Several important comments are in order here. Several important comments are in order here.
For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be classified as weakly correlated. For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be classified as weakly correlated.