We report a first generation of local, weight-dependent exchange-correlation density-functional approximations (DFAs) for molecules.
These density-functional approximations for ensembles (eDFAs) incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
They are specially designed for the computation of double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) to ensembles.
The resulting eDFAs, dubbed eLDA, which are based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence.
Their accuracy is illustrated by computing on the prototypical H$_2$ molecule.
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.
At a relatively low computational cost (at least compared to the other excited-state methods), TD-DFT can provide accurate transition energies for low-lying excited states in organic molecules.
Importantly, setting up a TD-DFT calculation for a given system is an almost pain-free process from the user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.
Indeed, TD-DFT is a in-principle exact theory which recast the many-body problem by transferring its complexity to the xc functional.
However, TD-DFT is far from being perfect, and, in practice, approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible substantial variations in the quality of the excitation energy for two different choices of xc functionals.
Moreover, because it was so popular, it has been studied in excruciated details, and researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
Practically, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation is that double excitations are completely absent from the TD-DFT spectra.
Moreover, TD-DFT has problems with charge-transfer and Rydberg excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the xc functional.
The paper is organised as follows.
In Sec.~\ref{sec:theo}, ...
Section \ref{sec:func} provides details about the construction of the weight-dependent exchange-correlation functional.
The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
Unless otherwise stated, atomic units are used throughout.
The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT.
As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system.
Based on highly-accurate calculations (see below), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R =1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
$-\e{c}{(I)}$ as a function of the radius of the ring $R$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014}
Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
This is a crucial point as we intend 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).