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@ -165,6 +165,7 @@ In the spirit of optimally-tuned range-separated hybrid functionals, a two-step
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%
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).
Importantly, within the widely-used adiabatic approximation, setting up a TD-DFT calculation for a given system is an
@ -236,6 +237,7 @@ Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}
%%%%%%%%%%%%%%%%%%%%
Let us consider a GOK ensemble of $\nEns$ electronic states with
individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and
@ -314,7 +316,7 @@ equals the exact ensemble one
n^{\bw}(\br)=\sum_{I=0}^{\nEns-1}
\ew{I}n_{\Psi_I}(\br).
\eeq
In practice, the minimizing KS density matrix operator
In practice, the minimising KS density matrix operator
$\hgam{\bw}\left[\n{}{\bw}\right]$
can be determined from the following KS reformulation of the
GOK variational principle, \cite{Gross_1988b,Senjean_2015}
@ -327,7 +329,7 @@ where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
result, the orbitals
$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
\nOrb}$ from which the KS
wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq
wave functions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq
I\leq \nEns-1}$ are constructed can be obtained by solving the following ensemble KS equation
\begin{equation}
\label{eq:eKS}
@ -371,8 +373,7 @@ densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to re
Nevertheless, these densities can still be extracted in principle
exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
In the following, we will work at the (weight-dependent) ensemble LDA (eLDA)
level of approximation, \ie
In the following, we will work at the (weight-dependent) ensemble LDA (eLDA) level of approximation, \ie
\beq
\E{\xc}{\bw}[\n{}{}]
&\overset{\rm eLDA}{\approx}&
@ -391,97 +392,12 @@ weight-dependent density-functional exchange and correlation energies
per particle, respectively.
The explicit construction of these functionals is discussed at length in Sec.~\ref{sec:res}.
%\section{Some thoughts illustrated with the Hubbard dimer model}
%
%The definition of an ensemble density functional relies on the concavity
%of the ensemble energy with respect to the external potential. In the
%case of the Hubbard dimer, the singlet triensemble non-interacting
%energy (which contains both singly- and doubly-excited states) reads
%\beq
%\begin{split}
%\mathcal{E}_{\rm KS}^{\bw}\left(\Delta
%v\right)=&(1-\ew{1}-\ew{2})\mathcal{E}_0\left(\Delta
%v\right)+\ew{1}\mathcal{E}_1\left(\Delta
%v\right)
%\\
%&+\ew{2}\mathcal{E}_2\left(\Delta
%v\right),
%\end{split}
%\eeq
%where $\mathcal{E}_0\left(\Delta
%v\right)=2\varepsilon_0\left(\Delta
%v\right)$, $\mathcal{E}_1\left(\Delta
%v\right)=0$, $\mathcal{E}_2\left(\Delta
%v\right)=-2\varepsilon_0\left(\Delta
%v\right)$, and
%\beq
%\varepsilon_0\left(\Delta
%v\right)=-\sqrt{t^2+\dfrac{\Delta v^2}{4}},
%\eeq
%thus leading to
%\beq
%\mathcal{E}_{\rm KS}^{\bw}\left(\Delta
%v\right)=-2\left(1-\ew{1}-2\ew{2}\right)\sqrt{t^2+\dfrac{\Delta
%v^2}{4}}.
%\eeq
%If we ignore the single excitation ($\ew{1}=0$) and denote
%$\ew{}=\ew{2}$, the ensemble energy becomes
%\beq
%\mathcal{E}_{\rm KS}^{\ew{}}\left(\Delta
%v\right)=-2(1-2\ew{})\sqrt{t^2+\dfrac{\Delta
%v^2}{4}}.
%\eeq
%As readily seen, it is concave only if $\ew{}\leq 1/2$. Outside the
%usual range of weight values, it is convex, thus preventing any density
%to be ensemble non-interacting $v$-representable. This statement is
%based on the Legendre--Fenchel transform expression of the
%non-interacting ensemble kinetic energy functional:
%\beq
%T^{\ew{}}_{\rm s}(n)=\sup_{\Delta
%v}\left\{\mathcal{E}_{\rm KS}^{\ew{}}\left(\Delta
%v\right)+\Delta
%v\times(n-1)\right\}.
