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Manuscript/FarDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-03 21:33:16 +0200
%% Created for Pierre-Francois Loos at 2020-05-10 19:45:36 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -17,7 +17,8 @@
Pages = {226405},
Title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
Volume = {109},
Year = {2012}}
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.226405}}
@article{Stein_2012,
Author = {Tamar Stein and Jochen Autschbach and Niranjan Govind and Leeor Kronik and Roi Baer},
@ -28,7 +29,8 @@
Pages = {3740},
Title = {Curvature and Frontier Orbital Energies in Density Functional Theory},
Volume = {3},
Year = {2012}}
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1021/jz3015937}}
@article{Stein_2010,
Author = {Tamar Stein and Helen Eisenberg and Leeor Kronik and Roi Baer},
@ -39,7 +41,8 @@
Pages = {266802},
Title = {Fundamental Gaps in Finite Systems from Eigenvalues of a Generalized Kohn-Sham Method},
Volume = {105},
Year = {2010}}
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.266802}}
@article{Stein_2009,
Author = {Tamar Stein and Leeor Kronik and Roi Baer},
@ -50,7 +53,8 @@
Pages = {2818},
Title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
Volume = {131},
Year = {2009}}
Year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1021/ja8087482}}
@article{Paragi_2001,
Author = {G. Paragi and I. K. Gyemnnt and V. E. VanDoren},
@ -4014,13 +4018,12 @@
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.123.016401}}
@article{Gould_2019_insights,
title={Density driven correlations in ensemble density functional theory: insights from simple excitations in atoms},
author={Tim Gould and Stefano Pittalis},
year={2020},
eprint={2001.09429},
archivePrefix={arXiv},
primaryClass={cond-mat.str-el}
}
Archiveprefix = {arXiv},
Author = {Tim Gould and Stefano Pittalis},
Eprint = {2001.09429},
Primaryclass = {cond-mat.str-el},
Title = {Density driven correlations in ensemble density functional theory: insights from simple excitations in atoms},
Year = {2020}}
@article{Gould_2013,
Author = {Gould, Tim and Dobson, John F.},
@ -9261,37 +9264,24 @@ title={Density driven correlations in ensemble density functional theory: insigh
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5043411}}
@article{Seunghoon_2018,
author = {Lee,Seunghoon and Filatov,Michael and Lee,Sangyoub and Choi,Cheol Ho },
title = {Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix},
journal = {J. Chem. Phys.},
volume = {149},
number = {10},
pages = {104101},
year = {2018},
doi = {10.1063/1.5044202},
@article{Lee_2018,
Author = {Lee, Seunghoon and Filatov, Michael and Lee, Sangyoub and Choi, Cheol Ho},
Date-Modified = {2020-05-10 19:44:18 +0200},
Doi = {10.1063/1.5044202},
Journal = {J. Chem. Phys.},
Number = {10},
Pages = {104101},
Title = {Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5044202}}
URL = {
https://doi.org/10.1063/1.5044202
},
eprint = {
https://doi.org/10.1063/1.5044202
}
}
@Article{TDDFTfromager2013,
author = {Emmanuel Fromager and Stefan Knecht and Hans J. {Aa. Jensen}},
title = {Multi-configuration time-dependent density-functional theory based on range separation},
year = {2013},
journal = {J. Chem. Phys.},
volume = {138},
pages = {084101},
URL = {
https://doi.org/10.1063/1.4792199
},
}
@article{TDDFTfromager2013,
Author = {Emmanuel Fromager and Stefan Knecht and Hans J. {Aa. Jensen}},
Journal = {J. Chem. Phys.},
Pages = {084101},
Title = {Multi-configuration time-dependent density-functional theory based on range separation},
Url = {https://doi.org/10.1063/1.4792199},
Volume = {138},
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4792199}}

131
Manuscript/FarDFT.tex
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@ -180,13 +180,12 @@ system and its interacting analog which have both
exactly the same one-electron density.
However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made.
