1558 lines
73 KiB
TeX
1558 lines
73 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
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]{hyperref}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\la}{\lambda}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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% operators
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hh}{\Hat{h}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\vne}{v_\text{ne}}
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\newcommand{\hWee}{\Hat{W}_\text{ee}}
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\newcommand{\WHF}{W_\text{HF}}
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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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\newcommand{\E}[2]{E_\text{#1}^{#2}}
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\newcommand{\bE}[2]{\overline{E}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\overline{f}_\text{#1}^{#2}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\dn}[2]{\Delta n_{#1}^{#2}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\Ecat}{E_\text{cat}}
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\newcommand{\Eneu}{E_\text{neu}}
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% matrices/operator
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\newcommand{\bw}{{\boldsymbol{w}}}
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\newcommand{\bGam}[1]{\boldsymbol{\Gamma}^{#1}}
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\newcommand{\bgam}[1]{\boldsymbol{\gamma}^{#1}}
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\newcommand{\opGam}[1]{\hat{\Gamma}^{#1}}
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\newcommand{\bh}{\boldsymbol{h}}
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\newcommand{\bF}[1]{\boldsymbol{F}^{#1}}
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\newcommand{\Ex}[2]{\Omega_\text{#1}^{#2}}
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% elements
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% units
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\newcommand{\SI}{\textcolor{blue}{supplementary material}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
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%%% added by Manu %%%
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\newcommand{\beq}{\begin{equation}}
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\newcommand{\eeq}{\end{equation}}
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\newcommand{\bmk}{\boldsymbol{\kappa}} % orbital rotation vector
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%%%%
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\begin{document}
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\title{Weight-dependent local density-functional approximations for ensembles}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Emmanuel Fromager}
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\email{fromagere@unistra.fr}
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\affiliation{\LCQ}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{abstract}
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We report a local, weight-dependent correlation density-functional approximation that incorporates information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
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This density-functional approximation for ensembles is specially designed for the computation of single and double excitations within Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for excited states), and can be seen as a natural extension of the ubiquitous local-density approximation in the context of ensembles.
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The resulting density-functional approximation for ensembles, based on both finite and infinite uniform electron gas models, automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence.
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Its accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
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Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
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\end{abstract}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Over the last two decades, density-functional theory (DFT)
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\cite{Hohenberg_1964,Kohn_1965,ParrBook} has become the method of choice for
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modeling the electronic structure of large molecular systems and
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materials.
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The main reason is that, within DFT, the quantum contributions to the
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electronic repulsion energy --- the so-called exchange-correlation (xc)
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energy --- is rewritten as a functional of the electron density $\n{}{} \equiv \n{}{}(\br{})$, 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
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density functional.
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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
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referred to as ``strong correlation problem'') still remains a
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challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017}
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Another issue, which is partly connected to the previous one, is the
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description of low-lying quasi-degenerate states.
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The standard approach for modeling excited states in a DFT framework is
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linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012}
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In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong.
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Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action} \cite{Vignale_2008} which is a
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functional of the time-dependent density $\n{}{} \equiv \n{}{}(\br,t)$ and, as
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such, it should incorporate memory effects. Standard implementations of TDDFT rely on
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the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} In other
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words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
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As a result, double electronic excitations (where two electrons are simultaneously promoted by a single photon) 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}
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When affordable (\ie, for relatively small molecules), time-independent
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state-averaged wave function methods
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\cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002,Helgakerbook} can be employed to fix the various issues mentioned above.
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The basic idea is to describe a finite (canonical) ensemble of ground
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and excited states altogether, \ie, with the same set of orbitals.
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Interestingly, a similar approach exists in DFT. Referred to as
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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
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of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
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In GOK-DFT, the ensemble xc energy is a functional of the
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density but also a
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function of the ensemble weights. Note that, unlike in conventional
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Boltzmann ensembles, \cite{Pastorczak_2013} the ensemble weights (each state in the ensemble
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is assigned a given and fixed weight) are allowed to vary
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independently in a GOK ensemble.
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The weight dependence of the xc functional plays a crucial role in the
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calculation of excitation energies.
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\cite{Gross_1988b,Yang_2014,Deur_2017,Deur_2019,Senjean_2018,Senjean_2020}
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It actually accounts for the derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
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%\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
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Even though GOK-DFT is in principle able to
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describe near-degenerate situations and multiple-electron excitation
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processes, it has not
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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}
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One of the reason is the lack, not to say the absence, of reliable
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density-functional approximations for ensembles.
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The most recent works dealing with this particular issue are still fundamental and
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exploratory, as they rely either on simple (but nontrivial) model
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systems
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\cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
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or atoms. \cite{Yang_2014,Yang_2017,Gould_2019_insights}
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Despite all these efforts, it is still unclear how weight dependencies
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can be incorporated into density-functional approximations. This problem is actually central not
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only in GOK-DFT but also in conventional (ground-state) DFT as the infamous derivative
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discontinuity problem that ocurs when crossing an integral number of
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electrons can be recast into a weight-dependent ensemble
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one. \cite{Senjean_2018,Senjean_2020}
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The present work is an attempt to address this problem,
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with the ambition to turn, in the forthcoming future, GOK-DFT into a
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(low-cost) practical computational method for modeling excited states in molecules and extended systems.
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Starting from the ubiquitous local-density approximation (LDA), we
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design a weight-dependent ensemble correction based on a finite uniform
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electron gas from which density-functional excitation energies can be
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extracted. The present density-functional approximation for ensembles, which can be seen as a natural
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extension of the LDA, will be referred to as eLDA in the remaining of this paper.
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As a proof of concept, we apply this general strategy to
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ensemble correlation energies (that we combine with
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ensemble exact exchange energies) in the particular case of
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\emph{strict} one-dimensional (1D)
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spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
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In other words, the Coulomb interaction used in this work corresponds to
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particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
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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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%Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
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This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
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In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
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The paper is organized as follows.
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Exact and approximate formulations of GOK-DFT are discussed in Sec.~\ref{sec:eDFT},
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with a particular emphasis on the calculation of individual energy levels.
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In Sec.~\ref{sec:eDFA}, we detail the construction of the
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weight-dependent local correlation functional specially designed for the
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computation of single and double excitations within GOK-DFT.
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Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
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In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA functional by computing single and double excitations in 1D many-electron systems in the weak, intermediate and strong correlation regimes.
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Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
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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:eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{GOK-DFT}\label{subsec:gokdft}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In this section we give a brief review of GOK-DFT and discuss the
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extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
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individual exchange energies.
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Let us start by introducing the GOK ensemble energy \cite{Gross_1988a}
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\beq\label{eq:exact_GOK_ens_ener}
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\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
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\eeq
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where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hH = \hh + \hWee$, where
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\beq
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\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{i}^2 + \vne(\br{i}) ]
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\eeq
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is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator.
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The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$.
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They are normalized, \ie,
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\beq\label{eq:weight_norm_cond}
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\ew{0} = 1 - \sum_{K>0} \ew{K},
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\eeq
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so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.
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For simplicity we will assume in the following that the energies are not degenerate.
