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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-01-28 08:55:21 +0100
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%% Created for Pierre-Francois Loos at 2019-09-05 12:13:36 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Perdew_1983,
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Author = {J. P. Perdew and M. Levy},
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Date-Added = {2019-09-05 12:04:19 +0200},
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Date-Modified = {2019-09-05 12:13:34 +0200},
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Journal = {Phys. Rev. Lett.},
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Pages = {1884},
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Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
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Volume = {51},
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Year = {1983}}
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@article{Schulz_1993,
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Author = {H. J. Schulz},
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Date-Added = {2018-12-11 15:14:49 +0100},
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@ -2473,11 +2483,6 @@
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Volume = {8},
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Year = {1987}}
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@article{cite-key,
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2018-10-24 22:38:52 +0200},
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Bdsk-Url-1 = {http://aip.scitation.org/doi/pdf/10.1063/1.4941606?class=pdf}}
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@article{Clima_2007,
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Author = {Clima, Sergiu and Hendrickx, Marc F.A.},
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Date-Added = {2018-10-24 22:38:52 +0200},
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@ -3377,16 +3382,15 @@
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Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0009261479802766},
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Bdsk-Url-2 = {http://dx.doi.org/10.1016/0009-2614(79)80276-6}}
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@article{Gould_2018b,
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@article{Gould_2019,
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Author = {Gould, Tim and Pittalis, Stefano},
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2018-12-11 14:01:03 +0100},
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File = {/Users/loos/Zotero/storage/R2I9XXUN/Gould and Pittalis - Correlation energies of many-electron ensembles ar.pdf},
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Journal = {arXiv},
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Pages = {5},
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Title = {Correlation Energies of Many-Electron Ensembles Are More than the Sum of Their Parts},
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Volume = {1808.04994},
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Year = {2018}}
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Date-Modified = {2019-09-05 12:09:05 +0200},
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Journal = {Phys. Rev. Lett.},
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Pages = {016401},
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Title = {Density-Driven Correlations in Many-Electron Ensembles: Theory and Application for Excited States},
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Volume = {123},
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Year = {2019}}
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@article{Gould_2013,
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Author = {Gould, Tim and Dobson, John F.},
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@ -3441,10 +3445,10 @@
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Year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.243001}}
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@article{Gould_2018a,
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@article{Gould_2018,
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Author = {Gould, Tim and Kronik, Leeor and Pittalis, Stefano},
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2018-12-09 11:39:36 +0100},
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Date-Modified = {2019-09-05 12:09:11 +0200},
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Doi = {10.1063/1.5022832},
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File = {/Users/loos/Zotero/storage/C5DEDGG2/Gould et al. - 2018 - Charge transfer excitations from exact and approxi.pdf},
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Issn = {0021-9606, 1089-7690},
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@ -27,7 +27,7 @@
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% functionals, potentials, densities, etc
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\newcommand{\eps}{\epsilon}
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\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
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\renewcommand{\v}[2]{v_\text{#1}^{#2}}
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\renewcommand{\v}[2]{E_\text{#1}^{#2}}
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\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}}
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\newcommand{\n}[1]{n^{#1}}
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@ -119,9 +119,9 @@ Their accuracy is illustrated by computing single and double excitations in one-
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
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The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the electronic wave function.
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The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the many-electron wave function.
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The complexity of the many-body problem is then transferred to the xc functional.
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Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}
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Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}
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The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge.
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Another issue, which is partly connected to the previous one, is the description of electronically-excited states.
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@ -132,28 +132,26 @@ The simplest and most widespread approximation in state-of-the-art electronic st
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In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time.
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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,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}
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When affordable (i.e.~for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
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The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e.,~with the same set of orbitals.
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When affordable (i.e., for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
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The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e., with the same set of orbitals.
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Interestingly, a similar approach exists in DFT.
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Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988, Gross_1988a, Oliveira_1988} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
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In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
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This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies.
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It actually accounts for the infamous derivative discontinuity contribution to energy gaps.
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It actually accounts for the infamous derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
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\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
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Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018a,Gould_2018b,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,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}
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Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until 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_2018a,Deur_2018b,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}
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The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature.
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Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed. \cite{Yang_2014,Yang_2017}
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In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open.
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A first step towards this goal is presented in this Letter with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
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A first step towards this goal is presented in this article with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
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\alert{Mention WIDFA?}
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In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
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Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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Atomic units are used throughout.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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\label{sec:theory}
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\section{Density-functional theory for ensembles}
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\label{sec:eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -161,7 +159,7 @@ Atomic units are used throughout.
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\label{sec:geKS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In eDFT, the ensemble energy
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\alert{In eDFT, the ensemble energy
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\begin{equation}
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\E{\bw} = (1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
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\end{equation}
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@ -170,24 +168,23 @@ In analogy with ground-state generalized KS-DFT, we consider the following parti
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\begin{equation}
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F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}
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\begin{equation}
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F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}
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\end{equation}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{KS-eDFT for excited states}
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\label{sec:KS-eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we explain how to perform a self-consistent KS calculation for ensembles (eKS) in the context of excited states.
