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new values of Req

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Pierre-Francois Loos 2020-02-04 16:17:49 +01:00
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5 changed files with 24 additions and 22 deletions
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BSE-PES.tex
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@ -108,6 +108,7 @@
%% bold in Table %% bold in Table
\newcommand{\bb}[1]{\textbf{#1}} \newcommand{\bb}[1]{\textbf{#1}}
\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} \newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
% excitation energies % excitation energies
\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
@ -303,15 +304,15 @@ where the excitation amplitudes are
\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}. \bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
\end{align} \end{align}
\end{subequations} \end{subequations}
With the Mulliken notation for the bare two-electron integrals With Mulliken's notation of 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 $\IS$ 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{}{\IS}(\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 BSE matrix elements read
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE-A} \label{eq:LR_BSE-A}
@ -328,20 +329,19 @@ In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is buil
\label{eq:wrpa} \label{eq:wrpa}
\begin{align} \begin{align}
\W{}{\IS}(\br{},\br{}') \W{}{\IS}(\br{},\br{}')
& = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}}, & = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
\\ \\
\epsilon_{\IS}(\br{},\br{}'; \omega) \epsilon_{\IS}(\br{},\br{}'; \omega)
& = \delta(\br{}-\br{}') - \IS \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}}, & = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
\end{align} \end{align}
\end{subequations} \end{subequations}
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 $W$.} 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 spatial 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}
\\ \\
\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 $\IS$ 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}.
@ -359,9 +359,10 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
\end{subequations} \end{subequations}
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 relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening
%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ %namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
so that $\W{}{\IS}$ 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: so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
In this 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}
@ -412,10 +413,10 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace. is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018} Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been named ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies. Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies.
Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added. Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added.
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. However, as commonly done within RPA and RPAx, \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}]. 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 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}. 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}.
@ -468,6 +469,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
\caption{ \caption{
Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively. The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
The values in parenthesis have been obtained via a Morse fit of the PES.
} }
\label{tab:Req} \label{tab:Req}
@ -490,17 +492,17 @@ The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold
& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\ & cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\ & cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & 2.700 \\ BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & 2.700 \\
& cc-pVTZ & 1.404 & 3.023 & & 2.410 & 2.068 & 2.116 & & \\ & cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{} &\rb{} &\rb{} &\rb{} &\rb{} &\rb{} \\ & cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)}&\rb{(2.408)}&\rb{(2.070)}&\rb{(2.130)}&\rb{(2.383)}&\rb{(2.640)} \\
RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & 2.691 \\ RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & 2.691 \\
& cc-pVTZ & 1.388 & 3.013 & & 2.408 & 2.065 & 2.114 & & \\ & cc-pVTZ & 1.388 & 3.013 & (2.965) & 2.408 & 2.065 & 2.114 & (2.370) & (2.584) \\
& cc-pVQZ & 1.382 & 3.013 & & & & & & \\ & cc-pVQZ & 1.382 & 3.013 & (2.965) & (2.389) & (2.045) & (2.110) & (2.367) & (2.571) \\
RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\ RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
& cc-pVTZ & 1.395 & 3.003 & 2.971 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\ & cc-pVTZ & 1.395 & 3.003 &\gb{2.971} & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
& cc-pVQZ & 1.394 & 3.011 & & & & & & \\ & cc-pVQZ & 1.394 & 3.011 & (2.943) & (2.393) & (2.041) & (2.105) & (2.367) & (2.563) \\
RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\ RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\ & cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
& cc-pVQZ & 1.386 & 2.994 & & & & & & \\ & cc-pVQZ & 1.386 & 2.994 & (2.946) & (2.385) & (2.042) & (2.104) & (2.365) & (2.571) \\
% FROZEN CORE VERSION % FROZEN CORE VERSION
% Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\ % Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
% \hline % \hline
@ -551,7 +553,7 @@ Here again, the BSE@{\GOWO}@HF equilibrium bond length (obtained with cc-pVQZ) i
\includegraphics[width=0.49\linewidth]{LiH_GS_VQZ} \includegraphics[width=0.49\linewidth]{LiH_GS_VQZ}
\caption{ \caption{
Ground-state PES of \ce{H2} (left) and \ce{LiH} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVQZ basis set. Ground-state PES of \ce{H2} (left) and \ce{LiH} (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-H2-LiH} \label{fig:PES-H2-LiH}
} }
\end{figure*} \end{figure*}
@ -572,7 +574,7 @@ Note that these irregularities would be genuine discontinuities in the case of {
\includegraphics[height=0.35\linewidth]{HCl_GS_VQZ} \includegraphics[height=0.35\linewidth]{HCl_GS_VQZ}
\caption{ \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 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}. %Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-LiF-HCl} \label{fig:PES-LiF-HCl}
} }
\end{figure*} \end{figure*}
@ -588,7 +590,7 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown
\includegraphics[height=0.26\linewidth]{BF_GS_VQZ} \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 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}
} }
\end{figure*} \end{figure*}
@ -603,7 +605,7 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
\includegraphics[width=\linewidth]{F2_GS_VQZ} \includegraphics[width=\linewidth]{F2_GS_VQZ}
\caption{ \caption{
Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the 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}. %Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-F2} \label{fig:PES-F2}
} }
\end{figure} \end{figure}

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