First, within the \manu{commonly used} linear-response \trashEF{approximation} \manu{regime}, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation).
First, within the commonly used linear-response regime, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation).
Second, the time dependence of the functional is usually treated at the local approximation level within the standard adiabatic approximation.
In other words, memory effects are absent from the xc functional which is assumed to be local in time
(the xc energy is in fact an xc action, not an energy functional). \cite{Vignale_2008}
Third and more importantly in the present context, a major issue of
TD-DFT actually originates directly from the choice of the
\manu{(ground-state)} xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
TD-DFT actually originates directly from the choice of the (ground-state) xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Because of its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional.
@ -197,8 +196,8 @@ Although these double excitations are usually experimentally dark (which means t
One possible solution to access double excitations within TD-DFT is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009}
However, spin contamination might be an issue. \cite{Huix-Rotllant_2010}
\manu{Note that a simple remedy based on a mixed reference reduced density
matrix has been recently introduced by Lee {\it et al.} \cite{Seunghoon_2018}}
Note that a simple remedy based on a mixed reference reduced density
matrix has been recently introduced by Lee \textit{ et al.} \cite{Lee_2018}
In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
In this approach the xc kernel is made frequency dependent, which allows to treat doubly-excited states. \cite{Romaniello_2009a,Sangalli_2011,Loos_2019}
@ -206,7 +205,7 @@ Maybe surprisingly, another possible way of accessing double excitations is to r
With a computational cost similar to traditional KS-DFT, DFT for
ensembles (eDFT)
\cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable
alternative \trashEF{following} \manu{that follows} such a strategy \manu{and is} currently under active development.\cite{Gidopoulos_2002,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,Smith_2016,Senjean_2018}
alternative that follows such a strategy and is currently under active development.\cite{Gidopoulos_2002,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,Smith_2016,Senjean_2018}
In the assumption of monotonically decreasing weights, eDFT for excited states has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, the so-called Gross--Oliveria--Kohn (GOK) variational principle. \cite{Gross_1988a}
In short, GOK-DFT (\ie, eDFT for neutral excitations) is the density-based analog of state-averaged wave function methods, and excitation energies can then be easily extracted from the total ensemble energy. \cite{Deur_2019}
Although the formal foundations of GOK-DFT have been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
@ -223,7 +222,7 @@ However, Loos and Gill have recently shown that there exists other UEGs which co
Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems.
Unlike any standard functional, this first-rung functional automatically incorporates ensemble derivative contributions thanks to its natural weight dependence, \cite{Levy_1995, Perdew_1983} and has shown to deliver accurate excitation energies for both single and double excitations.
\manu{In order to} extend this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT.
In order to extend this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT.
The paper is organised as follows.
In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is briefly presented.
@ -261,9 +260,9 @@ where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of
The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$.
Multiplet degeneracies can be easily handled by assigning the same
weight to the degenerate states. \cite{Gross_1988b}
One of the key feature of the GOK ensemble is that \trashEF{individual} excitation
One of the key feature of the GOK ensemble is that excitation
energies can be extracted from the ensemble energy via differentiation
with respect to the individual \manu{excited-state} weights \manu{$\ew{I}$ ($I>0$)}:
with respect to the individual excited-state weights $\ew{I}$ ($I>0$):
\begin{equation}\label{eq:diff_Ew}
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}.
