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Pierre-Francois Loos 2020-02-03 16:21:06 +01:00
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BSE-PES.tex
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@ -333,19 +333,19 @@ In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is bu
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}
\W{ij,ab}{\IS}(\omega) = \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:
where the 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}
\label{eq:LR_RPA}
\begin{align}
\label{eq:LR_RPA}
\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{jb},
\\
\BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{bj},
@ -357,8 +357,8 @@ The relation between the BSE formalism and the well-known RPAx approach can be o
%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}
\label{eq:LR_RPAx}
\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].