%\eeq
%In this simple example, ignoring the single excitation is fine. However,
%considering $1/2\leq \ew{}\leq 1$ is meaningless. Of course, if we
%employ approximate ground-state-based density-functional potentials and
%manage to converge the KS wavefunctions, one may obtain something
%interesting. But I have no idea how meaningful such a solution is.\\
%
%In the interacting case, the bi-ensemble (with the double excitation
%only) energy reads
%\beq
%%\begin{split}
%E^{\ew{}}\left(\Delta
%v\right)&=&(1-\ew{})E_0\left(\Delta
%v\right)+\ew{}E_2\left(\Delta
%v\right)
%\nonumber
%\\
%&=&(1-\ew{})E_0\left(\Delta
%v\right)+\ew{}\Big(2U-E_0\left(\Delta
%v\right)-E_1\left(\Delta
%v\right)\Big)
%\nonumber
%\\
%&=&(1-2\ew{})E_0\left(\Delta
%v\right)-\ew{}E_1\left(\Delta
%v\right)+2U\ew{}.
%%\end{split}
%\eeq
%In the vicinity of the symmetric regime ($\Delta
%v=0$), the excited-state energy is $E_1\left(\Delta
%v\right)\approx U$. In this case, the ensemble energy is concave if
%$\ew{}\leq 1/2$. One should check if $(1-2\ew{})E_0\left(\Delta
%v\right)-\ew{}E_1\left(\Delta
%v\right)$ remains concave away from this regime (I see no reason why it
%should be).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% COMPUTATIONAL DETAILS %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
@ -500,7 +416,7 @@ Note also that additional lower-in-energy single excitations may have to be incl
In the present exploratory work, we will simply exclude them from the ensemble and leave the more consistent (from a GOK point of view) description of all low-lying excitations to future work.
Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equi-weight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
(Note that the zero-weight limit corresponds to a conventional ground-state KS calculation.)
Let us finally mention that we will sometimes ``violate'' the GOK
@ -544,7 +460,7 @@ ground state of electronic configuration $1\sigma_g^2$, the lowest
singly-excited state of the same symmetry as the ground state with
configuration $1\sigma_g 2\sigma_g$, 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}).
and has an auto-ionising resonance nature \cite{Bottcher_1974}).
As mentioned previously, the lower-lying
singly-excited states like $1\sigma_g3\sigma_g$ and
$1\sigma_g4\sigma_g$, which should in principle be part of the ensemble
@ -589,16 +505,16 @@ linear ensemble energy and, hence, the same value of the excitation energy indep
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Second, in order to remove some of this spurious curvature of the ensemble
energy (which is mostly due to the ghost-interaction error \cite{Gidopoulos_2002}, but not only \cite{Loos_2020}),
energy (which is mostly due to the ghost-interaction error, \cite{Gidopoulos_2002} but not only \cite{Loos_2020}),
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 $0 \le \ew{2} \le 1$ 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).
Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
\begin{equation}\label{eq:ensemble_Slater_func}
\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\e{\ex}{\ew{2},\text{CC-S}}(\n{}{}) = \Cx{\ew{2}} \n{}{1/3},
\end{equation}
with
\begin{equation}
\label{eq:Cxw}
\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (\ew{} - 1/2)^2 ],
\frac{\Cx{\ew{2}}}{\Cx{}} = 1 - \ew{2} (1 - \ew{2})\qty[ \alpha + \beta (\ew{2} - 1/2) + \gamma (\ew{2} - 1/2)^2 ],
\end{equation}
and
\begin{subequations}
@ -611,30 +527,32 @@ and
\end{align}
\end{subequations}
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
It also allows to ``flatten the curve'' making 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}).\\
\manuf{One point is not clear to me at all. If I understood correctly,
the optimization of $\alpha$, $\beta$, and $\gamma$ is done for
$\ew{1}=0$. So, once the optimisation is done, we have a coefficient
$\Cx{\ew{2}}$ that is a function of $\ew{2}$. Then, how do you obtain
a coefficient $\Cx{\ew{}}$ that is supposed to describe a {\it
different} ensemble defined as $\ew{1}=\ew{2}=\ew{}$ (it says in the
computational details that, ultimately, this is what we are looking at)? Did you just
replace $\ew{2}$ by $\ew{}$? This should be clarified. Another point: in
order to apply Eq.~\eqref{eq:dEdw} for computing excitation energies,
you need $\ew{1}$ and $\ew{2}$ to be independent variables before
differentiating (and taking the value of the derivatives at
$\ew{1}=\ew{2}=\ew{}$). I do not see how you can do this (and generate
the results in Fig.~\ref{fig:Om_H2}) if the only ensemble functional you
have depends on $\ew{}$ and not on both $\ew{1}$ and $\ew{2}$. Regarding
Fig.~\ref{fig:Om_H2}, I would suspect
that you took $\ew{1}=0$, which is questionable and not clear at all from
the text.}
%
%\manuf{One point is not clear to me at all. If I understood correctly,
%the optimization of $\alpha$, $\beta$, and $\gamma$ is done for
%$\ew{1}=0$. So, once the optimisation is done, we have a coefficient
%$\Cx{\ew{2}}$ that is a function of $\ew{2}$. Then, how do you obtain
%a coefficient $\Cx{\ew{}}$ that is supposed to describe a {\it
%different} ensemble defined as $\ew{1}=\ew{2}=\ew{}$ (it says in the
%computational details that, ultimately, this is what we are looking at)? Did you just
%replace $\ew{2}$ by $\ew{}$? This should be clarified. Another point: in
%order to apply Eq.~\eqref{eq:dEdw} for computing excitation energies,
%you need $\ew{1}$ and $\ew{2}$ to be independent variables before
%differentiating (and taking the value of the derivatives at
%$\ew{1}=\ew{2}=\ew{}$). I do not see how you can do this (and generate
%the results in Fig.~\ref{fig:Om_H2}) if the only ensemble functional you
%have depends on $\ew{}$ and not on both $\ew{1}$ and $\ew{2}$. Regarding
%Fig.~\ref{fig:Om_H2}, I would suspect
%that you took $\ew{1}=0$, which is questionable and not clear at all from
%the text.}
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 behaviour in the whole range of weights in order to obtain accurate excitation energies.
Note that the S-GIC functional does only depend on $\ew{2}$, but not on $\ew{1}$, as it is specifically tuned for the double excitation.
Hence, only the double excitation includes a contribution from the ensemble derivative term [see Eq.~\eqref{eq:dEdw}].
The present procedure can be related to optimally-tuned range-separated hybrid functionals, \cite{Stein_2009} where the range-separation parameters (which control the amount of short- and long-range exact exchange) are determined individually for each system by iteratively tuning them in order to enforce non-empirical conditions related to frontier orbitals (\eg, ionisation potential, electron affinity, etc) or, more importantly here, the piecewise linearity of the ensemble energy for ensemble states described by a fractional number of electrons. \cite{Stein_2009,Stein_2010,Stein_2012,Refaely-Abramson_2012}
In this context, the analog of the ``ionisation potential theorem'' for the first
@ -644,25 +562,19 @@ Eqs.~\eqref{eq:exp_ens_ener}, \eqref{eq:diff_Ew}, and \eqref{eq:dEdw}]:
\beq
2\left(E^{\ew{1}=1/2}-E^{\ew{1}=0}\right)&\overset{0\leq \ew{1}\leq 1/2}{=}&\Eps{1}{\ew{1}} - \Eps{0}{\ew{1}} + \left.
\pdv{\E{\xc}{\ew{1}}[\n{}{}]}{\ew{1}} \right|_{\n{}{} =
\n{}{\ew{1}}}.%,\hspace{0.2cm}0\leq \ew{1}\leq 1/2.
\n{}{\ew{1}}}.
\eeq
We enforce this type of {\it exact} constraint (to the
maximum possible extent) when optimizing the parameters in
Eq.~\eqref{eq:Cxw} in order to minimize the curvature of the ensemble energy.
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.
