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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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\usepackage{natbib}
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\usepackage[extra]{tipa}
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\bibliographystyle{achemso}
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\AtBeginDocument{\nocite{achemso-control}}
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\usepackage{mathpazo,libertine}
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@ -130,7 +131,8 @@
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\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
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\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
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\newcommand{\tn}[2]{\widetilde{n}_{#1}^{#2}}
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\newcommand{\tn}[2]{\tilde{n}_{#1}^{#2}}
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\newcommand{\ttn}[2]{\mathring{n}_{#1}^{#2}}
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% energies
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@ -309,6 +311,7 @@ An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regu
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\end{equation}
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ensuring that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete.
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We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.
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\titou{From hereon, we drop the dependency in $\Bas$.}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Short-range correlation functionals}
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@ -318,66 +321,65 @@ We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,L
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The local-density approximation (LDA) of the ECMD complementary functional is defined as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\bE{\LDA}{}[\n{}{},\rsmu{}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\end{equation}
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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}.
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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.
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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}
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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 $\mu = 0$.
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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.
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An attempt to solve these problems has been proposed by some of the authors in the context of RS-DFT. \cite{FerGinTou-JCP-18}
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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 $\mu = 0$.
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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.
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\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.
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Therefore, based on the propositions of ~Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general functional form for the PBE linked complementary functional:
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\titou{Obviously, $\n{2}{}(\br{})$ cannot be accessed in practice} and must be approximated by a function referred here as $\tn{2}{}(\br{})$.
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Therefore, based on the proposition of Ref.~\onlinecite{FerGinTou-JCP-18}, we introduce the general form of the PBE complementary functional:
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBE,X}{\Bas}[\n{}{},\n{2}{X},\rsmu{}{}] =
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\bE{\PBE}{}[\n{}{},\tn{2}{},\rsmu{}{}] =
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\int \n{}{}(\br{})
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\\
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{X}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\end{multline}
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with
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\begin{subequations}
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\begin{gather}
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\label{eq:epsilon_cmdpbe}
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\be{\text{c,md}}{\sr,\PBE}(\n{}{},\n{2}{X},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},\n{2}{X},s,\zeta) \rsmu{}{3} },
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\be{\text{c,md}}{\sr,\PBE}(\n{}{},\tn{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},\tn{2}{},s,\zeta) \rsmu{}{3} },
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\\
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\label{eq:beta_cmdpbe}
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\beta^\PBE(\n{}{},\n{2}{X},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{X}/\n{}{}}.
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\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{}{}}.
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\end{gather}
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\end{subequations}
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In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the present authors introduced a version, here-referred as "PBE-UEG", where the exact on-top pair density $\n{2}{}(\br{})$ was approximated by that of the UEG. In practice, within the present notation, the PBE-UEG functional reads:
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In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a new PBE-based functional, here-referred as \titou{PBE-UEG},
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBE,\UEG}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{}] =
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\bE{\titou{\PBE\text{-}\UEG}}{}[\n{}{},\n{2}{\UEG},\rsmu{}{}] =
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\int \n{}{}(\br{})
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\\
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\end{multline}
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with
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in which the on-top pair density was approximated by its UEG version,
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\begin{equation}
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\n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})),
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\end{equation}
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where the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.}
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where $g_0(n)$ is the UEG on-top pair distribution function [see Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}].
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\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 $\extrfunc(\n{2}{},\mu)$
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\begin{equation}
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\extrfunc(\n{2}{},\mu) = \n{2}{} \qty(1+ \frac{2}{\sqrt{\pi}\mu})^{-1},
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\end{equation}
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we propose the following approximation for the on-top pair density:
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\begin{equation}
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\tn{2}{\Bas}(\br{}) = \extrfunc(\n{2}{\Bas}(\br{}),\rsmu{}{\Bas}(\br{})).
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\end{equation}
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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.
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Therefore, we propose the "PBE-ontop" (PBEot) functional which reads
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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.
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However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems.
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Therefore, we propose here the "PBE-ontop" (PBEot) functional,
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBEot}{\Bas}[\n{}{},\tn{2}{\Bas},\rsmu{}{}] =
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\bE{\PBEot}{}[\n{}{},\ttn{2}{},\rsmu{}{}] =
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\int \n{}{}(\br{})
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\\
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{\Bas}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}.
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\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\ttn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\end{multline}
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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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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}
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\begin{equation}
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\ttn{2}{}(\br{}) = \n{2}{}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}.
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\end{equation}
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%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.
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The sole distinction between \titou{PBE-UEG} and PBEot is the level of approximation of the exact on-top pair density.
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%%%%%%%%%%%%%%%%%%%%%%%%
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