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PCCP
Physical Chemistry Chemical Physics
Accepted Manuscript
Volume 19
Number 1
7 January 2017
This is an Accepted Manuscript, which has been through the
Pages 1-896
Royal Society of Chemistry peer review process and has been
accepted for publication.
PCCP
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PAPER
H.-P. Loock et al.
Determination of the thermal, oxidative and photochemical
shall the Royal Society of Chemistry be held responsible for any errors
degradation rates of scintillator liquid by fluorescence EEM
spectroscopy
or omissions in this Accepted Manuscript or any consequences arising
from the use of any information it contains.
rsc.li/pccpPage 1 of 85 Physical Chemistry Chemical Physics
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DOI: 10.1039/D2CP02827A
Department of Physics, Durham University
Gill, Peter; Australian National University, Research School of Chemistry
Gori-Giorgi, Paola; VU University, Theoretical Chemistry
Görling, Andreas; Friedrich-Alexander Universität Erlangen-Nürnberg,
Physical Chemistry Chemical Physics Accepted Manuscript
Lehrstuhl für Theoretische Chemie
Gould, Tim; Griffith University, Queensland Micro and Nanotechnology
Centre
Grimme, Stefan; University of Bonn, Mulliken Center for Theoretical
Chemistry
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Gritsenko, Oleg; Vrije Universiteit, Chemistry
Jensen, Hans Jørgen Aagaard; University of Southern Denmark ,
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
Department of Physics, Chemistry and Pharmacy
Johnson, Erin; Dalhousie University, Chemistry
Jones, Robert; Forschungszentrum Jülich GmbH, Institut fur
Fostkiorperforschung (IFF)
Kaupp, Martin; Technische Universität Berlin, Institut für Chemie, Sekr.
C7
Koster, Andreas; Ciudad de Mexico Avenida Instituto Politecnico
Nacional, El departamento de Quimica se encuentra ubicado al norte
Kronik, Leeor; Weizmann Institute of Science, Materials and Interfaces
Krylov, Anna; University of Southern California Department of
Chemistry, Department of Chemistry
Kvaal, Simen; Hylleraas Centre for Quantum Molecular Sciences
Laestadius, Andre; Hylleraas Centre for Quantum Molecular Sciences
Levy, Melvyn P.; Tulane University
Lewin, Mathieu; Université Paris Dauphine
Liu, SB; University of North Carolina System, Research Computing
Center
Loos, Pierre-François; Universite Toulouse III Paul Sabatier, Laboratoire
de Chimie et Physique Quantiques
Maitra, Neepa; Rutgers University Newark, Department of Physics
Neese, Frank; Max-Planck-Institut für Kohlenforschung
Perdew, John; Temple University,
Pernal, Katarzyna; Lodz University of Technology, Institute of Physics
Pernot, Pascal; Institut de Chimie Physique
Piecuch, P.; Michigan State University, Chemistry
Rebolini, Elisa; Institut Laue-Langevin
Reining, Lucia; CNRS, LSI
Romaniello, Pina; Université Toulouse III Paul Sabatier - Complexe
Scientifique de Rangueil, Laboratoire de Physique Théorique
Ruzsinszky, Adrienn ; Temple University
Salahub, Dennis ; University of Calgary, Chemistry
Scheffler, Matthias; Abt. Theorie, Fritz-Haber-Institut
Schwerdtfeger, Peter; Massey University - Albany Campus
Staroverov, Viktor; The University of Western Ontario, Department of
Chemistry
Sun, Jianwei; Tulane University, Physics
Tellgren, Erik; Hylleraas Centre for Quantum Molecular Sciences
Tozer, David; University of Durham, Department of Chemistry
Trickey, Samuel; Univ. of Florida, QTP, Physics and Chemistry
Ullrich, Carsten; University of Missouri, Department of Physics and
Astronomy
Vela, Alberto; CINVESTAV, Quimica
Vignale, Giovanni; University of Missouri
Wesolowski, Tomasz; University of Geneva, Physical Chemistry
Xu, Xin; Fudan University, Chemistry
Yang, Weitao; Duke University, Department of ChemistryOpen Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Physical Chemistry Chemical Physics
DOI: 10.1039/D2CP02827A
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Physical Chemistry Chemical Physics Accepted Manuscript
Page 2 of 85Page 3 of 85 Physical Chemistry Chemical Physics
View Article Online
DOI: 10.1039/D2CP02827A
Physical Chemistry Chemical Physics Accepted Manuscript
DFT Exchange: Sharing Perspectives on the Workhorse
of Quantum Chemistry and Materials Science †
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Andrew M. Teale,∗AMT Trygve Helgaker,∗T H Andreas Savin,∗AS Carlo Adamo,CA Bálint
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
Aradi,BA Alexei V. Arbuznikov,AVA Paul W. Ayers,PWA Evert Jan Baerends,EJB Vin-
cenzo Barone,V B Patrizia Calaminici,PC Eric Cancès,EC Emily A. Carter,EAC Pratim Ku-
mar Chattaraj,PKC Henry Chermette,HCh Ilaria Ciofini,IC T. Daniel Crawford,T DC Frank
De Proft,FDP John F. Dobson,JD Claudia Draxl,CD Thomas Frauenheim,T F Emmanuel
Fromager,EF Patricio Fuentealba,PF Laura Gagliardi,LG Giulia Galli,GG Jiali Gao,JG Paul
Geerlings,PG Nikitas Gidopoulos,NG Peter M. W. Gill,PMW G Paola Gori-Giorgi,PGG An-
dreas Görling,AG Tim Gould,T G Stefan Grimme,SG Oleg Gritsenko,OG Hans Jørgen Aa-
gaard Jensen,HJAaJ Erin R. Johnson,ERJ Robert O. Jones,ROJ Martin Kaupp,MK Andreas
M. Köster,AK Leeor Kronik,LK Anna I. Krylov,AIK Simen Kvaal,SK Andre Laestadius,AL Mel
Levy,MLe Mathieu Lewin,ML Shubin Liu,SL Pierre-François Loos,PFL Neepa T. Maitra,NM
Frank Neese,FN John P. Perdew,JPP Katarzyna Pernal,KP Pascal Pernot,PPe Piotr Piecuch,PPi
Elisa Rebolini,ER Lucia Reining,LR Pina Romaniello,PR Adrienn Ruzsinszky,AR Dennis R.
