\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
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\begin{document}
\title{A weight-dependent local correlation density-functional approximation for ensembles}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Fromager}
\email{fromagere@unistra.fr}
\affiliation{\LCQ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We report a local, weight-dependent correlation density-functional approximation that incorporates information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
This density-functional approximation for ensembles is specially
designed for the computation of single and double excitations within
Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for neutral
excitations), and can be seen as a natural extension of the ubiquitous local-density approximation in the context of ensembles.
The resulting density-functional approximation, based on both finite and infinite uniform electron gas models, automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence.
Its accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
Although the present weight-dependent functional has been specifically
designed for one-dimensional systems, the methodology proposed here is
general, \ie, directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Over the last two decades, density-functional theory (DFT)
\cite{Hohenberg_1964,Kohn_1965,ParrBook} has become the method of choice for
modeling the electronic structure of large molecular systems and
materials.
The main reason is that, within DFT, the quantum contributions to the
electronic repulsion energy --- the so-called exchange-correlation (xc)
energy --- is rewritten as a functional of the electron density $\n{}{}\equiv\n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function.
The complexity of the many-body problem is then transferred to the xc
density functional.
Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003,Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tapavicza_2008,Levine_2006}
The description of strongly multiconfigurational ground states (often
referred to as ``strong correlation problem'') still remains a
In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong.
Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action}\cite{Vignale_2008} which is a
functional of the time-dependent density $\n{}{}\equiv\n{}{}(\br,t)$ and, as
such, it should incorporate memory effects. Standard implementations of TDDFT rely on
the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} In other
words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
As a result, double electronic excitations (where two electrons are simultaneously promoted by a single photon) are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
When affordable (\ie, for relatively small molecules), time-independent
state-averaged wave function methods
\cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002,Helgakerbook} can be employed to fix the various issues mentioned above.
The basic idea is to describe a finite (canonical) ensemble of ground
and excited states altogether, \ie, with the same set of orbitals.
Interestingly, a similar approach exists in DFT. Referred to as
Gross--Oliveira--Kohn (GOK) DFT, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} it was proposed at the end of the 80's as a generalization
of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
In GOK-DFT, the ensemble xc energy is a functional of the
density {\it and} a
function of the ensemble weights. Note that, unlike in conventional
Boltzmann ensembles, \cite{Pastorczak_2013} the ensemble weights (each state in the ensemble
is assigned a given and fixed weight) are allowed to vary
independently in a GOK ensemble.
The weight dependence of the xc functional plays a crucial role in the
It actually accounts for the derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
Even though GOK-DFT is in principle able to
describe near-degenerate situations and multiple-electron excitation
processes, it has not
been given much attention until quite recently. \cite{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,Senjean_2018,Smith_2016}
One of the reason is the lack, not to say the absence, of reliable
density-functional approximations for ensembles.
The most recent works dealing with this particular issue are still fundamental and
exploratory, as they rely either on simple (but nontrivial) model
In other words, the Coulomb interaction used in this work corresponds to
particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
The paper is organized as follows.
Exact and approximate formulations of GOK-DFT are discussed in Sec.~\ref{sec:eDFT},
with a particular emphasis on the extraction of individual energy levels.
In Sec.~\ref{sec:eDFA}, we detail the construction of the
weight-dependent local correlation functional specially designed for the
computation of single and double excitations within GOK-DFT.
Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA functional by computing single and double excitations in 1D many-electron systems in the weak, intermediate and strong correlation regimes.
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
Atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:eDFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{GOK-DFT}\label{subsec:gokdft}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we give a brief review of GOK-DFT and discuss the
extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
individual exchange energies.
Let us start by introducing the GOK ensemble energy \cite{Gross_1988a}
\beq\label{eq:exact_GOK_ens_ener}
\E{}{\bw}=\sum_{K \geq 0}\ew{K}\E{}{(K)},
\eeq
where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hH=\hh+\hWee$, where
As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange (Hx) and correlation (c) energies are described with density functionals that are \textit{weight dependent}.
We focus in the following on the (exact) Hx part, which is defined as~\cite{Gould_2017}
The (approximate) description of the correlation part is discussed in
Sec.~\ref{sec:eDFA}.
In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example).
As pointed out recently in Ref.~\onlinecite{Deur_2019}, the latter can be extracted
exactly from a single ensemble calculation as follows:
is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Density-functional approximations for ensembles}
\label{sec:eDFA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Paradigm}
\label{sec:paradigm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Most of the standard local and semi-local density-functional approximations rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
One major drawback of the jellium paradigm, when it comes to develop density-functional approximations for ensembles, is that the ground and excited states are not easily accessible like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
Moreover, because the infinite uniform electron gas model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
From this point of view, using finite uniform electron gases, \cite{Loos_2011b,
Gill_2012} which have, like an atom, discrete energy levels and non-zero
gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
However, an obvious drawback of using finite uniform electron gases is
that the resulting density-functional approximation for ensembles
will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
Here, we propose to construct a weight-dependent LDA functional for the
calculation of excited states in 1D systems by combining finite uniform electron gases with the
usual infinite uniform electron gas paradigm.
