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Pierre-Francois Loos 2020-02-04 08:49:50 +01:00
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11 changed files with 29 additions and 33 deletions

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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-01-28 17:57:57 +0100 %% Created for Pierre-Francois Loos at 2020-02-03 16:35:45 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -13050,22 +13050,15 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113}, Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}} Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
%%%% Xavier
@misc{complexw,
note = {In the case of complex molecular orbitals, see Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$. }
}
@article{Holzer_2019, @article{Holzer_2019,
author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim }, Author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim},
title = {GW quasiparticle energies of atoms in strong magnetic fields}, Doi = {10.1063/1.5093396},
journal = { J. Chem. Phys. }, Eprint = {https://doi.org/10.1063/1.5093396},
volume = {150}, Journal = {J. Chem. Phys.},
number = {21}, Number = {21},
pages = {214112}, Pages = {214112},
year = {2019}, Title = {GW quasiparticle energies of atoms in strong magnetic fields},
doi = {10.1063/1.5093396}, Url = {https://doi.org/10.1063/1.5093396},
URL = { https://doi.org/10.1063/1.5093396}, Volume = {150},
eprint = { https://doi.org/10.1063/1.5093396} Year = {2019},
} Bdsk-Url-1 = {https://doi.org/10.1063/1.5093396}}

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@ -307,9 +307,9 @@ With the Mulliken notation for the bare two-electron integrals
\begin{equation} \begin{equation}
\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
\end{equation} \end{equation}
and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\lambda$ and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$
\begin{equation} \begin{equation}
\W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) W^{\lambda}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}', \W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}',
\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}
@ -318,20 +318,23 @@ the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read:
\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} \label{eq:LR_BSE-B}
\BBSE{ia,jb}{\IS} & = \lambda \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ], \BBSE{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ],
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\eGW{p}$ are the $GW$ quasiparticle energies. where $\eGW{p}$ are the $GW$ quasiparticle energies.
%In the standard BSE implementation, the screened Coulomb potential $W^{\lambda}$ is taken to be static $(\omega \rightarrow 0)$. %In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$.
In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is built within the direct RPA scheme: In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is built within the direct RPA scheme:
\begin{subequations} \begin{subequations}
\label{eq:wrpa} \label{eq:wrpa}
\begin{align} \begin{align}
W^{\lambda}({\bf r},{\bf r}') &= \int d{\bf r}_1 \; \frac{\epsilon_{\lambda}^{-1}({\bf r},{\bf r}_1; \omega=0) } { |{\bf r}_1-{\bf r}' | }, \\ \W{}{\IS}(\br{},\br{}')
\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) &= \delta({\bf r}-{\bf r}') - \lambda \int d{\bf r}_1 \; \frac{ \chi_{0}({\bf r},{\bf r}_1; \omega) }{ |{\bf r}_1 - {\bf r}'| }, & = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}},
\\
\epsilon_{\IS}(\br{},\br{}'; \omega)
& = \delta(\br{}-\br{}') - \IS \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}},
\end{align} \end{align}
\end{subequations} \end{subequations}
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} with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real molecular orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
\begin{multline} \begin{multline}
\label{eq:W} \label{eq:W}
\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \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}
@ -339,7 +342,7 @@ with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$
\times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \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} \end{multline}
where the spectral weights at coupling strength $\lambda$ read where the spectral weights at coupling strength $\IS$ read
\begin{equation} \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} \end{equation}
@ -357,8 +360,8 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
where $\eHF{p}$ are the HF orbital energies. 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 $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
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.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: so that $\W{}{\IS}$ 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}
\begin{align} \begin{align}
\label{eq:LR_RPAx-A} \label{eq:LR_RPAx-A}
@ -368,7 +371,7 @@ so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, t
\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}
%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. %This allows to understand that the strength parameter $\IS$ enters twice in the $\IS W^{\IS}$ contribution, one time to renormalize the screening efficiency, and a second time to renormalize the direct electron-hole interaction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Ground-state BSE energy} %\subsection{Ground-state BSE energy}
@ -582,9 +585,9 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown
\begin{figure*} \begin{figure*}
\includegraphics[height=0.26\linewidth]{N2_GS_VQZ} \includegraphics[height=0.26\linewidth]{N2_GS_VQZ}
\includegraphics[height=0.26\linewidth]{CO_GS_VQZ} \includegraphics[height=0.26\linewidth]{CO_GS_VQZ}
\includegraphics[height=0.26\linewidth]{BF_GS_VTZ} \includegraphics[height=0.26\linewidth]{BF_GS_VQZ}
\caption{ \caption{
Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set. Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (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}. Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-N2-CO-BF} \label{fig:PES-N2-CO-BF}
} }

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