Salahub,DRS Matthias Scheffler,MS Peter Schwerdtfeger,PSc Viktor N. Staroverov,V NS Jian-
wei Sun,JS Erik Tellgren,ET David J. Tozer,DJT Samuel B. Trickey,SBT Carsten A. Ullrich,CAU
Alberto Vela,AV Giovanni Vignale,GV Tomasz A. Wesolowski,TW Xin XuXX Weitao Yang,WY
In this paper, the history, present status, and future of density-functional theory (DFT) is informally
reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists,
method developers and practitioners. The format of the paper is that of a roundtable discussion, in
which the participants express and exchange views on DFT in the form of 302 individual contributions,
formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries,
the paper represents a broad snapshot of DFT, anno 2022.
AMT
School of Chemistry, University of Nottingham, University Park, Nottingham, NG7
2RD, United Kingdom.
TH
Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, Uni-
versity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
AS
Laboratoire de Chimie Théorique, CNRS and Sorbonne University, 4 Place Jussieu,
CEDEX 05, 75252 Paris, France.
CA
PSL University, CNRS, ChimieParisTech-PSL, Institute of Chemistry for Health and
Life Sciences, i-CLeHS, 11 rue P. et M. Curie, 75005 Paris, France
BA
Bremen Center for Computational Materials Science, University of Bremen, P.O.
Box 330440, D-28334 Bremen, Germany.
AVA
Institute of Chemistry, Technical University of Berlin, Straße des 17. Juni 135,
10623, Berlin, Germany
PWA
McMaster University, Hamilton, Ontario, Canada
EJB
Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije
Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.
VB
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56125 Pisa, Italy.
PC
Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cin-
vestav), CDMX, 07360, México
EC
CERMICS, Ecole des Ponts and Inria Paris, 6 Avenue Blaise Pascal, 77455 Marne-la-
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Vallée, France.
ERJ
EAC
Department of Mechanical and Aerospace Engineering and the Andlinger Center for Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, B3H 4R2
Energy and the Environment, Princeton University, Princeton, NJ 08544-5263 USA. Canada
ROJ
PKC
Department of Chemistry, Indian Institute of Technology Kharagpur, 721302, India. Peter Grünberg Institut PGI-1, Forschungszentrum Jülich, 52425 Jülich, Germany.
Physical Chemistry Chemical Physics Accepted Manuscript
MK
HCh
Institut Sciences Analytiques, Université Claude Bernard Lyon1, CNRS UMR 5280, Institute of Chemistry, Technical University of Berlin, Straße des 17. Juni 135,
69622 Villeurbanne, France. 10623, Berlin, Germany
AK
IC
PSL University, CNRS, ChimieParisTech-PSL, Institute of Chemistry for Health and Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cin-
Life Sciences, i-CLeHS, 11 rue P. et M. Curie, 75005 Paris, France vestav), CDMX, 07360, México
LK
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
T DC
Department of Chemistry, Virginia Tech, Blacksburg, VA 24061 USA, and the Molec- Department of Molecular Chemistry and Materials Science, Weizmann Institute of
ular Sciences Software Institute, Blacksburg, VA 24060 USA. Science, Rehovoth, Israel 76100.
AIK
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
FDP
Research group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Department of Chemistry, University of Southern California, Los Angeles, California
Pleinlaan 2, B-1050 Brussels, Belgium. 90089, USA
SK
JD
Griffith University, Nathan, Queensland 4111, Australia Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, Uni-
CD
Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, 12489 versity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
AL
Berlin, Germany. Fritz-Haber-Institut der Max-Planck-Gesellschaft, 14195 Berlin, Ger- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, Uni-
many. versity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
MLe
TF
Beijing Computational Science Research Center (CSRC), 100193 Beijing, China. Department of Chemistry, Tulane University, New Orleans, Louisiana, 70118, USA
ML
Shenzhen JL Computational Science and Applied Research Institute, 518110 Shen- CNRS & CEREMADE, Université Paris-Dauphine, PSL Research University, Place de
zhen, China. Bremen Center for Computational Materials Science, University of Lattre de Tassigny, 75016 Paris, France.
SL
Bremen, P.O. Box 330440, D-28334 Bremen, Germany. Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-
EF
Laboratoire de Chimie Quantique, Institut de Chimie, CNRS/Université de Stras- 3420; Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-
bourg, 4 rue Blaise Pascal, 67000 Strasbourg, France. 3290, USA.
PFL
PF
Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse,
Santiago, Chile. CNRS, UPS, France
NM
LG
Department of Chemistry, Pritzker School of Molecular Engineering, The James Department of Physics, Rutgers University at Newark, 101 Warren Street, Newark,
Franck Institute, and Chicago Center for Theoretical Chemistry, The University of NJ 07102, USA
FN
Chicago, Chicago, Illinois 60637, United States. Max Planck Institut für Kohlenforschung, Kaiser Wilhelm Platz 1, D-45470 Mülheim
GG
Pritzker School of Molecular Engineering and Department of Chemistry, The Univer- an der Ruhr, Germany
JPP
sity of Chicago, Chicago, Il, USA Departments of Physics and Chemistry, Temple University, Philadelphia, PA 19122,
JG
Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, Shenzhen USA.
KP
518055, China; Department of Chemistry, University of Minnesota, Minneapolis, MN Institute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924
55455, USA Lodz, Poland
PPe
PG
Research group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Plein- Institut de Chimie Physique, UMR8000, CNRS and Université Paris-Saclay, Bât. 349,
laan 2, B-1050 Brussels, Belgium. Campus d’Orsay, 91405 Orsay, France.
