2475 lines
134 KiB
TeX
2475 lines
134 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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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{\vne}{v_\text{ne}}
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\newcommand{\hWee}{\Hat{W}_\text{ee}}
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% functionals, potentials, densities, etc
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% energies
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% elements
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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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%%%%
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\begin{document}
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\title{A weight-dependent local correlation density-functional approximation 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
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designed for the computation of single and double excitations within
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Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for neutral
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excitations), 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, 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
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designed for one-dimensional systems, the methodology proposed here is
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general, \ie, 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 {\it and} 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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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 occurs 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 the ensemble weight dependence problem
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in GOK-DFT,
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with the ambition to turn the theory, in the forthcoming future, 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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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 extraction 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 conclusions 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
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\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
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\eeq
|
|
Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble
|
|
densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are obtained 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
|
|
correlation ensemble 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
|
|
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
|
|
while the ensemble density matrix
|
|
and the ensemble density read
|
|
\beq\label{eq:ens1RDM}
|
|
\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\label{eq:ens_dens_from_ens_1RDM}
|
|
\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
|
|
|
|
%%%%%%%%%%%%%%%
|
|
\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
|
|
|
|
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}].
|
|
The explicit linear weight dependence of the ensemble Hx energy is actually restored 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 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 \manurev{\textit{weight-dependent}} ensemble correlation
|
|
energy per particle \manurev{will have the general
|
|
expression}
|
|
\beq\label{eq:decomp_ens_correner_per_part}
|
|
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{}).
|
|
\eeq
|
|
\manurev{Note that, at this level of approximation, which is expected to
|
|
be exact for any \textit{uniform}
|
|
system, the
|
|
density-functional correlation components $\be{c}{(K)}(\n{}{})$ are
|
|
weight-\textit{independent}, unlike in the exact theory. \cite{Fromager_2020}
|
|
As discussed further in Sec.~\ref{sec:eDFA}, these components can be
|
|
extracted from a
|
|
finite uniform electron gas model for which density-functional correlation excitation
|
|
energies can be computed.
|
|
}\titou{Note also that, here, only the correlation part of the
|
|
energy will be treated at the
|
|
DFT level while we rely on HF for the exchange part.
|
|
This is different from the usual context where both exchange and
|
|
correlation are treated at the LDA level which provides key error compensation features.
|
|
As shown in Sec.~\ref{sec:res}, moving from the pure
|
|
ground-state picture to an equiensemble one can actually improve
|
|
the ground-state energy significantly within such a scheme, thus
|
|
highlighting a major difference between conventional and GOK DFT
|
|
calculations.}
|
|
|
|
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
|
|
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
|
|
Eq.~\eqref{eq:eLDA_corr_fun} leads to our final expression of the
|
|
KS-eLDA energy levels
|
|
\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, and
|
|
\begin{gather}
|
|
\begin{split}
|
|
\label{eq:Xic}
|
|
\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}
|
|
\\
|
|
\label{eq:Upsic}
|
|
\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}
|
|
\manurev{
|
|
One may naturally wonder about the physical content of the above correlation energy
|
|
expressions. It is in fact difficult to readily distinguish from
|
|
Eqs.~\eqref{eq:Xic} and \eqref{eq:Upsic} purely (uncoupled) individual
|
|
contributions from mixed ones. For that purpose, we may
|
|
consider a density regime which has a weak deviation from the uniform
|
|
one. In such a regime, where eLDA is a reasonable approximation, the
|
|
deviation of the individual densities from the ensemble one will be
|
|
small. As a result,
|
|
we can} Taylor expand the density-functional
|
|
correlation contributions
|
|
around the $I$th KS state density
|
|
$\n{\bGam{(I)}}{}(\br{})$, \manurev{so that} the
|
|
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}
|
|
\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. \manurev{This perturbation expansion
|
|
is of course less relevant for (more realistic) systems that exhibit significant
|
|
deviations from the uniform
|
|
density regime. Nevertheless, it
|
|
gives more insight into the eLDA approximation and it becomes useful when
|
|
it comes to rationalize its performance, as illustrated in Sec. \ref{sec:res}.\\}
|
|
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 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}
|
|
\Upsilon_\text{c}^{(I)}
|
|
=\int
|
|
\qty[\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
|
|
-
|
|
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
|
|
] \n{\bGam{\bw}}{}(\br{})
|
|
d\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}
|
|
\Upsilon_\text{c}^{(I)}
|
|
&=
|
|
\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 contribution $\Upsilon_\text{c}^{(I)}$ is, through zeroth order, to substitute the expected
|
|
individual correlation energy per particle for the ensemble one.
|
|
|
|
Let us finally mention 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} can be simplified as follows:
|
|
\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 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 ensembles
|
|
will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
|
|
Here, we propose to construct a weight-dependent LDA functional for the
|
|
calculation of excited states in 1D systems by combining finite uniform electron gases with the
|
|
usual infinite uniform electron gas paradigm.
|
|
|
|
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 GOK-DFT 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
|
|
correlation ensemble derivatives [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.
|
|
|
|
%%% 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}
|
|
\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}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=0.7\linewidth]{fig1}
|
|
\caption{
|
|
\label{fig:embedding}
|
|
\titou{Schematic view of the ``embedding'' scheme: the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
|
|
The electronic excitation occurs locally, \ie, on the impurity.}
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
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 \titou{``embedding''} scheme \titou{(illustrated in Fig.~\ref{fig:embedding})} 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 ensembles.
|
|
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}
|
|
\titou{The one-electron density in these two regimes of correlation is represented in Fig.~\ref{fig:rho}.}
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig2}
|
|
\caption{
|
|
\titou{Ground-state one-electron density $\n{}{}(x)$ of 4-boxium (\ie, $N = 4$) for $L = \pi/32$ (left) and $L = 32\pi$ (right).
|
|
In the weak correlation regime (small box length), the one-electron density is much more delocalized and uniform than in the strong correlation regime (large box length), where a Wigner crystal starts to appear. \cite{Rogers_2017,Rogers_2016}}
|
|
\label{fig:rho}
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
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}] of the KS-DFT self-consistent calculation is set
|
|
to $10^{-5}$.
|
|
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-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
|
|
\titou{Note that a zero-weight calculation does correspond to a ground-state KS calculation with $100\%$ exact exchange and LDA correlation.}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Results and discussion}
|
|
\label{sec:res}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{fig3}
|
|
\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 $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
|
|
\manurev{More precisely, we follow a continuous path that connects
|
|
ground-state [$\bw=(0,0)$] and equiensemble [$\bw=(1/3,1/3)$]
|
|
calculations. For convenience, we use two connected paths. The first
|
|
one, for which $\ew{2}=0$ and $0\leq \ew{1}\leq 1/3$, relies on the
|
|
biensemble while the second one is defined as follows:
|
|
$\ew{1}=1/3$ and $0\leq \ew{2}\leq 1/3$.}
|
|
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.
|
|
It is important to note that, even though the GIC removes the explicit
|
|
quadratic Hx terms from the ensemble energy, a non-negligible curvature
|
|
remains in the GIC-eLDA ensemble energy when the electron
|
|
correlation is strong. \manurev{The latter ensemble energy is computed
|
|
as the weighted
|
|
sum of the individual KS-eLDA energies [see
|
|
Eq.~\eqref{eq:Ew-eLDA}]. Therefore, its
|
|
curvature can only originate from the weight dependence of the
|
|
individual energies.
|
|
Note that such a dependence does not exist in the exact theory. Here,
|
|
the individual density-functional eLDA correlation energies exhibit an
|
|
explicit linear and quadratic dependence on the weights, as discussed
|
|
further in the next paragraph. Note also that the individual KS-eLDA energies
|
|
may gain an additional (implicit) dependence on the weights through the optimization of the
|
|
ensemble KS orbitals in the presence of ghost-interaction errors [see
|
|
Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}].
|
|
}
|
|
|
|
%%% FIG 2 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{fig4}
|
|
\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. The variations in the ensemble weights are essentially linear or quadratic.
|
|
\manurev{This can be rationalized as follows. As readily seen from
|
|
Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:ind_HF-like_ener}, the individual
|
|
HF-like energies do not depend explicitly on the weights, which means
|
|
that the above-mentioned variations originate from the eLDA correlation
|
|
functional [second and third terms on the right-hand side of
|
|
Eq.~\eqref{eq:EI-eLDA}]. If, for analysis purposes, we consider the
|
|
Taylor expansions around the uniform density regime in
|
|
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
|
|
\eqref{eq:Taylor_exp_DDisc_term}, contributions with an explicit weight
|
|
dependence still remain after summation. As both the ensemble density and
|
|
the ensemble correlation energy per particle vary linearly with the
|
|
weights $\bw$ [see Eqs.~\eqref{eq:ens1RDM},
|
|
\eqref{eq:ens_dens_from_ens_1RDM}, and
|
|
\eqref{eq:decomp_ens_correner_per_part}], the latter contributions will contain both linear and quadratic terms in
|
|
$\bw$, as evidenced by Eq.~\eqref{eq:Taylor_exp_DDisc_term} [see the second term on the right-hand
|
|
side].}\\
|
|
Interestingly, the
|
|
individual energies do not vary in the same way depending on the state
|
|
considered and the value of the weights.
