From bcbaaf5c691d1b759d37d50253c3b234209b3fe3 Mon Sep 17 00:00:00 2001 From: Emmanuel Giner Date: Tue, 25 Jun 2019 10:58:42 +0200 Subject: [PATCH] minor modifs --- Manuscript/Ex-srDFT.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex index 370403c..099dd97 100644 --- a/Manuscript/Ex-srDFT.tex +++ b/Manuscript/Ex-srDFT.tex @@ -264,8 +264,8 @@ In other words, the correction vanishes in the CBS limit, hence guaranteeing an \label{sec:rs} %%%%%%%%%%%%%%%%%%%%%%%% -As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference. \cite{TouGorSav-TCA-05} -The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, admits, for any $\n{}{}$, the following two limits +As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference \manu{taken from RS-DFT}. \cite{TouGorSav-TCA-05} +The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, \manu{depends on the range-separation parameter $\mu$ and} admits, for any $\n{}{}$, the following two limits \manu{as a function of $\mu$} \begin{align} \label{eq:large_mu_ecmd} \lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0, @@ -275,12 +275,12 @@ The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, admits, for any which correspond to the WFT limit ($\mu = \infty$) and the DFT limit ($\mu = 0$). In Eq.~\eqref{eq:large_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65} -The key ingredient, the range-separated function +The key ingredient \manu{that allows us to use the ECMD to correct for the basis set incompleteness error is} the range-separated function \begin{equation} \label{eq:def_mu} \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}) \end{equation} -automatically adapts to the spatial non-homogeneity of the basis set incompleteness error. +\manu{which} automatically adapts to the spatial non-homogeneity of the basis set incompleteness error. It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18} The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by \begin{equation} @@ -333,7 +333,7 @@ In this regime, the ECMD energy \label{eq:exact_large_mu} \bE{\text{c,md}}{\sr}[\n{2}{},\rsmu{}{}] \propto \frac{1}{\mu^3} \int \dbr{} \n{2}{}(\br{}) + \order*{\mu^{-4}} \end{align} -only depends on the \textit{exact} on-top pair density $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground state wave function $\Psi$ belonging to the Hilbert space spanned by the complete basis set. +only depends on the \textit{exact} on-top pair density $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground state wave function $\Psi$ belonging to the Hilbert space spanned by \manu{a} complete basis set. Obviously, an exact quantity such as $\n{2}{}(\br{})$ is out of reach in practical calculations and must be approximated by a function referred here as $\tn{2}{}(\br{})$. For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functional form in order to interpolate between $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and Eq.~\eqref{eq:exact_large_mu} as $\mu \to \infty$: \cite{FerGinTou-JCP-18} @@ -348,7 +348,7 @@ For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functi \end{subequations} As illustrated in the context of RS-DFT, \cite{FerGinTou-JCP-18} such a functional form is able to treat both weakly and strongly correlated systems thanks to the explicit inclusion of $\e{\text{c}}{\PBE}$ and $\tn{2}{}$, respectively. -Therefore, in the present context, we consider the explicit form of Eqs.~\eqref{eq:epsilon_cmdpbe} and \eqref{eq:beta_cmdpbe} with $\rsmu{}{\Bas}$ and introduce the general form of the PBE-based complementary functional: +Therefore, in the present context, we consider the explicit form of Eqs.~\eqref{eq:epsilon_cmdpbe} and \eqref{eq:beta_cmdpbe} with $\rsmu{}{\Bas}$ and introduce the general form of the PBE-based complementary functional \manu{for the basis set $\Bas$}: \begin{multline} \label{eq:def_pbe_tot} \bE{\PBE}{\Bas}[\n{}{},\tn{2}{},\rsmu{}{\Bas}] =