PPi
NG
Department of Physics, Durham University, South Road, Durham DH1 3LE, United Department of Chemistry, Michigan State University, East Lansing, Michigan
Kingdom. 48824, USA and Department of Physics and Astronomy, Michigan State University,
PMW G
School of Chemistry, University of Sydney, Camperdown NSW 2006, Australia East Lansing, Michigan 48824, USA
ER
PGG
Department of Chemistry and Pharmaceutical Sciences, Amsterdam Institute of Institut Laue Langevin, 71 avenue des Martyrs, 38000 Grenoble, France.
LR
Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boele- LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut Polytechnique de Paris,
laan 1083, 1081HV Amsterdam, The Netherlands. F-91120 Palaiseau, France and European Theoretical Spectroscopy Facility
PR
AG
Chair of Theoretical Chemistry, University of Erlangen-Nuremberg, Egerlandstrasse Laboratoire de Physique Théorique (UMR 5152), Université de Toulouse, CNRS, UPS,
3, 91058 Erlangen, Germany. France
AR
TG
Qld Micro- and Nanotechnology Centre, Griffith University, Gold Coast, Qld 4222, Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
DRS
Australia Department of Chemistry, Department of Physics and Astronomy, CMS – Centre
SG
Mulliken Center for Theoretical Chemistry, University of Bonn, Beringstrasse 4, for Molecular Simulation, IQST – Institute for Quantum Science and Technology,
53115 Bonn, Germany. Quantum Alberta, University of Calgary, 2500 University Drive NW, Calgary, Alberta,
OG
Department of Chemistry and Pharmaceutical Sciences, Amsterdam Institute of Canada T2N 1N4
MS
Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boele- The NOMAD Laboratory at FHI, Max Planck Society, Faradayweg 4-6, D-14195,
laan 1083, 1081HV Amsterdam, The Netherlands. Germany
PSc
HJAaJ
Department of Physics, Chemistry and Pharmacy, University of Southern Den- Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Ad-
mark, DK-5230 Odense M, Denmark. vanced Study, Massey University Auckland, 0632 Auckland, New Zealand.
V NS
Department of Chemistry, The University of Western Ontario, London, Ontario N6A
5B7, Canada.
JS
Department of Physics and Engineering Physics, Tulane University, New Orleans, LA
70118, USA.
ET
Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, Uni-
versity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
DJT
Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, UK
SBT
Quantum Theory Project, Dept. of Physics, Univ. of Florida, Gainesville FL 32611
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document with the preliminary questions and answers. A total of
67 accepted the invitation, bringing the total number of authors
to 70.
In a process involving all authors, the preliminary questions
were revised and preliminary answers removed. A final set of
Physical Chemistry Chemical Physics Accepted Manuscript
26 questions was agreed upon: five questions for Density func-
1 Introduction tional Theory (DFT), nine for Density-Functional Approximations
(DFAs), eight for The Future of DFT and DFAs, and four for Com-
What is the status of DFT? Where is DFT heading? What are the
municating and Sharing Our Results.
important new developments in DFT and what are the points of
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
All authors were then invited by the initiators to contribute to
contention? What is DFT?
the discussion by providing answers to the questions and also
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
Such questions are discussed whenever developers and users of comments to answers over a six-week period, encouraging dis-
DFT meet — in conferences and workshops, during coffee breaks cussions among the authors. Guidelines were provided to ensure
and over dinners. We do not expect short, clear answers to such a smooth collaborative process. The end result was an extensive
questions but the discussions and conversations they give rise to first draft of the manuscript, running over sixty pages and with
are often informative and entertaining — and different to discus- several hundred references. After a two-week internal review in-
sions in publications and presentations. We learn about new ideas volving all authors, an additional two weeks were allotted for re-
and developments and about failed attempts — a casual remark sponses to the internal review. The purpose of the internal review
may trigger new research or lead to new collaborations. These was solely to improve clarity of expression – not to restrict in any
discussions are an important reason for travelling to conferences way the freedom of the authors to express their opinions.
and something we have missed during the pandemic. The final draft was edited by the three initiators, with the aim
This article is an attempt to bring such discussions to the of improving the organization of the manuscript by reordering
printed format — to let prominent workers in the field exchange contributions and comments, reducing, where possible, repetition
views and thoughts about DFT in an open informal manner, mim- and ensuring a certain level of uniformity in notation and clarity
icking the format of a roundtable discussion, but backing up their of presentation. However, to retain the spontaneity of the discus-
statements by arguments and references to the literature. The sion and reflect the multitude of views presented, reorganization
end result should be a lively guide to DFT and its development. was kept to a minimum. As a consequence, some themes may
The format of the present article is an unusual one, resembling be revisited in different contexts throughout the paper – much as
most closely the Faraday Discussions but not anchored to the talks would happen in a lively round table discussion.
presented at a conference. It is to our knowledge the first paper Having received a final go-ahead from all co-authors, the final
of its kind in PCCP and the first such paper on DFT. Given its manuscript was submitted to the journal. All work on the paper
unusual format, we here describe how it came about. was carried out with LATEX, using the Overleaf platform 1 for ease
The initiative for the article was taken by three of the authors, of collaboration.
Andy Teale, Trygve Helgaker, and Andreas Savin. Having received The final manuscript provides an interesting snapshot of where
a go-ahead for the project from the publisher, the three initiators DFT stands today and where it is moving. It covers much of DFT
compiled an initial list of questions about DFT and some tentative with an extensive bibliography, but coverage is nevertheless not
answers. A letter of invitation was then sent out to about hundred exhaustive — classical DFT and multicomponent DFT are not dis-
workers in the field, inviting them “to participate in what will cussed, for example. The topics covered in the paper reflect the
hopefully be an open, thought provoking and informal discussion interests of the authors. Also, the views stated are those of the
about density-functional theory and its applications”. To clarify individual authors — as such, the paper has no conclusion. In the
the format of the article, the invitation contained a link to the spirit of the paper, you are instead encouraged to continue this
exchange of views, by contacting the authors.