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Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states.\cite{Gross_1988b}
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In the KS formulation of GOK-DFT, {which is simply referred to as
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KS ensemble DFT (KS-eDFT) in the following}, the ensemble energy is determined variationally as follows:\cite{Gross_1988b}
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\beq\label{eq:var_ener_gokdft}
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\E{}{\bw}
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= \min_{\opGam{\bw}}
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\qty{
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\Tr[\opGam{\bw} \hh] + \E{Hx}{\bw} \qty[\n{\opGam{\bw}}{}] + \E{c}{\bw} \qty[\n{\opGam{\bw}}{}]
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},
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\eeq
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where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads
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\beq
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\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
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\eeq
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The KS determinants [or configuration state functions~\cite{Gould_2017}]
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$\Det{(K)}$ are all constructed from the same set of ensemble KS
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orbitals that are variationally optimized.
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The trial ensemble density in Eq.~\eqref{eq:var_ener_gokdft} is simply
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the weighted sum of the individual KS densities, \ie,
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\beq\label{eq:KS_ens_density}
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\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K)}}{}(\br{}).
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\eeq
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As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange (Hx) and correlation (c) energies are described with density functionals that are \textit{weight dependent}.
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We focus in the following on the (exact) Hx part, which is defined as~\cite{Gould_2017}
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\beq\label{eq:exact_ens_Hx}
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\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bw}[\n{}{}]}{\hWee}{\Det{(K),\bw}[\n{}{}]},
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\eeq
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where the KS wavefunctions fulfill the ensemble density constraint
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\beq
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\sum_{K\geq 0} \ew{K} \n{\Det{(K),\bw}[\n{}{}]}{}(\br{}) = \n{}{}(\br{}).
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\eeq
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The (approximate) description of the correlation part is discussed in
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Sec.~\ref{sec:eDFA}.
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In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example).
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As pointed out recently in Ref.~\onlinecite{Deur_2019}, the latter can be extracted
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exactly from a single ensemble calculation as follows:
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\beq\label{eq:indiv_ener_from_ens}
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\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
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\pdv{\E{}{\bw}}{\ew{K}},
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\eeq
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where, according to the normalization condition of Eq.~\eqref{eq:weight_norm_cond},
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\beq
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\pdv{\E{}{\bw}}{\ew{K}}= \E{}{(K)} -
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\E{}{(0)}\equiv\Ex{}{(K)}
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\eeq
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corresponds to the $K$th excitation energy.
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According to the {\it variational} ensemble energy expression of
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Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$
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can be evaluated from the minimizing weight-dependent KS wavefunctions
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$\Det{(K)} \equiv \Det{(K),\bw}$ as follows:
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\beq\label{eq:deriv_Ew_wk}
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\begin{split}
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\pdv{\E{}{\bw}}{\ew{K}}
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& = \mel*{\Det{(K)}}{\hh}{\Det{(K)}}-\mel*{\Det{(0)}}{\hh}{\Det{(0)}}
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\\
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& + \Bigg\{\int \fdv{\E{Hx}{\bw}[\n{}{}]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{}
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+ \pdv{\E{Hx}{\bw} [\n{}{}]}{\ew{K}}
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\\
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& + \int \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{}
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+ \pdv{\E{c}{\bw}[n]}{\ew{K}}
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\Bigg\}_{\n{}{} = \n{\opGam{\bw}}{}}.
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\end{split}
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\eeq
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The Hx contribution from Eq.~\eqref{eq:deriv_Ew_wk} can be recast as
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\beq\label{eq:_deriv_wk_Hx}
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\left.
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\pdv{}{\xi_K} \qty(\E{Hx}{\bxi} [\n{}{\bxi,\bxi}]
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- \E{Hx}{\bw}[\n{}{\bw,\bxi}] )
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\right|_{\bxi=\bw},
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\eeq
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where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and the
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auxiliary double-weight ensemble density reads
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\beq
|
|
\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
|
|
\eeq
|
|
Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
|
|
from the exact expression in Eq.~\eqref{eq:exact_ens_Hx} that
|
|
\beq
|
|
\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
|
|
\eeq
|
|
and
|
|
\beq
|
|
\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}.
|
|
\eeq
|
|
This yields, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx}, the simplified expression
|
|
\beq\label{eq:deriv_Ew_wk_simplified}
|
|
\begin{split}
|
|
\pdv{\E{}{\bw}}{\ew{K}}
|
|
& = \mel*{\Det{(K)}}{\hH}{\Det{(K)}}
|
|
- \mel*{\Det{(0)}}{\hH}{\Det{(0)}}
|
|
\\
|
|
& + \qty{
|
|
\int \fdv{\E{c}{\bw}[\n{}{}]}{\n{}{}({\br{}})}
|
|
\qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
|
|
+
|
|
\pdv{\E{c}{\bw} [\n{}{}]}{\ew{K}}
|
|
}_{\n{}{} = \n{\opGam{\bw}}{}} d\br{}.
|
|
\end{split}
|
|
\eeq
|
|
Since, according to Eqs.~\eqref{eq:var_ener_gokdft} and \eqref{eq:exact_ens_Hx}, the ensemble energy can be evaluated as
|
|
\beq
|
|
\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGam{\bw}}{}],
|
|
\eeq
|
|
with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$],
|
|
we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
|
|
\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\onlinecite{Fromager_2020} for the $I$th energy level:
|
|
\beq\label{eq:exact_ener_level_dets}
|
|
\begin{split}
|
|
\E{}{(I)}
|
|
& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGam{\bw}}{}]
|
|
\\
|
|
& + \int \fdv{\E{c}{\bw}[\n{\opGam{\bw}}{}]}{\n{}{}(\br{})}
|
|
\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{}
|
|
\\
|
|
&+
|
|
\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
|
|
\left.
|
|
\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
|
|
\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
|
|
\end{split}
|
|
\eeq
|
|
Note that, when $\bw=0$, the ensemble correlation functional reduces to the
|
|
conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
|
|
result, the regular KS-DFT expression is recovered from
|
|
Eq.~\eqref{eq:exact_ener_level_dets} for the ground-state energy
|
|
\beq
|
|
\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
|
|
\E{c}{}[\n{\Det{(0)}}{}],
|
|
\eeq
|
|
or, equivalently,
|
|
\beq\label{eq:gs_ener_level_gs_lim}
|
|
\E{}{(0)}=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}}
|
|
,
|
|
\eeq
|
|
where the density-functional Hamiltonian reads
|
|
\beq\label{eq:dens_func_Hamilt}
|
|
\hat{H}[n]=\hH+
|
|
\sum^N_{i=1}\left(\fdv{\E{c}{}[n]}{\n{}{}(\br{i})}
|
|
+C_{\rm c}[n]
|
|
\right),
|
|
\eeq
|
|
and
|
|
\beq\label{eq:corr_LZ_shift}
|
|
C_{\rm c}[n]=\dfrac{\E{c}{}[n]
|
|
-\int
|
|
\fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}}
|
|
\eeq
|
|
is the correlation component of
|
|
Levy--Zahariev's constant shift in potential.\cite{Levy_2014}
|
|
Similarly, the excited-state ($I>0$) energy level expressions
|
|
can be recast as follows:
|
|
\beq\label{eq:excited_ener_level_gs_lim}
|
|
\E{}{(I)}
|
|
= \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}}
|
|
+
|
|
\left.
|
|
\pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}}
|
|
\right|_{\bw=0}.
|
|
\eeq
|
|
As readily seen from Eqs.~\eqref{eq:dens_func_Hamilt} and
|
|
\eqref{eq:corr_LZ_shift}, introducing any constant shift $\delta
|
|
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta
|
|
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ into the correlation
|
|
potential leaves the density-functional Hamiltonian $\hat{H}[n]$ (and
|
|
therefore the individual energy levels) unchanged. As a result, in
|
|
this context,
|
|
the correlation derivative discontinuities induced by the
|
|
excitation process~\cite{Levy_1995} will be fully described by the ensemble
|
|
correlation derivatives [second term on the right-hand side of
|
|
Eq.~\eqref{eq:excited_ener_level_gs_lim}].
