new graphs

This commit is contained in:
Pierre-Francois Loos 2020-02-03 16:19:33 +01:00
parent d77017e8d5
commit 9a9da4873b
9 changed files with 14 additions and 16 deletions

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@ -315,9 +315,9 @@ 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],
\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \W{ij,ab}{\IS} ],
\\
\BBSE{ia,jb}{\IS} & = \lambda \left[ \ERI{ia}{bj} - \W{ib,aj}{\IS} \right],
\BBSE{ia,jb}{\IS} & = \lambda \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ],
\end{align}
\end{subequations}
where $\eGW{p}$ are the $GW$ quasiparticle energies.
@ -333,22 +333,22 @@ 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} + 2 \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})
\times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \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 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}
\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}) + \IS \ERI{ia}{jb},
\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb},
\\
\BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{bj},
\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj},
\end{align}
\end{subequations}
where $\eHF{p}$ are the HF orbital energies.
@ -359,15 +359,13 @@ so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, t
\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],
\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \ERI{ij}{ab} ],
\\
\BRPAx{ia,jb}{\IS} & = \IS \left[ \ERI{ia}{bj} - \ERI{ib}{aj} \right].
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Ground-state BSE energy}
%\label{sec:BSE_energy}
@ -564,9 +562,9 @@ Note that these irregularities would be genuine discontinuities in the case of {
%%% FIG 2 %%%
\begin{figure*}
\includegraphics[height=0.35\linewidth]{LiF_GS_VQZ}
\includegraphics[height=0.35\linewidth]{HCl_GS_VTZ}
\includegraphics[height=0.35\linewidth]{HCl_GS_VQZ}
\caption{
Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set.
Ground-state PES of \ce{LiF} (left) and \ce{HCl} (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-LiF-HCl}
}
@ -595,9 +593,9 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{F2_GS_VTZ}
\includegraphics[width=\linewidth]{F2_GS_VQZ}
\caption{
Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set.
Ground-state PES of \ce{F2} around its 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-F2}
}

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