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Manuscript/Ex-srDFT.tex
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@ -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}] =