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{One-electron reduced density matrix formulation}
|
|
%%%%%%%%%%%%%%%%
|
|
For implementation purposes, we will use in the rest of this work
|
|
(one-electron reduced) density matrices
|
|
as basic variables, rather than Slater determinants.
|
|
As the theory is applied later on to \textit{spin-polarized}
|
|
systems, we drop spin indices in the density matrices, for convenience.
|
|
If we expand the
|
|
ensemble KS orbitals (from which the determinants are constructed) in an atomic orbital (AO) basis,
|
|
\beq
|
|
\MO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
|
|
\eeq
|
|
\iffalse%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\titou{\beq
|
|
\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
|
|
\eeq
|
|
where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatial degrees of freedom, and
|
|
\beq
|
|
s(\omega)
|
|
=
|
|
\begin{cases}
|
|
\alpha(\omega), & \text{for spin-up electrons,} \\
|
|
\text{or} \\
|
|
\beta(\omega), & \text{for spin-down electrons,}
|
|
\end{cases}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%%%%%%%%%%%%%%%
|
|
then the density matrix of the
|
|
determinant $\Det{(K)}$ can be expressed as follows in the AO basis
|
|
\beq
|
|
\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
|
|
\eeq
|
|
where the summation runs over the orbitals that are occupied in $\Det{(K)}$.
|
|
The electron density of the $K$th KS determinant can then be evaluated
|
|
as follows:
|
|
\beq
|
|
\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}),
|
|
\eeq
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% Manu's derivation %%%
|
|
\iffalse%%
|
|
\blue{
|
|
\beq
|
|
n_{\bmg^{(K)}}(\br{})&=&\sum_\sigma\left\langle\hat{\Psi}^\dagger(\br{}\sigma)\hat{\Psi}(\br{}\sigma)\right\rangle^{(K)}
|
|
\nonumber\\
|
|
&=&\sum_\sigma\sum_{pq}\varphi^\sigma_p(\br{})\varphi^\sigma_q(\br{})\left\langle\hat{a}_{p^\sigma,\sigma}^\dagger\hat{a}_{q^\sigma,\sigma}\right\rangle^{(K)}
|
|
\nonumber\\
|
|
&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\left(\varphi^\sigma_p(\br{})\right)^2
|
|
\nonumber\\
|
|
&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\sum_{\mu\nu}c^\sigma_{{\mu
|
|
p}}c^\sigma_{{\nu p}}\AO{\mu}(\br{})\AO{\nu}(\br{})
|
|
\nonumber\\
|
|
&=&\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{})\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}c^\sigma_{{\mu
|
|
p}}c^\sigma_{{\nu p}}
|
|
\eeq
|
|
}
|
|
\fi%%%
|
|
%%%% end Manu
|
|
while the ensemble density matrix
|
|
and the ensemble density read
|
|
\beq
|
|
\bGam{\bw}
|
|
= \sum_{K\geq 0} \ew{K} \bGam{(K)}
|
|
\equiv \eGam{\mu\nu}{\bw}
|
|
= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)},
|
|
\eeq
|
|
and
|
|
\beq
|
|
\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
|
|
\eeq
|
|
respectively.
|
|
The exact individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
|
|
\beq\label{eq:exact_ind_ener_rdm}
|
|
\begin{split}
|
|
\E{}{(I)}
|
|
& =\Tr[\bGam{(I)} \bh]
|
|
+ \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
|
|
+ \E{c}{{\bw}}[\n{\bGam{\bw}}{}]
|
|
\\
|
|
& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
|
|
\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ] d\br{}
|
|
\\
|
|
& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
|
|
\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bGam{\bw}}{}}
|
|
,
|
|
\end{split}
|
|
\eeq
|
|
where
|
|
\beq
|
|
\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}
|
|
\eeq
|
|
denotes the matrix of the one-electron integrals.
|
|
The exact individual Hx energies are obtained from the following trace formula
|
|
\beq
|
|
\Tr[\bGam{(K)} \bG \bGam{(L)}]
|
|
= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},
|
|
\eeq
|
|
where the antisymmetrized two-electron integrals read
|
|
\beq
|
|
\bG
|
|
\equiv G_{\mu\nu\la\si}
|
|
= \dbERI{\mu\nu}{\la\si}
|
|
= \ERI{\mu\nu}{\la\si} - \ERI{\mu\si}{\la\nu},
|
|
\eeq
|
|
with
|
|
\beq
|
|
\ERI{\mu\nu}{\la\si} = \iint \frac{\AO{\mu}(\br{1}) \AO{\nu}(\br{1}) \AO{\la}(\br{2}) \AO{\si}(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}.
|
|
\eeq
|
|
%Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
|
|
%polarized systems in which $\eGam{\mu\nu}{(K)\beta}=0$ and
|
|
%$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
|
|
%-(\mu\omega\vert\lambda\nu)$.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%% Hx energy ...
|
|
%%% Manu's derivation
|
|
\iffalse%%%%
|
|
\blue{
|
|
\beq
|
|
&&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
|
|
RS\rangle\eGam{PR}^{(K)}\eGam{QS}^{(L)}
|
|
\nonumber\\
|
|
&&
|
|
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
|
|
\nonumber\\
|
|
&&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
|
|
p^\sigma\sigma q^\tau\tau
|
|
\vert SR\rangle
|
|
\Big)\Gamma^{(K)}_{p^\sigma\sigma,R}\Gamma^{(L)}_{q^\tau\tau, S}
|
|
\nonumber\\
|
|
&&
|
|
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
|
|
\nonumber\\
|
|
&&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
|
|
\Gamma^{(K)\sigma}_{p^\sigma r^\sigma}\Gamma^{(L)\tau}_{q^\tau s^\tau}
|
|
\nonumber\\
|
|
&& -\sum_{s^\sigma r^\tau}\langle
|
|
p^\sigma q^\tau
|
|
\vert s^\sigma r^\tau\rangle
|
|
\delta_{\sigma\tau}\Gamma^{(K)\sigma}_{p^\sigma
|
|
r^\sigma}\Gamma^{(L)\sigma}_{q^\sigma s^\sigma}\Big)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
|
|
\nonumber\\
|
|
&&\left(\langle p^\sigma q^\tau\vert p^\sigma q^\tau\rangle
|
|
n_{p^\sigma}^{(K)\sigma}n_{q^\tau}^{(L)\tau}
|
|
-\delta_{\sigma\tau}\langle p^\sigma q^\sigma\vert q^\sigma p^\sigma \rangle
|
|
n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\sigma}
|
|
\Big)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
|
|
\Big)
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big[({\mu}{\nu}\vert{\lambda}{\omega})
|
|
-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
|
|
\Big]
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%
|
|
%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%
|
|
\iffalse%%%% Manu's derivation ...
