getting there

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Pierre-Francois Loos 2019-04-05 14:18:33 +02:00
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@ -414,6 +414,101 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12
Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$. Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set. The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
%=================================================================
\subsection{Complementary functional}
%=================================================================
\label{sec:ecmd}
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
\label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
\\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline}
where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
\begin{equation}
\label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation}
with
\begin{equation}
\label{eq:weemu}
\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
\end{equation}
and
\begin{equation}
\label{eq:erf}
\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
\end{equation}
is the long-range part of the Coulomb operator.
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
\begin{subequations}
\begin{align}
\label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
\\
\label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
\end{align}
\end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model.
%--------------------------------------------
%\subsubsection{Local density approximation}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation}
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
%--------------------------------------------
%\subsubsection{New PBE functional}
%--------------------------------------------
The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
\begin{subequations}
\begin{gather}
\label{eq:epsilon_cmdpbe}
\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
\\
\label{eq:epsilon_cmdpbe}
\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather}
\end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
\begin{equation}
\label{eq:ueg_ontop}
\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
\end{equation}
where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
Therefore, the PBE complementary function reads
\begin{equation}
\label{eq:def_lda_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
\begin{equation}
\label{eq:approx_ecfuncbasis}
\ecompmodel \approx \ecmuapproxmurmodel
\end{equation}
Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
\begin{equation}
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
\end{equation}
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
%================================================================= %=================================================================
\subsection{Effective Coulomb operator} \subsection{Effective Coulomb operator}
%================================================================= %=================================================================
@ -502,99 +597,6 @@ and therefore
%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . %\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
\end{equation} \end{equation}
%=================================================================
\subsection{Complementary functional}
%=================================================================
\label{sec:ecmd}
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
\label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
\\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline}
where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
\begin{equation}
\label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation}
with
\begin{equation}
\label{eq:weemu}
\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
\end{equation}
and
\begin{equation}
\label{eq:erf}
\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
\end{equation}
is the long-range part of the Coulomb operator.
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
\begin{subequations}
\begin{align}
\label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
\\
\label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
\end{align}
\end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model.
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
\begin{equation}
\label{eq:approx_ecfuncbasis}
\ecompmodel \approx \ecmuapproxmurmodel
\end{equation}
Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
\begin{equation}
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
\end{equation}
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
%--------------------------------------------
%\subsubsection{Local density approximation}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation}
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
%--------------------------------------------
%\subsubsection{New PBE functional}
%--------------------------------------------
The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
\begin{subequations}
\begin{gather}
\label{eq:epsilon_cmdpbe}
\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
\\
\label{eq:epsilon_cmdpbe}
\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather}
\end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
\begin{equation}
\label{eq:ueg_ontop}
\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
\end{equation}
where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
Therefore, the PBE complementary function reads
\begin{equation}
\label{eq:def_lda_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
%================================================================= %=================================================================
\subsection{Valence effective interaction} \subsection{Valence effective interaction}
%================================================================= %=================================================================