diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 48fd165..8f86be1 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -160,15 +160,16 @@ Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1 The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017} -Another issue, which is partly connected to the previous one, is the description of electronically-excited states. +Another issue, which is partly connected to the previous one, is the +description of low-lying quasi-degenerate states. -The standard approach for modeling excited states in DFT is +The standard approach for modeling excited states in a DFT framework is linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012} -In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong. -Moreover, in exact TDDFT, the xc energy is in fact an xc action \cite{Vignale_2008} which is a +In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong. +Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action} \cite{Vignale_2008} which is a functional of the time-dependent density $n\equiv n(\br,t)$ and, as -such, has memory. Standard implementations of TDDFT rely on -the adiabatic approximation where memory effects are neglected. In other +such, it should incorporate memory effects. Standard implementations of TDDFT rely on +the adiabatic approximation where these effects are neglected. In other words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012} As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019} @@ -180,11 +181,11 @@ and excited states altogether, \ie, with the same set of orbitals. Interestingly, a similar approach exists in DFT. Referred to as Gross--Oliveira--Kohn (GOK) DFT\cite{Gross_1988a,Gross_1988b,Oliveira_1988}, it was proposed at the end of the 80's as a generalization of Theophilou's DFT for equiensembles. \cite{Theophilou_1979} -In exact GOK-DFT, the ensemble xc energy not only a functional of the +In GOK-DFT, the ensemble xc energy is a functional of the density but also a function of the ensemble weights. Note that, unlike in conventional -Boltzmann ensembles~\cite{Pastorczak_2013}, the weights (each state in the ensemble -is assigned a given and fixed weight) are allowed to vary +Boltzmann ensembles~\cite{Pastorczak_2013}, the ensemble weights [each state in the ensemble +is assigned a given and fixed weight] are allowed to vary independently in a GOK ensemble. The weight dependence of the xc functional plays a crucial role in the calculation of excitation energies. @@ -193,12 +194,12 @@ It actually accounts for the derivative discontinuity contribution to energy gap %\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?} Even though GOK-DFT is in principle able to -tackle near-degenerate situations and multiple-electron excitation +describe near-degenerate situations and multiple-electron excitation processes, it has not been given much attention until quite recently. \cite{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,Senjean_2018,Smith_2016} One of the reason is the lack, not to say the absence, of reliable -density-functional approximations for ensembles (eDFAs) in the literature. -The most recent works on this topic are still fundamental and +density-functional approximations for ensembles (eDFAs). +The most recent works dealing with this particular issue are still fundamental and exploratory, as they rely either on simple (but nontrivial) model systems \cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019} @@ -210,7 +211,7 @@ discontinuity problem that ocurs when crossing an integral number of electrons can be recast into a weight-dependent ensemble one. \cite{Senjean_2018,Senjean_2020} -The present work is an attempt to answer this question, +The present work is an attempt to address this problem, with the ambition to turn, in the forthcoming future, GOK-DFT into a (low-cost) practical computational method for modeling excited states in molecules and extended systems. Starting from the ubiquitous local-density approximation (LDA), we @@ -220,8 +221,8 @@ extracted. The present eDFA, \trashEF{is specially designed for the computation single and double excitations within GOK-DFT}, which can be seen as a natural extension of LDA, will be referred to as eLDA in the remaining of this paper. As a proof of concept, we apply this general strategy to -ensemble correlation energies only [we use the orbital-dependent exact -ensemble exchange energy for convenience] in the particular case of +ensemble correlation energies [that we combine with +ensemble exact exchange energies] in the particular case of \emph{strict} one-dimensional (1D) and spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b} In other words, the Coulomb interaction used in this work describes @@ -234,9 +235,8 @@ In these extreme conditions, where magnetic effects compete with Coulombic force The paper is organized as follows. Exact and approximate formulations of GOK-DFT are discussed in Section -\ref{sec:eDFT}, with a particular emphasis on the extraction of -individual energy levels and the calculation of individual exact exchange -energies. +\ref{sec:eDFT}, with a particular emphasis on the calculation of +individual energy levels. In Sec.~\ref{sec:eDFA}, we detail the construction of the weight-dependent local correlation functional specially designed for the computation of single and double excitations within GOK-DFT.