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still working on the notations

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Pierre-Francois Loos 2019-04-11 21:30:23 +02:00
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34
Manuscript/G2-srDFT.tex
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@ -385,9 +385,9 @@ Of course, there exists \textit{a priori} an infinite set of functions in $\math
\label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
\end{equation}
for all points $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems.
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems.
%=================================================================
%\subsection{Range-separation function}
@ -402,6 +402,7 @@ such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
\begin{equation}
\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
\end{equation}
\PFL{This expression looks like a cheap spherical average.}
coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$
\begin{equation}
\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}).
@ -521,7 +522,7 @@ Therefore, the PBE complementary functional reads
\label{eq:def_pbe_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
\end{equation}
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modY}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
%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 $\mu$.
@ -546,35 +547,34 @@ We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor
%According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$.
We therefore define the valence-only effective interaction
\begin{equation}
% \label{eq:Wval}
\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \left\{
\begin{array}{ll}
\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2})\ne 0\\
\,\,\,\,+\infty & \mbox{otherwise. }
\end{array}
\right.
\W{\Bas}{\Val}(\br{1},\br{2}) =
\begin{cases}
\f{\Bas}{\Val}(\br{1},\br{2})/\n{2}{\Val}(\br{1},\br{2}), & \text{if $\n{2}{\Val}(\br{1},\br{2})\ne 0$},
\\
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
with
\begin{subequations}
\begin{gather}
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})
\f{\Bas}{\Val}(\br{1},\br{2})
= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
\\
\n{2}{\wf{}{\Bas},\Val}(\br{1},\br{2})
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2},
\n{2}{\Val}(\br{1},\br{2})
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather}
\end{subequations}
and the corresponding valence range separation function $\rsmu{\wf{}{\Bas}}{\Val}(\br{})$
and the corresponding valence range separation function $\rsmu{\Bas}{\Val}(\br{})$
\begin{equation}
\label{eq:muval}
\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\br{}).
\rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}).
\end{equation}
%\begin{equation}
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
%\end{equation}
%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
%Defining the valence one-body spin density matrix as
@ -597,7 +597,7 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
% \label{eq:def_lda_tot}
% \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
%\end{equation}
Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the model $\modY$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modY}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
Regarding now the main computational source of the present approach, it consists in the evaluation
of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.