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Pierre-Francois Loos 2019-04-06 12:13:33 +02:00
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Manuscript/G2-srDFT.tex
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@ -407,6 +407,7 @@ Contrary to conventional RS-DFT schemes which require a range-separated paramete
The first step of our basis set correction consists in obtaining the 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 first step of our basis set correction consists in obtaining the 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.
In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$. In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$.
The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. %Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
%First, 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{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . %First, 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{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). %Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}).
@ -469,13 +470,13 @@ Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-1
%================================================================= %=================================================================
\subsection{Range-separation function} \subsection{Range-separation function}
%================================================================= %=================================================================
To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}. To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction to a long-range interaction characterised by a range-separation function $\rsmu{}{}(\br{})$, i.e~varying in space.
More precisely, if we define the value of the interaction at coalescence as More precisely, if we define the value of the interaction at coalescence as
\begin{equation} \begin{equation}
\label{eq:def_wcoal} \label{eq:def_wcoal}
\wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}). \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
\end{equation} \end{equation}
where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$, where $(\bx{},\Bar{\bx{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction: we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
\begin{equation} \begin{equation}
\wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{}) \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
@ -487,7 +488,7 @@ where the long-range-like interaction is defined as
Equation \eqref{eq:def_wcoal} is equivalent to the following condition Equation \eqref{eq:def_wcoal} is equivalent to the following condition
\begin{equation} \begin{equation}
\label{eq:mu_of_r} \label{eq:mu_of_r}
\rsmu{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\Bas}}{}(\br{}) \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{2}{\sqrt{\pi}} \W{\wf{}{\Bas}}{}(\br{})
\end{equation} \end{equation}
%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as %As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
%\begin{equation} %\begin{equation}