|
|
\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
|
|
the ground and \titou{second} excited-state increase with respect to the
|
|
first-excited-state weight $\ew{1}$, thus showing that, in this
|
|
case, we
|
|
``deteriorate'' these states by optimizing the orbitals for the
|
|
ensemble, rather than for each state separately.
|
|
\titou{The singly excited state is, on the other hand, stabilized in the biensemble, which is reasonable as the weight associated with this state increases.
|
|
For the triensemble, as $\ew{2}$ increases, the energy of the ground state increases, while the energy of the first excited state remains stable with a slight increase at large $L$.
|
|
The second excited state is obviously stabilized by the increase of its weight in the ensemble.
|
|
\manurev{
|
|
These are all very sensible observations.\\
|
|
Let us finally stress that the (well-known) poor performance of the
|
|
combined 100\% HF-exchange/LDA correlation scheme in
|
|
ground-state [\ie, $\bw=(0,0)$] DFT, where the correlation energy is
|
|
overestimated, is substantially improved for the
|
|
ground state within the equiensemble [$\bw=(1/3,1/3)$]} (see the {\SI} for
|
|
further details).
|
|
This is a
|
|
remarkable and promising result. A similar improvement is observed for
|
|
the first excited state, at least in the weak correlation regime,
|
|
without deteriorating too much the second-excited-state energy.}
|
|
|
|
%%% FIG 3 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig5}
|
|
\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 lengths $L$.
|
|
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 lengths 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 disputable. \cite{Loos_2019}
|
|
This can be clearly evidenced by the weights of the different
|
|
configurations in the FCI wave function.
|
|
|
|
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, in the strong correlation regime, for the single excitation
|
|
which is significantly improved by using equal weights.
|
|
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}).
|
|
\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}.}
|
|
|
|
%%% FIG 4 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{fig6}
|
|
\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
|
|
(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]{fig7}
|
|
\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}, 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 level with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
|
|
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
|
|
always the case for larger numbers of electrons.
|
|
For 3-boxium, in the zero-weight limit, the correlation ensemble derivative is
|
|
significantly larger for the single
|
|
excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
|
|
case.
|
|
However, for 5- and 7-boxium, it 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 eLDA.
|
|
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
|
|
the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
|
|
Incorporating a $\nEl$-dependence in the functional through the
|
|
curvature of the Fermi hole, in the spirit of Ref.~\onlinecite{Loos_2017a}, would be
|
|
valuable in this respect. This is left for future work.
|
|
Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble
|
|
derivative is substantially reduced when switching from a zero-weight to
|
|
an equal-weight calculation, while giving similar excitation energies,
|
|
even in the strongly correlated regime. A possible interpretation is
|
|
that, at least for the single excitation, equiensemble orbitals partially remove the burden
|
|
of modelling properly the correlation ensemble derivative.
|
|
This conclusion does not hold for larger
|
|
numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-functional correlation ensemble
|
|
derivatives from a two-electron uniform electron gas, as mentioned previously.
|
|
For the double excitation, the ensemble derivative remains important, even in
|
|
the equiensemble case.
|
|
To summarize, the equiensemble calculation
|
|
is always more accurate than a zero-weight
|
|
(\ie, a conventional ground-state DFT) one, with or without including the ensemble
|
|
derivative correction. Note that the second term on the right-hand side
|
|
of
|
|
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 (results not
|
|
shown).
|
|
|
|
%%% FIG 6 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig8}
|
|
\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 the
|
|
correlation ensemble derivative has a rather significant impact on the double
|
|
excitation (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 (which decreases with the
|
|
number of electrons) 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 correlation ensemble 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 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 on which eLDA is
|
|
(partially) based. The KS-eLDA scheme, where exact individual
|
|
exchange energies are
|
|
combined with the eLDA correlation functional , 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 correlation ensemble derivative 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 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.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Data availability statement}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The data that supports the findings of this study are available within the article [and its supplementary material].
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\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}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
|
|
%Control: key (0)
|
|
%Control: author (8) initials jnrlst
|
|
%Control: editor formatted (1) identically to author
|
|
%Control: production of article title (-1) disabled
|
|
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|
|
%Control: year (1) truncated
|
|
%Control: production of eprint (0) enabled
|
|
\begin{thebibliography}{118}%
|
|
\makeatletter
|
|
\providecommand \@ifxundefined [1]{%
|
|
\@ifx{#1\undefined}
|
|
}%
|
|
\providecommand \@ifnum [1]{%
|
|
\ifnum #1\expandafter \@firstoftwo
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|
\else \expandafter \@secondoftwo
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|
\fi
|
|
}%
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|
\providecommand \@ifx [1]{%
|
|
\ifx #1\expandafter \@firstoftwo
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|
\else \expandafter \@secondoftwo
|
|
\fi
|
|
}%
|
|
\providecommand \natexlab [1]{#1}%
|
|
\providecommand \enquote [1]{``#1''}%
|
|
\providecommand \bibnamefont [1]{#1}%
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|
\providecommand \bibfnamefont [1]{#1}%
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\providecommand \citenamefont [1]{#1}%
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\providecommand \href [0]{\begingroup \@sanitize@url \@href}%
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\providecommand \@href[1]{\@@startlink{#1}\@@href}%
|
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\providecommand \@@href[1]{\endgroup#1\@@endlink}%
|
|
\providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode
|
|
`\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax}%
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\providecommand \@@startlink[1]{}%
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\providecommand \url [0]{\begingroup\@sanitize@url \@url }%
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\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }}%
|
|
\providecommand \urlprefix [0]{URL }%
|
|
\providecommand \Eprint [0]{\href }%
|
|
\providecommand \doibase [0]{http://dx.doi.org/}%
|
|
\providecommand \selectlanguage [0]{\@gobble}%
|
|
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|
|
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|
|
\providecommand \translation [1]{[#1]}%
|
|
\providecommand \BibitemOpen [0]{}%
|
|
\providecommand \bibitemStop [0]{}%
|
|
\providecommand \bibitemNoStop [0]{.\EOS\space}%
|
|
\providecommand \EOS [0]{\spacefactor3000\relax}%
|
|
\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}%
|
|
\let\auto@bib@innerbib\@empty
|
|
%</preamble>
|
|
\bibitem [{\citenamefont {Hohenberg}\ and\ \citenamefont
|
|
{Kohn}(1964)}]{Hohenberg_1964}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
|
{Hohenberg}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
|
{Kohn}},\ }\href {\doibase 10.1103/PhysRev.136.B864} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {136}},\
|
|
\bibinfo {pages} {B864} (\bibinfo {year} {1964})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Kohn}\ and\ \citenamefont {Sham}(1965)}]{Kohn_1965}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
|
{Kohn}}\ and\ \bibinfo {author} {\bibfnamefont {L.~J.}\ \bibnamefont
|
|
{Sham}},\ }\href {\doibase 10.1103/PhysRev.140.A1133} {\bibfield {journal}
|
|
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {140}},\
|
|
\bibinfo {pages} {A1133} (\bibinfo {year} {1965})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Parr}\ and\ \citenamefont {Yang}(1989)}]{ParrBook}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~G.}\ \bibnamefont
|
|
{Parr}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}},\
|
|
}\href@noop {} {\emph {\bibinfo {title} {Density-functional theory of atoms
|
|
and molecules}}}\ (\bibinfo {publisher} {Oxford},\ \bibinfo {address}
|
|
{Clarendon Press},\ \bibinfo {year} {1989})\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Woodcock}, \citenamefont {Schaefer},\ and\
|
|
\citenamefont {Schreiner}(2002)}]{Woodcock_2002}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~L.}\ \bibnamefont
|
|
{Woodcock}}, \bibinfo {author} {\bibfnamefont {H.~F.}\ \bibnamefont
|
|
{Schaefer}}, \ and\ \bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
|
|
{Schreiner}},\ }\href {\doibase 10.1021/jp0212895} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Phys. Chem. A}\ }\textbf {\bibinfo {volume} {106}},\
|
|
\bibinfo {pages} {11923} (\bibinfo {year} {2002})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Tozer}(2003)}]{Tozer_2003}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
|
|
{Tozer}},\ }\href {\doibase 10.1063/1.1633756} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {119}},\
|
|
\bibinfo {pages} {12697} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Tozer}\ \emph {et~al.}(1999)\citenamefont {Tozer},
|
|
\citenamefont {Amos}, \citenamefont {Handy}, \citenamefont {Roos},\ and\
|
|
\citenamefont {{Serrano-Andres}}}]{Tozer_1999}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
|
|
{Tozer}}, \bibinfo {author} {\bibfnamefont {R.~D.}\ \bibnamefont {Amos}},
|
|
\bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont {Handy}}, \bibinfo
|
|
{author} {\bibfnamefont {B.~O.}\ \bibnamefont {Roos}}, \ and\ \bibinfo
|
|
{author} {\bibfnamefont {L.}~\bibnamefont {{Serrano-Andres}}},\ }\href
|
|
{\doibase 10.1080/00268979909482888} {\bibfield {journal} {\bibinfo
|
|
{journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo {pages}
|
|
{859} (\bibinfo {year} {1999})}\BibitemShut {NoStop}%
|
|
\bibitem [{\citenamefont {Dreuw}, \citenamefont {Weisman},\ and\ \citenamefont
|
|
{{Head-Gordon}}(2003)}]{Dreuw_2003}%
|
|
\BibitemOpen
|
|
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
|
{Dreuw}}, \bibinfo {author} {\bibfnamefont {J.~L.}\ \bibnamefont {Weisman}},
|
|
\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{Head-Gordon}}},\
|
|
}\href {\doibase 10.1063/1.1590951} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {119}},\ \bibinfo
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{pages} {2943} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Sobolewski}\ and\ \citenamefont