\end{equation}
@ -372,16 +371,15 @@ 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) \manu{ensemble}
LDA \manu{(eLDA)}
In the following, we will work at the (weight-dependent) ensemble LDA (eLDA)
level of approximation, \ie
\beq
\E{\xc}{\bw}[\n{}{}]
&\overset{\rm \manu{e}LDA}{\approx}&
&\overset{\rm eLDA}{\approx}&
\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{},
\\
\fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
&\overset{\rm \manu{e}LDA}{\approx}&
&\overset{\rm eLDA}{\approx}&
\left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\eeq
We will also adopt the usual decomposition, and write down the weight-dependent xc functional as
@ -493,42 +491,27 @@ Numerical quadratures are performed with the \texttt{numgrid} library \cite{numg
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
Assuming that the singly-excited state is lower in energy than the doubly-excited state, one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
If the doubly-excited state \manu{(whose weight is denoted $\ew{2}$
throughout this work)} is lower in energy than the singly-excited state
\manu{(with weight $\ew{1}$)}, which can be the case as one would notice
If the doubly-excited state (whose weight is denoted $\ew{2}$
throughout this work) is lower in energy than the singly-excited state
(with weight $\ew{1}$), which can be the case as one would notice
later, then one has to swap $\ew{1}$ and $\ew{2}$ in the above
inequalities. \manu{Note also that additional lower-in-energy single
excitations may have to be included into the ensemble
before
incorporating the double excitation of interest. 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.}
inequalities.
Note also that additional lower-in-energy single excitations may have to be included into the ensemble before incorporating the double excitation of interest.
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.
(Note that the zero-weight limit corresponds to a conventional ground-state KS calculation.)
Let us \manu{finally} mention that we will sometimes ``violate'' the GOK
Let us finally mention that we will sometimes ``violate'' the GOK
variational principle in order to build our weight-dependent functionals
by considering the extended range of weights $0 \le \ew{2} \le 1$.
The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is
\trashEF{nonetheless} of particular interest as it is, like the
The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is of particular interest as it is, like the
(ground-state) zero-weight limit, a genuine saddle point of the
restricted KS equations [see Eqs. (\ref{eq:min_KS_DM}) and
(\ref{eq:eKS})], and \manu{it matches} perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\manu{From a GOK-DFT perspective, considering a (stationary) pure-excited-state limit
can be seen as a way to construct density-functional approximations to
individual exchange and state-driven correlation within an ensemble.
\cite{Gould_2019,Gould_2019_insights,Fromager_2020}}
However, \trashEF{let us stress that we will not compute excitation
energies with these ensembles inconsistent with GOK theory}
\manu{when it comes to compute excitation
energies, we will exclusively consider ensembles where the largest
weight is assigned to the ground state.}\\
\manuf{We do not completely
follow GOK when we ignore lower-lying singles...}
restricted KS equations [see Eqs.~\eqref{eq:min_KS_DM} and
\eqref{eq:eKS}], and it matches perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
From a GOK-DFT perspective, considering a (stationary) pure-excited-state limit can be seen as a way to construct density-functional approximations to individual exchange and state-driven correlation within an ensemble. \cite{Gould_2019,Gould_2019_insights,Fromager_2020}
However, when it comes to compute excitation energies, we will exclusively consider ensembles where the largest weight is assigned to the ground state.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
@ -549,7 +532,7 @@ This procedure is applied to various two-electron systems in order to extract ex
First, we compute the ensemble energy of the \ce{H2} molecule at
equilibrium bond length (\ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ
basis set and the \manu{conventional (weight-independent)} LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
basis set and the conventional (weight-independent) LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
\begin{align}
\label{eq:Slater}
\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
@ -562,22 +545,22 @@ 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}).
\manu{As mentioned previously, the lower-lying
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
(see Fig.~3 in Ref.~\onlinecite{TDDFTfromager2013}),
have been excluded, for simplicity.}
have been excluded, for simplicity.
The deviation from linearity of the ensemble energy $\E{}{\ew{}}$
\manu{[we recall that $\ew{1}=\ew{2}=\ew{}$]} is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
[we recall that $\ew{1}=\ew{2}=\ew{}$] is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
Because the Slater exchange functional defined in Eq.~\eqref{eq:Slater} does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that
the excitation energy associated with the doubly-excited state obtained
via the derivative of the ensemble energy \manu{with respect to $\ew{2}$
(and taken at $\ew{2}=\ew{}=\ew{1}$)} revaries significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
via the derivative of the ensemble energy with respect to $\ew{2}$
(and taken at $\ew{2}=\ew{}=\ew{1}$) varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/3$.