\manuf{again, when reading the text and looking at the figure, I feel
like $\ew{}$ on the $x$ axis is in fact $\ew{2}$, and $\ew{1}$ is set to
zero. Nothing to do with the $\ew{1}=\ew{2}=\ew{}$ case expected from
the computational details. This is very confusing.}
We enforce this type of \textit{exact} constraint (to the maximum possible extent) when optimising the parameters in Eq.~\eqref{eq:Cxw} in order to minimise the curvature of the ensemble energy.
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 $\Cx{\ew{2}}$ 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 that one needs to catch in order to get accurate excitation energies in the zero-weight limit.
Finally, let us mention that, around $\ew{2} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that one needs to catch in order to get accurate excitation energies in the zero-weight limit.
We shall come back to this point later on.
%%% FIG 3 %%%
\begin{figure}
\includegraphics[width=\linewidth]{Cxw}
\caption{
$\Cx{\ew{}}/\Cx{}$ 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).
$\Cx{\ew{2}}/\Cx{}$ as a function of $\ew{2}$ [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).
\label{fig:Cxw}
}
\end{figure}
@ -714,7 +626,7 @@ Combining these, we build a three-state weight-dependent correlation functional:
\label{eq:ecw}
\e{\co}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{\co}{(0)}(\n{}{}) + \ew{1} \e{\co}{(1)}(\n{}{}) + \ew{2} \e{\co}{(2)}(\n{}{}),
\end{equation}
where, unlike in the exact theory~\cite{Fromager_2020}, the individual components are weight \textit{independent}.
where, unlike in the exact theory, \cite{Fromager_2020} the individual components are weight \textit{independent}.
%%% FIG 4 %%%
\begin{figure}
@ -811,27 +723,25 @@ We note also that, by construction, we have
= \e{\co}{(I)}(n) - \e{\co}{(0)}(n),
\end{equation}
showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
Contrary to the CC-S exchange functional which only depends on $\ew{1}$, the eVWN5 correlation functional depends on both weights.
As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is very slightly less
As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 ensemble energy (as a function of $\ew{}$) is very slightly less
concave than its CC-SVWN5 counterpart and it also improves (not by much)
the excitation energy (see purple curve in Fig.~\ref{fig:Om_H2}).\\
\manuf{Again, which value of $\ew{1}$ has been used for generating the
results in this Figures (see my previous comments)? $\ew{1}=0$? If so, we should not claim that we follow GOK
theory because, for H$_2$ at equilibrium, the single excitation is
missing in the ensemble.}
the excitation energy (see purple curve in Fig.~\ref{fig:Om_H2}).
For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
In particular, we report the excitation energies obtained with GOK-DFT
in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$).
These excitation energies are computed using
Eq.~\eqref{eq:dEdw}.\manuf{OK but, again, how do you compute the exchange ensemble
derivative for both excited states when it seems like the functional in
Eqs.~\eqref{eq:ensemble_Slater_func} and \eqref{eq:Cxw}
only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.}
Eq.~\eqref{eq:dEdw}.
%\manuf{OK but, again, how do you compute the exchange ensemble
%derivative for both excited states when it seems like the functional in
%Eqs.~\eqref{eq:ensemble_Slater_func} and \eqref{eq:Cxw}
%only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.}
For comparison, we also report results obtain
with the linear interpolation method (LIM). \cite{Senjean_2015,Senjean_2016}
For comparison, we also report results obtained with the linear interpolation method (LIM). \cite{Senjean_2015,Senjean_2016}
The latter simply consists in extracting the excitation energies (which are
weight-independent, by construction) from the equiensemble energies, as
weight-independent, by construction) from the equi-ensemble energies, as
follows:
\begin{subequations}
\begin{align}
@ -851,7 +761,10 @@ follows:
%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
%}\\
As readily seen, it requires three successive calculations.
MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles.
Note that two calculations are needed to get the first LIM excitation energy, but only one is required for each higher excitation.