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Here, we explain how to perform a self-consistent KS calculation for ensembles (KS-eDFT) in the context of excited states.
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In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
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i) the ground state ($I=0$), ii) the first singly-excited state ($I=1$), and iii) the first doubly-excited state ($I=2$).
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(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$).
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Generalization to a larger number of states is straightforward and is left for future work.
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By definition, the ensemble energy is
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\begin{equation}
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\E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
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\end{equation}
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$\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the to the single excitation and double excitation, respectively.
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The $\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the single and double excitation, respectively.
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To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
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$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
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Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
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@ -201,7 +198,7 @@ In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combinat
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\MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br).
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\end{equation}
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Within the self-consistent eKS process, one is looking for the following weight-dependent density matrix:
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Within the self-consistent KS-eDFT process, one is looking for the following weight-dependent density matrix:
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\begin{equation}
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\label{eq:Gamma}
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\bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)},
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@ -231,16 +228,18 @@ The one-electron ensemble density is
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\n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
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\end{equation}
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with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads
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\begin{multline}
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\begin{equation}
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\label{eq:Ew}
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\E{\bw}
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= \Tr(\bGamma{\bw} \, \bHc)
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+ \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw})
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\\
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+ \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
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\end{multline}
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% \\
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% + \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
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+ \int \v{c}{\bw}[\n{\bw}(\br)] d\br.
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\end{equation}
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The self-consistent process described above is carried on until $\max \abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} < \tau$, where $\tau$ is a user-defined threshold and $\eS{\mu\nu} = \braket{\AO{\mu}}{\AO{\nu}}$ are elements of the overlap matrix $\bS$.
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\alert{Note that the weight-dependent energy given by Eq.~\eqref{eq:Ew} is polluted by the so-called ghost interaction. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} which makes the ensemble energy non linear.
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Below, we propose a ghost-interaction correction in order to minimize this error.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Extracting individual energies}
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@ -274,7 +273,7 @@ From this point of view, using finite uniform electron gases \cite{Loos_2011b, G
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Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
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Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
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As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e.,~a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
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As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
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The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
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As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
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This is a necessary condition for being able to model derivative discontinuities.
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@ -299,15 +298,20 @@ This is a necessary condition for being able to model derivative discontinuities
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Density-functional approximations for ensembles}
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\section{Density-functional approximations for ensembles}
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\label{sec:eDFA}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent DFAs}
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\label{sec:wDFAs}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
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As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
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All these states have the same (uniform) density $\n{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
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We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
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Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e.~per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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\begin{equation}
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\label{eq:ec}
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\e{c}{(I)}(\n{}) = \frac{a^{(I)}\,\n{}}{\n{} + b^{(I)} \sqrt{\n{}} + c^{(I)}},
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@ -362,7 +366,7 @@ Finally, we note that, by construction,
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\begin{equation}
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\left. \pdv{\be{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} = \be{c}{(J)}[\n{\bw}(\br)] - \be{c}{(0)}[\n{\bw}(\br)].
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\end{equation}
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\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2018b}}
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\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
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%%% FIG 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{EvsL_5}
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@ -370,7 +374,6 @@ Finally, we note that, by construction,
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\label{fig:EvsL}
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Error with respect to FCI in single and double excitation energies for 5-boxium for various methods and box length $L$.
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Graphs for additional values of $\Nel$ can be found as {\SI}.
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\alert{T2: we could combine Figs. 1 and 2 into a single figure.}
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}
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\end{figure}
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%%% %%% %%%
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@ -419,14 +422,14 @@ Concerning the eKS calculations, two sets of weight have been tested: the zero-w
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e.~$\Nel = 5$).
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In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$).
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Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
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In the weakly correlated regime (i.e.~small $L$), all methods provide accurate estimates of the excitation energies.
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In the weakly correlated regime (i.e., small $L$), all methods provide accurate estimates of the excitation energies.
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When the box gets larger, they start to deviate.
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For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
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TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
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TDA-TDLDA slightly corrects this trend thanks to error compensation.
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Concerning the eLDA functional, our results clearly evidences that the equi-weights [i.e. $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [i.e. $\bw = (0,0)$].
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Concerning the eLDA functional, our results clearly evidences that the equi-weights [i.e., $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [i.e., $\bw = (0,0)$].
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This is especially true for the single excitation which is significantly improved by using state-averaged weights.
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The effect on the double excitation is less pronounced.
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Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
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@ -456,7 +459,7 @@ It would highlight the contribution of the derivative discontinuity.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Concluding remarks}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In the present Letter, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
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In the present article, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
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This eDFA delivers accurate excitation energies for both single and double excitations.
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Generalization to a larger number of states is straightforward and will be investigated in future work.
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Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
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|
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