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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, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, 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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\begin{document}
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\title{Excitation Energies Near The Complete Basis Set Limit}
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\title{Chemically-Accurate Excitation Energies With a Small Basis Set}
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\author{Emmanuel Giner}
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\affiliation{\LCT}
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@ -164,7 +164,8 @@
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\affiliation{\LCPQ}
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\begin{abstract}
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By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
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By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with, typically, augmented double-$\zeta$ basis sets.
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The present study clearly evidences that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set.
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\end{abstract}
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\maketitle
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@ -212,6 +213,195 @@ Contrary to our recent study on atomization and correlation energies, \cite{LooP
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%Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis-set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
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%The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%Let us assume that we have reasonable approximations of the FCI energy and density of a $\Ne$-electron system in an incomplete basis set $\Bas$, say the CCSD(T) energy $\E{\CCSDT}{\Bas}$ and the Hartree-Fock (HF) density $\n{\HF}{\Bas}$.
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%According to Eq.~(15) of Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, the exact ground-state energy $\E{}{}$ may be approximated as
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%\begin{equation}
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% \label{eq:e0basis}
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% \titou{\E{}{}
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% \approx \E{\CCSDT}{\Bas}
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% + \bE{}{\Bas}[\n{\HF}{\Bas}],}
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%\end{equation}
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%where
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%\begin{equation}
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% \label{eq:E_funcbasis}
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% \bE{}{\Bas}[\n{}{}]
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% = \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
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% - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
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%\end{equation}
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%is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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%In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
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%Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
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%Importantly, in the CBS limit (which we refer to as $\Bas \to \CBS$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \CBS} \bE{}{\Bas}[\n{}{}] = 0$.
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%This implies that
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%\begin{equation}
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% \label{eq:limitfunc}
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% \titou{\lim_{\Bas \to \CBS} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{\CBS} \approx \E{}{},}
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%\end{equation}
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%where \titou{$\E{\CCSDT}{\CBS}$ is the $\CCSDT$ energy} in the CBS limit.
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%Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.
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%In the case where \titou{$\CCSDT$ is replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{\CBS} = \E{}{}$.
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%Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the approximate nature of the $\CCSDT$ and $\HF$ methods, and the lack of self-consistency of the present scheme.
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%
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%The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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%Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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%for the lack of cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$ at the e-e coalescence points, a universal condition of exact wave functions.
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%Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
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%Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
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%Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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%
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%The first step of the present basis-set correction consists in obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$.
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%In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$.
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%As a final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
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%
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%We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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%\begin{equation}
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% \label{eq:def_weebasis}
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% \W{}{\Bas}(\br{1},\br{2}) =
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% \begin{cases}
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% \f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
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% \\
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% \infty, & \text{otherwise,}
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% \end{cases}
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%\end{equation}
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%where
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%\begin{equation}
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% \label{eq:n2basis}
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% \n{2}{\Bas}(\br{1},\br{2})
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% = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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%\end{equation}
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%and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
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%\begin{equation}
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% \label{eq:fbasis}
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% \f{}{\Bas}(\br{1},\br{2})
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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 $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
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%With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\citenum{GinPraFerAssSavTou-JCP-18})
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%\begin{equation}
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% \iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
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% \iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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%\end{equation}
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%which intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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%Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis-set incompleteness error originating from the e-e cusp.
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%As already discussed in Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
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%Thanks to its definition one can show that (see Appendix B of Ref.~\citenum{GinPraFerAssSavTou-JCP-18})
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%\begin{equation}
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% \label{eq:lim_W}
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% \lim_{\Bas \to \CBS}\W{}{\Bas}(\br{1},\br{2}) = \frac{1}{r_{12}},
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%\end{equation}
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%for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
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%A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
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%which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing.
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%Because $\W{}{\Bas}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
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%Although this choice is not unique, we choose here the range-separation function
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%\begin{equation}
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% \label{eq:mu_of_r}
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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 with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$.