Additionally, MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\begin{subequations}
\begin{align}
\Ex{\MOM}{(1)} & = \E{}{\bw{}=(1,0)} - \E{}{\bw{}=(0,0)}, \label{eq:MOM1}
@ -859,10 +772,7 @@ MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}, \label{eq:MOM2}
\end{align}
\end{subequations}
which also require three separate calculations at a different set of
ensemble weights, have been computed, for comparison.
For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles.
Note that two calculations are needed to get the first LIM excitation energy, but only one is required for each higher excitation.
which also require three separate calculations at a different set of ensemble weights, have been computed for further comparisons.
As readily seen in Eqs.~\eqref{eq:LIM1} and \eqref{eq:LIM2}, LIM is a recursive strategy where the first excitation energy has to be determined
in order to compute the second one.
@ -884,14 +794,8 @@ Eqs.~\eqref{eq:MOM1} and \eqref{eq:MOM2}.
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights 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.
Interestingly, the CC-S functional
leads to a substantial improvement of the LIM
excitation energy, getting closer to the reference value
(with an error as small as $0.24$ eV) when no correlation
functional is used. When correlation functionals are
added (\ie, VWN5 or eVWN5), LIM tends to overestimate
the excitation energy by about $1$ eV but still performs
better than when no correction of the curvature is considered.
Interestingly, the CC-S functional leads to a substantial improvement of the LIM excitation energy, getting closer to the reference value (with an error as small as $0.24$ eV) when no correlation functional is used. When correlation functionals are added (\ie, VWN5 or eVWN5), LIM tends to overestimate
the excitation energy by about $1$ eV but still performs better than when no correction of the curvature is considered.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}).
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 the ghost-interaction-free pure-state limits.
@ -963,15 +867,16 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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 defined in Sec.~\ref{sec:H2}.
Although we could safely restrict ourselves to a bi-ensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same tri-ensemble defined in Sec.~\ref{sec:H2}.
Nonetheless, one should just be careful when reading the equations reported above, as they correspond to the case where the singly-excited state is lower in energy than the doubly-excited state.
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.
It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
The weight dependence of $\Cx{\ew{}}$ is illustrated in
Fig.~\ref{fig:Cxw} (green curve).\manuf{Again, it would be nice to say
explicitly if you construct a functional, function of $\ew{1}$ and
$\ew{2}$ (how then) or just $\ew{}$ (how to compute the separate
derivatives then?)}
The weight dependence of $\Cx{\ew{2}}$ is illustrated in
Fig.~\ref{fig:Cxw} (green curve).
%\manuf{Again, it would be nice to say
%explicitly if you construct a functional, function of $\ew{1}$ and
%$\ew{2}$ (how then) or just $\ew{}$ (how to compute the separate
%derivatives then?)}
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.
%In other words, the curvature ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
@ -1054,7 +959,7 @@ The excitation energies associated with this double excitation computed with var
Before analysing the results, we would like to highlight the fact that there is a large number of singly-excited states lying in between the $1s2s$ and $2s^2$ states.
Therefore, the present ensemble is not consistent with GOK theory.
However, it is impossible, from a practical point of view, to take into account all these single excitations.
We then restrict ourselves to a triensemble keeping in mind the possible theoretical loopholes of such a choice.
We then restrict ourselves to a tri-ensemble keeping in mind the possible theoretical loopholes of such a choice.
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 results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
@ -1065,6 +970,7 @@ zero-weight are more accurate than at equi-weight, while the opposite
conclusion was 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. Here again, the LIM excitation energy when the CC-S functional is used is very accurate with only a 22 millihartree error compared to the reference value, while adding the correlation contribution to the functional tends to overestimate the excitation energy.
Hence, in the light of the results obtained in this paper, it seems that the weight-dependent curvature correction to the exchange functional has the largest impact on the accuracy of the excitation energies.
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.
%%% TABLE V %%%
@ -1107,7 +1013,10 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\label{sec:ccl}
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.
In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system- and excitation-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy and improves excitation energies.
The present weight-dependent exchange functional, CC-S, specifically tailored for double excitations, only depends on the weight of the doubly-excited state, CC-S being independent on the weight of the singly-excited state.
We are currently investigating a generalisation of the present procedure in order to include a dependency on both weights in the exchange functional.
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