USA. 2 Density-Functional Theory (DFT)
CAU
Department of Physics and Astronomy, University of Missouri, Columbia, MO
65211, USA
2.1 What is DFT?
AV
Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cin- (2.1.1) Savin : Density-functional theory (DFT) is more than
vestav), CDMX, 07360, México existence theorems. I like to make the distinction between
GV
Department of Physics, University of Missouri, Columbia, MO 65203, USA
TW
Department of Physical Chemistry, Université de Genève, 30 Quai Ernest-Ansermet, 1. a density functional, a number obtained from the density;
1211 Genève, Switzerland 2. DFT, the collection of theorems useful for obtaining exact
XX
Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Col-
results with procedures using density functionals, without
laborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for
Computational Physical Science, Department of Chemistry, Fudan University, Shanghai
having to solve the exact many-body problem;
200433, China 3. the methods using them – for example, the Kohn–Sham
WY
Department of Chemistry and Physics, Duke University, Durham, NC 27516, USA. method; and
∗
Corresponding author email: andrew.teale@nottingham.ac.uk
∗
Corresponding author email: trygve.helgaker@kjemi.uio.no 4. density-functional approximations (DFAs), the approxima-
∗
Corresponding author email: andreas.savin@lct.jusieu.fr tions (or models).
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The latter can originate from a choice of a “closed form”, as is needed to evaluate the functionals for the various observables,
mentioned in contribution (2.1.4), or from controllable ones, as without calculating the density from the many-body wave func-
related to the numerical treatment and discussed in contribu- tion. Otherwise, DFT could probably not compete with other ap-
tion (4.6.7). proaches, not even as an idea – for example, also the external
potential is a sufficient descriptor (for given particle number or
(2.1.2) Levy : Federico Zahariev and I have recently shown
Physical Chemistry Chemical Physics Accepted Manuscript
chemical potential), it is simple, and it has the advantage that we
in ref. 2 that it is useful and variationally valid to employ spin-
(think we) know it. The variational character also has the benefit
free wave functions in the constrained-search formulation when
that a slightly wrong density may still lead to a reasonable energy
deriving certain properties of a functional for the purpose of its
(whereas this may not hold for other observables).
approximation.
This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
So, we may consider DFT as one possibility: one possible way
In the constrained-search formulation of pure-state (or ensem-
to formulate the calculation of observables in a many-body sys-
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
ble) DFT, the kinetic plus electron-electron repulsion energy of
tem. There are many such ways, and we know that for most
a density is the expectation value of the wave function (or en-
systems we will never be able to obtain the exact answer. There-
semble) that yields this density and minimizes the kinetic plus
fore, once we agree that those various ways are in principle exact,
electron-electron repulsion expectation value. That is,
the true question is: how suitable are they as starting points for
approximations? And so, for our purpose here: in which way is
Z
EGS = min v(r)ρ(r)dr + F[ρ] , (1)
ρ DFT a good starting point for approximations?
where, with the use of pure-state wave functions, (2.1.4) Scheffler : Since the development of the quantum
mechanics of atoms and polyatomic systems, it was clear that in-
F(ρ) = min hΨ|T +W |Ψi, (2) spection of the ground-state electron density ρ(r) provides the
Ψ→ρ
information on the total number of electrons, N, the positions of
where the wave functions are here spin-free, but antisymmetric the atoms, {RI }, and from ρ(RI ) the nuclear charges 3,4 . Thus,
in the first M spatial coordinates and separately antisymmetric in ρ(r) determines N, {RI }, {ZI } – that is, the many-electron Hamil-
last (N − M) spatial coordinates. The generalization of F[ρ] to tonian, and therefore, it determines everything. This is the algo-
ensembles should be clear. This generalization ensures convexity. rithm that defines how to go from the ground-state density to the
energy.
The theorem of Hohenberg and Kohn 5 and the works by
(2.1.3) Reining : One may distinguish different possible as-
Levy 6,7 and Lieb 8 are beautiful mathematical treatments. Impor-
pects in this question: What is the message of DFT? Why has it been
tantly, the basic concept that the ground-state electron density de-
successful? How is it used today? What distinguishes it from other
termines everything often enables decisive physical insight. The
theories that deal with the many-body problem? Some are treated
often misleading assumption is that the above laid out, exact algo-
later, so I think we should focus on the first aspect here. I also
rithm “ρ(r) → ground-state energy (and even everything)” can be
think that, in answering this and many other questions, a glance
expressed in terms of a closed mathematical expression. Approxi-
at other possible theoretical approaches is healthy, because we al-
mating the algorithm by a mathematical functional, i.e., by a DFA,
ways learn from comparison, so let us try to have such a point of
suffers from the severe problem that the range of validity of this
view whenever possible.
functional is typically unclear: We can test its accuracy only by
The term DFT expresses the fact that observables in the ground
comparing results with experiments or high-level wave-function
state at zero temperature can be considered as functionals of the
theories. We trust the reliability for systems that we believe (!)
ground-state density. This can then be extended to thermal equi-
are “similar” to the tested ones, but we don’t know about the ac-
librium etc., as others point out. So, it means that the density is
curacy for untested systems. And the term “similar” is not even
a sufficient descriptor. It is important to say “can be considered as
defined.
a functional of the density” and not “is a functional of the den-
Let me add: I am not aware of a proof that the exact exchange–
sity”, because this is a choice: observables can also be considered
correlation-functional exists, beyond the noted algorithm which
as functionals of the many-body ground-state wave function, or
requires to solve the many-body Schrödinger equation. How-
the one-body Green’s function, or many other possible choices.
ever, and most importantly, the works by Hohenberg and Kohn
The functional of the many-body ground-state wave function is
and Kohn and Sham have shown the way to develop density-
very simple (whereas the wave function is not, of course), and a
functional approximations which revolutionized the description
density functional will in most cases be exceedingly complicated
and understanding of poly-atomic systems.