Note that the exact xc ensemble functional would yield a perfectly
linear \manu{ensemble} energy and, hence, the same value of the excitation energy independently of the ensemble weights.
linear ensemble energy and, hence, the same value of the excitation energy independently of the ensemble weights.
%%% FIG 1 %%%
\begin{figure}
@ -640,7 +623,7 @@ 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.~(\ref{eq:dEdw}) for computing excitation energies,
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
@ -654,10 +637,10 @@ The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have
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.
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}
\manu{In this context, the analog of the ``IP theorem'' for the first
In this context, the analog of the ``ionisation potential theorem'' for the first
(neutral)
excitation, for example, would read as follows [see
Eqs.~(\ref{eq:exp_ens_ener}), (\ref{eq:diff_Ew}), and (\ref{eq:dEdw})]:
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{}{} =
@ -665,7 +648,7 @@ Eqs.~(\ref{eq:exp_ens_ener}), (\ref{eq:diff_Ew}), and (\ref{eq:dEdw})]:
\eeq
We enforce this type of {\it exact} constraint (to the
maximum possible extent) when optimizing the parameters in
Eq.~(\ref{eq:Cxw}) in order to minimize the curvature of the ensemble energy.}
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
@ -690,9 +673,9 @@ We shall come back to this point later on.
\subsubsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Third, we add up correlation effects via the \manu{conventional} VWN5 local correlation functional. \cite{Vosko_1980}
Third, we add up correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the \manu{(weight-independent)} Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
The combination of the (weight-independent) Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Weight-dependent correlation functional}
@ -731,8 +714,8 @@ 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}
\manu{where, unlike in the exact theory~\cite{Fromager_2020}, the individual components are
weight-{\it independent}.}
where, unlike in the exact theory~\cite{Fromager_2020}, the individual components are weight \textit{independent}.
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=0.8\linewidth]{fig1}
@ -796,7 +779,7 @@ As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure ca
The weight-dependence of the correlation functional is then carried
exclusively by the impurity [\ie, the functional defined in
\eqref{eq:ecw}], while the remaining effects are produced by the bath
(\ie, the usual \manu{ground-state} LDA correlation functional).
(\ie, the usual ground-state LDA correlation functional).
Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
\begin{equation}
@ -842,15 +825,14 @@ in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} =
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.~(\ref{eq:ensemble_Slater_func}) and (\ref{eq:Cxw})
Eqs.~\eqref{eq:ensemble_Slater_func} and \eqref{eq:Cxw}
only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.}
For comparison \trashEF{purposes}, we also report \manu{results obtain
with }the linear interpolation method (LIM) \trashEF{excitation
energies.} \cite{Senjean_2015,Senjean_2016} \manu{The latter simply
consists in extracting the excitation energies (which are
For comparison, we also report results obtain
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
follows:}
follows:
\begin{subequations}
\begin{align}
\Ex{\LIM}{(1)} & = 2 \qty[\E{}{\bw{}=(1/2,0)} - \E{}{\bw{}=(0,0)}], \label{eq:LIM1}
@ -868,7 +850,8 @@ follows:}
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
%}\\
\manu{As readily seen, it requires three successive calculations.} MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
As readily seen, it requires three successive calculations.
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}
@ -903,11 +886,11 @@ The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitati
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 close to the reference value
(with an error of up to 0.24 eV) when no correlation
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
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.
@ -1080,13 +1063,8 @@ 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 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 makes the biggest impact in providing
accurate excitation energies.
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 %%%
@ -1134,10 +1112,7 @@ Although the weight-dependent correlation functional developed in this paper (eV
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
of the self-consistent one.
\manu{Exploring the impact of both density- and state-driven
correlations}
\cite{Gould_2019,Gould_2019_insights,Fromager_2020} \trashEF{can also be
calculated} \manu{may provide} additional insights about the present results.
Exploring the impact of both density- and state-driven correlations \cite{Gould_2019,Gould_2019_insights,Fromager_2020} may provide additional insights about the present results.
This is left for future work.
In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future, and we hope to be able to report further on this in the near future.