As a finite uniform electron gas, 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}
The most appealing feature of ringium regarding the development of
functionals in the context of GOK-DFT is the fact that both ground- and
excited-state densities are uniform, and therefore {\it equal}.
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
This is a necessary condition for being able to model the
correlation ensemble derivatives [last term
on the right-hand side of Eq.~\eqref{eq:exact_ener_level_dets}].
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous infinite uniform electron gas paradigm. \cite{Loos_2013,Loos_2013a}
Let us stress that, in a finite uniform electron gas like ringium, the interacting and
noninteracting densities match individually for all the states within the
ensemble
(these densities are all equal to the uniform density), which means that
so-called density-driven correlation
effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.
Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
The present weight-dependent density-functional approximation is specifically designed for the
calculation of excited-state energies within GOK-DFT.
To take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
To ensure the GOK variational principle, \cite{Gross_1988a} the
triensemble weights must fulfil the following conditions: \cite{Deur_2019}
$0\le\ew{2}\le1/3$ and $\ew{2}\le\ew{1}\le(1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.
All these states have the same (uniform) density $\n{}{}=2/(2\pi R)$, where $R$ is the radius of the ring on which the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
Generalization to a larger number of states is straightforward and is left for future work.
Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (\ie, per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
\titou{Schematic view of the ``embedding'' scheme: the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
The electronic excitation occurs locally, \ie, on the impurity.}
One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
Obviously, the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the finite uniform electron gas.
However, one can partially cure this dependency by applying a simple \titou{``embedding''} scheme \titou{(illustrated in Fig.~\ref{fig:embedding})} in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} as follows:
Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
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.
In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2\le\nEl\le7$.
These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
\titou{The one-electron density in these two regimes of correlation is represented in Fig.~\ref{fig:rho}.}
%%% FIG 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{rho}
\caption{
\titou{Ground-state one-electron density $\n{}{}(x)$ of 4-boxium (\ie, $N =4$) for $L =\pi/32$ (left) and $L =32\pi$ (right).
In the weak correlation regime (small box length), the one-electron density is much more delocalized than in the strong correlation regime (large box length).}
We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
\begin{equation}
\AO{\mu}(x) =
\begin{cases}
\sqrt{2/L}\cos(\mu\pi x/L), &\mu\text{ is odd,}
\\
\sqrt{2/L}\sin(\mu\pi x/L), &\mu\text{ is even,}
\end{cases}
\end{equation}
with $\mu=1,\ldots,\nBas$ and $\nBas=30$ for all calculations.
The convergence threshold $\tau=\max{\abs{\bF{\bw}\bGam{\bw}
\bS - \bS\bGam{\bw}\bF{\bw}}}$[see Eq.~\eqref{eq:commut_F_AO}] of the KS-DFT self-consistent calculation is set
to $10^{-5}$.
In order to compute the various density-functional
integrals that cannot be performed in closed form,
a 51-point Gauss-Legendre quadrature is employed.
In order to test the present eLDA functional we perform various sets of calculations.
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}.
For the single excitations, we also perform time-dependent LDA (TDLDA)
calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}].
Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered. \cite{Dreuw_2005}
Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
(ground-state) limit where $\bw=(0,0)$ and the
equi-triensemble (or equal-weight state-averaged) limit where $\bw=(1/3,1/3)$.
Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl=5$) with a box of length $L =\pi/8$ (left), $L =\pi$ (center), and $L =8\pi$ (right).
}
\end{figure*}
%%% %%% %%%
First, we discuss the linearity of the computed (approximate)
ensemble energies.
To do so, we consider 5-boxium with box lengths of $L =\pi/8$, $L =\pi$, and $L =8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
The deviation from linearity of the three-state ensemble energy
$\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
linearly-interpolated ensemble energy) is represented
in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
fulfilling the restrictions on the ensemble weights to ensure the GOK
variational principle [\ie, $0\le\ew{2}\le1/3$ and $\ew{2}\le\ew{1}\le(1-\ew{2})/2$].
To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
ensemble energy becomes less and less linear as $L$
gets larger, while the GIC reduces the curvature of the ensemble energy
drastically.
It is important to note that, even though the GIC removes the explicit
quadratic Hx terms from the ensemble energy, a non-negligible curvature
remains in the GIC-eLDA ensemble energy when the electron
KS-eLDA individual energies, $\E{eLDA}{(0)}$ (black), $\E{eLDA}{(1)}$ (red), and $\E{eLDA}{(2)}$ (blue), as functions of the weights $\ew{1}$ (solid) and $\ew{2}$ (dashed) for 5-boxium (\ie, $\nEl=5$) with a box of length $L =\pi/8$ (left), $L =\pi$ (center), and $L =8\pi$ (right).}
\end{figure*}
%%% %%% %%%
Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
Unlike in the exact theory, we do not obtain
straight horizontal lines when plotting these
energies, which is in agreement with
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
individual energies do not vary in the same way depending on the state
considered and the value of the weights.
We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
the ground and first excited-state increase with respect to the
first-excited-state weight $\ew{1}$, thus showing that, in this
case, we
``deteriorate'' these states by optimizing the orbitals for the
ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble
as $\ew{2}$ increases. The variations in the ensemble
Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box lengths $L$.
