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@ -724,7 +724,7 @@ However, an obvious drawback of using FUEGs is that the resulting eDFA will inex
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Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems by combining these FUEGs with the usual IUEG to construct a weigh-dependent LDA functional for ensembles (eLDA).
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As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
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The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
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The most appealing feature of ringium regarding the development of functionals in the context of eDFT is the fact that both ground- and excited-state densities are uniform.
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As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
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This is a necessary condition for being able to model derivative discontinuities.
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Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a}
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@ -824,21 +824,19 @@ In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
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Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
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Finally, we note that, by construction,
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\begin{equation}
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\left. \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}}\right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} = \be{c}{(J)}(\n{\bGam{\bw}}{}(\br{})) - \be{c}{(0)}(\n{\bGam{\bw}}{}(\br{})).
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\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})).
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\end{equation}
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%\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:comp_details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
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Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium.
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Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium in the following.
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In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$.
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%\titou{Comment on the quality of these density: density- and functional-driven errors?}
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These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
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For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
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We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
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\begin{equation}
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\AO{\mu}(x) =
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@ -849,13 +847,13 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
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\end{cases}
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\end{equation}
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with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
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For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ been set to $10^{-5}$.
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For KS-DFT, KS-eDFT and TDDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
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For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ is set to $10^{-5}$.
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For KS-DFT, KS-eDFT and TDDFT calculations, a 51-point Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
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In order to test the present eLDA functional we have performed various sets of calculations.
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To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
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For the single excitations, we have also performed time-dependent LDA (TDLDA) calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}], and the effect of the Tamm-Dancoff approximation (TDA) has been also investigated. \cite{Dreuw_2005}
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Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
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In order to test the present eLDA functional we perform various sets of calculations.
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To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
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For the single excitations, we also perform time-dependent LDA (TDLDA) calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}], and the effect of the Tamm-Dancoff approximation (TDA) has been also investigated. \cite{Dreuw_2005}
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Concerning the KS-eDFT and eHF calculations, two sets of weight are tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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@ -930,7 +928,7 @@ As a rule of thumb, in the weak and intermediate correlation regimes, we see tha
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Moreover, we note that, in the strong correlation regime (left graph of Fig.~\ref{fig:EvsN}), the single excitation energies obtained at the state-averaged KS-eLDA level remain in good agreement with FCI and are much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
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This also applies to double excitations, the discrepancy between FCI and KS-eLDA remaining of the order of a few percents in the strong correlation regime.
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These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations.
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This is definitely a very pleasing outcome.
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This is definitely a very pleasing outcome, which additionally shows that, even though we have designed the eLDA functional based on a two-electron model system, the present methodology is applicable to any 1D electronic system.
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%%% FIG 4 %%%
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\begin{figure}
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