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{Domcke}(2003)}]{Sobolewski_2003}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~L.}\ \bibnamefont
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{Sobolewski}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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{Domcke}},\ }\href {\doibase 10.1016/S0301-0104(03)00388-4} {\bibfield
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{journal} {\bibinfo {journal} {Chem. Phys.}\ }\textbf {\bibinfo {volume}
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{294}},\ \bibinfo {pages} {73} (\bibinfo {year} {2003})}\BibitemShut
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\bibitem [{\citenamefont {Dreuw}\ and\ \citenamefont
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{Head-Gordon}(2004)}]{Dreuw_2004}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Dreuw}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Head-Gordon}},\ }\href {\doibase 10.1021/ja039556n} {\bibfield {journal}
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{\bibinfo {journal} {J. Am. Chem. Soc.}\ }\textbf {\bibinfo {volume}
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{126}},\ \bibinfo {pages} {4007} (\bibinfo {year} {2004})}\BibitemShut
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\bibitem [{\citenamefont {Tozer}\ and\ \citenamefont
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{Handy}(1998)}]{Tozer_1998}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Tozer}}\ and\ \bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
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{Handy}},\ }\href {\doibase 10.1063/1.477711} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {109}},\ \bibinfo
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{pages} {10180} (\bibinfo {year} {1998})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Tozer}\ and\ \citenamefont
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{Handy}(2000)}]{Tozer_2000}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Tozer}}\ and\ \bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
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{Handy}},\ }\href {\doibase 10.1039/a910321j} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Chem. Chem. Phys.}\ }\textbf {\bibinfo {volume} {2}},\
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\bibinfo {pages} {2117} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Casida}\ \emph {et~al.}(1998)\citenamefont {Casida},
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\citenamefont {Jamorski}, \citenamefont {Casida},\ and\ \citenamefont
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{Salahub}}]{Casida_1998}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont
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{Casida}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Jamorski}},
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\bibinfo {author} {\bibfnamefont {K.~C.}\ \bibnamefont {Casida}}, \ and\
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\bibinfo {author} {\bibfnamefont {D.~R.}\ \bibnamefont {Salahub}},\ }\href
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{\doibase 10.1063/1.475855} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {108}},\ \bibinfo {pages} {4439}
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(\bibinfo {year} {1998})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Casida}\ and\ \citenamefont
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{Salahub}(2000)}]{Casida_2000}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont
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{Casida}}\ and\ \bibinfo {author} {\bibfnamefont {D.~R.}\ \bibnamefont
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{Salahub}},\ }\href {\doibase 10.1063/1.1319649} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {113}},\
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\bibinfo {pages} {8918} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Tapavicza}\ \emph {et~al.}(2008)\citenamefont
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{Tapavicza}, \citenamefont {Tavernelli}, \citenamefont {Rothlisberger},
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\citenamefont {Filippi},\ and\ \citenamefont {Casida}}]{Tapavicza_2008}%
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\BibitemOpen
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{Tapavicza}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
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{Tavernelli}}, \bibinfo {author} {\bibfnamefont {U.}~\bibnamefont
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{Rothlisberger}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Filippi}}, \ and\ \bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont
|
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{Casida}},\ }\href {\doibase 10.1063/1.2978380} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {129}},\
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\bibinfo {pages} {124108} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Levine}\ \emph {et~al.}(2006)\citenamefont {Levine},
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\citenamefont {Ko}, \citenamefont {Quenneville},\ and\ \citenamefont
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{Mart\'Inez}}]{Levine_2006}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~G.}\ \bibnamefont
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{Levine}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Ko}}, \bibinfo
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{author} {\bibfnamefont {J.}~\bibnamefont {Quenneville}}, \ and\ \bibinfo
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{author} {\bibfnamefont {T.~J.}\ \bibnamefont {Mart\'Inez}},\ }\href
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{\doibase 10.1080/00268970500417762} {\bibfield {journal} {\bibinfo
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{journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo {pages}
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{1039} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
|
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{Seidl}(2010)}]{Gori-Giorgi_2010}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Seidl}},\ }\href {\doibase 10.1039/c0cp01061h} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Chem. Chem. Phys.}\ }\textbf {\bibinfo {volume}
|
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{12}},\ \bibinfo {pages} {14405} (\bibinfo {year} {2010})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Fromager}(2015)}]{Fromager_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Fromager}},\ }\href {https://doi.org/10.1080/00268976.2014.993342}
|
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{\bibfield {journal} {\bibinfo {journal} {Mol. Phys.}\ }\textbf {\bibinfo
|
|
{volume} {113}},\ \bibinfo {pages} {419} (\bibinfo {year}
|
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{2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gagliardi}\ \emph {et~al.}(2017)\citenamefont
|
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{Gagliardi}, \citenamefont {Truhlar}, \citenamefont {Manni}, \citenamefont
|
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{Carlson}, \citenamefont {Hoyer},\ and\ \citenamefont
|
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{Bao}}]{Gagliardi_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
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{Gagliardi}}, \bibinfo {author} {\bibfnamefont {D.~G.}\ \bibnamefont
|
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{Truhlar}}, \bibinfo {author} {\bibfnamefont {G.~L.}\ \bibnamefont {Manni}},
|
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\bibinfo {author} {\bibfnamefont {R.~K.}\ \bibnamefont {Carlson}}, \bibinfo
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{author} {\bibfnamefont {C.~E.}\ \bibnamefont {Hoyer}}, \ and\ \bibinfo
|
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{author} {\bibfnamefont {J.~L.}\ \bibnamefont {Bao}},\ }\href {\doibase
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10.1021/acs.accounts.6b00471} {\bibfield {journal} {\bibinfo {journal}
|
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{Acc. Chem. Res.}\ }\textbf {\bibinfo {volume} {50}},\ \bibinfo {pages} {66}
|
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(\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Runge}\ and\ \citenamefont
|
|
{Gross}(1984)}]{Runge_1984}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Runge}}\ and\ \bibinfo {author} {\bibfnamefont {E.~K.~U.}\ \bibnamefont
|
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{Gross}},\ }\href {\doibase 10.1103/PhysRevLett.52.997} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {52}},\
|
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\bibinfo {pages} {997} (\bibinfo {year} {1984})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Casida}(1995)}]{Casida}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont
|
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{Casida}},\ }\enquote {\bibinfo {title} {Recent advances in density
|
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functional methods},}\ \ (\bibinfo {publisher} {World Scientific,
|
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Singapore},\ \bibinfo {year} {1995})\ p.\ \bibinfo {pages} {155}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Casida}\ and\ \citenamefont
|
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{Huix-Rotllant}(2012)}]{Casida_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Casida}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Huix-Rotllant}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
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{Annu. Rev. Phys. Chem.}\ }\textbf {\bibinfo {volume} {63}},\ \bibinfo
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{pages} {287} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Vignale}(2008)}]{Vignale_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
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{Vignale}},\ }\href {\doibase 10.1103/PhysRevA.77.062511} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
|
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{77}},\ \bibinfo {pages} {062511} (\bibinfo {year} {2008})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Dreuw}\ and\ \citenamefont
|
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{Head-Gordon}(2005)}]{Dreuw_2005}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
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{Dreuw}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{Head-Gordon}},\ }\href {\doibase 10.1021/cr0505627} {\bibfield {journal}
|
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{\bibinfo {journal} {Chem. Rev.}\ }\textbf {\bibinfo {volume} {105}},\
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\bibinfo {pages} {4009} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Maitra}\ \emph {et~al.}(2004)\citenamefont {Maitra},
|
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\citenamefont {Zhang}, \citenamefont {Cave},\ and\ \citenamefont
|
|
{Burke}}]{Maitra_2004}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~T.}\ \bibnamefont
|
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{Maitra}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Zhang}},
|
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\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont {Cave}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
|
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{\doibase 10.1063/1.1651060} {\bibfield {journal} {\bibinfo {journal} {J.
