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Pierre-Francois Loos 2019-06-17 22:16:59 +02:00
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Manuscript/Ex-srDFT.tex
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@ -209,7 +209,7 @@ Unless otherwise stated, atomic units are used.
\label{sec:theory} \label{sec:theory}
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The present basis set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}$ and $\n{0}{\Bas}$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{},\br{})$ and $\n{2,k}{\Bas}(\br{},\br{})$, The present basis set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}$ and $\n{0}{\Bas}$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{})$ and $\n{2,k}{\Bas}(\br{})$,
Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18} Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18}
\begin{align} \begin{align}
\label{eq:ECBS} \label{eq:ECBS}
@ -325,9 +325,9 @@ where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarizat
The ECMD LDA functional \eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEG-based quantities are hardly transferable when the system becomes strongly correlated or multi-configurational. The ECMD LDA functional \eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEG-based quantities are hardly transferable when the system becomes strongly correlated or multi-configurational.
An attempt to solve these problems has been proposed by some of the authors in the context of the RS-DFT. \cite{FerGinTou-JCP-18} An attempt to solve these problems has been proposed by some of the authors in the context of the RS-DFT. \cite{FerGinTou-JCP-18}
They proposed to interpolate between the exact large-$\mu$ behavior \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} and the Perdew-Burke-Ernzerhof (PBE) functional \cite{PerBurErn-PRL-96} at $\mu = 0$. They proposed to interpolate between the exact large-$\mu$ behavior \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} and the Perdew-Burke-Ernzerhof (PBE) functional \cite{PerBurErn-PRL-96} at $\mu = 0$.
The PBE correlation functional has clearly shown to improve the results for small $\mu$, and the exact behavior at large $\mu$ naturally introduces the \textit{exact} on-top pair density $\n{2}{}(\br{})$ which contains information about the level of strong correlation. The PBE correlation functional has clearly shown to improve the results for small $\mu$, and the exact behavior at large $\mu$ naturally introduces the \textit{exact} on-top pair density $\n{2}{}(\br{},\br{}) \equiv \n{2}{}(\br{})$ which contains information about the level of strong correlation.
Obviously, $\n{2}{}(\br{})$ cannot be accessed in practice and must be approximated by a function referred here as $\n{2}{X}$, $X$ standing for the label of a given approximation. \manu{Obviously, $\n{2}{}(\br{})$ cannot be accessed in practice and must be approximated by a function referred here as $\n{2}{X}$, $X$ standing for the label of a given approximation.
Therefore, based on the propositions of ~Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general functional form for the PBE linked complementary functional: Therefore, based on the propositions of ~Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general functional form for the PBE linked complementary functional:
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
@ -358,8 +358,9 @@ with
\begin{equation} \begin{equation}
\n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})), \n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})),
\end{equation} \end{equation}
where the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. \\ where the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.}
As shown in Ref~\cite{LooPraSceTouGin-JPCL-19}, the link with the usual PBE functional has shown to improve the results over LDA for weakly correlated systems, but the remaining on-top pair density obtained from the UEG might not be suited for the treatment of excited states and/or strongly-correlated systems. Therefore, we propose here a variant inspired by the work of ~Ref\cite{FerGinTou-JCP-18} where we obtain an approximation of the exact on-top pair density based on an extrapolation proposed by Gori-Giorgi \textit{et. al} in Ref.~\onlinecite{GorSav-PRA-06}. Introducing the extrapolation function $\extrfunc(\n{2}{},\mu)$
\manu{As shown in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, the link with the usual PBE functional has shown to improve the results over LDA for weakly correlated systems, but the remaining on-top pair density obtained from the UEG might not be suited for the treatment of excited states and/or strongly-correlated systems. Therefore, we propose here a variant inspired by the work of ~Ref\cite{FerGinTou-JCP-18} where we obtain an approximation of the exact on-top pair density based on an extrapolation proposed by Gori-Giorgi \textit{et. al} in Ref.~\onlinecite{GorSav-PRA-06}. Introducing the extrapolation function $\extrfunc(\n{2}{},\mu)$
\begin{equation} \begin{equation}
\extrfunc(\n{2}{},\mu) = \n{2}{} \qty(1+ \frac{2}{\sqrt{\pi}\mu})^{-1}, \extrfunc(\n{2}{},\mu) = \n{2}{} \qty(1+ \frac{2}{\sqrt{\pi}\mu})^{-1},
\end{equation} \end{equation}
@ -376,7 +377,7 @@ Therefore, we propose the "PBE-ontop" (PBEot) functional which reads
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{\Bas}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}. \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{\Bas}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}.
\end{multline} \end{multline}
Therefore, the unique difference between the PBE-UEG and PBEot functionals are the approximation for the exact on-top pair density. Therefore, the unique difference between the PBE-UEG and PBEot functionals are the approximation for the exact on-top pair density.}
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