|
|
\blue{
|
|
\beq
|
|
n^{\bw}({\br{}})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
|
w}_Kn^{(K)}({\bfx})
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
|
w}_K\sum_{pq}\varphi_p({\bfx})\varphi_q({\bfx})\Gamma_{pq}^{(K)}
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{\sigma=\alpha,\beta}
|
|
\sum_{K\geq 0}
|
|
{\tt
|
|
w}_K\sum_{p\in (K)}\varphi^2_p({\bfx})
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{\sigma=\alpha,\beta}
|
|
\sum_{K\geq 0}
|
|
{\tt
|
|
w}_K
|
|
\sum_{\mu\nu}
|
|
\sum_{p\in (K)}c_{\mu p}c_{\nu p}\AO{\mu}({\bfx})\AO{\nu}({\bfx})
|
|
\nonumber\\
|
|
&=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%% end
|
|
%%%%%%%%%%%%%%%
|
|
%\subsection{Hybrid GOK-DFT}
|
|
%%%%%%%%%%%%%%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%
|
|
\subsection{Approximations}\label{subsec:approx}
|
|
%%%%%%%%%%%%%%%
|
|
|
|
|
|
In the following, GOK-DFT will be applied
|
|
to 1D
|
|
spin-polarized systems where
|
|
Hartree and exchange energies cannot be separated.
|
|
For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
|
|
\beq\label{eq:eHF-dens_mat_func}
|
|
\WHF[\bGam{}] = \frac{1}{2} \Tr[\bGam{} \bG \bGam{}],
|
|
\eeq
|
|
for the Hx density-functional energy in the variational energy
|
|
expression of Eq.~\eqref{eq:var_ener_gokdft}, thus leading to the
|
|
following approximation:
|
|
\beq\label{eq:min_with_HF_ener_fun}
|
|
\bGam{\bw}
|
|
\rightarrow \argmin_{\bgam{\bw}}
|
|
\qty{
|
|
\Tr[\bgam{\bw} \bh ] + \WHF[ \bgam{\bw}] + \E{c}{\bw}[\n{\bgam{\bw}}{}]
|
|
}.
|
|
\eeq
|
|
The minimizing ensemble density matrix in Eq.~\eqref{eq:min_with_HF_ener_fun} fulfills the following
|
|
stationarity condition
|
|
\beq\label{eq:commut_F_AO}
|
|
\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
|
|
\eeq
|
|
where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the
|
|
overlap matrix and the ensemble Fock-like matrix reads
|
|
\beq
|
|
\bF{\bw} \equiv \eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} +
|
|
\sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw},
|
|
\eeq
|
|
with
|
|
\beq
|
|
\eh{\mu\nu}{\bw}
|
|
= \eh{\mu\nu}{} + \int \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}) d\br{}.
|
|
\eeq
|
|
%%%%%%%%%%%%%%%
|
|
\iffalse%%%%%%
|
|
% Manu's derivation %%%%
|
|
\color{blue}
|
|
I am teaching myself ...\\
|
|
Stationarity condition
|
|
\beq
|
|
&&0=\sum_{K\geq 0}w_K\sum_{t^\sigma}\Big(f_{p^\sigma\sigma,t^\sigma\sigma}\Gamma^{(K)\sigma}_{t^\sigma
|
|
q^\sigma}-\Gamma^{(K)\sigma}_{p^\sigma
|
|
t^\sigma}f_{t^\sigma\sigma,q^\sigma\sigma}\Big)
|
|
\nonumber\\
|
|
&&=\sum_{K\geq 0}w_K
|
|
\Big(f_{p^\sigma\sigma,q^\sigma\sigma}n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}f_{p^\sigma\sigma,q^\sigma\sigma}\Big)
|
|
\nonumber\\
|
|
&&
|
|
=\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
|
|
p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)
|
|
\eeq
|
|
thus leading to
|
|
\beq
|
|
&&0=\sum_{p^\sigma q^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\omega q}\left(\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
|
|
p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)\right)
|
|
\nonumber\\
|
|
&&=\sum_{\mu\nu}\sum_{K\geq 0}w_K
|
|
F_{\mu\nu}^\sigma\left(\Gamma^{(K)\sigma}_{\nu\omega}\sum_{p^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\mu
|
|
p}-\Gamma^{(K)\sigma}_{\mu\lambda}\sum_{q^\sigma}c^\sigma_{\omega q}c^\sigma_{\nu q}\right)
|
|
\nonumber\\
|
|
\eeq
|
|
If we denote $M^\sigma_{\lambda\mu}=\sum_{p^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\mu
|
|
p}$ it comes
|
|
\beq
|
|
S_{\mu\nu}=\sum_{\lambda\omega}S_{\mu\lambda}M^\sigma_{\lambda\omega}S_{\omega\nu}
|
|
\eeq
|
|
which simply means that
|
|
\beq
|
|
{\bm S}={\bm S}{\bm M}{\bm S}
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
{\bm M}={\bm S}^{-1}.
|
|
\eeq
|
|
The stationarity condition simply reads
|
|
\beq
|
|
\sum_{\mu\nu}F_{\mu\nu}^\sigma\left(\Gamma^{\bw\sigma}_{\nu\omega}
|
|
\left[{\bm S}^{-1}\right]_{\lambda\mu}
|
|
-\Gamma^{\bw\sigma}_{\mu\lambda}\left[{\bm S}^{-1}\right]_{\omega\nu}\right)
|
|
=0
|
|
\eeq
|
|
thus leading to
|
|
\beq
|
|
{\bm S}^{-1}{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}={\bm \Gamma}^{\bw\sigma}{{\bm F}^\sigma}{\bm S}^{-1}
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
|
|
\Gamma}^{\bw\sigma}{{\bm F}^\sigma}.
|
|
\eeq
|
|
%%%%%
|
|
|
|
Fock operator:\\
|
|
\beq
|
|
&&f_{p^\sigma\sigma,q^\sigma\sigma}-\langle\varphi_p^\sigma\vert\hat{h}\vert\varphi_q^\sigma\rangle
|
|
\nonumber\\
|
|
&&=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau s^\tau}
|
|
\nonumber\\
|
|
&&
|
|
\Big(\langle p^\sigma r^\tau\vert
|
|
q^\sigma s^\tau\rangle
|
|
-\delta_{\sigma\tau}\langle p^\sigma r^\sigma\vert
|
|
s^\sigma q^\sigma\rangle
|
|
\Big)
|
|
\Gamma^{(L)\tau}_{r^\tau
|
|
s^\tau}
|
|
\nonumber\\
|
|
&&
|
|
=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau}\Big(\langle p^\sigma r^\tau\vert
|
|
q^\sigma r^\tau\rangle
|
|
-\delta_{\sigma\tau}\langle p^\sigma r^\tau\vert
|
|
r^\tau q^\sigma\rangle
|
|
\Big)
|
|
n^{(L)\tau}_{r^\tau}
|
|
\nonumber\\
|
|
&&=\sum_{L\geq 0}w_L
|
|
\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
|
|
p^\sigma\lambda\vert q^\sigma\omega\rangle
|
|
-\delta_{\sigma\tau}
|
|
\langle
|
|
p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
|
|
\Gamma^{(L)\tau}_{\lambda\omega}
|
|
\nonumber\\
|
|
&&=
|
|
\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
|
|
p^\sigma\lambda\vert q^\sigma\omega\rangle
|
|
-\delta_{\sigma\tau}
|
|
\langle
|
|
p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
|
|
\Gamma^{\bw\tau}_{\lambda\omega}
|
|
\nonumber\\
|
|
&&=\sum_{\mu\nu\lambda\omega}\sum_{\tau}
|
|
\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
|
|
\Big)\Gamma^{\bw\tau}_{\lambda\omega}c^\sigma_{\mu p}c^\sigma_{\nu q}
|
|
\nonumber\\
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
f_{p^\sigma\sigma,q^\sigma\sigma}=\sum_{\mu\nu}F_{\mu\nu}^\sigma c^\sigma_{\mu p}c^\sigma_{\nu q}
|
|
\eeq
|
|
where
|
|
\beq
|
|
F_{\mu\nu}^\sigma=h_{\mu\nu}+\sum_{\lambda\omega}\sum_\tau
|
|
G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
|
|
\eeq
|
|
and
|
|
\color{black}
|
|
\\
|
|
\fi%%%%%%%%%%%
|
|
%%%%% end Manu
|
|
%%%%%%%%%%%%%%%%%%%%
|
|
Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the ensemble density matrix is
|
|
optimized with a non-local exchange potential rather than a
|
|
density-functional local one, as expected from
|
|
Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
|
|
applicable to not-necessarily spin polarized and real (higher-dimensional) systems.