|
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {120}},\ \bibinfo {pages} {5932}
|
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(\bibinfo {year} {2004})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Cave}\ \emph {et~al.}(2004)\citenamefont {Cave},
|
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\citenamefont {Zhang}, \citenamefont {Maitra},\ and\ \citenamefont
|
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{Burke}}]{Cave_2004}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
|
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{Cave}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Zhang}},
|
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\bibinfo {author} {\bibfnamefont {N.~T.}\ \bibnamefont {Maitra}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
|
|
{\doibase 10.1016/j.cplett.2004.03.051} {\bibfield {journal} {\bibinfo
|
|
{journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {389}},\ \bibinfo
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{pages} {39} (\bibinfo {year} {2004})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Mazur}\ and\ \citenamefont
|
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{W\l{}odarczyk}(2009)}]{Mazur_2009}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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{Mazur}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
|
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{W\l{}odarczyk}},\ }\href {\doibase 10.1002/jcc.21102} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Comput. Chem.}\ }\textbf {\bibinfo {volume} {30}},\
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\bibinfo {pages} {811} (\bibinfo {year} {2009})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Romaniello}\ \emph {et~al.}(2009)\citenamefont
|
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{Romaniello}, \citenamefont {Sangalli}, \citenamefont {Berger}, \citenamefont
|
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{Sottile}, \citenamefont {Molinari}, \citenamefont {Reining},\ and\
|
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\citenamefont {Onida}}]{Romaniello_2009a}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Romaniello}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
|
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{Sangalli}}, \bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
|
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{Berger}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Sottile}},
|
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\bibinfo {author} {\bibfnamefont {L.~G.}\ \bibnamefont {Molinari}}, \bibinfo
|
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{author} {\bibfnamefont {L.}~\bibnamefont {Reining}}, \ and\ \bibinfo
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{author} {\bibfnamefont {G.}~\bibnamefont {Onida}},\ }\href {\doibase
|
|
10.1063/1.3065669} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
|
Phys.}\ }\textbf {\bibinfo {volume} {130}},\ \bibinfo {pages} {044108}
|
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(\bibinfo {year} {2009})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Sangalli}\ \emph {et~al.}(2011)\citenamefont
|
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{Sangalli}, \citenamefont {Romaniello}, \citenamefont {Onida},\ and\
|
|
\citenamefont {Marini}}]{Sangalli_2011}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
|
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{Sangalli}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
|
{Romaniello}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Onida}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Marini}},\ }\href
|
|
{\doibase 10.1063/1.3518705} {\bibfield {journal} {\bibinfo {journal} {J.
|
|
Chem. Phys.}\ }\textbf {\bibinfo {volume} {134}},\ \bibinfo {pages} {034115}
|
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(\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Mazur}\ \emph {et~al.}(2011)\citenamefont {Mazur},
|
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\citenamefont {Makowski}, \citenamefont {W\l{}odarczyk},\ and\ \citenamefont
|
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{Aoki}}]{Mazur_2011}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
|
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{Mazur}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Makowski}},
|
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\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {W\l{}odarczyk}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Aoki}},\ }\href {\doibase
|
|
10.1002/qua.22876} {\bibfield {journal} {\bibinfo {journal} {Int. J.
|
|
Quantum Chem.}\ }\textbf {\bibinfo {volume} {111}},\ \bibinfo {pages} {819}
|
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(\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{Huix-Rotllant}}\ \emph {et~al.}(2011)\citenamefont
|
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{{Huix-Rotllant}}, \citenamefont {Ipatov}, \citenamefont {Rubio},\ and\
|
|
\citenamefont {Casida}}]{Huix-Rotllant_2011}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{{Huix-Rotllant}}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
|
{Ipatov}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Rubio}}, \
|
|
and\ \bibinfo {author} {\bibfnamefont {M.~E.}\ \bibnamefont {Casida}},\
|
|
}\href {\doibase 10.1016/j.chemphys.2011.03.019} {\bibfield {journal}
|
|
{\bibinfo {journal} {Chem. Phys.}\ }\textbf {\bibinfo {volume} {391}},\
|
|
\bibinfo {pages} {120} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Elliott}\ \emph {et~al.}(2011)\citenamefont
|
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{Elliott}, \citenamefont {Goldson}, \citenamefont {Canahui},\ and\
|
|
\citenamefont {Maitra}}]{Elliott_2011}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{Elliott}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Goldson}},
|
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\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Canahui}}, \ and\
|
|
\bibinfo {author} {\bibfnamefont {N.~T.}\ \bibnamefont {Maitra}},\ }\href
|
|
{\doibase 10.1016/j.chemphys.2011.03.020} {\bibfield {journal} {\bibinfo
|
|
{journal} {Chem. Phys.}\ }\textbf {\bibinfo {volume} {391}},\ \bibinfo
|
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{pages} {110} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Maitra}(2012)}]{Maitra_2012}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~T.}\ \bibnamefont
|
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{Maitra}},\ }\enquote {\bibinfo {title} {Memory: History , initial-state
|
|
dependence , and double-excitations},}\ in\ \href {\doibase
|
|
10.1007/978-3-642-23518-4_8} {\emph {\bibinfo {booktitle} {Fundamentals of
|
|
Time-Dependent Density Functional Theory}}},\ Vol.\ \bibinfo {volume} {837},\
|
|
\bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont {M.~A.}\
|
|
\bibnamefont {Marques}}, \bibinfo {editor} {\bibfnamefont {N.~T.}\
|
|
\bibnamefont {Maitra}}, \bibinfo {editor} {\bibfnamefont {F.~M.}\
|
|
\bibnamefont {Nogueira}}, \bibinfo {editor} {\bibfnamefont {E.}~\bibnamefont
|
|
{Gross}}, \ and\ \bibinfo {editor} {\bibfnamefont {A.}~\bibnamefont
|
|
{Rubio}}}\ (\bibinfo {publisher} {Springer Berlin Heidelberg},\ \bibinfo
|
|
{address} {Berlin, Heidelberg},\ \bibinfo {year} {2012})\ pp.\ \bibinfo
|
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{pages} {167--184}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Sundstrom}\ and\ \citenamefont
|
|
{{Head-Gordon}}(2014)}]{Sundstrom_2014}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~J.}\ \bibnamefont
|
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{Sundstrom}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
|
{{Head-Gordon}}},\ }\href {\doibase 10.1063/1.4868120} {\bibfield {journal}
|
|
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\
|
|
\bibinfo {pages} {114103} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ \emph {et~al.}(2019)\citenamefont {Loos},
|
|
\citenamefont {Boggio-Pasqua}, \citenamefont {Scemama}, \citenamefont
|
|
{Caffarel},\ and\ \citenamefont {Jacquemin}}]{Loos_2019}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
|
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{Loos}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Boggio-Pasqua}},
|
|
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \bibinfo
|
|
{author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}}, \ and\ \bibinfo
|
|
{author} {\bibfnamefont {D.}~\bibnamefont {Jacquemin}},\ }\href {\doibase
|
|
10.1021/acs.jctc.8b01205} {\bibfield {journal} {\bibinfo {journal} {J.
|
|
Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages}
|
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{1939} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {B.~O.~Roos}\ \emph {et~al.}(1996)\citenamefont
|
|
{B.~O.~Roos}, \citenamefont {Fulscher}, \citenamefont {Malmqvist},\ and\
|
|
\citenamefont {{Serrano-Andres}}}]{Roos}%
|
|
\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~A.}\ \bibnamefont
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{B.~O.~Roos}}, \bibinfo {author} {\bibfnamefont {M.~P.}\ \bibnamefont
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{Fulscher}}, \bibinfo {author} {\bibfnamefont {P.-A.}\ \bibnamefont
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{Malmqvist}}, \ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
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{{Serrano-Andres}}},\ }\enquote {\bibinfo {title} {Adv. chem. phys.}}\ \
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(\bibinfo {publisher} {Wiley, New York},\ \bibinfo {year} {1996})\ pp.\
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\bibinfo {pages} {219--331}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Andersson}\ \emph {et~al.}(1990)\citenamefont
|
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{Andersson}, \citenamefont {Malmqvist}, \citenamefont {Roos}, \citenamefont
|
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{Sadlej},\ and\ \citenamefont {Wolinski}}]{Andersson_1990}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Andersson}}, \bibinfo {author} {\bibfnamefont {P.~A.}\ \bibnamefont
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{Malmqvist}}, \bibinfo {author} {\bibfnamefont {B.~O.}\ \bibnamefont {Roos}},
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\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont {Sadlej}}, \ and\
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Wolinski}},\ }\href
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{\doibase 10.1021/j100377a012} {\bibfield {journal} {\bibinfo {journal} {J.