Graphs for additional values of $\nEl$ can be found as {\SI}.
}
\end{figure}
%%% %%% %%%
Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box lengths in the case of 5-boxium (\ie, $\nEl=5$).
Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
For small $L$, the single and double excitations can be labeled as
``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
However, when the box gets larger (\ie, as $L$ increases), there is a strong mixing between the different excitation degrees.
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more disputable. \cite{Loos_2019}
This can be clearly evidenced by the weights of the different
As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
When the box gets larger, they start to deviate.
For the single excitation, TDLDA is extremely accurate up to $L =2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
TDA-TDLDA slightly corrects this trend thanks to error compensation.
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw=(1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw=(0,0)$].
This is especially true, in the strong correlation regime, for the single excitation
which is significantly improved by using equal weights.
The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
This conclusion is verified for smaller and larger numbers of electrons
\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}).}
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right).
}
\end{figure*}
%%% %%% %%%
For the same set of methods, Fig.~\ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI) as a function of $\nEl$ for three values of $L$ ($\pi/8$, $\pi$, and $8\pi$).
We draw similar conclusions as above: irrespectively of the number of
electrons, the eLDA functional with equal
weights is able to accurately model single and double excitations, with
a very significant improvement brought by the
equiensemble KS-eLDA orbitals as compared to their zero-weight
(\ie, conventional ground-state) analogs.
As a rule of thumb, in the weak and intermediate correlation regimes, we
see that the single
excitation obtained from equiensemble KS-eLDA is of
the same quality as the one obtained in the linear response formalism
(such as TDLDA). On the other hand, the double
excitation energy only deviates
from the FCI value by a few tenth of percent.
Moreover, we note that, in the strong correlation regime
(right graph of Fig.~\ref{fig:EvsN}), the single excitation
energy obtained at the equiensemble KS-eLDA level remains in good
agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60\%$.
This also applies to the double excitation, the discrepancy
between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
These observations nicely illustrate the robustness of the
GOK-DFT scheme in any correlation regime for both single and double excitations.
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, \ie, a system that has more than two
electrons.
%%% FIG 5 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{EvsL_DD}
\caption{
\label{fig:EvsL_DD}
Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium (left), 5-boxium (center), and 7-boxium (right) at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
Zero-weight (\ie, $\ew{1}=\ew{2}=0$, red lines) and equiweight (\ie, $\ew{1}=\ew{2}=1/3$, blue lines) calculations are reported.
}
\end{figure*}
%%% %%% %%%
It is also interesting to investigate the influence of the
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl=3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
on the excitation energies obtained at the KS-eLDA level with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
We first stress that although for $\nEl=3$ both single and double excitation energies are
systematically improved (as the strength of electron correlation
increases) when
taking into account
the correlation ensemble derivative, this is not
always the case for larger numbers of electrons.
For 3-boxium, in the zero-weight limit, the correlation ensemble derivative is
significantly larger for the single
excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
case.
However, for 5- and 7-boxium, it hardly
influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
This non-systematic behavior in terms of the number of electrons might
be a consequence of how we constructed eLDA.
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
Zero-weight (\ie, $\ew{1}=\ew{2}=0$, red lines) and equiweight (\ie, $\ew{1}=\ew{2}=1/3$, blue lines) calculations are reported.
}
\end{figure}
%%% %%% %%%
Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
The difference between the solid and dashed curves
undoubtedly show that the
correlation ensemble derivative has a rather significant impact on the double
excitation (around $10\%$) with a slight tendency of worsening the excitation energies
in the case of equal weights, as the number of electrons
increases. It has a rather large influence (which decreases with the
number of electrons) on the single
excitation energies obtained in the zero-weight limit, showing once
again that the usage of equal weights has the benefit of significantly reducing the magnitude of the correlation ensemble derivative.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding remarks}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A local and ensemble-weight-dependent correlation density-functional approximation
(eLDA) has been constructed in the context of GOK-DFT for spin-polarized
triensembles in
1D. The approach is general and can be extended to real
model excited states in molecules and solids. Work is currently in
progress in this direction.
Unlike any standard functional, eLDA incorporates derivative
discontinuities through its weight dependence. The latter originates
from the finite uniform electron gas on which eLDA is
(partially) based. The KS-eLDA scheme, where exact individual
exchange energies are
combined with the eLDA correlation functional , delivers accurate excitation energies for both
single and double excitations, especially when an equiensemble is used.
In the latter case, the same weights are assigned to each state belonging to the ensemble.
The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
We have observed that, although the correlation ensemble derivative has a
non-negligible effect on the excitation energies (especially for the
single excitations), its magnitude can be significantly reduced by
performing equiweight calculations instead of zero-weight
calculations.
Let us finally stress that the present methodology can be extended to other types of ensembles like, for example, the
$\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
calculation of ionization potentials, electron affinities, and
fundamental gaps.
Like in the present
eLDA, such a functional would incorporate the infamous derivative
discontinuity contribution to the fundamental gap through its explicit weight
dependence. We hope to report on this in the near future.
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\section*{Supplementary material}
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See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
