modified theory section
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@ -150,6 +150,8 @@
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\newcommand{\si}{\sigma}
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\newcommand{\sis}{\sigma^\star}
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\newcommand{\extrfunc}[0]{\epsilon}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
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@ -272,7 +274,12 @@ which correspond to the WFT limit ($\mu = \infty$) and the DFT limit ($\mu = 0$)
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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}
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The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
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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}
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It is defined as
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\begin{equation}
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\label{eq:def_mu}
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{})
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\end{equation}
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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}
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The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by
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\begin{equation}
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\label{eq:def_weebasis}
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@ -296,8 +303,7 @@ and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals.
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An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{-1}$ as $\Bas \to \CBS$ , and
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ensures that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete.
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An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{-1}$ as $\Bas \to \CBS$ , which implies that $\lim_{\Bas \rightarrow \CBS} \rsmu{}{\Bas}(\br{}) = \infty$ and therefore ensures 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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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -312,50 +318,60 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
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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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To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional\cite{LooPraSceTouGin-JPCL-19},
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\begin{equation}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{}] =
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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\end{equation}
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where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
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$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ and the exact large-$\rsmu{}{}$ behavior where the on-top pair density naturally appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
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In order to avoid the computation of the exact on-top pair density, we proposed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} to use an approximated alternative which yields
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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{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},s,\zeta) \rsmu{}{3} },
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\\
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\label{eq:beta_cmdpbe}
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\beta^\PBE(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)/\n{}{}}.
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\end{gather}
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\end{subequations}
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where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ is the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.
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We will refer to this functional as the PBE ECMD functional.
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The functional purely based on the LDA presents two different defects: i) at small $\mu$ it tends to overestimate the correlation energy, and ii) the quantities based on the UEG 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~\cite{FerGinTou-JCP-18} in the context of the RS-DFT where they proposed to connect between the exact behaviour at large $\mu$ and the Perdew-Burke-Ernzerhof (PBE) functional at $\mu=0$. The use of the PBE correlation functional has clearly shown to improve the results for small $\mu$, and the exact behaviour at large $\mu$ naturally introduces the \textit{exact} on-top pair density $n_2(\br{})$ which contains information about the level of strong correlation of the system.
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As the UEG on-top pair density might not be suitable to treat strongly correlated systems, \titou{we propose here to use an extrapolation of the exact on-top pair density based on the on-top pair density}
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\begin{equation}
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\tn{2}{}(\br{},\n{2}{},\rsmu{}{}) = \n{2}{}(\br{},\br{}) \qty(1+ \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}.
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\end{equation}
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Therefore, we introduce the ``on top'' PBE (PBEot) ECMD functional which reads
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Obviously, the exact on-top pair density 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\cite{FerGinTou-JCP-18}, we introduce the general functional form for the PBE linked complementary functional:
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBEot}{\Bas}[\n{}{},\n{2}{},\rsmu{}{}] =
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\bE{\PBE,X}{\Bas}[\n{}{},\n{2}{X},\rsmu{}{}] =
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\int \n{}{}(\br{})
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\\
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\times \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),\n{2}{}(\br{},\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
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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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\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,\PBEot}(\n{}{},\n{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBEot(\n{}{},\n{2}{},s,\zeta) \rsmu{}{3} },
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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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\\
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\label{eq:beta_cmdpbe}
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\beta^\PBEot(\n{}{},\n{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}(\n{2}{},\rsmu{}{})/\n{}{}}.
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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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\end{gather}
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\end{subequations}
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In Ref~\cite{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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\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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\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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\end{multline}
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with
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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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As shown in Ref~\cite{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 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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\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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\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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\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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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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@ -402,9 +418,9 @@ Total energies for each state can be found in the {\SI}.
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Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.015 eV.
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From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and converge steadily to the CBS limit when one increases the size of the basis set.
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Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies.
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The performance of the PBE and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive.
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The performance of the PBE-UEG and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive.
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Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy.
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Note that the results for the PBE functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below).
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Note that the results for the PBE-UEG functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below).
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It is also quite evident that, the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state.
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This trend is quite systematic as we shall see below.
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@ -458,7 +474,7 @@ This trend is quite systematic as we shall see below.
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& $1.378$ [$-0.011$]
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& $2.489$ [$-0.016$] \\
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\\
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exFCI+PBE & AVDZ
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exFCI+PBE-UEG & AVDZ
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& $0.308$ [$-0.080$]
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& $1.388$ [$-0.002$]
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& $2.560$ [$+0.056$] \\
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@ -555,7 +571,7 @@ However, these results also clearly evidence that special care has to be taken f
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\cline{5-16}
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& & & & \mc{3}{c}{exFCI}
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& \mc{3}{c}{exFCI+PBEot}
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& \mc{3}{c}{exFCI+PBE}
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& \mc{3}{c}{exFCI+PBE-UEG}
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& \mc{3}{c}{exFCI+LDA}
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\\
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\cline{5-7} \cline{8-10} \cline{11-13} \cline{14-16}
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@ -715,7 +731,7 @@ They have been recently studied with state-of-the-art methods, and have been sho
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The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
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An interesting point here is that one really needs to consider the PBEot functional to get chemically-accurate absorption energies with the AVDZ atomic basis set.
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We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
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In other words, the UEG on-top density used in the LDA and PBE functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system.
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In other words, the UEG on-top density used in the LDA and PBE-UEG functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system.
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%%% FIG 5 %%%
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\begin{figure}
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@ -758,7 +774,7 @@ In the present context, ethylene is a particularly interesting system as it cont
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Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
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Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
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%(Note that one cannot afford exFCI/AVQZ calculations for ethylene.)
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Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
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Consistently with the previous examples, the LDA and PBE-UEG functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
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%%% FIG 6 %%%
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\begin{figure}
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@ -777,7 +793,7 @@ Consistently with the previous examples, the LDA and PBE functionals are slightl
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We have shown that, by employing the recently proposed density-based basis set correction developed by some of the authors, \cite{GinPraFerAssSavTou-JCP-18} one can obtain, using sCI methods, chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
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This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets.
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The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set.
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Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE functionals relying on the UEG on-top density.
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Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE-UEG functionals relying on the UEG on-top density.
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We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future.
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%%%%%%%%%%%%%%%%%%%%%%%%
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