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Phys. Chem.}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} {5483}
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(\bibinfo {year} {1990})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Angeli}, \citenamefont {Cimiraglia},\ and\
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\citenamefont {Malrieu}(2001)}]{Angeli_2001a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Angeli}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Cimiraglia}},
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\ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont {Malrieu}},\
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}\href {\doibase 10.1016/S0009-2614(01)01303-3} {\bibfield {journal}
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{\bibinfo {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume}
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{350}},\ \bibinfo {pages} {297} (\bibinfo {year} {2001})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Angeli}\ \emph {et~al.}(2001)\citenamefont {Angeli},
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\citenamefont {Cimiraglia}, \citenamefont {Evangelisti}, \citenamefont
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{Leininger},\ and\ \citenamefont {Malrieu}}]{Angeli_2001b}%
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\BibitemOpen
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{Angeli}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Cimiraglia}},
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\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Evangelisti}}, \bibinfo
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{author} {\bibfnamefont {T.}~\bibnamefont {Leininger}}, \ and\ \bibinfo
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{author} {\bibfnamefont {J.-P.}\ \bibnamefont {Malrieu}},\ }\href {\doibase
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10.1063/1.1361246} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
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Phys.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo {pages} {10252}
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(\bibinfo {year} {2001})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Angeli}, \citenamefont {Cimiraglia},\ and\
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\citenamefont {Malrieu}(2002)}]{Angeli_2002}%
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{Angeli}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Cimiraglia}},
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\ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont {Malrieu}},\
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}\href {\doibase 10.1063/1.1515317} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {117}},\ \bibinfo
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{pages} {9138} (\bibinfo {year} {2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Helgaker}, \citenamefont {J{\o}rgensen},\ and\
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\citenamefont {Olsen}(2013)}]{Helgakerbook}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Helgaker}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{J{\o}rgensen}}, \ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Olsen}},\ }\href@noop {} {\emph {\bibinfo {title} {Molecular
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Electronic-Structure Theory}}}\ (\bibinfo {publisher} {John Wiley \& Sons,
|
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Inc.},\ \bibinfo {year} {2013})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gross}, \citenamefont {Oliveira},\ and\ \citenamefont
|
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{Kohn}(1988{\natexlab{a}})}]{Gross_1988a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~K.~U.}\
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\bibnamefont {Gross}}, \bibinfo {author} {\bibfnamefont {L.~N.}\ \bibnamefont
|
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{Oliveira}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
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{Kohn}},\ }\href {\doibase 10.1103/PhysRevA.37.2805} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {37}},\
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\bibinfo {pages} {2805} (\bibinfo {year} {1988}{\natexlab{a}})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Gross}, \citenamefont {Oliveira},\ and\ \citenamefont
|
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{Kohn}(1988{\natexlab{b}})}]{Gross_1988b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~K.~U.}\
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\bibnamefont {Gross}}, \bibinfo {author} {\bibfnamefont {L.~N.}\ \bibnamefont
|
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{Oliveira}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
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{Kohn}},\ }\href {\doibase 10.1103/PhysRevA.37.2809} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {37}},\
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\bibinfo {pages} {2809} (\bibinfo {year} {1988}{\natexlab{b}})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Oliveira}, \citenamefont {Gross},\ and\ \citenamefont
|
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{Kohn}(1988)}]{Oliveira_1988}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.~N.}\ \bibnamefont
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{Oliveira}}, \bibinfo {author} {\bibfnamefont {E.~K.~U.}\ \bibnamefont
|
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{Gross}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Kohn}},\
|
|
}\href {\doibase 10.1103/PhysRevA.37.2821} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {37}},\ \bibinfo
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{pages} {2821} (\bibinfo {year} {1988})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Theophilou}(1979)}]{Theophilou_1979}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~K.}\ \bibnamefont
|
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{Theophilou}},\ }\href {\doibase 10.1088/0022-3719/12/24/013} {\bibfield
|
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{journal} {\bibinfo {journal} {J. Phys. C}\ }\textbf {\bibinfo {volume}
|
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{12}},\ \bibinfo {pages} {5419} (\bibinfo {year} {1979})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Pastorczak}, \citenamefont {Gidopoulos},\ and\
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\citenamefont {Pernal}(2013)}]{Pastorczak_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Pastorczak}}, \bibinfo {author} {\bibfnamefont {N.~I.}\ \bibnamefont
|
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{Gidopoulos}}, \ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
|
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{Pernal}},\ }\href {\doibase 10.1103/PhysRevA.87.} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {87}},\
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\bibinfo {pages} {062501} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Yang}\ \emph {et~al.}(2014)\citenamefont {Yang},
|
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\citenamefont {Trail}, \citenamefont {{Pribram-Jones}}, \citenamefont
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{Burke}, \citenamefont {Needs},\ and\ \citenamefont {Ullrich}}]{Yang_2014}%
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\BibitemOpen
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{Yang}}, \bibinfo {author} {\bibfnamefont {J.~R.}\ \bibnamefont {Trail}},
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {{Pribram-Jones}}},
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}}, \bibinfo
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{author} {\bibfnamefont {R.~J.}\ \bibnamefont {Needs}}, \ and\ \bibinfo
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{author} {\bibfnamefont {C.~A.}\ \bibnamefont {Ullrich}},\ }\href {\doibase
|
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10.1103/PhysRevA.90.042501} {\bibfield {journal} {\bibinfo {journal} {Phys.
|
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Rev. A}\ }\textbf {\bibinfo {volume} {90}},\ \bibinfo {pages} {042501}
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(\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Deur}, \citenamefont {Mazouin},\ and\ \citenamefont
|
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{Fromager}(2017)}]{Deur_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Deur}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Mazouin}}, \
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and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href
|
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{\doibase 10.1103/PhysRevB.95.035120} {\bibfield {journal} {\bibinfo
|
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{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo
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{pages} {035120} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Deur}\ and\ \citenamefont
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{Fromager}(2019)}]{Deur_2019}%
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\BibitemOpen
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{Deur}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Fromager}},\ }\href {\doibase 10.1063/1.5084312} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {150}},\
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\bibinfo {pages} {094106} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Senjean}\ and\ \citenamefont
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{Fromager}(2018)}]{Senjean_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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{Senjean}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Fromager}},\ }\href {\doibase 10.1103/PhysRevA.98.022513} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
|
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{98}},\ \bibinfo {pages} {022513} (\bibinfo {year} {2018})}\BibitemShut
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\bibitem [{\citenamefont {Senjean}\ and\ \citenamefont
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{Fromager}()}]{Senjean_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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{Senjean}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Fromager}},\ }\href {\doibase 10.1002/qua.26190} {\bibinfo {journal} {Int.
|
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J. Quantum Chem.}\ ,\ \bibinfo {pages} {e26190}}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Levy}(1995)}]{Levy_1995}%
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\BibitemOpen
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\bibfield {journal} { }\bibfield {author} {\bibinfo {author} {\bibfnamefont
|
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{M.}~\bibnamefont {Levy}},\ }\href {\doibase 10.1103/PhysRevA.52.R4313}
|
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
|
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{volume} {52}},\ \bibinfo {pages} {R4313} (\bibinfo {year}
|
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{1995})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Perdew}\ and\ \citenamefont
|
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{Levy}(1983)}]{Perdew_1983}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont
|
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{Perdew}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Levy}},\
|
|
}\href {\doibase 10.1103/PhysRevLett.51.1884} {\bibfield {journal} {\bibinfo
|
|
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {51}},\ \bibinfo
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{pages} {1884} (\bibinfo {year} {1983})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Franck}\ and\ \citenamefont
|
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{Fromager}(2014)}]{Franck_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
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{Franck}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
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{Fromager}},\ }\href {\doibase 10.1080/00268976.2013.858191} {\bibfield
|
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{journal} {\bibinfo {journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume}
|
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{112}},\ \bibinfo {pages} {1684} (\bibinfo {year} {2014})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Borgoo}, \citenamefont {Teale},\ and\ \citenamefont
|
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{Helgaker}(2015)}]{Borgoo_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
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{Borgoo}}, \bibinfo {author} {\bibfnamefont {A.~M.}\ \bibnamefont {Teale}}, \
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and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}},\ }\href
|
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{\doibase 10.1063/1.4938857} {\bibfield {journal} {\bibinfo {journal} {AIP
|
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Conf. Proc.}\ }\textbf {\bibinfo {volume} {1702}},\ \bibinfo {pages} {090049}
|
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(\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kazaryan}, \citenamefont {Heuver},\ and\
|
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\citenamefont {Filatov}(2008)}]{Kazaryan_2008}%
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\BibitemOpen
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{Kazaryan}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Heuver}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Filatov}},\ }\href
|
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{\doibase 10.1021/jp8033837} {\bibfield {journal} {\bibinfo {journal} {J.
