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Manuscript/FarDFT.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-18 14:01:54 +0100
%% Created for Pierre-Francois Loos at 2019-11-21 21:56:23 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2019,
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
Date-Added = {2019-11-21 21:56:17 +0100},
Date-Modified = {2019-11-21 21:56:23 +0100},
Doi = {10.1021/acs.jctc.8b01205},
Eprint = {https://doi.org/10.1021/acs.jctc.8b01205},
Journal = {J. Chem. Theory Comput.},
Number = {3},
Pages = {1939--1956},
Title = {Reference Energies for Double Excitations},
Url = {https://doi.org/10.1021/acs.jctc.8b01205},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
@article{Vosko_1980,
Author = {Vosko, S. H. and Wilk, L. and Nusair, M.},
Date-Added = {2019-11-17 21:47:25 +0100},

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Manuscript/FarDFT.tex
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage[
colorlinks=true,
citecolor=blue,
@ -120,11 +124,12 @@
\affiliation{\LCPQ}
\begin{abstract}
We report a first generation of local, weight-dependent exchange-correlation density-functional approximations (DFAs) for molecules.
These density-functional approximations for ensembles (eDFAs) incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
They are specially designed for the computation of double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) to ensembles.
The resulting eDFAs, dubbed eLDA, which are based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence.
Their accuracy is illustrated by computing the double excitation in the prototypical H$_2$ molecule.
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 weights of the excited states.
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.
In the present article, we report a first-rung (\ie, local), weight-dependent exchange-correlation density-functional approximation for atoms and molecules specially 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.
\end{abstract}
\maketitle
@ -134,32 +139,33 @@ Their accuracy is illustrated by computing the double excitation in the prototyp
%%%%%%%%%%%%%%%%%%%%
\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,Ulrich_2012}
At a relatively low computational cost (at least compared to the other excited-state 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, setting up a TD-DFT calculation for a given system is an almost pain-free process from the user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.
At a relatively low 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, setting up a TD-DFT calculation for a given system is an almost pain-free process from a user perspective as the only (yet essential) input variable is the choice of the so-called 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 foundation relies on the Runge-Gross theorem. \cite{Runge_1984}
The Kohn-Sham (KS) formalism 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 with the same one-electron density.
The Kohn-Sham (KS) formalism 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 the exact same one-electron density.
However, TD-DFT is far from being perfect as, in practice, approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Moreover, because it was so popular, it has been studied in excruciated details, and researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the 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.
Moreover, because it is so popular, it has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
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 xc functional.
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.
The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004}
From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Elliott_2011}.
One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Elliott_2011}
Although these double excitations are usually experimentally dark (which means they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007,Loos_2019}
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}
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 \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
Maybe surprisingly, a possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
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 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 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 in the context of eDFT.
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.
@ -184,9 +190,9 @@ Unless otherwise stated, atomic units are used throughout.
\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
\begin{equation}
\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{I}{}
\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,
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,
\begin{align}
& \sum_{I=0}^{\Nens-1} \ew{I} = 1,
&
@ -194,7 +200,7 @@ built from an ensemble of $\Nens$ electronic states with individual energies $\E
\end{align}
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}
\pdv{\E{}{\bw}}{\ew{I}} = \E{I}{} - \E{0}{},
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)},
\end{equation}
where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
@ -224,7 +230,7 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
\begin{equation}
\label{eq:dEdw}
\pdv{\E{}{\bw}}{\ew{I}}
= \E{I}{} - \E{0}{}
= \E{}{(I)} - \E{}{(0)}
= \eps{I}{\bw} - \eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
\end{equation}
where $\eps{I}{\bw}$ is the $I$th KS orbital energy.
@ -394,7 +400,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
Our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
Hence, we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
The weight-dependence of the xc functional solely originates from the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
The weight-dependence of the xc functional then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference
\begin{equation}