diff --git a/BF_GS_VTZ.pdf b/BF_GS_VTZ.pdf index 8e45c2e..b614af2 100644 Binary files a/BF_GS_VTZ.pdf and b/BF_GS_VTZ.pdf differ diff --git a/BSE-PES.tex b/BSE-PES.tex index 98224ed..38b8f03 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -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 +333,24 @@ 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) = \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 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} \label{eq:LR_RPA} \begin{align} - \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{jb}, + \label{eq:LR_RPA} + \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 +361,14 @@ so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, t \begin{subequations} \label{eq:LR_RPAx} \begin{align} - \ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \left[ \ERI{ia}{jb} - \ERI{ij}{ab} \right], + \label{eq:LR_RPAx} + \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 +565,9 @@ 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} } @@ -589,15 +590,15 @@ Additional graphs for other basis sets and within the frozen-core approximation \end{figure*} %%% %%% %%% -The \ce{F2} molecule is a notoriously difficult case to treat due to the relative weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence the relatively long equilibrium bond length observed for \ce{F2}. +The \ce{F2} molecule is a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level). Similarly to what we have observed for \ce{LiF} and \ce{BF}, there is an irregularities near the minimum of the {\GOWO}-based curves. 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} } diff --git a/CO_GS_VQZ.pdf b/CO_GS_VQZ.pdf index cecba64..dd129ed 100644 Binary files a/CO_GS_VQZ.pdf and b/CO_GS_VQZ.pdf differ diff --git a/F2_GS_VQZ.pdf b/F2_GS_VQZ.pdf new file mode 100644 index 0000000..6982878 Binary files /dev/null and b/F2_GS_VQZ.pdf differ diff --git a/F2_GS_VTZ.pdf b/F2_GS_VTZ.pdf deleted file mode 100644 index bd7063e..0000000 Binary files a/F2_GS_VTZ.pdf and /dev/null differ diff --git a/H2_GS_VQZ.pdf b/H2_GS_VQZ.pdf index 78c23de..f0678ab 100644 Binary files a/H2_GS_VQZ.pdf and b/H2_GS_VQZ.pdf differ diff --git a/HCl_GS_VQZ.pdf b/HCl_GS_VQZ.pdf new file mode 100644 index 0000000..80f4051 Binary files /dev/null and b/HCl_GS_VQZ.pdf differ diff --git a/HCl_GS_VTZ.pdf b/HCl_GS_VTZ.pdf deleted file mode 100644 index a5bab9d..0000000 Binary files a/HCl_GS_VTZ.pdf and /dev/null differ diff --git a/LiF_GS_VQZ.pdf b/LiF_GS_VQZ.pdf index 18695a2..1c87260 100644 Binary files a/LiF_GS_VQZ.pdf and b/LiF_GS_VQZ.pdf differ diff --git a/LiH_GS_VQZ.pdf b/LiH_GS_VQZ.pdf index b05739c..3aae5ac 100644 Binary files a/LiH_GS_VQZ.pdf and b/LiH_GS_VQZ.pdf differ diff --git a/N2_GS_VQZ.pdf b/N2_GS_VQZ.pdf index f19efcc..20dffc3 100644 Binary files a/N2_GS_VQZ.pdf and b/N2_GS_VQZ.pdf differ