clean up theory
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-05-31 09:36:15 +0200
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%% Created for Pierre-Francois Loos at 2019-06-12 14:59:56 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -789,11 +789,12 @@
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@article{QP2,
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Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
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Date-Added = {2019-04-07 13:54:16 +0200},
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Date-Modified = {2019-05-28 22:19:25 +0200},
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Date-Modified = {2019-06-12 14:59:52 +0200},
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Doi = {10.1021/acs.jctc.9b00176},
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Journal = {J. Chem. Theory Comput.},
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Pages = {3591},
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Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
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Volume = {in press},
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Volume = {15},
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
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@ -130,7 +130,7 @@
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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{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
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\newcommand{\tn}[2]{\widetilde{n}_{#1}^{#2}}
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% energies
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@ -268,7 +268,8 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limits
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&
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\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
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\end{align}
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where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
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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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@ -307,48 +308,51 @@ 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{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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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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\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{}{\Bas}] =
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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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 $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior where the on-top pair density appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} In order to avoid the computation of the exact on-top pair density, the authors proposed to use that of the UEG which leads to
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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(\n{}{},s,\zeta) \rsmu{}{3} },
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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(\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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\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 $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with 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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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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As the UEG on-top pair density might not be suitable to treat strongly correlated systems, 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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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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\n{2}{\text{extr}}(\br{},\n{2}{\Bas},\rsmu{}{\Bas}) = \frac{\n{2}{\Bas}(\br{},\br{})}{1+ \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})}}.
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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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\begin{equation}
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Therefore, we introduce the ``on top'' PBE (PBEot) ECMD 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{}{},\rsmu{}{\Bas}] =
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\n{2}{\Bas}(\br{},\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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\end{equation}
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\bE{\PBEot}{\Bas}[\n{}{},\n{2}{},\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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\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{}{},s,\zeta,\n{2}{\Bas},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta,\n{2}{}) \rsmu{}{3} },
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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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\\
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\label{eq:beta_cmdpbe}
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\beta(\n{}{},s,\zeta,\n{2}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\text{extr}}(\n{2}{\Bas},\rsmu{}{\Bas})/\n{}{}}.
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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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\end{gather}
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\end{subequations}
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@ -540,7 +544,7 @@ However, these results also clearly evidence that special care has to be taken f
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\begin{squeezetable}
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\begin{table*}
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\caption{
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Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, carbon monoxyde, ethylene and water for various methods and basis sets.
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Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, ethylene and water for various methods and basis sets.
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The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries.
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See the {\SI} for raw data.}
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\label{tab:Mol}
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@ -598,12 +602,12 @@ However, these results also clearly evidence that special care has to be taken f
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& 0.11 & 0.02 & 0.00
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\\
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\\
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Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02
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& 0.05 & 0.00 & 0.00
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& 0.07 & 0.01 & 0.02
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& 0.07 & 0.00 & 0.02
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\\
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\\
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% Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02
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% & 0.05 & 0.00 & 0.00
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% & 0.07 & 0.01 & 0.02
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% & 0.07 & 0.00 & 0.02
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% \\
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% \\
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Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 &
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& -0.05 & -0.01 &
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& -0.04 & -0.01 &
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@ -733,13 +737,10 @@ iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen ato
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%%% FIG 4 %%%
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\begin{figure*}
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\includegraphics[height=0.45\linewidth]{C2_mu}
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\hspace{0.5cm}
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\includegraphics[height=0.45\linewidth]{C2_PBEot}
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\includegraphics[height=0.45\linewidth]{C2_PBE.pdf}
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\hspace{0.5cm}
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\includegraphics[height=0.45\linewidth]{C2_n2.pdf}
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\caption{$\rsmu{}{\Bas}$ (top left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (top right), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBE}$ (bottom left) and $\n{2}{\Bas}$ (bottom right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
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\includegraphics[height=0.35\linewidth]{C2_mu}
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\includegraphics[height=0.35\linewidth]{C2_PBEot}
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\includegraphics[height=0.35\linewidth]{C2_n2}
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\caption{$\rsmu{}{\Bas}$ (left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (center) and $\n{2}{\Bas}$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
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The two electronic states are both of $\Sigma_g^+$ symmetry.
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The carbon nuclei are located at $z= \pm 1.180$ bohr and represented by the thin black lines.}
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\label{fig:C2_mu}
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