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%Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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%As in Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05}
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%\begin{equation}
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% \label{eq:ec_md_mu}
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% \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
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% = \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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% - \mel*{\wf{}{\rsmu{}{}}[n]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[n]},
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%\end{equation}
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%where $\wf{}{\rsmu{}{}}[n]$ is defined by the constrained minimization
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%\begin{equation}
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%\label{eq:argmin}
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% \wf{}{\rsmu{}{}}[n] = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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%\end{equation}
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%with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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%The ECMD functionals admit, for any $\n{}{}$, the following two limits
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%\begin{align}
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% \label{eq:large_mu_ecmd}
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% \lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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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 KS-DFT.
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%The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
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%Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
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%Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
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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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%\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 short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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%In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
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%Inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
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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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%\end{equation}
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%where \titou{$\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and} $s=\abs{\nabla \n{}{}}/\n{}{4/3}$ 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, \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} yielding
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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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% \\
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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)}.
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%\end{gather}
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%\end{subequations}
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%The difference between the ECMD functional defined in Ref.~\citenum{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its \titou{uniform electron gas \cite{LooGil-WIRES-16} (UEG)} version, i.e.~$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, 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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%This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
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%The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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%As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
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%We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
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% \begin{equation}
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% \label{eq:WFC}
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% \WFC{}{\Bas}(\br{1},\br{2}) =
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% \begin{cases}
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% \fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
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% \\
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% \infty, & \text{otherwise,}
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% \end{cases}
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% \end{equation}
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%with
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%\begin{subequations}
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%\begin{gather}
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% \label{eq:fbasisval}
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% \fFC{}{\Bas}(\br{1},\br{2})
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% = \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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% \\
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% \nFC{2}{\Bas}(\br{1},\br{2})
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% = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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%\end{gather}
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%\end{subequations}
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%and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
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%It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
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%Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$.
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%The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
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%In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
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%The computational cost can be reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ with no memory footprint when $\wf{}{\Bas}$ is a single Slater determinant.
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%As shown in Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
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%Hence, we will stick to this choice throughout the present study.
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%In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
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%Nevertheless, this step usually has to be performed for most correlated WFT calculations.
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%To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
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%i) can be applied to any WFT method that provides an energy and a density,
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%ii) does not correct one-electron systems, and
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%iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method.
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -223,11 +413,11 @@ The present methodology is identical to the one described in Ref.~\onlinecite{Lo
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
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We refer the interested reader to Refs.~\citenum{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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The one-electron density and on-top density is computed from a very large CIPSI expansion containing several million determinants.
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All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
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For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
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The geometries have been extracted from Refs.~\citenum{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
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The geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
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They are also reported in the {\SI}.
|
||||
Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
|
||||
The FC density-based correction is used consistently with the FC approximation in WFT methods.
|
||||
@ -237,21 +427,14 @@ The FC density-based correction is used consistently with the FC approximation i
|
||||
\section{Results}
|
||||
\label{sec:res}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%=======================
|
||||
\subsection{Water}
|
||||
\label{sec:H2O}
|
||||
%=======================
|
||||
|
||||
%=======================
|
||||
\subsection{Formaldehyde}
|
||||
\label{sec:CH2O}
|
||||
%=======================
|
||||
|
||||
%=======================
|
||||
\subsection{Methylene}
|
||||
\label{sec:CH2}
|
||||
%=======================
|
||||
|
||||
As a first test of the present basis set correction, we consider the adiabatic transitions of methylene which have been thourhoughly studied in the literature with high-level ab initio methods.
|
||||
|
||||
%%% TABLE 1 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
@ -355,12 +538,27 @@ The FC density-based correction is used consistently with the FC approximation i
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{CH2}
|
||||
\caption{Error in adiabatic excitation energies $\Ead$ (in eV) of methylene for various basis sets and methods.}
|
||||
\label{fig:CH2}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%=======================
|
||||
\subsection{Rydberg States of Water and Ammonia}
|
||||
\label{sec:H2O-NH3}
|
||||
%=======================
|
||||
|
||||
Water and ammonia are two interesting molecules with Rydberge excited states which are highly sensitive to the radial completeness of the one-electron basis set.