|
|
As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
|
|
ensemble density matrix into the HF interaction energy functional
|
|
introduces unphysical \textit{ghost-interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
|
|
as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
|
|
\beq\label{eq:WHF}
|
|
\begin{split}
|
|
\WHF[\bGam{\bw}]
|
|
& = \frac{1}{2} \sum_{K\geq 0} \ew{K}^2 \Tr[\bGam{(K)} \bG \bGam{(K)}]
|
|
\\
|
|
& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr[\bGam{(K)} \bG \bGam{(L)}].
|
|
\end{split}
|
|
\eeq
|
|
The ensemble energy is of course expected to vary linearly with the ensemble
|
|
weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
|
|
These errors are essentially removed when evaluating the individual energy
|
|
levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
|
|
|
|
Turning to the density-functional ensemble correlation energy, the
|
|
following ensemble local-density \textit{approximation} (eLDA) will be employed
|
|
\beq\label{eq:eLDA_corr_fun}
|
|
\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
|
|
\eeq
|
|
where the ensemble correlation energy per particle
|
|
\beq\label{eq:decomp_ens_correner_per_part}
|
|
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
|
|
\eeq
|
|
is \titou{explicitly} \textit{weight dependent}.
|
|
As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
|
|
from a finite uniform electron gas model.
|
|
%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
|
|
%What do you think?}
|
|
|
|
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
|
|
reads
|
|
\beq\label{eq:Ew-GIC-eLDA}
|
|
\E{eLDA}{\bw}=\Tr[\bGam{\bw}\bh] + \WHF[\bGam{\bw}] +\int
|
|
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{\bw}}{}(\br{}) d\br{}.
|
|
\eeq
|
|
%Manu, would it be useful to add this equation and the corresponding text?
|
|
%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
|
|
%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
|
|
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our \titou{final expression of the KS-eLDA energy level}
|
|
\titou{\beq\label{eq:EI-eLDA}
|
|
\begin{split}
|
|
\E{{eLDA}}{(I)}
|
|
=
|
|
\E{HF}{(I)}
|
|
+ \Xi_\text{c}^{(I)}
|
|
+ \Upsilon_\text{c}^{(I)},
|
|
\end{split}
|
|
\eeq}
|
|
where
|
|
\beq\label{eq:ind_HF-like_ener}
|
|
\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
|
|
\eeq
|
|
is the analog for ground and excited states (within an ensemble) of the HF energy, \titou{and
|
|
\begin{gather}
|
|
\begin{split}
|
|
\Xi_\text{c}^{(I)}
|
|
& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
\\
|
|
&
|
|
+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
|
|
\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
\\
|
|
\end{split}
|
|
\\
|
|
\Upsilon_\text{c}^{(I)}
|
|
= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
|
|
\end{gather}}
|
|
|
|
If, for analysis purposes, we Taylor expand the density-functional
|
|
correlation contributions
|
|
around the $I$th KS state density
|
|
$\n{\bGam{(I)}}{}(\br{})$, the sum of
|
|
the \titou{second term} on the right-hand side
|
|
of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
|
|
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
|
\beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
|
|
\titou{\Xi_\text{c}^{(I)}}
|
|
= \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
|
|
\eeq
|
|
Therefore, it can be identified as
|
|
an individual-density-functional correlation energy where the density-functional
|
|
correlation energy per particle is approximated by the ensemble one for
|
|
all the states within the ensemble.
|
|
|
|
|
|
Let us stress that, to the best of our knowledge, eLDA is the first
|
|
density-functional approximation that incorporates ensemble weight
|
|
dependencies explicitly, thus allowing for the description of derivative
|
|
discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
|
|
comment that follows] {\it via} the \titou{third term} on the right-hand side
|
|
of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
|
|
the ensemble
|
|
correlation energy per particle in Eq.
|
|
\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
|
|
\begin{equation}
|
|
\titou{\Upsilon_\text{c}^{(I)}}
|
|
%&=
|
|
%\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
%\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
|
|
%-
|
|
%\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
|
|
%\Big)
|
|
%d\br{}
|
|
%\\
|
|
=\int
|
|
\qty[\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
|
|
-
|
|
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
|
|
] \n{\bGam{\bw}}{}(\br{})
|
|
d\br{},
|
|
%\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
|
|
\end{equation}
|
|
thus leading to the following Taylor expansion through first order in
|
|
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
|
\beq\label{eq:Taylor_exp_DDisc_term}
|
|
\begin{split}
|
|
\titou{\Upsilon_\text{c}^{(I)}}
|
|
%& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
%\\
|
|
&=
|
|
\int \qty[ \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) - \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) ] \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
\\
|
|
&+\int \Bigg[
|
|
\n{\bGam{(I)}}{}(\br{})
|
|
\left.\left(
|
|
\pdv{\be{c}{{(I)}}(\n{}{})}{\n{}{}}
|
|
-
|
|
\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}
|
|
\right)\right|_{\n{}{} =
|
|
\n{\bGam{(I)}}{}(\br{})}
|
|
\\
|
|
&+\be{c}{(I)}(\n{\bGam{(I)}}{}(\br{}))
|
|
-
|
|
\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]
|
|
\qty[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]
|
|
d\br{}
|
|
\\
|
|
&
|
|
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
|
|
\end{split}
|
|
\eeq
|
|
As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the
|
|
role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$}
|
|
\trashPFL{[last term on the right-hand side of Eq.~\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
|
|
individual correlation energy per particle for the ensemble one.
