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Emmanuel Giner 2019-03-28 11:46:02 +01:00
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@ -202,17 +202,17 @@ where $\efuncbasis$ is the complementary density functional defined in equation
\efuncbasisfci = & \min_{\Psi \rightarrow \denfci} \elemm{\Psi}{\kinop + \weeop}{\Psi} \\&- \min_{\psibasis \rightarrow \denfci} \elemm{\psibasis}{\kinop + \weeop}{\psibasis}, \efuncbasisfci = & \min_{\Psi \rightarrow \denfci} \elemm{\Psi}{\kinop + \weeop}{\Psi} \\&- \min_{\psibasis \rightarrow \denfci} \elemm{\psibasis}{\kinop + \weeop}{\psibasis},
\end{aligned} \end{aligned}
\end{equation} \end{equation}
where $\Psi$ is a general wave function being obtained in a complete basis. Provided that functional $\efuncbasis$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\denfci$ coincides with the exact ground state density. where $\Psi$ is a general wave function being obtained in a complete basis. Provided that functional $\efuncbasis$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\denfci$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.
The functional $\efuncbasisfci$ is not universal as it depends on the basis set $\basis$ used. A simple analytical form for such a functional is of course not known and we approximate it in two-steps. First, we define a real-space representation of the coulomb interaction truncated in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \ref{sec:weff}). The functional $\efuncbasisfci$ is not universal as it depends on the basis set $\basis$ used. A simple analytical form for such a functional is of course not known and we approximate it in two-steps. First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \ref{sec:weff}).
Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al}\cite{Toulouse2005_ecmd}, that we evaluate with the range-separation parameter $\mu(r)$ varying in space at the FCI density $\denfci$ (see \ref{sec:ecmd}). Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al}\cite{Toulouse2005_ecmd}, that we evaluate at the FCI density $\denfci$ (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.
\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$} \subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$}
One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also come from a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point. One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also come from a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point.
The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which: The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
\begin{itemize} \begin{itemize}
\item is non-divergent at the electron coalescence point as long as an incomplete basis set $\basis$ is used \item is non-divergent at the electron coalescence point as long as an incomplete basis set $\basis$ is used,
\item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\basis$. \item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\basis$.
\end{itemize} \end{itemize}
\subsubsection{General definition of an effective interaction for the basis set $\basis$} \subsubsection{General definition of an effective interaction for the basis set $\basis$}
@ -229,7 +229,7 @@ After a few mathematical work (see appendix A of \cite{GinPraFerAssSavTou-JCP-18
\label{eq:expectweeb} \label{eq:expectweeb}
\elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis, \elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis,
\end{equation} \end{equation}
where the function $\fbasis$ is defined as: where the function $\fbasis$ is
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\begin{aligned} \begin{aligned}
@ -550,13 +550,13 @@ Regarding the wave function chosen to define the local range-separation paramete
& ex (FC)FCI+PBE & 225.8 & 227.6 & 228.4 & 228.3 & \\ & ex (FC)FCI+PBE & 225.8 & 227.6 & 228.4 & 228.3 & \\
& ex (FC)FCI+PBE-val & 227.5 & 227.7 & 228.4 & 228.0 & \\ & ex (FC)FCI+PBE-val & 227.5 & 227.7 & 228.4 & 228.0 & \\
\hline \hline
& ex FCI & 202.2 & 218.5 & 224.4 & 225.4 & \\ & ex FCI & 202.2 & 218.5 & 224.4 & ----- & \\
\hline \hline
& ex FCI+LDA & 218.0 & 226.8 & 229.1 & 228.2 & \\ & ex FCI+LDA & 218.0 & 226.8 & 229.1 & ----- & \\
& ex FCI+LDA-val & 219.1 & 226.9 & 229.0 & 227.7 & \\ & ex FCI+LDA-val & 219.1 & 226.9 & 229.0 & ----- & \\
\hline \hline
& ex FCI+PBE & 226.4 & 228.2 & 229.1 & 228.0 & \\ & ex FCI+PBE & 226.4 & 228.2 & 229.1 & ----- & \\
& ex FCI+PBE -val & 228.0 & 228.2 & 229.1 & 227.6 & \\ & ex FCI+PBE -val & 228.0 & 228.2 & 229.1 & ----- & \\
\\ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}