|
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Phys. Chem. A}\ }\textbf {\bibinfo {volume} {112}},\ \bibinfo {pages} {12980}
|
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(\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}\ and\ \citenamefont
|
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{Dobson}(2013)}]{Gould_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Gould}}\ and\ \bibinfo {author} {\bibfnamefont {J.~F.}\ \bibnamefont
|
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{Dobson}},\ }\href {\doibase 10.1063/1.4773284} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\
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\bibinfo {pages} {014103} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}\ and\ \citenamefont
|
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{Toulouse}(2014)}]{Gould_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Gould}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
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{Toulouse}},\ }\href {\doibase 10.1103/PhysRevA.90.050502} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
|
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{90}},\ \bibinfo {pages} {050502} (\bibinfo {year} {2014})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Filatov}, \citenamefont {{Huix-Rotllant}},\ and\
|
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\citenamefont {Burghardt}(2015)}]{Filatov_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Filatov}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
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{{Huix-Rotllant}}}, \ and\ \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
|
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{Burghardt}},\ }\href {\doibase 10.1063/1.4919773} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {142}},\
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\bibinfo {pages} {184104} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Filatov}(2015{\natexlab{a}})}]{Filatov_2015b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Filatov}},\ }\enquote {\bibinfo {title} {Ensemble {{DFT Approach}} to
|
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{{Excited States}} of {{Strongly Correlated Molecular Systems}}},}\ in\ \href
|
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{\doibase 10.1007/128_2015_630} {\emph {\bibinfo {booktitle}
|
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{Density-{{Functional Methods}} for {{Excited States}}}}},\ Vol.\ \bibinfo
|
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{volume} {368},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
|
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{\bibfnamefont {N.}~\bibnamefont {Ferr\'e}}, \bibinfo {editor} {\bibfnamefont
|
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{M.}~\bibnamefont {Filatov}}, \ and\ \bibinfo {editor} {\bibfnamefont
|
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{M.}~\bibnamefont {{Huix-Rotllant}}}}\ (\bibinfo {publisher} {{Springer
|
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International Publishing}},\ \bibinfo {address} {Cham},\ \bibinfo {year}
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{2015})\ pp.\ \bibinfo {pages} {97--124}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Filatov}(2015{\natexlab{b}})}]{Filatov_2015c}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Filatov}},\ }\href {\doibase 10.1002/wcms.1209} {\bibfield {journal}
|
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{\bibinfo {journal} {WIREs Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
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{5}},\ \bibinfo {pages} {146} (\bibinfo {year}
|
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{2015}{\natexlab{b}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}\ and\ \citenamefont
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{Pittalis}(2017)}]{Gould_2017}%
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\BibitemOpen
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{Gould}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
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{Pittalis}},\ }\href {\doibase 10.1103/PhysRevLett.119.243001} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {119}},\ \bibinfo {pages} {243001} (\bibinfo {year}
|
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{2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}, \citenamefont {Kronik},\ and\ \citenamefont
|
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{Pittalis}(2018)}]{Gould_2018}%
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{Gould}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Kronik}}, \
|
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and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Pittalis}},\ }\href
|
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{\doibase 10.1063/1.5022832} {\bibfield {journal} {\bibinfo {journal} {J.
|
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {148}},\ \bibinfo {pages} {174101}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}\ and\ \citenamefont
|
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{Pittalis}(2019)}]{Gould_2019}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Gould}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
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{Pittalis}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
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{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {123}},\ \bibinfo {pages}
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{016401} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Sagredo}\ and\ \citenamefont
|
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{Burke}(2018)}]{Sagredo_2018}%
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\BibitemOpen
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{Sagredo}}\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
|
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{Burke}},\ }\href {\doibase 10.1063/1.5043411} {\bibfield {journal}
|
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibinfo {pages} {134103} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ayers}, \citenamefont {Levy},\ and\ \citenamefont
|
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{Nagy}(2018)}]{Ayers_2018}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~W.}\ \bibnamefont
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{Ayers}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Levy}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Nagy}},\ }\href {\doibase
|
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10.1007/s00214-018-2352-7} {\bibfield {journal} {\bibinfo {journal} {Theor.
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Chem. Acc.}\ ,\ \bibinfo {pages} {137}} (\bibinfo {year} {2018})}\BibitemShut
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\bibitem [{\citenamefont {Deur}\ \emph {et~al.}(2018)\citenamefont {Deur},
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\citenamefont {Mazouin}, \citenamefont {Senjean},\ and\ \citenamefont
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{Fromager}}]{Deur_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Deur}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Mazouin}},
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\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Senjean}}, \ and\
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\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href
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{\doibase 10.1140/epjb/e2018-90124-7} {\bibfield {journal} {\bibinfo
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{journal} {Eur. Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo
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{pages} {162} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kraisler}\ and\ \citenamefont
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{Kronik}(2013)}]{Kraisler_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Kraisler}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Kronik}},\ }\href {\doibase 10.1103/PhysRevLett.110.126403} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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{volume} {110}},\ \bibinfo {pages} {126403} (\bibinfo {year}
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{2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Kraisler}\ and\ \citenamefont
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{Kronik}(2014)}]{Kraisler_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Kraisler}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Kronik}},\ }\href {\doibase 10.1063/1.4871462} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\
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\bibinfo {pages} {18A540} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Alam}, \citenamefont {Knecht},\ and\ \citenamefont
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{Fromager}(2016)}]{Alam_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont
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{Alam}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Knecht}}, \ and\
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\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href
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{\doibase 10.1103/PhysRevA.94.012511} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo
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{pages} {012511} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Alam}\ \emph {et~al.}(2017)\citenamefont {Alam},
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\citenamefont {Deur}, \citenamefont {Knecht},\ and\ \citenamefont
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{Fromager}}]{Alam_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont
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{Alam}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Deur}}, \bibinfo
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{author} {\bibfnamefont {S.}~\bibnamefont {Knecht}}, \ and\ \bibinfo {author}
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{\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href {\doibase
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10.1063/1.4999825} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {147}},\ \bibinfo {pages} {204105}
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(\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Nagy}(1998)}]{Nagy_1998}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Nagy}},\ }\href {\doibase
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10.1002/(SICI)1097-461X(1998)69:3<247::AID-QUA4>3.0.CO;2-V} {\bibfield
|
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{journal} {\bibinfo {journal} {Int. J. Quantum Chem.}\ }\textbf {\bibinfo
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{volume} {69}},\ \bibinfo {pages} {247} (\bibinfo {year} {1998})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Nagy}(2001)}]{Nagy_2001}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
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{Nagy}},\ }\href {\doibase 10.1088/0953-4075/34/12/305} {\bibfield {journal}
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{\bibinfo {journal} {J. Phys. B At. Mol. Opt. Phys.}\ }\textbf {\bibinfo
|
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{volume} {34}},\ \bibinfo {pages} {2363} (\bibinfo {year}
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{2001})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Nagy}, \citenamefont {Liu},\ and\ \citenamefont
|
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{Bartolloti}(2005)}]{Nagy_2005}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Nagy}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Liu}}, \ and\
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\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Bartolloti}},\ }\href
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{\doibase 10.1063/1.1871933} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages} {134107}
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(\bibinfo {year} {2005})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Pastorczak}\ and\ \citenamefont
|
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{Pernal}(2014)}]{Pastorczak_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Pastorczak}}\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Pernal}},\ }\href {\doibase 10.1063/1.4866998} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\
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\bibinfo {pages} {18A514} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{Pribram-Jones}}\ \emph {et~al.}(2014)\citenamefont
|
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{{Pribram-Jones}}, \citenamefont {Yang}, \citenamefont {Trail}, \citenamefont
|
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{Burke}, \citenamefont {Needs},\ and\ \citenamefont
|
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{Ullrich}}]{Pribram-Jones_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{{Pribram-Jones}}}, \bibinfo {author} {\bibfnamefont {Z.-h.}\ \bibnamefont
|
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{Yang}}, \bibinfo {author} {\bibfnamefont {J.~R.}\ \bibnamefont {Trail}},
|
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}}, \bibinfo
|
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{author} {\bibfnamefont {R.~J.}\ \bibnamefont {Needs}}, \ and\ \bibinfo
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{author} {\bibfnamefont {C.~A.}\ \bibnamefont {Ullrich}},\ }\href {\doibase
|
|
10.1063/1.4872255} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
|
Phys.}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo {pages} {18A541}
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(\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Yang}, \citenamefont {{Mori-S\'anchez}},\ and\
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\citenamefont {Cohen}(2013)}]{Yang_2013a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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{Yang}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
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{{Mori-S\'anchez}}}, \ and\ \bibinfo {author} {\bibfnamefont {A.~J.}\
|
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\bibnamefont {Cohen}},\ }\href {\doibase 10.1063/1.4817183} {\bibfield
|
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{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
|
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{139}},\ \bibinfo {pages} {104114} (\bibinfo {year} {2013})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Yang}\ \emph {et~al.}(2017)\citenamefont {Yang},
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\citenamefont {{Pribram-Jones}}, \citenamefont {Burke},\ and\ \citenamefont
|
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{Ullrich}}]{Yang_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Z.-H.}\ \bibnamefont
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{Yang}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{{Pribram-Jones}}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
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{Burke}}, \ and\ \bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
|
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{Ullrich}},\ }\href {\doibase 10.1103/PhysRevLett.119.033003} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {119}},\ \bibinfo {pages} {033003} (\bibinfo {year}
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{2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Senjean}\ \emph {et~al.}(2015)\citenamefont
|
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{Senjean}, \citenamefont {Knecht}, \citenamefont {Jensen},\ and\
|
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\citenamefont {Fromager}}]{Senjean_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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{Senjean}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Knecht}},
