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Manuscript/Ex-srDFT.tex
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@ -334,10 +334,10 @@ The PBE correlation functional has clearly shown to improve energetics for small
Therefore, based on the proposition of Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general form of the PBE complementary functional: Therefore, based on the proposition of Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general form of the PBE complementary functional:
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBE}{}[\n{}{},\tn{2}{},\rsmu{}{}] = \bE{\PBE}{\manu{\Bas}}[\n{}{},\tn{2}{},\rsmu{}{\manu{\Bas}}] =
\int \n{}{}(\br{}) \int \n{}{}(\br{})
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
\end{multline} \end{multline}
with with
\begin{subequations} \begin{subequations}
@ -353,10 +353,10 @@ with
In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a new PBE-based functional, here-referred as \titou{PBE-UEG}, In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a new PBE-based functional, here-referred as \titou{PBE-UEG},
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\titou{\PBE\text{-}\UEG}}{}[\n{}{},\n{2}{\UEG},\rsmu{}{}] = \bE{\titou{\PBE\text{-}\UEG}}{\manu{\Bas}}[\n{}{},\n{2}{\UEG},\rsmu{}{\manu{\Bas}}] =
\int \n{}{}(\br{}) \int \n{}{}(\br{})
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
\end{multline} \end{multline}
in which the on-top pair density was approximated by its UEG version, in which the on-top pair density was approximated by its UEG version,
\begin{equation} \begin{equation}
@ -369,14 +369,14 @@ However, the underlying UEG on-top pair density might not be suited for the trea
Therefore, we propose here the "PBE-ontop" (PBEot) functional, Therefore, we propose here the "PBE-ontop" (PBEot) functional,
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBEot}{}[\n{}{},\ttn{2}{},\rsmu{}{}] = \bE{\PBEot}{\manu{\Bas}}[\n{}{},\ttn{2}{\manu{\Bas}},\rsmu{}{\manu{\Bas}}] =
\int \n{}{}(\br{}) \int \n{}{}(\br{})
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{\manu{\Bas}}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
\end{multline} \end{multline}
a variant inspired by the work of Ref.~\onlinecite{FerGinTou-JCP-18} where the exact on-top pair density is approximated by the extrapolation formula proposed by Gori-Giorgi and Savin:\cite{GorSav-PRA-06} a variant inspired by the work of Ref.~\onlinecite{FerGinTou-JCP-18} where the exact on-top pair density is approximated by the extrapolation formula proposed by Gori-Giorgi and Savin:\cite{GorSav-PRA-06}\manu{. In the present context, we use the on-top pair density in the basis set $\Bas$ $ \n{2}{\manu{\Bas}}(\br{})$ together with the associated range separation function $\rsmu{}{\Bas}(\br{})$, which leads to the following approximated on-top pair density: }
\begin{equation} \begin{equation}
\ttn{2}{}(\br{}) = \n{2}{}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}. \ttn{2}{\manu{\Bas}}(\br{}) = \n{2}{\manu{\Bas}}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\manu{\Bas}}(\br{})})^{-1}.
\end{equation} \end{equation}
%Such formula relies on the association between $\n{2}{\Bas}(\br{ },\br{ })$, which is the on-top pair density computed in the basis set $\Bas$ in a given point, and the local value of the range-separation parameter $\rsmu{}{\Bas}$ at the same point. %Such formula relies on the association between $\n{2}{\Bas}(\br{ },\br{ })$, which is the on-top pair density computed in the basis set $\Bas$ in a given point, and the local value of the range-separation parameter $\rsmu{}{\Bas}$ at the same point.
The sole distinction between \titou{PBE-UEG} and PBEot is the level of approximation of the exact on-top pair density. The sole distinction between \titou{PBE-UEG} and PBEot is the level of approximation of the exact on-top pair density.