|
||||
|
||||
%%% TABLE 2 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{
|
||||
Vertical absorption energies $\Eabs$ (in eV) of excited states of water, carbon dimer and ammonia for various methods and basis sets.}
|
||||
Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, water and ethylene for various methods and basis sets.
|
||||
The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19}.}
|
||||
\begin{ruledtabular}{}
|
||||
\begin{tabular}{lllddddddddddddd}
|
||||
& & & & \mc{12}{c}{Deviation with respect to TBE}
|
||||
@ -471,190 +669,181 @@ The FC density-based correction is used consistently with the FC approximation i
|
||||
& 0.07 & 0.02 & 0.03
|
||||
& 0.09 & 0.03 & 0.03
|
||||
& 0.06 & 0.03 & 0.04
|
||||
\\
|
||||
\\
|
||||
% Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00
|
||||
% & 0.07 & 0.00
|
||||
% & 0.11 & 0.00
|
||||
% & 0.11 & 0.00
|
||||
% \\
|
||||
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00
|
||||
% & 0.04 & -0.01
|
||||
% & 0.12 & 0.02
|
||||
% & 0.11 & 0.02
|
||||
% \\
|
||||
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03
|
||||
% & 0.07 & 0.02
|
||||
% & 0.04 & 0.00
|
||||
% & 0.02 & 0.00
|
||||
% \\
|
||||
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00
|
||||
% & 0.10 & 0.02
|
||||
% & 0.14 & 0.03
|
||||
% & 0.12 & 0.03
|
||||
% \\
|
||||
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01
|
||||
% & 0.08 & 0.00
|
||||
% & 0.16 & 0.04
|
||||
% & 0.14 & 0.04
|
||||
% \\
|
||||
% \\
|
||||
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -0.12 & -0.04 &
|
||||
& -0.05 & -0.01 &
|
||||
& -0.04 & -0.01 &
|
||||
& -0.02 & 0.00 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92 & 0.01 & 0.01 &
|
||||
& 0.00 & 0.00 &
|
||||
& 0.06 & 0.03 &
|
||||
& 0.06 & 0.03 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10 & -0.1 & -0.02 &
|
||||
& -0.03 & 0.00 &
|
||||
& -0.02 & 0.00 &
|
||||
& 0.00 & 0.01 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54 & 0.01 & 0.00 &
|
||||
& 0.07 & 0.03 &
|
||||
& 0.10 & 0.04 &
|
||||
& 0.08 & 0.04 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28 & -0.12 & -0.04 &
|
||||
& -0.03 & 0.00 &
|
||||
& 0.00 & 0.00 &
|
||||
& 0.00 & 0.02 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00 & -0.07 & -0.01 &
|
||||
& 0.01 & 0.03 &
|
||||
& 0.04 & 0.03 &
|
||||
& 0.05 & 0.04 &
|
||||
\\
|
||||
\\
|
||||
% Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Val. & 3.97 & 0.02 & 0.01 &
|
||||
% & 0.05 & 0.02 &
|
||||
% & 0.03 & 0.02 &
|
||||
% & 0.02 & 0.01 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & Ryd. & 7.30 & -0.19 & -0.07 &
|
||||
% & 0.00 & 0.00 &
|
||||
% & -0.02 & 0.00 &
|
||||
% & -0.04 & 0.00 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & Ryd. & 8.14 & -0.10 & -0.01 &
|
||||
% & 0.09 & 0.07 &
|
||||
% & 0.08 & 0.06 &
|
||||
% & 0.05 & 0.06 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.27 & -0.15 & -0.04 &
|
||||
% & 0.03 & 0.04 &
|
||||
% & 0.02 & 0.03 &
|
||||
% & 0.00 & 0.03 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Val. & 3.58 & 0.00 & 0.00 &
|
||||
% & 0.09 & 0.05 &
|
||||
% & 0.11 & 0.06 &
|
||||
% & 0.07 & 0.04 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Val. & 6.07 & 0.03 & 0.01 &
|
||||
% & 0.13 & 0.04 &
|
||||
% & 0.15 & 0.05 &
|
||||
% & 0.11 & 0.04 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & Ryd. & 7.14 & -0.19 & -0.08 &
|
||||
% & 0.01 & 0.01 &
|
||||
% & 0.02 & 0.01 &
|
||||
% & -0.01 & 0.00 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & Ryd. & 7.96 & -0.09 & -0.02 &
|
||||
% & 0.13 & 0.08 &
|
||||
% & 0.14 & 0.08 &
|
||||
% & 0.10 & 0.07 &
|
||||
% \\
|
||||