(whereas the density is simple). Actually, I chose to say “can be
considered as”, because this does not imply that there must be an (2.1.5) Kvaal : I agree with Savin in contribution (2.1.1) –
explicit expression. in particular. with respect to the claim that a distinction between
A second important point: the density is not known a priori but exact DFT and approximate DFT is useful. In my opinion, they are
is needed as input to evaluate our density functionals for a given both conceptually and mathematically different. They share the
system and observable. So, as a second aspect of DFT, we also use of the density and potential as dual basic variables, but oth-
have to invoke the variational character of the energy as func- erwise the similarities disappear for me. For instance, a DFA will
tional of the density, because it allows us to find the density that have much nicer mathematical properties than the exact univer-
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sal functional, as they are built from simple, explicit ingredients, tionals, nonlocal and also nonsemilocal, are SIE-free by construc-
at least partially necessitated by the need for efficient numerical tion for any one-electron system and perform as well on ther-
evaluation and optimization in order to be useful. On the other modynamics benchmarks as hybrid functionals, they still retain
hand, the exact universal density functional has a complicated significant errors in the dissociation of molecular ions, band gaps
implicit definition, leading to a highly complicated functional. A of molecules, and polymer polarizability problems, much like the
Physical Chemistry Chemical Physics Accepted Manuscript
concrete formulation of this is due to Schuch and Verstraete, 9 hybrid functional of B3LYP. The only significant improvement ob-
who demonstrated that, if an efficient evaluation of the universal served is in the prediction of reaction barriers. Thus the system-
functional could be done, all NP hard problems would be solvable atic error is clearly not the SIE.
in polynomial time. This is highly unlikely. On the other hand, To describe the systematic error of DFAs, the concept of the de-
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DFAs are necessarily computable! (It is of course one of the mar- localization error has been developed, and it can be understood
vels of DFT, that it is even possible to obtain such good results from the perspective of fractional charges. 16,17 For systems of
Open Access Article. Published on 10 August 2022. Downloaded on 10/5/2022 5:20:34 PM.
with so little computational effort.) small or moderate physical sizes, conventional DFAs usually have
Thus, approximate and exact density-functionals are mathe- good accuracy in total energies for an integer number of elec-
matically quite different. The noncomputability of the exact func- trons. For a fractional number of electrons, conventional DFAs,
tional indicates that systematically improvable DFAs are probably however, violate the Perdew−Parr−Levy−Balduz (PPLB) linear-
possible, in the sense of mathematical a priori error estimation – ity condition 18–20 , which states that the exact ground-state en-
that is, mathematical statements towards an approximation’s ac- ergy E(N) is a linear function of the fractional electron numbers
curacy in terms of its adjustable parameters, such as basis size. connecting adjacent integer points. Inconsistent with the require-
Therefore, I would like to go out on a limb and say that approx- ment of the PPLB linearity condition, E(N) curves from conven-
imate density functionals are not really approximations to exact tional DFAs are usually convex, with drastic underestimation to
density functionals. They are instead largely independent and, the ground-state energies of fractional systems. The convex devi-
to a variable extent, semiempirical models that have the common ation of conventional DFAs decreases when the systems become
use of the density as a basic variable as a characteristic. The latter larger and vanishes at the bulk limit. However, the delocalization
aspect is for me an answer to the question “What is DFT?” error is exhibited in another way, in which the error manifests it-
self in too low relative ground-state energies of ionized systems
(2.1.6) Savin : Let me comment on the difficulty of obtain-
and incorrect linear E(N) curves with wrong slopes at the bulk
ing exact functionals in a (semi)local form by choosing a simpler
limit. 16,17,21
example. The Hartree density functional,
To reduce or eliminate the delocalization error, enormous ef-
1
Z Z
EH = ρ(r1 )ρ(r2 )/|r1 − r2 | dr1 dr2 , (3) forts have been devoted to the development of new exchange–
2 R3 R3 correlation functionals. None of these developments are based
is universal, and not only known but also simple. However, I on a semilocal form. All have nonlocal features in the functionals
don’t see how to replace it by a (semi)local form.* One can argue – see the development of the scaling approaches. 22–25
that this does not lead to problems, as we compute EH explicitly. In addition to the delocalization error characterized by frac-
However, this argument is not valid if we choose to express the tional charges, commonly used DFAs also have a significant sys-
exchange functional, Ex , in a (semi)local form: for one-electron tematic static correlation error characterized by the violation of
systems, Ex = −EH . the constancy conditions on fractional spins. 17,20,26 The combi-
nation of the exact fractional charge condition 18 and the exact
(2.1.7) Yang : I agree with Savin on the difficulty of semilo- fractional spin condition 20,26 leads to the general flat-plane con-
cal functionals. The example of the interaction energy of a one- dition, 27 the satisfaction of which is a necessary condition for
electron system is a clear case: the exact exchange–correlation describing the band gap of strongly correlated Mott insulators.
energy has to cancel the classical Coulomb energy. 10 Otherwise, The flat-plane condition also leads to the conclusion that the ex-
the functional has a self-interaction error (SIE). act exchange–correlation functional cannot be a continuous func-
For many years, the SIE had been assumed to be the main tional of the electron density or the density matrix of the nonin-
systematic error in DFAs, related to the incorrect dissociation of teracting reference system everywhere. 27 . To reduce or eliminate
molecular ions, the underestimation of chemical reaction barri- the static correlation error, one has to use nonlocal functionals 28
ers and band gaps of molecules and bulk materials, the overesti-
mation of polymer polarizability, and many other failure of com-
monly used DFAs. 11,12 However, the development of two SIE-free (2.1.8) Savin : Warren Pickett said during a talk (Brisbane,
functionals, the Becke05 13 and the MCY2 14 functionals, changed 1996): “True, the density gives the potential, and this makes the
the understanding. 15 While these two exchange–correlation func- Hohenberg–Kohn theorem sound so empty, because the potential,
we know it anyhow”. We do not need to start with an unknown
function, ρ(r), when it is equivalent to using a known function of
the position r – namely, the external potential, v(r).
