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Manuscript/Ex-srDFT.tex
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@ -321,25 +321,21 @@ We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,L
The local-density approximation (LDA) of the ECMD complementary functional is defined as The local-density approximation (LDA) of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{}[\n{}{},\rsmu{}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, \bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}. where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
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 $\be{\text{c,md}}{\sr,\LDA}$ used in \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 RS-DFT. \cite{FerGinTou-JCP-18} An attempt to solve these problems has been proposed by some of the authors in the context of 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} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$. They proposed to interpolate between the usual Perdew-Burke-Ernzerhof (PBE) correlation functional \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and the exact large-$\mu$ behavior known from RS-DFT\cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}.
The PBE correlation functional has clearly shown to improve energetics for small $\mu$, \cite{LooPraSceTouGin-JPCL-19} 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. \manu{In the context of RS-DFT, the large-$\mu$ behavior corresponds to an extremely short-range interaction, and in that regime the ECMD energy only depends on the \textit{exact} on-top pair density $\n{2}{}(\br{},\br{})$
\begin{align}
\titou{Obviously, $\n{2}{}(\br{})$ cannot be accessed in practice} and must be approximated by a function referred here as $\tn{2}{}(\br{})$. \label{eq:exact_large_mu}
Therefore, based on the proposition of Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general form of the PBE complementary functional: \bE{\text{c,md}}{\sr}[\n{2}{},\rsmu{}{}] \propto \frac{1}{\mu^3} \int \dbr{} \n{2}{}(\br{},\br{}) + O(\frac{1}{\mu^4}),
\begin{multline} \end{align}
\label{eq:def_pbe_tot} where $\n{2}{}(\br{},\br{})$ is obtained from the exact ground state wave function $\Psi$ developed in an complete basis set.
\bE{\PBE}{\manu{\Bas}}[\n{}{},\tn{2}{},\rsmu{}{\manu{\Bas}}] = Obviously, an exact quantity such as $\n{2}{}(\br{},\br{})$ is out of reach in practical calculations and must be approximated by a function generally referred here as $\tn{2}{}(\br{},\br{})$.
\int \n{}{}(\br{}) For a given choice of $\tn{2}{}(\br{},\br{})$, the authors in Ref~\cite{FerGinTou-JCP-18} proposed to use the following functional form for the correlation energy density in order to interpolate between the PBE at $\mu = 0$ and the equation \eqref{eq:exact_large_mu}:
\\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
\end{multline}
with
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
@ -349,8 +345,18 @@ with
\beta^\PBE(\n{}{},\tn{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}/\n{}{}}. \beta^\PBE(\n{}{},\tn{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}/\n{}{}}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
As illustrated in the context of RS-DFT in Ref~\cite{FerGinTou-JCP-18}, such a functional form is able to treat both the weak correlation regime (thanks to the use of the $\e{\text{c}}{\PBE}$) and the strong correlation regime (thanks to good approximations of the on-top pair density $\tn{2}{}$).\\
Therefore, in the context of the basis set correction we use the explicit form of equations ~\eqref{eq:epsilon_cmdpbe}~and~\eqref{eq:beta_cmdpbe} with $\rsmu{}{\manu{\Bas}}$ and introduce the general form of the PBE-based complementary functional: }
\begin{multline}
\label{eq:def_pbe_tot}
\bE{\PBE}{\manu{\Bas}}[\n{}{},\tn{2}{},\rsmu{}{\manu{\Bas}}] =
\int \n{}{}(\br{})
\\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\br{})) \dbr{},
\end{multline}
\manu{which depends on the approximation used for $\tn{2}{}$.}
In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a new PBE-based functional, here-referred as \titou{PBE-UEG}, \manu{In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a version of the 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}}{\manu{\Bas}}[\n{}{},\n{2}{\UEG},\rsmu{}{\manu{\Bas}}] = \bE{\titou{\PBE\text{-}\UEG}}{\manu{\Bas}}[\n{}{},\n{2}{\UEG},\rsmu{}{\manu{\Bas}}] =
@ -358,15 +364,22 @@ In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a n
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\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, \manu{\textit{i.e.} $\tn{2}{}(\br{}) = \n{2}{\UEG}(\br{})$, 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 $g_0(n)$ is the UEG on-top pair distribution function [see Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}]. and where $g_0(n)$ is the UEG on-top pair distribution function [see Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}].
\manu{As illustrated in Ref~\cite{LooPraSceTouGin-JPCL-19} for weakly correlated systems, this PBE-based functional has clearly shown to improve energetics over the pure UEG-based functional $\bE{\LDA}{\Bas}$ (see \eqref{eq:def_lda_tot}) thanks to the inclusion of the PBE functional.}
As mentioned earlier, the incorporation of the PBE functional as a limiting form at $\mu = 0$ [see Eqs.~\eqref{eq:epsilon_cmdpbe} and \eqref{eq:beta_cmdpbe}] has shown to significantly improve the energetic properties over the LDA for weakly correlated systems.
However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems. However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems.
Therefore, we propose here the "PBE-ontop" (PBEot) functional, \manu{Also, in the context of the basis set correction, we have to compute the on-top pair density $\n{2}{\manu{\Bas}}(\br{})$ to compute $\rsmu{}{\manu{\Bas}}(\br{})$ (see equations~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}), and therefore we can use $\n{2}{\manu{\Bas}}(\br{})$ to have an approximation of the exact on-top pair density.
More precisely, we propose an approximation of the on-top pair density $\ttn{2}{\manu{\Bas}}(\br{})$ defined as
\begin{equation}
\ttn{2}{\manu{\Bas}}(\br{}) = \n{2}{\manu{\Bas}}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\manu{\Bas}}(\br{})})^{-1}
\end{equation}
which is inspired by the extrapolation at large $\mu$ of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RS-DFT.
}
Therefore, we propose here the "PBE-ontop" (PBEot) functional which uses the on-top pair density computed in the basis set $\Bas$ as ingredient:
\begin{multline} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBEot}{\manu{\Bas}}[\n{}{},\ttn{2}{\manu{\Bas}},\rsmu{}{\manu{\Bas}}] = \bE{\PBEot}{\manu{\Bas}}[\n{}{},\ttn{2}{\manu{\Bas}},\rsmu{}{\manu{\Bas}}] =
@ -374,10 +387,6 @@ Therefore, we propose here the "PBE-ontop" (PBEot) functional,
\\ \\
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{\manu{\Bas}}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\manu{\Bas}}(\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}\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}
\ttn{2}{\manu{\Bas}}(\br{}) = \n{2}{\manu{\Bas}}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\manu{\Bas}}(\br{})})^{-1}.
\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.