From 36392fd8514afd5fff848ad2e2e7b0ae5b7fac9c Mon Sep 17 00:00:00 2001 From: Julien Toulouse Date: Tue, 7 Jan 2020 18:17:19 +0100 Subject: [PATCH] changes in beginning of theory section --- Manuscript/srDFT_SC.tex | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 3a5deee..5505442 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -323,30 +323,30 @@ Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the p \section{Theory} \label{sec:theory} %%%%%%%%%%%%%%%%%%%%%%%% -As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary density functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Sec.~\ref{sec:def_func}. +As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary density functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}. \subsection{Basic equations} \label{sec:basic} -The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over one-electron densities $\denr$ +The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over $N$-representable one-electron densities $\denr$ \begin{equation} \label{eq:levy} E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, \end{equation} -where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83} +where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83} \begin{equation} \label{eq:levy_func} F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi}, \end{equation} where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$. -The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$. +The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, we always implicitly consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$. In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as \begin{equation} \label{eq:def_levy_bas} F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, \end{equation} -where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and +where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, and \begin{equation} \begin{aligned} \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi} @@ -368,7 +368,7 @@ As a simple non-self-consistent version of this approach, we can approximate the \label{eq:e0approx} E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI, \end{equation} -where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation. Two key requirements for this purpose are i) size consistency, and ii) spin-multiplet degeneracy. +where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency. \subsection{Effective interaction in a finite basis} \label{sec:wee} @@ -412,7 +412,7 @@ The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the c \subsection{Local range-separation parameter} \label{sec:mur} \subsubsection{General definition} -As the effective interaction within a finite basis, $\wbasis$ is bounded and resembles the long-range interaction used in RSDFT +The effective interaction within a finite basis, $\wbasis$, is bounded and resembles the long-range interaction used in RSDFT \begin{equation} \label{eq:weelr} w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}},