From 308830cf2058bcde77348dd81b9d14acc3e77f0e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 11 Apr 2019 21:30:23 +0200 Subject: [PATCH] still working on the notations --- Manuscript/G2-srDFT.tex | 34 +++++++++++++++++----------------- 1 file changed, 17 insertions(+), 17 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 143b33e..2c45544 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -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.