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\bibinfo {author} {\bibfnamefont {H.~J.~A.}\ \bibnamefont {Jensen}}, \ and\
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\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href
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{\doibase 10.1103/PhysRevA.92.012518} {\bibfield {journal} {\bibinfo
|
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {92}},\ \bibinfo
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{pages} {012518} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Senjean}\ \emph {et~al.}(2016)\citenamefont
|
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{Senjean}, \citenamefont {Hedeg\aa{}rd}, \citenamefont {Alam}, \citenamefont
|
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{Knecht},\ and\ \citenamefont {Fromager}}]{Senjean_2016}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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{Senjean}}, \bibinfo {author} {\bibfnamefont {E.~D.}\ \bibnamefont
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{Hedeg\aa{}rd}}, \bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont
|
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{Alam}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Knecht}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Fromager}},\ }\href
|
|
{\doibase 10.1080/00268976.2015.1119902} {\bibfield {journal} {\bibinfo
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{journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo {pages}
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{968} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Smith}, \citenamefont {{Pribram-Jones}},\ and\
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\citenamefont {Burke}(2016)}]{Smith_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont
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{Smith}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
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{{Pribram-Jones}}}, \ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
|
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{Burke}},\ }\href {\doibase 10.1103/PhysRevB.93.245131} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {93}},\
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\bibinfo {pages} {245131} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont
|
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\
|
|
\citenamefont {Burke}}]{Carrascal_2015}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
|
|
\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\
|
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
|
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{\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo
|
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{journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\
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\bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Fromager}(2020)}]{Fromager_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Fromager}},\ }\href@noop {} {\ (\bibinfo {year} {2020})},\ \Eprint
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{http://arxiv.org/abs/2001.08605} {arXiv:2001.08605 [physics.chem-ph]}
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\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gould}\ and\ \citenamefont
|
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{Pittalis}(2020)}]{Gould_2019_insights}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
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{Gould}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
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{Pittalis}},\ }\href@noop {} {\ (\bibinfo {year} {2020})},\ \Eprint
|
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{http://arxiv.org/abs/2001.09429} {arXiv:2001.09429 [cond-mat.str-el]}
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\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont {Gill}(2012)}]{Loos_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
|
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{Gill}},\ }\href {\doibase 10.1103/PhysRevLett.108.083002} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {108}},\ \bibinfo {pages} {083002} (\bibinfo {year}
|
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{2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont {Gill}(2013)}]{Loos_2013a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
|
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{Gill}},\ }\href {\doibase 10.1063/1.4802589} {\bibfield {journal} {\bibinfo
|
|
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo
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{pages} {164124} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}(2014)}]{Loos_2014a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1103/PhysRevA.89.052523} {\bibfield {journal}
|
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{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {89}},\
|
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\bibinfo {pages} {052523} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}, \citenamefont {Ball},\ and\ \citenamefont
|
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{Gill}(2014)}]{Loos_2014b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}, \bibinfo {author} {\bibfnamefont {C.~J.}\ \bibnamefont {Ball}}, \
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and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont {Gill}},\
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|
}\href {\doibase 10.1063/1.4867910} {\bibfield {journal} {\bibinfo
|
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo
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{pages} {18A524} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Schulz}(1993)}]{Schulz_1993}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
|
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{Schulz}},\ }\href {\doibase 10.1103/PhysRevLett.71.1864} {\bibfield
|
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {71}},\ \bibinfo {pages} {1864} (\bibinfo {year}
|
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{1993})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Fogler}(2005)}]{Fogler_2005a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont
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{Fogler}},\ }\href {\doibase 10.1103/PhysRevLett.94.056405} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
|
{volume} {94}},\ \bibinfo {pages} {056405} (\bibinfo {year}
|
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{2005})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Bockrath}\ \emph {et~al.}(1999)\citenamefont
|
|
{Bockrath}, \citenamefont {Cobden}, \citenamefont {Lu}, \citenamefont
|
|
{Rinzler}, \citenamefont {Smalley}, \citenamefont {Balents},\ and\
|
|
\citenamefont {McEuen}}]{Bockrath_1999}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Bockrath}}, \bibinfo {author} {\bibfnamefont {D.~H.}\ \bibnamefont
|
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{Cobden}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Lu}}, \bibinfo
|
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{author} {\bibfnamefont {A.~G.}\ \bibnamefont {Rinzler}}, \bibinfo {author}
|
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{\bibfnamefont {R.~E.}\ \bibnamefont {Smalley}}, \bibinfo {author}
|
|
{\bibfnamefont {L.}~\bibnamefont {Balents}}, \ and\ \bibinfo {author}
|
|
{\bibfnamefont {P.~L.}\ \bibnamefont {McEuen}},\ }\href {\doibase
|
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10.1038/17569} {\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf
|
|
{\bibinfo {volume} {397}},\ \bibinfo {pages} {598} (\bibinfo {year}
|
|
{1999})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Ishii}\ \emph {et~al.}(2003)\citenamefont {Ishii},
|
|
\citenamefont {Kataura}, \citenamefont {Shiozawa}, \citenamefont {Yoshioka},
|
|
\citenamefont {Otsubo}, \citenamefont {Takayama}, \citenamefont {Miyahara},
|
|
\citenamefont {Suzuki}, \citenamefont {Achiba}, \citenamefont {Nakatake},
|
|
\citenamefont {Narimura}, \citenamefont {Higashiguchi}, \citenamefont
|
|
{Shimada}, \citenamefont {Namatame},\ and\ \citenamefont
|
|
{Taniguchi}}]{Ishii_2003}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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{Ishii}}, \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Kataura}},
|
|
\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Shiozawa}}, \bibinfo
|
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{author} {\bibfnamefont {H.}~\bibnamefont {Yoshioka}}, \bibinfo {author}
|
|
{\bibfnamefont {H.}~\bibnamefont {Otsubo}}, \bibinfo {author} {\bibfnamefont
|
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{Y.}~\bibnamefont {Takayama}}, \bibinfo {author} {\bibfnamefont
|
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{T.}~\bibnamefont {Miyahara}}, \bibinfo {author} {\bibfnamefont
|
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{S.}~\bibnamefont {Suzuki}}, \bibinfo {author} {\bibfnamefont
|
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{Y.}~\bibnamefont {Achiba}}, \bibinfo {author} {\bibfnamefont
|
|
{M.}~\bibnamefont {Nakatake}}, \bibinfo {author} {\bibfnamefont
|
|
{T.}~\bibnamefont {Narimura}}, \bibinfo {author} {\bibfnamefont
|
|
{M.}~\bibnamefont {Higashiguchi}}, \bibinfo {author} {\bibfnamefont
|
|
{K.}~\bibnamefont {Shimada}}, \bibinfo {author} {\bibfnamefont
|
|
{H.}~\bibnamefont {Namatame}}, \ and\ \bibinfo {author} {\bibfnamefont
|
|
{M.}~\bibnamefont {Taniguchi}},\ }\href {\doibase 10.1038/nature02074}
|
|
{\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo
|
|
{volume} {426}},\ \bibinfo {pages} {540} (\bibinfo {year}
|
|
{2003})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Deshpande}\ and\ \citenamefont
|
|
{Bockrath}(2008)}]{Deshpande_2008}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.~V.}\ \bibnamefont
|
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{Deshpande}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
|
{Bockrath}},\ }\href {\doibase 10.1038/nphys895} {\bibfield {journal}
|
|
{\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {4}},\
|
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\bibinfo {pages} {314} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Meyer}\ and\ \citenamefont
|
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{Matveev}(2009)}]{Meyer_2009}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
|
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{Meyer}}\ and\ \bibinfo {author} {\bibfnamefont {K.~A.}\ \bibnamefont
|
|
{Matveev}},\ }\href {\doibase 10.1088/0953-8984/21/2/023203} {\bibfield
|
|
{journal} {\bibinfo {journal} {J. Phys.: Condens. Matter}\ }\textbf
|
|
{\bibinfo {volume} {21}},\ \bibinfo {pages} {023203} (\bibinfo {year}
|
|
{2009})}\BibitemShut {NoStop}%
|
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\bibitem [{\citenamefont {Deshpande}\ \emph {et~al.}(2010)\citenamefont
|
|
{Deshpande}, \citenamefont {Bockrath}, \citenamefont {Glazman},\ and\
|
|
\citenamefont {Yacoby}}]{Deshpande_2010}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.~V.}\ \bibnamefont
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{Deshpande}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Bockrath}},
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\bibinfo {author} {\bibfnamefont {L.~I.}\ \bibnamefont {Glazman}}, \ and\
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Yacoby}},\ }\href
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{\doibase 10.1038/nature08918} {\bibfield {journal} {\bibinfo {journal}
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{Nature}\ }\textbf {\bibinfo {volume} {464}},\ \bibinfo {pages} {209}
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(\bibinfo {year} {2010})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Schmelcher}\ and\ \citenamefont
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{Cederbaum}(1990)}]{Schmelcher_1990}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Schmelcher}}\ and\ \bibinfo {author} {\bibfnamefont {L.~S.}\ \bibnamefont
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{Cederbaum}},\ }\href {\doibase 10.1103/PhysRevA.41.4936} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
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{41}},\ \bibinfo {pages} {4936} (\bibinfo {year} {1990})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Lange}\ \emph {et~al.}(2012)\citenamefont {Lange},
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\citenamefont {Tellgren}, \citenamefont {Hoffmann},\ and\ \citenamefont
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{Helgaker}}]{Lange_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~K.}\ \bibnamefont
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{Lange}}, \bibinfo {author} {\bibfnamefont {E.~I.}\ \bibnamefont {Tellgren}},
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\bibinfo {author} {\bibfnamefont {M.~R.}\ \bibnamefont {Hoffmann}}, \ and\
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\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}},\ }\href
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{\doibase 10.1126/science.1219703} {\bibfield {journal} {\bibinfo {journal}
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{Science}\ }\textbf {\bibinfo {volume} {337}},\ \bibinfo {pages} {327}
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(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Schmelcher}(2012)}]{Schmelcher_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Schmelcher}},\ }\href {\doibase 10.1126/science.1224869} {\bibfield
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{journal} {\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume}
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{337}},\ \bibinfo {pages} {302} (\bibinfo {year} {2012})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Schmelcher}\ and\ \citenamefont
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{Cederbaum}(1997)}]{Schmelcher_1997}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Schmelcher}}\ and\ \bibinfo {author} {\bibfnamefont {L.~S.}\ \bibnamefont
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{Cederbaum}},\ }\href {\doibase
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10.1002/(SICI)1097-461X(1997)64:5<501::AID-QUA3>3.0.CO;2-%23} {\bibfield
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{journal} {\bibinfo {journal} {Int. J. Quantum Chem.}\ }\textbf {\bibinfo
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{volume} {64}},\ \bibinfo {pages} {501} (\bibinfo {year} {1997})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Tellgren}, \citenamefont {Soncini},\ and\
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\citenamefont {Helgaker}(2008)}]{Tellgren_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~I.}\ \bibnamefont
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{Tellgren}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Soncini}}, \
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and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}},\ }\href
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{\doibase 10.1063/1.2996525} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {129}},\ \bibinfo {pages} {154114}
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(\bibinfo {year} {2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Tellgren}, \citenamefont {Helgaker},\ and\
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\citenamefont {Soncini}(2009)}]{Tellgren_2009}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~I.}\ \bibnamefont
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{Tellgren}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}},
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\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Soncini}},\ }\href
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{\doibase 10.1039/B822262B} {\bibfield {journal} {\bibinfo {journal} {Phys.
