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Some small corrections.

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Bruno Senjean 2020-05-11 17:02:15 +02:00
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Manuscript/FarDFT.tex
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@ -173,11 +173,11 @@ almost pain-free process from a user perspective as the only (yet
essential) input variable is the choice of the
ground-state exchange-correlation (xc) functional.
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}
Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations rely on the Runge-Gross theorem. \cite{Runge_1984}
The Kohn-Sham (KS) formulation of TD-DFT transfers the
complexity of the many-body problem to the xc functional thanks to a
judicious mapping between a time-dependent non-interacting reference
system and its interacting analog which have both
system and its interacting analog which both have
exactly the same one-electron density.
However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made.
@ -371,7 +371,7 @@ $\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb}
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
exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
exactly from the KS ensemble as shown by one of the author. \cite{Fromager_2020}
In the following, we will work at the (weight-dependent) ensemble LDA (eLDA) level of approximation, \ie
\beq
@ -399,7 +399,7 @@ The explicit construction of these functionals is discussed at length in Sec.~\r
\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.
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} freely available on \texttt{github}, 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.
For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994}
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}
@ -455,12 +455,12 @@ basis set and the conventional (weight-independent) LDA Slater exchange function
&
\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
\end{align}
In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$
In the case of \ce{H2}, the ensemble is composed by the
ground state of electronic configuration $1\sigma_g^2$, the lowest
singly-excited state of the same symmetry as the ground state with
singly-excited state of
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 auto-ionising resonance nature \cite{Bottcher_1974}).
of configuration $1\sigma_u^2$ (which has an auto-ionising resonance nature \cite{Bottcher_1974})
which all are of symmetry $\Sigma_g^+$.
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
@ -551,7 +551,7 @@ 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 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.
Note that the CC-S functional depends on $\ew{2}$ only, and not $\ew{1}$, 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}
@ -723,7 +723,7 @@ 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.
Contrary to the CC-S exchange functional which only depends on $\ew{2}$, the eVWN5 correlation functional depends on both weights.
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)
@ -760,9 +760,8 @@ follows:
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
%}\\
As readily seen, it requires three successive calculations.
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.
Note that two calculations are needed to get the first LIM excitation energy, with an additional equi-ensemble calculation for each higher excitation energy.
Additionally, MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\begin{subequations}
@ -779,7 +778,7 @@ in order to compute the second one.
In the above equations, we
assumed that the singly-excited state (with weight $\ew{1}$) is lower
in energy than the doubly-excited state (with weight $\ew{2}$).
If the ordering changes (like in the case of the stretched \ce{H2} molecule, see below), then one should substitute $\E{}{\bw{}=(0,1/2)}$
If the ordering changes (like in the case of the stretched \ce{H2} molecule, see below), one should substitute $\E{}{\bw{}=(0,1/2)}$
by $\E{}{\bw{}=(1/2,0)}$ in Eqs.~\eqref{eq:LIM1} and \eqref{eq:LIM2} which then correspond to the excitation energies of the
doubly-excited and singly-excited states, respectively.
The same holds for the MOM excitation energies in
@ -794,7 +793,7 @@ 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
Interestingly, the CC-S functional leads to a substantial improvement of the LIM excitation energy, getting closer to the reference value 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.
@ -968,7 +967,7 @@ The CC-S exchange functional attenuates significantly this dependence, and when
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 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.
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 using the CC-S functional 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.
@ -1011,9 +1010,9 @@ Excitation energies (in hartree) associated with the lowest double excitation of
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\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 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 capable of extracting 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- and excitation-specific, weight-dependent local exchange functional can significantly reduce 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.