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Pierre-Francois Loos 2020-02-03 16:33:24 +01:00
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BSE-PES.tex
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@ -313,10 +313,11 @@ With the Mulliken notation for the bare two-electron integrals
\end{equation}
the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read:
\begin{subequations}
\label{eq:LR_BSE}
\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} ],
\\
\label{eq:LR_BSE-B}
\BBSE{ia,jb}{\IS} & = \lambda \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ],
\end{align}
\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
\begin{subequations}
\label{eq:LR_RPA}
\begin{align}
\label{eq:LR_RPA-A}
\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},
\end{align}
\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
%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}
\label{eq:LR_RPAx}
\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} ],
\\
\label{eq:LR_RPAx-B}
\BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ].
\end{align}
\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.
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.
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.
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.