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Pierre-Francois Loos 2020-02-10 10:30:10 +01:00
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24
FarDFT.nb
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Manuscript/FarDFT.tex
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\author{Clotilde \surname{Marut}}
\affiliation{\LCPQ}
\author{Emmanuel \surname{Fromager}}
\affiliation{\LCQ}
%\author{Emmanuel \surname{Fromager}}
%\affiliation{\LCQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
@ -122,7 +122,7 @@
\begin{abstract}
Density-functional theory for ensembles (eDFT) is a time-independent formalism which allows to compute individual excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within eDFT.
However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contributions to the excitation energies.
However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contribution to the excitation energies.
In the present article, we report a first-rung (\ie, local), weight-dependent exchange-correlation density-functional approximation for atoms and molecules specifically designed for the computation of double excitations within eDFT.
This density-functional approximation for ensembles, based on finite and infinite uniform electron gas models, incorporate information about both ground and excited states.
Its accuracy is illustrated by computing the double excitation in the prototypical H$_2$ molecule.
@ -158,23 +158,23 @@ In this approach the xc kernel is made frequency dependent \cite{Romaniello_2009
Maybe surprisingly, another possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
DFT for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable alternative following such a strategy 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 short, eDFT is the density-based analog of state-averaged wave function methods.
In the assumption of monotonically decreasing weights, eDFT has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be easily extracted from the total ensemble energy. \cite{Deur_2019}
Although the formal foundation of eDFT has been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} the practical developments of eDFT have been rather slow.
We believe that it is due to the lack of accurate approximations for eDFT.
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 excited states) 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 foundation of GOK-DFT has been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
We believe that it is partly due to the lack of accurate approximations for GOK-DFT.
In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules.
The present contribution is a first step towards this goal.
When one talks about constructing functionals, the local-density approximation (LDA) has always a special place.
When one talks about constructing functionals, the local-density approximation (LDA) is never far away.
The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
Although the Hohenberg-Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
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}
Here, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA functional for ensembles (eLDA).
The present eLDA functional is specifically designed to compute double excitations within eDFT, and it automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
The present eLDA functional is specifically designed to compute double excitations within GOK-DFT, and it automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
The paper is organised as follows.
In Sec.~\ref{sec:theo}, the theory behind eDFT is presented.
In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
Section \ref{sec:func} provides details about the construction of the weight-dependent xc LDA functional.
The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
@ -185,12 +185,12 @@ Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}
As mentioned above, eDFT is based on the so-called Gross-Oliveria-Kohn (GOK) variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy
As mentioned above, eDFT for excited states is based on the GOK variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy
\begin{equation}
\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{}{(I)},
\end{equation}
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and (normalized) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\Nens-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
Degeneracies can be easily handled.
Multiplet degeneracies can be easily handled by assigning the same ensemble weight to the degenerate states.
One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
\begin{equation}