|
|
|
|
Let us finally note that, while the weighted sum of the
|
|
individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of
|
|
the KS-eLDA ensemble energy, \ie,
|
|
\beq\label{eq:Ew-eLDA}
|
|
\begin{split}
|
|
\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
|
|
\\
|
|
&=
|
|
\E{eLDA}{\bw}
|
|
-\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}],
|
|
\end{split}
|
|
\eeq
|
|
the excitation energies computed from the KS-eLDA individual energy level
|
|
expressions in Eq. \eqref{eq:EI-eLDA} simply reads
|
|
\beq\label{eq:Om-eLDA}
|
|
\begin{split}
|
|
\Ex{eLDA}{(I)}
|
|
&=
|
|
\Ex{HF}{(I)}
|
|
\\
|
|
&+ \int
|
|
\qty[\e{c}{{\bw}}(\n{}{})+n\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}]
|
|
_{\n{}{} =
|
|
\n{\bGam{\bw}}{}(\br{})}
|
|
\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{}
|
|
\\ & + \DD{c}{(I)},
|
|
\end{split}
|
|
\eeq
|
|
where the HF-like excitation energies, $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
|
\E{HF}{(0)}$, are determined from a single set of ensemble KS orbitals and
|
|
\beq\label{eq:DD-eLDA}
|
|
\DD{c}{(I)}
|
|
= \int \n{\bGam{\bw}}{}(\br{})
|
|
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
\eeq
|
|
is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Density-functional approximations for ensembles}
|
|
\label{sec:eDFA}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Paradigm}
|
|
\label{sec:paradigm}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Most of the standard local and semi-local density-functional approximations rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
|
|
One major drawback of the jellium paradigm, when it comes to develop density-functional approximations for ensembles, is that the ground and excited states are not easily accessible like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
|
|
Moreover, because the infinite uniform electron gas model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
|
|
From this point of view, using finite finite uniform electron gases, \cite{Loos_2011b,
|
|
Gill_2012} which have, like an atom, discrete energy levels and non-zero
|
|
gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
|
|
However, an obvious drawback of using finite uniform electron gases is that the resulting density-functional approximation for ensemble
|
|
will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
|
|
Here, we propose to construct a weight-dependent eLDA for the
|
|
calculations of excited states in 1D systems by combining finite uniform electron gases with the
|
|
usual infinite uniform electron gas.
|
|
|
|
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
|
|
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, and therefore {\it equal}.
|
|
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
|
|
This is a necessary condition for being able to model the ensemble
|
|
correlation derivatives with respect to the weights [last term
|
|
on the right-hand side of Eq.~\eqref{eq:exact_ener_level_dets}].
|
|
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous infinite uniform electron gas paradigm. \cite{Loos_2013,Loos_2013a}
|
|
Let us stress that, in a finite uniform electron gas like ringium, the interacting and
|
|
noninteracting densities match individually for all the states within the
|
|
ensemble
|
|
(these densities are all equal to the uniform density), which means that
|
|
so-called density-driven correlation
|
|
effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.
|
|
Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
|
|
|
|
The present weight-dependent density-functional approximation is specifically designed for the
|
|
calculation of excited-state energies within GOK-DFT.
|
|
To take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
|
|
(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
|
|
To ensure the GOK variational principle, \cite{Gross_1988a} the
|
|
triensemble weights must fulfil the following conditions: \cite{Deur_2019}
|
|
$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.
|
|
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring on which the electrons are confined.
|
|
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
|
|
Generalization to a larger number of states is straightforward and is left for future work.
|
|
%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
|
|
%to consider an equi-bi-ensemble
|
|
%for which $\ew{1}=1/2$. This possibility is excluded with your
|
|
%inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
|
|
%and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
|
|
%$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.}
|
|
%\manu{
|
|
%Just in case, starting from
|
|
%\beq
|
|
%\begin{split}
|
|
%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
|
|
%\\
|
|
%\end{split}
|
|
%\eeq
|
|
%we obtain
|
|
%\beq
|
|
%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2
|
|
%\eeq
|
|
%which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
|
|
%1/3$.
|
|
%}
|
|
%%% TABLE 1 %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OG_func}
|
|
Parameters of the weight-dependent correlation density-functional approximations defined in Eq.~\eqref{eq:ec}.}
|
|
% \begin{ruledtabular}
|
|
\begin{tabular}{lcddd}
|
|
\hline\hline
|
|
State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\
|
|
\hline
|
|
Ground state & $0$ & -0.0137078 & 0.0538982 & 0.0751740 \\
|
|
Singly-excited state & $1$ & -0.0238184 & 0.00413142 & 0.0568648 \\
|
|
Doubly-excited state & $2$ & -0.00935749 & -0.0261936 & 0.0336645 \\
|
|
\hline\hline
|
|
\end{tabular}
|
|
% \end{ruledtabular}
|
|
\end{table*}
|
|
%%% %%% %%% %%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Weight-dependent correlation functional}
|
|
\label{sec:Ec}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (\ie, per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
|
|
\begin{equation}
|
|
\label{eq:ec}
|
|
\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
|
|
\end{equation}
|
|
where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
|
|
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
|
Equation \eqref{eq:ec} provides three state-specific correlation density-functional approximations based on a two-electron system.
|
|
Combining these, one can build the following three-state weight-dependent correlation density-functional approximation:
|
|
\begin{equation}
|
|
\label{eq:ecw}
|
|
%\e{c}{\bw}(\n{}{})
|
|
\Tilde{\epsilon}_{\rm c}^\bw(n)= (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
|
|
\end{equation}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{LDA-centered functional}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
|
|
Obviously, the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the finite uniform electron gas.
|
|
However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
|
|
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
|
|
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} as follows:
|
|
\begin{equation}
|
|
\label{eq:becw}
|
|
\Tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
|
|
\end{equation}
|
|
In the following, we will use the LDA correlation functional that has been specifically designed for 1D systems in
|
|
Ref.~\onlinecite{Loos_2013}:
|
|
\begin{equation}
|
|
\label{eq:LDA}
|
|
\e{c}{\text{LDA}}(\n{}{})
|
|
= a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
|
|
\end{equation}
|
|
where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
|
|
\begin{subequations}
|
|
\begin{align}
|
|
a_1^\text{LDA} & = - \frac{\pi^2}{360},
|
|
\\
|
|
a_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
|
|
\\
|
|
a_3^\text{LDA} & = 2.408779.
|
|
\end{align}
|
|
\end{subequations}
|
|
Note that the strategy described in Eq.~\eqref{eq:becw} is general and
|
|
can be applied to real (higher-dimensional) systems. In order to make the
|
|
connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may
|
|
recast Eq.~\eqref{eq:becw} as
|
|
\begin{equation}
|
|
\label{eq:eLDA}
|
|
\begin{split}
|
|
{\e{c}{\bw}(\n{}{})}
|
|
& = \e{c}{\text{LDA}}(\n{}{})
|
|
\\
|
|
& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
|
|
\end{split}
|
|
\end{equation}
|
|
or, equivalently,
|
|
\begin{equation}
|
|
\label{eq:eLDA_gace}
|
|
{\e{c}{\bw}(\n{}{})}
|
|
= \e{c}{\text{LDA}}(\n{}{})
|
|
+ \sum_{K>0}\int_0^{\ew{K}}
|
|
\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K,
|
|
\end{equation}
|
|
where the $K$th correlation excitation energy (per electron) is integrated over the
|
|
ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$.
|
|
Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present density-functional approximation for ensemble.
|
|
In particular, ${\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$.
|
|
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
|
|
Finally, we note that, by construction,
|
|
\begin{equation}
|
|
{\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).}
|
|
\end{equation}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Computational details}
|
|
\label{sec:comp_details}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
|
|
Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium in the following.
|
|
In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$.
|
|
These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
|
|
For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
|
|
|
|
We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
|
|
\begin{equation}
|
|
\AO{\mu}(x) =
|
|
\begin{cases}
|
|
\sqrt{2/L} \cos(\mu \pi x/L), & \mu \text{ is odd,}
|
|
\\
|
|
\sqrt{2/L} \sin(\mu \pi x/L), & \mu \text{ is even,}
|
|
\end{cases}
|
|
\end{equation}
|
|
with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
|
|
The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
|
|
\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] is set
|
|
to $10^{-5}$. For comparison, regular HF and KS-DFT calculations
|
|
are performed with the same threshold.