* Note that there is a (semi)local form for short-range interactions, e.g., δ (r1 − r2 ),
1
Z Z
1
Z (2.1.9) Trickey : The Pickett remark quoted by Savin is a
ρ(r1 )ρ(r2 )δ (r1 − r2 ) dr1 dr2 = ρ(r)2 dr
2 R3 R3 2 R3 paraphrase of the analysis that Per-Olov Löwdin had attributed
earlier to E. Bright Wilson 29 . The density cusps tell you the nu-
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clear charges, hence the external potential v, hence the Hamilto- The Hohenberg–Kohn theorem, 5 often thought of as the cor-
nian. Also see Krylov’s contribution (2.1.22) below. nerstone of DFT, is easy to prove (apart from some subtleties)
but perhaps not so easy to understand intuitively. Hohenberg and
(2.1.10) Yang : The Hohenberg–Kohn work established the
Kohn’s original formulation of DFT is therefore not only restric-
principles for describing a many-electron system from the reduced
tive in scope (in that it assumes v-representability) but may also
variable of its electron density and the Kohn–Sham work provided
Physical Chemistry Chemical Physics Accepted Manuscript
appear a little mysterious.
the formulation to use a noninteracting reference system to repre-
Levy’s constrained-search formulation 6 took the mystery out
sent the electron density of a many-electron system. These works
of DFT and brought clarity and generality to the field – a major
are the solid foundation of DFT. However, they do not lead to
step forward, indeed. Lieb’s convex formulation, 8 on the other
any systematic pathway to the approximation of the density func-
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hand, gave DFT beauty and elegance by identifying the density
tional; see contribution (2.1.8). The specific approximations for
functional with the Legendre transform (convex conjugate) of the
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the density functionals are the key to all applications.
ground-state energy, thereby placing DFT in a broader mathemat-
(2.1.11) Helgaker : I suppose the nontrivial result is that (for ical framework 32 .
a given number of electrons) the potential and density are dual It is an important and nontrivial result in DFT that the ensemble
variables – what you can calculate from one, you can calculate constrained-search functional and the Legendre-transform func-
from the other. In particular, we can calculate the energy directly tional are the same – they are merely complementary formula-
from the density, bypassing the potential. tions of the same thing. 8 Together, they constitute the solid foun-
dation of DFT.
(2.1.12) Yang : Indeed, the dual formulation of DFT is the
potential-functional theory (PFT). 30 PFT establishes two results: (2.1.16) Scheffler : I somehow disagree with the last sen-
the dual of the Hohenberg–Kohn theorem in terms of the external tence of contribution (2.1.13). Clearly, Kohn–Sham theory has
potential as the basic variable and the dual of the Kohn–Sham the- provided us with significant understanding, for polyatomic sys-
orem in terms of the potential of the noninteracting reference sys- tems, mostly for cases where the physics is largely governed by
tem. The first result provides a solution to the v-representability the independent-particle kinetic-energy operator (or its orbitals).
problem in the original Hohenberg–Kohn work. The second result However, in general, I would hesitate to call Kohn–Sham theory
provides the theoretical foundation for the optimized-effective- together with the known DFAs “(sufficiently) accurate”. A key sci-
potential approach for Kohn–Sham calculations with functionals entific problem is that the range of validity of the known DFAs is
of orbitals. unknown, and a reliable estimate of the accuracy and a systematic
convergence of the accuracy are not possible. Our own pragmatic
(2.1.13) Helgaker : I like to think of DFT in terms of approach is to perform calculations with different DFAs, and if
Legendre–Fenchel transforms. 8,31 In short, from the concavity the results are similar, we tend to accept them. Otherwise, we are
and continuity of the ground-state energy v 7→ E[v] as a function worried. And, if possible, we check final results by a higher-level
of the external potential v ∈ L3/2 (R3 ) + L∞ (R3 ) follows the exis- theory – by, for example, coupled-cluster theory.
tence of a universal density functional ρ 7→ F[ρ] as a function of
the electron density ρ ∈ L3 (R3 ) ∩ L1 (R3 ) such that (2.1.17) Kvaal : It is interesting to note, that in Lieb’s convex
formulation of exact DFT, the essence of which is succinctly de-
E[v] = inf (F[ρ] + (v | ρ)) ← HK variation principle 5 (4) scribed in contribution (2.1.13), does not rely in any way on the
ρ
classical Hohenberg–Kohn theorems to establish duality of ρ and
F[ρ] = sup (E[v] − (v | ρ)) ← Lieb variation principle 8 (5) v. Neither are the theorems necessary for the derivation of exact
v Kohn–Sham theory. While the original Hohenberg–Kohn theo-
R
where (v | ρ) = v(r)ρ(r)dr. Since E and F can be calculated from rems are now established rigorously, albeit with mild assumptions
each other, they contain the same information, only expressed in on the potential, 33 it is my opinion much easier to say that the
different ways. However, although the Lieb variation is a power- Legendre transform of E[v] is the essence and foundation of DFT,
ful tool for analysis and method development, it is not a practical from both a mathematical and a physical point of view. Lammert
tool for computation. Instead, the power of DFT derives from has pointed out that the Hohenberg–Kohn density-potential cor-
Kohn–Sham theory, making it possible to approximate F[ρ] (suf- respondence map is quite ill-behaved. 34 Nearby v-representable
ficiently) accurately and inexpensively for densities ρ of interest densities may have wildly different potentials, and thus funda-
to us by introducing orbitals. mental arguments that rely on, for example, some kind of differ-
entiation of v as a function of ρ are not useful, at least for exact
(2.1.14) Levy : In contribution (2.1.13), Helgaker states DFT. 34
that he prefers the Legendre-transform formulation. However,
(2.1.18) Laestadius : With recent development of unique-
it has been shown that the Legendre-transform formulation is
continuation from sets of measure zero, in particular by Gar-
equivalent to the ensemble constrained search. 8
rigue, 35 I regard the Hohenberg–Kohn theorem as rigorous, al-
(2.1.15) Helgaker : It is of course correct that the ensem- beit with some limitations. In particular, certain L p spaces need
ble constrained-search functional is identical to Lieb’s functional. to be consider for the potentials – for example, Theorem 30 in
With respect to the different formulations of DFT, my view is the ref. 33 is a Hohenberg–Kohn result with all previous gaps filled,
following. although it is not given for L3/2 + L∞ .
