From 0edb84acea036688ce55063af0cc15646f1a095b Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 23 Apr 2019 09:05:57 +0200 Subject: [PATCH] change of notations --- Manuscript/G2-srDFT.tex | 123 +++++++++++++++------------------------- 1 file changed, 46 insertions(+), 77 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index e6c75a8..b5545b6 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -109,7 +109,13 @@ \newcommand{\dbr}[1]{d\br{#1}} \newcommand{\ra}{\rightarrow} -\newcommand{\De}{D_\text{e}} + +% frozen core +\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}} +\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}} +\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}} +\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}} + \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France} @@ -219,11 +225,11 @@ Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, for the lack of \titou{cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$} at the e-e coalescence points, a universal condition of exact wave functions. Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence. Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction. -Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$. +Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$. -The first step of the present basis-set correction consists of obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$. -In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$. -In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range-separation function. +The first step of the present basis-set correction consists of obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$. +In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$. +In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function. %================================================================= %\subsection{Effective Coulomb operator} @@ -231,9 +237,9 @@ In the final step, we employ short-range density functionals \cite{TouGorSav-TCA We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18} \begin{equation} \label{eq:def_weebasis} - \W{\Bas}{}(\br{1},\br{2}) = + \W{}{\Bas}(\br{1},\br{2}) = \begin{cases} - \f{\Bas}{}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,} + \f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,} \\ \infty, & \text{otherwise,} \end{cases} @@ -247,70 +253,60 @@ where and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO), \begin{equation} \label{eq:fbasis} - \f{\Bas}{}(\br{1},\br{2}) + \f{}{\Bas}(\br{1},\br{2}) = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \end{equation} and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. -With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) +With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:int_eq_wee} - \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2}, + \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2}, \end{equation} where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as \begin{equation} \alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} = - \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},} + \iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},} \end{equation} -it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. -Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp. +it intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction. +Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp. -As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries. +As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries. Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:lim_W} - \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \titou{r_{12}^{-1} } + \lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = \titou{r_{12}^{-1} } \end{equation} -for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. +for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$. %================================================================= %\subsection{Range-separation function} %================================================================= -A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{\Bas}{}(\br{},{\br{}})$, -%\begin{equation} -% \label{eq:wcoal} -% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), -%\end{equation} +A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$, which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing. -Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator. +Because $\W{}{\Bas}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator. Although this choice is not unique, we choose here the range-separation function \begin{equation} \label{eq:mu_of_r} - \rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{},\br{}), + \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}), \end{equation} -such that the long-range interaction of RS-DFT +such that the long-range interaction of RS-DFT, \titou{$\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$}, %\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}} } +% \w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}} %\end{equation} -\begin{equation} - \w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}} -\end{equation} -coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}(\br{})}(0) = \W{\Bas}{}(\br{},\br{})$ at any point $\br{}$. +coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any point $\br{}$. %================================================================= %\subsection{Short-range correlation functionals} %================================================================= -Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$. +Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05} -%\begin{multline} \begin{equation} \label{eq:ec_md_mu} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} -% \\ - \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}}, -%\end{multline} \end{equation} where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization \begin{equation} @@ -318,55 +314,32 @@ where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization \wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \end{equation} with $\hWee{\lr,\rsmu{}{}} = \sum_{i