However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous ensemble derivative contribution to the excitation energies.
In the present article, we discuss the construction of first-rung (\textit{i.e.}, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
In the spirit of optimally-tuned range-separated hybrid functionals, a two-step system-dependent procedure is proposed to obtain accurate energies associated with double excitations.
Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida_1995,Ulrich_2012,Loos_2020a}
At a moderate computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984}
First, within the linear-response approximation, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation).
Second, the time dependence of the functional is usually treated at the local approximation level within the standard adiabatic approximation.
In other words, memory effects are absent from the xc functional which is assumed to be local in time
(the xc energy is in fact an xc action, not an energy functional). \cite{Vignale_2008}
Third and more importantly in the present context, a major issue of TD-DFT actually originates directly from the choice of the xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Because its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional.
One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Elliott_2011}
Although these double excitations are usually experimentally dark (which means they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007} They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018,Loos_2019,Loos_2020b}
One possible solution to access double excitations within TD-DFT is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009}
However, spin contamination might be an issue. \cite{Huix-Rotllant_2010}
In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
In this approach the xc kernel is made frequency dependent, which allows to treat doubly-excited states. \cite{Romaniello_2009a,Sangalli_2011,Loos_2019}
Maybe surprisingly, another possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
With a computational cost similar to traditional KS-DFT, DFT for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable alternative following such a strategy currently under active development.\cite{Gidopoulos_2002,Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018,Deur_2019,Kraisler_2013,Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Smith_2016,Senjean_2018}
In the assumption of monotonically decreasing weights, eDFT for excited states has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, the so-called Gross--Oliveria--Kohn (GOK) variational principle. \cite{Gross_1988a}
In short, GOK-DFT (\ie, eDFT for neutral excitations) is the density-based analog of state-averaged wave function methods, and excitation energies can then be easily extracted from the total ensemble energy. \cite{Deur_2019}
Although the formal foundations of GOK-DFT have been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct ground-state functionals as shown in Refs.~\onlinecite{Loos_2014a,Loos_2014b,Loos_2017a}, where the authors proposed generalised LDA exchange and correlation functionals.
Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems.
Unlike any standard functional, this first-rung functional incorporates derivative discontinuities thanks to its natural weight dependence, and has shown to deliver accurate excitation energies for both single and double excitations.
Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous ensemble derivative contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)}\le\ldots\le\E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw=(\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1}\ew{I}$, and $\ew{0}\ge\ldots\ge\ew{\nEns-1}$.
where $\lbrace\overline{\Psi}^{(I)}\rbrace_{0\le I \le\nEns-1}$ is a set of $\nEns$ orthonormal trial wave functions.
The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace\overline{\Psi}^{(I)}\rbrace_{0\le I \le\nEns-1}=\lbrace\Psi^{(I)}\rbrace_{0\le I \le\nEns-1}$.
Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988a}
not necessarily match the \textit{exact} (interacting) individual-state
densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
Nevertheless, these densities can still be extracted in principle
is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$].
The latters are determined by solving the ensemble KS equation
The self-consistent GOK-DFT calculations [see Eqs.~\eqref{eq:eKS} and \eqref{eq:ens_KS_dens}] have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented.
For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \cite{Becke_1988b,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{}=\n{\downarrow}{}=\n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns=3$) where the ground state ($I=0$ with weight $1-\ew{1}-\ew{2}$), the first singly-excited state ($I=1$ with weight $\ew{1}$), as well as the first doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0\le\ew{2}\le1/3$ and $\ew{2}\le\ew{1}\le(1-\ew{2})/2$ to ensure the GOK variational principle.
%Taking a generic two-electron system as an example, the individual one-electron densities read
%\begin{subequations}
%\begin{align}
% \n{}{(0)} & = 2 \HOMO{2},
% \\
% \n{}{(1)} & = \HOMO{2} + \LUMO{2},
% \\
% \n{}{(2)} & = 2 \LUMO{2},
%\end{align}
%\end{subequations}
%and they can be combined to produce the ensemble density
%\begin{equation}
% \label{eq:nw1w2}
% \n{}{(\ew{1},\ew{2})}
% = (1 - \ew{1} - \ew{2}) \n{}{(0)}
% + \ew{1} \n{}{(1)} + \ew{2} \n{}{(2)}.
%\end{equation}
%For analysis purposes, Eq.~\eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{}\equiv\ew{1}=\ew{2}$), and we consider the zero-weight limit (\ie, $\ew{}\equiv\ew{1}=\ew{2}=0$), and the equiweight ensemble (\ie, $\ew{}\equiv\ew{1}=\ew{2}=1/3$).
Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
The pure-state limit, $\ew{1}=0\land\ew{2}=1$, is nonetheless of particular interest as it is a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
%Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
%These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
%Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.
In this Section, we propose a two-step procedure to design, first, a weight- and system-dependent local exchange functional in order to remove the curvature of the ensemble energy.
First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bond length (\ie, $\RHH=1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ of the same symmetry as the ground state (\ie, $\Sigma_g^+$), and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $7$ eV from $\ew{}=0$ to $1/3$.
Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights.
\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) of the doubly-excited state $\Ex{}{(2)}$ as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
one can easily reverse-engineer (for this particular system, geometry, basis set, and excitation) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0\le\ew{2}\le1$ assuming a perfect linearity between the pure-state limits $\ew{1}=\ew{2}=0$ (ground state) and $\ew{1}=0\land\ew{2}=1$ (doubly-excited state).
makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by construction), and removes some of the curvature of $\E{}{\ew{}}$ (see yellow curve in Fig.~\ref{fig:Ew_H2}).
It also makes the excitation energy much more stable (with respect to $\ew{}$), and closer to the FCI reference (see yellow curve in Fig.~\ref{fig:Om_H2}).
The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have been obtained via a least-square fit of the non-linear component of the ensemble energy computed between $\ew{2}=0$ and $\ew{2}=1$ by steps of $0.025$.
Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure a correct behavior on the whole range of weights in order to obtain accurate excitation energies.
As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{1}=\ew{2}=0$ and $\ew{1}=0\land\ew{2}=1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above.
Finally, let us mention that, around $\ew{}=0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature one needs to catch in order to get accurate excitation energies in the zero-weight limit.
$\Cx{\ew{}}/\Cx{}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH=1.4$ bohr (red), and $\RHH=3.7$ bohr (green).
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
To build this correlation functional, we consider the singlet ground state, the first singly-excited state, as well as the first doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome). \cite{Loos_2009a,Loos_2009c,Loos_2010e}
Notably, these three states have the same (uniform) density $\n{}{}=2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
%Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
%As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron in its ground state, thus yielding an electron density that is uniform on the 3-sphere.
Thanks to highly-accurate calculations \cite{Loos_2009a,Loos_2009c,Loos_2010e} and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0}, \eqref{eq:eHF_1}, and \eqref{eq:eHF_2}, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following simple Pad\'e approximant \cite{Sun_2016,Loos_2020}
Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R =1/(\pi^2\n{}{})^{1/3}$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the two-electron FUEG.
$-\e{\co}{(I)}$ as a function of the radius of the glome $R =1/(\pi^2\n{}{})^{1/3}$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the two-electron FUEG.
Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
Because our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons), we employ a simple ``embedding'' scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014}
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
In the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
%We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{}=1/3$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{}=1$ (\textit{vide supra}).
\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{}=1/4$ and $\ew{}=1/2$, respectively, instead of performing an interpolation between two different calculations.}
Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\ew{}=0$ and $\ew{}=1/2$.
Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH=1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH=3.7$ bohr).
Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble as defined in Sec.~\ref{sec:H2}
We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH=3.7$ bohr.
One clearly sees that the correction brought by CC-S is much more gentle than at $\RHH=1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH=3.7$ bohr.
Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}.
As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
For $\RHH=3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-weights.
%\manu{We did not mention HF exchange neither in the theory section nor
%in the computational details. We should be clear about this. Is this an
%ad-hoc correction, like in our previous work on ringium? Is HF exchange
%used for the full ensemble energy (i.e. the HF interaction energy is
%computed with the ensemble density matrix and therefore with
%ghost-interaction errors) or for
%individual energies (that you state-average then), like in our previous
%work. I guess the latter option is what you did. We need to explain more
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
As expected from the linearity of the ensemble energy, the CC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
For additional comparison, we provide the excitation energy calculated by short-range multiconfigurational DFT in Ref.~\onlinecite{Senjean_2015}, using the (weight-independent) srLDA functional \cite{Toulouse_2004} and setting the range-separation parameter to $\mu=0.4$ bohr$^{-1}$.
The excitation energy improves by $1$ eV compared to the weight-independent SVWN5 functional, thus showing that treating the long-range part of the electron-electron repulsion by wave function theory plays a significant role.
%\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?}
Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} at $\RHH=3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals.
Similar to \ce{H2}, our ensemble contains the ground state of configuration $1s^2$, the lowest singlet excited state of configuration $1s2s$, and the first doubly-excited state of configuration $2s^2$.
In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963}
In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree for this $1s^2\rightarrow2s^2$ transition.
The excitation energies associated with this double excitation computed with various methods and combinations of xc functionals are gathered in Table \ref{tab:BigTab_He}.
The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha=+1.912\,574$, $\beta=+2.715\,267$, and $\gamma=+2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{}=0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} obtained with the d-aug-cc-pVQZ basis set for various methods and combinations of xc functionals.
In the present article, we have discussed the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron systems (\ce{He} and \ce{H2}) specifically designed for the computation of double excitations within GOK-DFT, a time-\textit{independent} formalism thanks to which one can extract excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing some of the curvature of the ensemble energy), and improves excitation energies.
Although the weight-dependent correlation functional developed in this paper (eVWN5) performs systematically better than their weight-independent counterpart (VWN5), the improvement remains rather small.
To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure,
\ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead
of the self-consistent one.
Density- and state-driven errors \cite{Gould_2019,Fromager_2020} can also be calculated to provide additional insights about the present results.
In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future, and we hope to be able to report further on this in the near future.
%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