|
|
In order to compute the various density-functional
|
|
integrals that cannot be performed in closed form,
|
|
a 51-point Gauss-Legendre quadrature is employed.
|
|
|
|
In order to test the present eLDA functional we perform various sets of calculations.
|
|
To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
|
|
For the single excitations, we also perform time-dependent LDA (TDLDA)
|
|
calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}].
|
|
Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered. \cite{Dreuw_2005}
|
|
|
|
Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
|
|
(ground-state) limit where $\bw = (0,0)$ and the
|
|
equi-tri-ensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Results and discussion}
|
|
\label{sec:res}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsW_n5}
|
|
\caption{
|
|
\label{fig:EvsW}
|
|
Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
First, we discuss the linearity of the computed (approximate)
|
|
ensemble energies.
|
|
To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
|
|
The deviation from linearity of the three-state ensemble energy
|
|
$\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
|
|
linearly-interpolated ensemble energy) is represented
|
|
in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
|
|
fulfilling the restrictions on the ensemble weights to ensure the GOK
|
|
variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
|
|
To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
|
|
As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
|
|
ensemble energy becomes less and less linear as $L$
|
|
gets larger, while the GIC reduces the curvature of the ensemble energy
|
|
drastically.
|
|
%\manu{This
|
|
%is a strong statement I am not sure about. The nature of the excitation
|
|
%should also be invoked I guess (charge transfer or not, etc ...). If we look at the GIE:
|
|
%\beq
|
|
%\WHF[
|
|
%\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
|
|
%\eeq
|
|
%For a bi-ensemble ($w_1=w$) it can be written as
|
|
%\beq
|
|
%\dfrac{1}{2}\left[(w^2-1)W_0+w(w-2)W_1\right]+w(1-w)W_{01}
|
|
%\eeq
|
|
%If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error
|
|
%reduces to $-W/2$, which is weight-independent (it fits for example with
|
|
%what you see in the weakly correlated regime). Such an assumption depends on the nature of the
|
|
%excitation, not only on the correlation strength, right? Nevertheless,
|
|
%when looking at your curves, this assumption cannot be made when the
|
|
%correlation is strong. It is not clear to me which integral ($W_{01}?$)
|
|
%drives the all thing.\\}
|
|
It is important to note that, even though the GIC removes the explicit
|
|
quadratic terms from the ensemble energy, a non-negligible curvature
|
|
remains in the GIC-eLDA ensemble energy due to the optimization of the
|
|
ensemble KS orbitals in the presence of ghost-interaction error {[see Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:Ew-eLDA}]}.
|
|
%However, this orbital-driven error is small (in our case at
|
|
%least) \trashEF{as the correlation part of the ensemble KS potential $\delta
|
|
%\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
|
|
%to the Hx contribution}.\manu{Manu: well, I guess that the problem arises
|
|
%from the density matrices (or orbitals) that are used to compute
|
|
%individual Coulomb-exchange energies (I would not expect the DFT
|
|
%correlation part to have such an impact, as you say). The best way to check is to plot the
|
|
%ensemble energy without the correlation functional.}\\
|
|
%\\
|
|
%\manu{Manu: another idea. As far as I can see we do
|
|
%not show any individual energies (excitation energies are plotted in the
|
|
%following). Plotting individual energies (to be compared with the FCI
|
|
%ones) would immediately show if there is some curvature (in the ensemble
|
|
%energy). The latter would
|
|
%be induced by any deviation from the expected horizontal straight lines.}
|
|
|
|
%%% FIG 2 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EIvsW_n5}
|
|
\caption{
|
|
\label{fig:EIvsW}
|
|
KS-eLDA individual energies, $\E{eLDA}{(0)}$ (black), $\E{eLDA}{(1)}$ (red), and $\E{eLDA}{(2)}$ (blue), as functions of the weights $\ew{1}$ (solid) and $\ew{2}$ (dashed) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
|
|
Unlike in the exact theory, we do not obtain
|
|
straight horizontal lines when plotting these
|
|
energies, which is in agreement with
|
|
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
|
|
individual energies do not vary in the same way depending on the state
|
|
considered and the value of the weights.
|
|
\titou{We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
|
|
the ground and first excited-state increase with respect to the first-excited-state weight $\ew{1}$, thus showing that we
|
|
``deteriorate'' these states by optimizing the orbitals. The reverse actually occurs for the ground state in the triensemble
|
|
as $\ew{2}$ increases.} The variations in the ensemble
|
|
weights are essentially linear or quadratic. They are induced by the
|
|
eLDA functional, as readily seen from
|
|
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
|
|
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
|
|
\titou{excited-state energy} is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
|
|
systematically enhances the weight dependence, due to the lowering of the
|
|
ground-state energy in this case, as $\ew{2}$ increases.
|
|
The reverse is observed for the second \titou{excited state}.
|
|
|
|
%%% FIG 3 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{EvsL_5}
|
|
\caption{
|
|
\label{fig:EvsL}
|
|
Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box length $L$.
|
|
Graphs for additional values of $\nEl$ can be found as {\SI}.
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
|
|
Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
|
|
For small $L$, the single and double excitations can be labeled as
|
|
``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
|
|
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
|
|
However, when the box gets larger (\ie, as $L$ increases), there is a strong mixing between the different excitation degrees.
|
|
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more \titou{disputable}. \cite{Loos_2019}
|
|
This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
|
|
% TITOU: shall we keep the paragraph below?
|
|
%Therefore, it is paramount to construct a two-weight correlation functional
|
|
%(\ie, a triensemble functional, as we have done here) which
|
|
%allows the mixing of singly- and doubly-excited configurations.
|
|
%Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
|
|
%\titou{Titou might add results for the biensemble to illustrate this.}
|
|
%\manu{Well, neglecting the second excited state is not the same as
|
|
%considering the $w_2=0$ limit. I thought you were referring to an
|
|
%approximation where the triensemble calculation is performed with
|
|
%the biensemble functional. This is not the same as taking $w_2=0$
|
|
%because, in this limit, you may still have a derivative discontinuity
|
|
%correction. The latter is absent if you truly neglect the second excited
|
|
%state in your ensemble functional. This should be clarified.}\\
|
|
%\manu{Are the results in the supp mat? We could just add "[not
|
|
%shown]" if not. This is fine as long as you checked that, indeed, the
|
|
%results deteriorate ;-)}
|
|
%\manu{Should we add that, in the bi-ensemble case, the ensemble
|
|
%correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
|
|
%is neglected (if this is really what you mean (?)). I guess that this is the reason why
|
|
%the second excitation energy would not be well described (?)}
|
|
|
|
As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
|
|
When the box gets larger, they start to deviate.
|
|
For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
|
|
TDA-TDLDA slightly corrects this trend thanks to error compensation.
|
|
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
|
|
This is especially true for the single excitation
|
|
which is significantly improved by using equal weights, \titou{especially in the strong correlation regime}.
|
|
The effect on the double excitation is less pronounced.
|
|
Overall, one clearly sees that, with
|
|
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
|
|
This conclusion is verified for smaller and larger numbers of electrons
|
|
(see {\SI}).
|
|
%\\
|
|
%\manu{Manu: now comes the question that is, I believe, central in this
|
|
%work. How important are the
|
|
%ensemble correlation derivatives $\partial \epsilon^\bw_{\rm
|
|
%c}(n)/\partial w_I$ that, unlike any functional
|
|
%in the literature, the eLDA functional contains. We have to discuss this
|
|
%point... I now see, after reading what follows that this question is
|
|
%addressed later on. We should say something here and then refer to the
|
|
%end of the section, or something like that ...}
|
|
|
|
|
|
%%% FIG 4 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsN}
|
|
\caption{
|
|
\label{fig:EvsN}
|
|
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right).