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Furthermore, comparing the situation with paramagnetic- cannot have a theory into another one.
current DFT, where the lack of a (corresponding) Hohenberg–
(2.1.22) Krylov : I first learned about the key ideas behind
Kohn theorem has been established by Capelle and Vignale, 36
DFT before its modern incarnation was developed. Back in the
it is striking that although (ρ, jp ) determines the nondegenerate
eighties, chemists were using the Xα method, which was re-
ground state, if degeneracies are allowed, then the level of degen-
garded by ab initio theorists as semiempirical and, therefore, in-
Physical Chemistry Chemical Physics Accepted Manuscript
eracy is not determined. 37 A given (ρ, jp ) can therefore be asso-
ferior to then-gold-standard – the full Hartree–Fock method. We
ciated with two different Hamiltonians (in fact, infinitely many)
were struggling to understand why an inferior method would give
that may have different numbers of degenerate ground states. (Of
more accurate results. I think the real insight was to understand
course, this doesn’t stop the constrained search, which remains
that the Wilson conjecture – the observation that the one-electron
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well defined.) In DFT, the extra layer of a Hohenberg–Kohn theo-
density contains all the information needed to reconstruct the
rem (not just the first part of a constrained search) rules out such
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many-body Hamiltonian (and, therefore, to find the exact solu-
situations. I view the Hohenberg–Kohn theorem as a gold reserve
tion of the Schrödinger equation) – provides a physical justifica-
– it is perhaps unexciting and just sits in the vault but is, on the
tion for the existence of a mapping between the density and the
other hand, good to have in certain extreme situations.
exact energy of the system. The Hohenberg–Kohn theorems in-
(2.1.19) Helgaker : Regarding the role of the Hohenberg– form us that this mapping is unique.
Kohn theorem in DFT, it is interesting to see what role it plays With such justification, one can approach the problem of find-
within the Legendre–Fenchel formulation of DFT. The condition ing this mapping in a completely different way – not by building
for a minimizing density in the Hohenberg–Kohn variation prin- approximations to the known exact solution (as done in the wave-
ciple as given in contribution (2.1.13) is −v ∈ ∂ F[ρ] where ∂ F[ρ] function theory), but by parameterizing an empirical representa-
is the subdifferential of F at ρ – that is, the collection of poten- tion of the mapping device, the functional. Most DFAs are built
tials with ground-state density ρ. Likewise, the condition for a upon mathematical representations of the functional grounded in
maximizing potential in the Lieb variation principle is ρ ∈ ∂ E[v], our physical understanding of what it should look like (based on
where the subdifferential of E at v is the collection of all ground- exact results for model systems), but one can envision finding the
state densities of v. In fact, the two conditions are equivalent: mapping without any such help from physics – for example, by
brute-force training of a neural network (machine learning). 38
E[v] = F[ρ] + (v|ρ) ⇐⇒ −v ∈ ∂ F[ρ] ⇐⇒ ρ ∈ ∂ E[v]. (6) One can, therefore, think of DFT as an empirical method that can
be made exact.
By the Hohenberg–Kohn theorem, the optimality condition of the
While the blind brute-force (e.g., via ML) discovery of the
Hohenberg–Kohn variation principle takes the form
density-energy mapping is, in principle, possible, it has impor-
tant limitations compared to physically motivated DFAs. First,
(
{−v + c | c ∈ R}, ρ is v-representable,
∂ F[ρ] = (7) without any constrains due to physics, such brute-force search
0,
/ ρ is not v-representable.
is going to be computationally wasteful. Second, having discov-
This uniqueness of the potential (up to an additive constant) is ered the mapping between energy and density, one still has no
not mission critical for DFT but tells us that there is a unique recipe for computing energy derivatives with respect to various
maximizing potential in the Lieb variation principle (if any). perturbations (i.e., properties), unless properties (or various en-
The optimality conditions in eqn (6) gives some additional in- ergy derivatives) were included in the training. In contrast, us-
sight: the ground-state energy E and the universal density func- ing a physically motivated form of the functional opens access to
tional F are functions whose subdifferential mappings (“func- properties (although the quality is not guaranteed, as illustrated
tional derivatives”) are each other’s inverses. Loosely speaking, by the developments of magnetic DFAs 39 ).
therefore, E and F may be obtained from each other by differen- (2.1.23) Helgaker : I am not so fond of the Wilson conjec-
tiation followed by inversion and integration. ture – it works only if we already know that the potential is a
Coulomb potential. It is a striking observation, but to some ex-
(2.1.20) Salahub : Savin’s answer in contribution (2.1.1)
tent it trivializes DFT. The Hohenberg–Kohn theorem makes no
to “what is DFT?” appeals to me because of its breadth. DFT
such assumptions regarding the potential.
appeals to different people for different reasons, from the joy of
pure theory, to the satisfying hard work of DFAs, to the romp (2.1.24) Jones : A fixation on exact energies appears to be so
of applications across disciplines (when it works), to the agony strong among chemists that it justifies any amount of data fitting,
when it doesn’t (appealing to masochists, but also affording the so reducing DFT to a “semiempirical” or “empirical” method. With
possibility of looping back for improvements). So “DFT” is like an their focus on extended systems, materials scientists know that
excellent marketing logo, as recognizable to scientists as the Nike new knowledge can result from DFT calculations, even if all the
logo is to the general public. Reasons for buying into DFT are calculated energies are wrong. See also contribution (2.2.23).
numerous and varied, as reflected in the sections of this paper.
