Theory
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@ -186,34 +186,10 @@ Although they have been extremely successful to speed up convergence of ground s
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Instead of F12 methods, here we propose to follow a different philosophy and investigate the performances of the recently proposed universal density-based basis set
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incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18}
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This density-based correction relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory \cite{TouColSav-PRA-04, AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15, LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} (RS-DFT) to estimate the basis-set incompleteness error.
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This choice is motivated by the much faster convergence of these methods with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
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Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular systems which is a much tougher test for the reasons mentioned above.
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RS-DFT combines rigorously density-functional theory (DFT) and wave function theory (WFT) via a decomposition of the electron-electron interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
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As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the convergence of these methods with respect to the size of the basis set is significantly improved. \cite{FraMusLupTou-JCP-15}
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%Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
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%Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
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%
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%WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new WFT \textit{ans\"atze}.
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%The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems.
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%By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually high.
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%One of the most fundamental drawbacks of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
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%This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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%To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
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%The resulting F12 methods yield a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
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%For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
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%To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
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%
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%Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
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%The attractiveness of DFT originates from its very favorable accuracy/cost ratio as it often provides reasonably accurate energies and properties at a relatively low computational cost.
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%Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
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%Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing Perdew's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
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%In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
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%
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%Progress toward unifying WFT and DFT are on-going.
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%In particular, range-separated DFT (RS-DFT) (see Ref.~\citenum{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
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%As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
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%Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
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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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Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular electronically-excited systems which is a much tougher test for the reasons mentioned above.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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@ -221,7 +197,7 @@ Contrary to our recent study on atomization and correlation energies, \cite{LooP
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%%%%%%%%%%%%%%%%%%%%%%%%
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The present basis set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}$ and $\n{0}{\Bas}$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{},\br{})$ and $\n{2,k}{\Bas}(\br{},\br{})$,
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Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as
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Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18}
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\begin{align}
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\label{eq:ECBS}
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\E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}],
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@ -259,129 +235,46 @@ is the excitation energy in $\Bas$ and
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\DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = \bE{}{\Bas}[\n{k}{\Bas}] - \bE{}{\Bas}[\n{0}{\Bas}]
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\end{equation}
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its basis set correction.
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An important of the present correction is
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = 0.
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\end{equation}
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In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit.
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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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%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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In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference, \cite{TouGorSav-TCA-05} $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$.
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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 key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- is defined 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 $\br{}$, with \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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and
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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 $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ 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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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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\subsection{Short-range correlation functionals}
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@ -409,7 +302,7 @@ $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between
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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}{\Bas}(\br{},\br{})}.
|
||||
\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{}(\br{},\br{})/\n{}{}}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional.
|
||||
|
||||
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Reference in New Issue
Block a user