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Chem. Chem. Phys..}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo {pages}
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{5489} (\bibinfo {year} {2009})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Boblest}, \citenamefont {Schimeczek},\ and\
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\citenamefont {Wunner}(2014)}]{Boblest_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Boblest}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Schimeczek}},
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\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Wunner}},\ }\href
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{\doibase 10.1103/PhysRevA.89.012505} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo
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{pages} {012505} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Stopkowicz}\ \emph {et~al.}(2015)\citenamefont
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{Stopkowicz}, \citenamefont {Gauss}, \citenamefont {Lange}, \citenamefont
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{Tellgren},\ and\ \citenamefont {Helgaker}}]{Stopkowicz_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Stopkowicz}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Gauss}},
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\bibinfo {author} {\bibfnamefont {K.~K.}\ \bibnamefont {Lange}}, \bibinfo
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{author} {\bibfnamefont {E.~I.}\ \bibnamefont {Tellgren}}, \ and\ \bibinfo
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{author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}},\ }\href {\doibase
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10.1063/1.4928056} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {143}},\ \bibinfo {pages} {074110}
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(\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Levy}\ and\ \citenamefont
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{Zahariev}(2014)}]{Levy_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Levy}}\ and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
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{Zahariev}},\ }\href {\doibase 10.1103/PhysRevLett.113.113002} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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{volume} {113}},\ \bibinfo {pages} {113002} (\bibinfo {year}
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{2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gidopoulos}, \citenamefont {Papaconstantinou},\ and\
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\citenamefont {Gross}(2002)}]{Gidopoulos_2002}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~I.}\ \bibnamefont
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{Gidopoulos}}, \bibinfo {author} {\bibfnamefont {P.~G.}\ \bibnamefont
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{Papaconstantinou}}, \ and\ \bibinfo {author} {\bibfnamefont {E.~K.~U.}\
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\bibnamefont {Gross}},\ }\href {\doibase 10.1103/PhysRevLett.88.033003}
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
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{\bibinfo {volume} {88}},\ \bibinfo {pages} {033003} (\bibinfo {year}
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{2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont {Gill}(2016)}]{Loos_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1002/wcms.1257} {\bibfield {journal} {\bibinfo
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{journal} {Wiley Interdiscip. Rev. Comput. Mol. Sci.}\ }\textbf {\bibinfo
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{volume} {6}},\ \bibinfo {pages} {410} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Gill}\ and\ \citenamefont {Loos}(2012)}]{Gill_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~M.~W.}\
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\bibnamefont {Gill}}\ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\
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\bibnamefont {Loos}},\ }\href {\doibase 10.1007/s00214-011-1069-7} {\bibfield
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{journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo
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{volume} {131}},\ \bibinfo {pages} {1069} (\bibinfo {year}
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{2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Agboola}\ \emph {et~al.}(2015)\citenamefont
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{Agboola}, \citenamefont {Knol}, \citenamefont {Gill},\ and\ \citenamefont
|
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{Loos}}]{Agboola_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Agboola}}, \bibinfo {author} {\bibfnamefont {A.~L.}\ \bibnamefont {Knol}},
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\bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont {Gill}}, \ and\
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\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont {Loos}},\ }\href
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{\doibase 10.1063/1.4929353} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {143}},\ \bibinfo {pages} {084114}
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(\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}(2017)}]{Loos_2017a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1063/1.4978409} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {146}},\ \bibinfo
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{pages} {114108} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont {Gill}(2011)}]{Loos_2011b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1063/1.3665393} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {135}},\ \bibinfo
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{pages} {214111} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}(2013)}]{Loos_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1063/1.4790613} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo
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{pages} {064108} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Olver}\ \emph {et~al.}(2010)\citenamefont {Olver},
|
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\citenamefont {Lozier}, \citenamefont {Boisvert},\ and\ \citenamefont
|
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{Clark}}]{NISTbook}%
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\BibitemOpen
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\bibinfo {editor} {\bibfnamefont {F.~W.~J.}\ \bibnamefont {Olver}}, \bibinfo
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{editor} {\bibfnamefont {D.~W.}\ \bibnamefont {Lozier}}, \bibinfo {editor}
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{\bibfnamefont {R.~F.}\ \bibnamefont {Boisvert}}, \ and\ \bibinfo {editor}
|
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{\bibfnamefont {C.~W.}\ \bibnamefont {Clark}},\ eds.,\ \href@noop {} {\emph
|
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{\bibinfo {title} {NIST Handbook of Mathematical Functions}}}\ (\bibinfo
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{publisher} {Cambridge University Press},\ \bibinfo {address} {New York},\
|
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\bibinfo {year} {2010})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rogers}\ and\ \citenamefont
|
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{Loos}(2017)}]{Rogers_2017}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.~J.}\ \bibnamefont
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{Rogers}}\ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1063/1.4974839} {\bibfield {journal} {\bibinfo
|
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {146}},\ \bibinfo
|
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{pages} {044114} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Rogers}, \citenamefont {Ball},\ and\ \citenamefont
|
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{Loos}(2016)}]{Rogers_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.~J.~M.}\
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\bibnamefont {Rogers}}, \bibinfo {author} {\bibfnamefont {C.~J.}\
|
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\bibnamefont {Ball}}, \ and\ \bibinfo {author} {\bibfnamefont {P.-F.}\
|
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\bibnamefont {Loos}},\ }\href {\doibase 10.1103/PhysRevB.93.235114}
|
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo
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{volume} {93}},\ \bibinfo {pages} {235114} (\bibinfo {year}
|
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{2016})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Knowles}\ and\ \citenamefont
|
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{Handy}(1989)}]{Knowles_1989}%
|
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont
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{Knowles}}\ and\ \bibinfo {author} {\bibfnamefont {N.~C.}\ \bibnamefont
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{Handy}},\ }\href {\doibase 10.1016/0010-4655(89)90033-7} {\bibfield
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{journal} {\bibinfo {journal} {Comput. Phys. Commun.}\ }\textbf {\bibinfo
|
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{volume} {54}},\ \bibinfo {pages} {75} (\bibinfo {year} {1989})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
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{Gill}(2009{\natexlab{a}})}]{Loos_2009}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1063/1.3275519} {\bibfield {journal} {\bibinfo
|
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {131}},\ \bibinfo
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{pages} {241101} (\bibinfo {year} {2009}{\natexlab{a}})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
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{Gill}(2009{\natexlab{b}})}]{Loos_2009c}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1103/PhysRevLett.103.123008} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
|
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{volume} {103}},\ \bibinfo {pages} {123008} (\bibinfo {year}
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{2009}{\natexlab{b}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
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{Gill}(2010{\natexlab{a}})}]{Loos_2010}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1016/j.cplett.2010.09.019} {\bibfield
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{journal} {\bibinfo {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo
|
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{volume} {500}},\ \bibinfo {pages} {1} (\bibinfo {year}
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{2010}{\natexlab{a}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}\ and\ \citenamefont
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{Gill}(2010{\natexlab{b}})}]{Loos_2010d}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
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{Gill}},\ }\href {\doibase 10.1103/PhysRevLett.105.113001} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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{volume} {105}},\ \bibinfo {pages} {113001} (\bibinfo {year}
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{2010}{\natexlab{b}})}\BibitemShut {NoStop}%
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\end{thebibliography}%
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\end{document}
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