getting there

Manuscript/Ex-srDFT.tex

@@ -255,7 +255,7 @@ An important property of the present correction is
   \label{eq:limitfunc}
         \lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}]  = 0.
 \end{equation}
-In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit. \cite{LooPraSceTouGin-JPCL-19}
+In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered limit. \cite{LooPraSceTouGin-JPCL-19}
 %In the following, we will drop the state index $k$ and focus on the quantity $\bE{}{\Bas}[\n{}{}]$.
 %Mention that the present approach is a one-shot procedure, i.e.~there is no self-consistent procedure involved.
 
@@ -311,7 +311,6 @@ An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regu
 \end{equation} 
 ensuring that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete.
 We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.
-\titou{From hereon, we drop the dependency in $\Bas$.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Short-range correlation functionals}
@@ -326,16 +325,18 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
 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 $\be{\text{c,md}}{\sr,\LDA}$ 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 was suggested by some of the authors in the context of RS-DFT. \cite{FerGinTou-JCP-18}  
-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}. 
+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. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
-\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  $\n{2}{}(\br{},\br{})$
+In the context of RS-DFT, the large-$\mu$ behavior corresponds to an extremely short-range interaction. 
+In this regime, the ECMD energy
 \begin{align}
   \label{eq:exact_large_mu}
-  \bE{\text{c,md}}{\sr}[\n{2}{},\rsmu{}{}] \propto \frac{1}{\mu^3} \int \dbr{} \n{2}{}(\br{},\br{}) + O(\frac{1}{\mu^4}),
+  \bE{\text{c,md}}{\sr}[\n{2}{},\rsmu{}{}] \propto \frac{1}{\mu^3} \int \dbr{} \n{2}{}(\br{}) + \order*{\mu^{-4}}
 \end{align}
-where $\n{2}{}(\br{},\br{})$ is obtained from the exact ground state wave function $\Psi$ developed in an 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 the complete basis set.
-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 $\tn{2}{}(\br{},\br{})$. 
+
-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}:
+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} 
 \begin{subequations}
 \begin{gather}
 	\label{eq:epsilon_cmdpbe}
@@ -345,49 +346,48 @@ For a given choice of $\tn{2}{}(\br{},\br{})$, the authors in Ref~\cite{FerGinTo
 	\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{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 $\tn{2}{}$).\\ 
+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 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 $\tn{2}{}$.}
 
-\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}:}
+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: 
 \begin{multline}
 	\label{eq:def_pbe_tot}
-	\bE{\titou{\PBE\text{-}\UEG}}{\manu{\Bas}}[\n{}{},\n{2}{\UEG},\rsmu{}{\manu{\Bas}}] =
+	\bE{\PBE}{\Bas}[\n{}{},\tn{2}{},\rsmu{}{\Bas}] =
 	\int \n{}{}(\br{}) 
 	\\
-	\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{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
 \end{multline}
-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}
+which depends on the approximation level of $\tn{2}{}$.
+
+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{equation}
+	\label{eq:def_pbe_tot}
+	\bE{\titou{\PBE\text{-}\UEG}}{\Bas} 
+	\equiv
+	\bE{\PBE}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{\Bas}],
+\end{equation}
+in which the on-top pair density was approximated by its UEG version, i.e., $\tn{2}{}(\br{}) = \n{2}{\UEG}(\br{})$, with
 \begin{equation}
  \n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})), 
 \end{equation}
 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 illustrated in Ref.~\onlinecite{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.
 
 However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems. 
-\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.   
+Besides, in the context of the present basis set correction, we have to compute $\n{2}{\Bas}(\br{})$ to obtain $\rsmu{}{\Bas}(\br{})$ [see Eqs.~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}]. 
-More precisely, we propose an approximation of the on-top pair density $\ttn{2}{\manu{\Bas}}(\br{})$ defined as
+Therefore we can use $\n{2}{\Bas}(\br{})$ to have an approximation of the exact on-top pair density.}
+More precisely, we propose an approximation of the on-top pair density $\ttn{2}{\Bas}(\br{})$ 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}
+  \ttn{2}{\Bas}(\br{}) = \n{2}{\Bas}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\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. 
+which is inspired by the large-$\mu$ extrapolation 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: 
+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{equation}
 	\label{eq:def_pbe_tot}
-	\bE{\PBEot}{\manu{\Bas}}[\n{}{},\ttn{2}{\manu{\Bas}},\rsmu{}{\manu{\Bas}}] =
+	\bE{\PBEot}{\Bas}
-	\int \n{}{}(\br{}) 
+	\equiv
-	\\
+	\bE{\PBE}{\Bas}[\n{}{},\ttn{2}{\Bas},\rsmu{}{\Bas}].
-	\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{equation}
-\end{multline}
-%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.