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Ryd. & 8.15 & -0.14 & -0.05 &
|
||||
% & 0.07 & 0.05 &
|
||||
% & 0.07 & 0.04 &
|
||||
% & 0.04 & 0.04 &
|
||||
% \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.}
|
||||
\fnt[2]{CT stands for charge transfer.}
|
||||
% \fnt[2]{CT stands for charge transfer.}
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% TABLE 3 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{
|
||||
Vertical absorption energies $\Eabs$ (in eV) of excited states of acetylene, ethylene and formaldehyde for various methods and basis sets.}
|
||||
\begin{ruledtabular}{}
|
||||
\begin{tabular}{lllddddddddd}
|
||||
& & & & \mc{8}{c}{Deviation with respect to TBE}
|
||||
\\
|
||||
\cline{5-12}
|
||||
& & & & \mc{2}{c}{exFCI}
|
||||
& \mc{2}{c}{exFCI+PBEot}
|
||||
& \mc{2}{c}{exFCI+PBE}
|
||||
& \mc{2}{c}{exFCI+LDA}
|
||||
\\
|
||||
\cline{5-6} \cline{7-8} \cline{9-10} \cline{11-12}
|
||||
Molecule & Transition & Nature & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ}
|
||||
\\
|
||||
\hline
|
||||
Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00
|
||||
& 0.07 & 0.00
|
||||
& 0.11 & 0.00
|
||||
& 0.11 & 0.00
|
||||
\\
|
||||
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00
|
||||
& 0.04 & -0.01
|
||||
& 0.12 & 0.02
|
||||
& 0.11 & 0.02
|
||||
\\
|
||||
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03
|
||||
& 0.07 & 0.02
|
||||
& 0.04 & 0.00
|
||||
& 0.02 & 0.00
|
||||
\\
|
||||
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00
|
||||
& 0.10 & 0.02
|
||||
& 0.14 & 0.03
|
||||
& 0.12 & 0.03
|
||||
\\
|
||||
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01
|
||||
& 0.08 & 0.00
|
||||
& 0.16 & 0.04
|
||||
& 0.14 & 0.04
|
||||
\\
|
||||
\\
|
||||
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -0.12 & -0.04
|
||||
& -0.05 & -0.01
|
||||
& -0.04 & -0.01
|
||||
& -0.02 & 0.00
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92 & 0.01 & 0.01
|
||||
& 0.00 & 0.00
|
||||
& 0.06 & 0.03
|
||||
& 0.06 & 0.03
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10 & -0.1 & -0.02
|
||||
& -0.03 & 0.00
|
||||
& -0.02 & 0.00
|
||||
& 0.00 & 0.01
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54 & 0.01 & 0.00
|
||||
& 0.07 & 0.03
|
||||
& 0.10 & 0.04
|
||||
& 0.08 & 0.04
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28 & -0.12 & -0.04
|
||||
& -0.03 & 0.00
|
||||
& 0.00 & 0.00
|
||||
& 0.00 & 0.02
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00 & -0.07 & -0.01
|
||||
& 0.01 & 0.03
|
||||
& 0.04 & 0.03
|
||||
& 0.05 & 0.04
|
||||
\\
|
||||
\\
|
||||
Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Val. & 3.97 & 0.02 & 0.01
|
||||
& 0.05 & 0.02
|
||||
& 0.03 & 0.02
|
||||
& 0.02 & 0.01
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & Ryd. & 7.30 & -0.19 & -0.07
|
||||
& 0.00 & 0.00
|
||||
& -0.02 & 0.00
|
||||
& -0.04 & 0.00
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & Ryd. & 8.14 & -0.10 & -0.01
|
||||
& 0.09 & 0.07
|
||||
& 0.08 & 0.06
|
||||
& 0.05 & 0.06
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.27 & -0.15 & -0.04
|
||||
& 0.03 & 0.04
|
||||
& 0.02 & 0.03
|
||||
& 0.00 & 0.03
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Val. & 3.58 & 0.00 & 0.00
|
||||
& 0.09 & 0.05
|
||||
& 0.11 & 0.06
|
||||
& 0.07 & 0.04
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Val. & 6.07 & 0.03 & 0.01
|
||||
& 0.13 & 0.04
|
||||
& 0.15 & 0.05
|
||||
& 0.11 & 0.04
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & Ryd. & 7.14 & -0.19 & -0.08
|
||||
& 0.01 & 0.01
|
||||
& 0.02 & 0.01
|
||||
& -0.01 & 0.00
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & Ryd. & 7.96 & -0.09 & -0.02
|
||||
& 0.13 & 0.08
|
||||
& 0.14 & 0.08
|
||||