(2.1.25) Ayers : Arguably, any electronic structure theory
(2.1.21) Fuentealba : The first time I heard about DFT was in method can be reformulated as a DFA by substituting its asso-
the eighties in Germany, and people called it “Density Functional ciated energy functional into the Legendre transform or its as-
Method”, because the theory is the quantum mechanics and one sociated wave-function ansatz into the constrained search. So
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Hartree–Fock may be legitimately considered a DFT (a gener- of an electronic system. This statement is at least misleading if
alized Kohn–Sham DFT). Is Hartree–Fock theory and its analy- not wrong because most DFT methods used in practice are Kohn–
sis therefore DFT? Clearly, many coupled-cluster and propagator Sham or generalized Kohn–Sham methods, which require orbitals
methods are also frequently analysed as DFT. I would not like to and thus one-electron wave functions to calculate crucial parts of
define DFT as “the sort of stuff that is done by density-functional the total energy.
Physical Chemistry Chemical Physics Accepted Manuscript
theorists” but some work that is marketed as DFT (cf. contribu-
(2.1.27) Gidopoulos : I believe the distinction in the lit-
tion (2.1.20)) is not presented in the context of the mathematical
erature between wave-function methods and DFT is slightly dif-
framework of DFT (cf. contribution (2.1.1)).
ferent. In my understanding, the distinction is not that in DFT
To me, only orbital-free DFT is unequivocally DFT; everything
the energy is actually calculated from the density, once we know
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else can also be fruitfully viewed from an alternative perspective.
the density, because the question remains how to find the density.
Indeed, some theoretical approaches and computational methods
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Rather, the distinction is that in DFT the solution to the electronic-
can legitimately be considered wave-function theories/methods,
structure problem is obtained by minimizing a total energy as a
density-matrix theories/methods, propagator theories/methods,
functional of the density, while in wave-function theory the solu-
and density-functional theories/methods. I do not wish to take a
tion is obtained by solving Schrödinger’s equation. So, calculating
hard line and proclaim that these types of theories/methods are
the energy from the density does not mean literally plugging the
not DFT because the philosophy (especially the emphasis on ex-
density into some orbital-free expression, but the process of mini-
plicitly defining and characterizing the functional that is being ap-
mization of the total-energy density functional to obtain the min-
proximated), traditions (especially the openness to pragmatic pa-
imum value, which is the total energy of the interacting system.
rameterization and approximation), and tools of DFT can be use-
ful even for theories/methods that are “not just DFT”. But other,
non-DFT, approaches could sometimes be even more useful. (2.1.28) Chattaraj : Any theory that applies density to un-
derstand a many-particle system, without using the exact wave
(2.1.26) Görling : While the electron density certainly is a function, can be termed as DFT. 47–49 According to Hohenberg–
key quantity in DFT, I feel that there is a too strong focus on it Kohn theorems, 5 DFT is a theory that legitimizes the use of the
– in particular, on the idea of getting the total energy or other density to calculate all possible properties. The Hohenberg–Kohn
information directly from the density. While this is the idea be- theorems are just existence theorems and do not provide any
hind certain flavours of orbital-free DFT, it is not the idea behind know-how for an explicit form of the energy as a functional of
the most commonly used DFT approaches, which are the Kohn– the density as well as functional forms of other properties.
Sham or generalized Kohn–Sham methods. For these methods, a
quite different view on DFT can be taken: To consider the elec- (2.1.29) Trickey : The foregoing discussion seems a bit
tron density as the quantity that enables one to associate the real parochial – for example, the identification in contribution (2.1.4)
electronic system with a model system that has the same ground- of DFT with “ground state”. That restriction seems to have been
state density, which makes it possible to describe the ground-state accepted by subsequent commentators in this section. But there
energy and other properties of the real system via the model sys- are several instances of what generically is a DFT. There is, for
tem, i.e., via its orbitals and eigenvalues. From the Kohn–Sham example, a well-developed classical DFT. Closer to the focus of
orbitals, traditionally, only the ‘noninteracting’ kinetic energy is this discussion (many-fermion systems), there is free-energy DFT
calculated exactly, while the exchange–correlation energy is ap- (also known as finite-temperature DFT) 50 . It inexorably involves
proximated by an explicit functional of the density. excited states. There has been progress on free-energy DFAs. 51–56
Another ensemble DFT is the Gross–Kohn–Oliveira (GOK) ap-
But this is just one strategy. It is possible to determine addi-
proach for excited states at T = 0 K (see other commentators be-
tionally other contributions to the energy from the orbitals – for
low).
example, parts of the exchange energy in hybrid methods – or
The common theme of these DFTs is the reduction of the inher-
even to calculate all contributions to the energy exactly from the
ent complexity of the direct description of a many-body system to
occupied orbitals, except the correlation energy. The latter can
the comparative simplicity of functionals of the density – either
then be approximated by orbital-dependent functionals. 40 In the
explicitly, or implicitly in terms of auxiliary functions such as or-
latter case, the density is not needed at all in the calculation of
bitals. The strategy, in the time-independent case at least, is to
the total DFT energy. If, furthermore, the orbitals are obtained via
obtain the relevant physics (hence also chemistry) by an appro-
the optimized-effective-potential (OEP) method 40–46 or within an
priate minimization procedure on a functional of the density itself
appropriate generalized Kohn–Sham approach, then DFT meth-
(whether it be pure-state or ensemble).
ods results that do not require at any point the calculation of the
density. The density is then only required in the underlying for- (2.1.30) Galli : In the Hohenberg–Kohn formulation, DFT is
malism. an exact theory of ground and excited states, entirely based on
I feel, that the perception of DFT has been somewhat blurred by the electron density. That is, the density determines uniquely the
a questionable statement that, one way or another, is frequently potential, hence both ground and excited state properties of the
found in textbooks and articles. This is the statement that DFT is system may in principle be derived. However there is no prac-
distinguished from wave-function methods by using the electron tical recipe on how to derive such potential and hence on how
density instead of a wave function to calculate the total energy to derive neither ground or excited state properties. The Kohn–
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