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
For the same set of methods, Fig.~\ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI) as a function of $\nEl$ for three values of $L$ ($\pi/8$, $\pi$, and $8\pi$).
|
|
We draw similar conclusions as above: irrespectively of the number of
|
|
electrons, the eLDA functional with equal
|
|
weights is able to accurately model single and double excitations, with
|
|
a very significant improvement brought by the
|
|
equiensemble KS-eLDA orbitals as compared to their zero-weight
|
|
(\ie, conventional ground-state) analogs.
|
|
As a rule of thumb, in the weak and intermediate correlation regimes, we
|
|
see that the single
|
|
excitation obtained from equiensemble KS-eLDA is of
|
|
the same quality as the one obtained in the linear response formalism
|
|
(such as TDLDA). On the other hand, the double
|
|
excitation energy only deviates
|
|
from the FCI value by a few tenth of percent.
|
|
Moreover, we note that, in the strong correlation regime
|
|
(\titou{right} graph of Fig.~\ref{fig:EvsN}), the single excitation
|
|
energy obtained at the equiensemble KS-eLDA level remains in good
|
|
agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
|
|
This also applies to the double excitation, the discrepancy
|
|
between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
|
|
These observations nicely illustrate the robustness of the
|
|
GOK-DFT scheme in any correlation regime for both single and double excitations.
|
|
This is definitely a very pleasing outcome, which additionally shows
|
|
that, even though we have designed the eLDA functional based on a
|
|
two-electron model system, the present methodology is applicable to any
|
|
1D electronic system, \ie, a system that has more than two
|
|
electrons.
|
|
|
|
%%% FIG 5 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsL_DD}
|
|
\caption{
|
|
\label{fig:EvsL_DD}
|
|
Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium (left), 5-boxium (center), and 7-boxium (right) at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
|
|
Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
It is also interesting to investigate the influence of the
|
|
correlation ensemble derivative contribution $\DD{c}{(I)}$
|
|
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
|
|
In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
|
|
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, \titou{for $\nEl = 3$, $5$, and $7$}, the error percentage (with respect to FCI) as a function of the box length $L$
|
|
on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
|
|
%\manu{Manu: there is something I do not understand. If you want to
|
|
%evaluate the importance of the ensemble correlation derivatives you
|
|
%should only remove the following contribution from the $K$th KS-eLDA
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|
%excitation energy:
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|
%\beq\label{eq:DD_term_to_compute}
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|
%\int \n{\bGam{\bw}}{}(\br{})
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|
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
%\eeq
|
|
%%rather than $E^{(I)}_{\rm HF}$
|
|
%}
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|
\titou{We first stress that although for $\nEl=3$ both single and double excitation energies are
|
|
systematically improved, as the strength of electron correlation
|
|
increases, when
|
|
taking into account
|
|
the correlation ensemble derivative, this is not systematically the case for larger number of electrons.}
|
|
\trashPFL{This statement holds in both zero-weight and equal-weight limits.}
|
|
The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
|
|
\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single
|
|
excitation; the reverse is observed in the equal-weight triensemble
|
|
case.
|
|
However, for 5- and 7-boxium, the correlation ensemble derivative hardly
|
|
influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
|
|
This non-systematic behavior in terms of the number of electrons might be a consequence of how we constructed the weight-dependent functional.
|
|
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of the eLDA functional is based on a two-electron finite uniform electron gas.
|
|
Therefore, it might be more appropriate to model the derivative discontinuity in few-electron systems.}
|
|
Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
|
|
derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
|
|
compared to the zero-weight calculations).
|
|
%\manu{Manu: well, this is not
|
|
%really the case for the double excitation, right? I would remove this
|
|
%sentence or mention the single excitation explicitly.}
|
|
This could explain why equiensemble calculations are clearly more
|
|
accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
|
|
for a given method, equiensemble orbitals partially remove the burden
|
|
of modelling properly the ensemble correlation derivative.
|
|
%\manu{Manu: I
|
|
%do not like this statement. As I wrote above, the ensemble derivative is
|
|
%still substantial in the strongly correlated limit of the equi
|
|
%triensemble for the double
|
|
%excitation.
|
|
%}
|
|
Note also that, in our case, the second term in
|
|
Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation
|
|
potential and the density difference between ground and excited states,
|
|
has a negligible effect on the excitation energies \titou{(results not shown)}.
|
|
%\manu{Manu: Is this
|
|
%something that you checked but did not show? It feels like we can see
|
|
%this in the Figure but we cannot, right?}
|
|
%\manu{Manu: well, we
|
|
%would need the exact derivative value to draw such a conclusion. We can
|
|
%only speculate. Let us first see how important the contribution in
|
|
%Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
|
|
%updated in the light of the new results.}
|
|
|
|
%%% FIG 6 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{EvsN_DD}
|
|
\caption{
|
|
\label{fig:EvsN_DD}
|
|
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
|
|
Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
|
|
The difference between the solid and dashed curves
|
|
undoubtedly show that, even in the strong correlation regime, the
|
|
ensemble correlation derivative has \titou{a rather significant impact} on the double
|
|
excitations \titou{(around $10\%$)} with a slight tendency of worsening the excitation energies
|
|
in the case of equal weights, as the number of electrons
|
|
increases. It has a rather large influence on the single
|
|
excitation energies obtained in the zero-weight limit, showing once
|
|
again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Concluding remarks}
|
|
\label{sec:conclusion}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
A local and ensemble-weight-dependent correlation density-functional approximation
|
|
(eLDA) has been constructed in the context of GOK-DFT for spin-polarized
|
|
triensembles in
|
|
1D. The approach is actually general and can be extended to real
|
|
(three-dimensional)
|
|
systems~\cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a}
|
|
and larger ensembles in order to
|
|
model excited states in molecules and solids. Work is currently in
|
|
progress in this direction.
|
|
|
|
Unlike any standard functional, eLDA incorporates derivative
|
|
discontinuities through its weight dependence. The latter originates
|
|
from the finite uniform electron gas \titou{on which} eLDA is
|
|
(partially) based on. The KS-eLDA scheme, where exact exchange is
|
|
combined with eLDA, delivers accurate excitation energies for both
|
|
single and double excitations, especially when an equiensemble is used.
|
|
In the latter case, the same weights are assigned to each state belonging to the ensemble.
|
|
The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
|
|
We have observed that, although the ensemble correlation discontinuity has a
|
|
non-negligible effect on the excitation energies (especially for the
|
|
single excitations), its magnitude can be significantly reduced by
|
|
performing equiweight calculations instead of zero-weight
|
|
calculations.
|
|
|
|
Let us finally stress that the present methodology can be extended
|
|
straightforwardly to other types of ensembles like, for example, the
|
|
$\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
|
|
calculation of ionization potentials, electron affinities, and
|
|
fundamental gaps.
|
|
Like in the present
|
|
eLDA, such a functional would incorporate the infamous derivative
|
|
discontinuity contribution to the fundamental gap through its explicit weight
|
|
dependence. We hope to report on this in the near future.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Supplementary material}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
|
|
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\bibliography{eDFT}
|
|
|
|
\end{document}
|