& 0.10 & 0.07
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Ryd. & 8.15 & -0.14 & -0.05
|
||||
& 0.07 & 0.05
|
||||
& 0.07 & 0.04
|
||||
& 0.04 & 0.04
|
||||
\\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{CH2}
|
||||
\caption{Adiabatic excitation energies of methylene for various basis sets and methods.}
|
||||
\label{fig:CH2}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{NH3}
|
||||
\caption{Vertical excitation energies of ammonia for various basis sets and methods.}
|
||||
\label{fig:NH3}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{C2}
|
||||
\caption{Vertical excitation energies for two doubly-excited states of the carbon dimer for various basis sets and methods.}
|
||||
\label{fig:C2}
|
||||
\end{figure}
|
||||
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{H2O}
|
||||
\caption{Vertical excitation energies of water for various basis sets and methods.}
|
||||
\caption{Error in vertical excitation energies (in eV) of water for various basis sets and methods.}
|
||||
\label{fig:H2O}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 4 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{C2H2}
|
||||
\caption{Vertical excitation energies of acetylene for various basis sets and methods.}
|
||||
\label{fig:C2H2}
|
||||
\includegraphics[width=\linewidth]{NH3}
|
||||
\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.}
|
||||
\label{fig:NH3}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%=======================
|
||||
\subsection{Doubly-Excited States of the Carbon Dimer}
|
||||
\label{sec:C2}
|
||||
%=======================
|
||||
It is also interesting to study doubly-excited states.
|
||||
In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJAc-JCTC-19}
|
||||
|
||||
%%% FIG 4 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{C2}
|
||||
\caption{Error in vertical excitation energies $\Eabs$ (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods.}
|
||||
\label{fig:C2}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
|
||||
%=======================
|
||||
\subsection{Ethylene}
|
||||
\label{sec:C2H4}
|
||||
%=======================
|
||||
|
||||
Ethylene is an interesting molecules as it contains both valence and Rydberg excited states.
|
||||
|
||||
%\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{C2H2}
|
||||
% \caption{Error in vertical excitation energies (in eV) of acetylene for various basis sets and methods.}
|
||||
% \label{fig:C2H2}
|
||||
%\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{C2H4}
|
||||
\caption{Vertical excitation energies of ethylene for various basis sets and methods.}
|
||||
\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.}
|
||||
\label{fig:C2H4}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{CH2O}
|
||||
\caption{Vertical excitation energies of formaldehyde for various basis sets and methods.}
|
||||
\label{fig:CH2O}
|
||||
\end{figure}
|
||||
%\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{CH2O}
|
||||
% \caption{Error in vertical excitation energies $\Eabs$ (in eV) of formaldehyde for various basis sets and methods.}
|
||||
% \label{fig:CH2O}
|
||||
%\end{figure}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusion}
|
||||
\label{sec:ccl}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
We have shown that by employing the recently proposed density-based basis set correction developed by some of the authors, one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
|
||||
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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\section*{Supporting Information Available}
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