Plasmonic systems, such as metal nanoparticles, are widely used in different areas of application, going from biology to photovoltaics. The modeling of the optical response of such systems is of fundamental importance to analyze their behavior and to design new systems with required properties. When the characteristic sizes/distances reach a few nanometers, nonlocal and spill-out effects become relevant and conventional classical electrodynamics models are no more appropriate. Methods based on the Time-Dependent Density Functional Theory (TD-DFT) represent the current reference for the description of quantum effects. However, TD-DFT is based on knowledge of all occupied orbitals, whose calculation is computationally prohibitive to model large plasmonic systems of interest for applications. On the other hand, methods based on the orbital-free (OF) formulation of TD-DFT can scale linearly with the system size. In this Review, OF methods ranging from semiclassical models to the Quantum Hydrodynamic Theory will be derived from the linear response TD-DFT, so that the key approximations and properties of each method can be clearly highlighted. The accuracy of the various approximations will then be validated for the linear optical properties of jellium nanoparticles, the most relevant model system in plasmonics. OF methods can describe the collective excitations in plasmonic systems with great accuracy and without system-tuned parameters. The accuracy of these methods depends only on the accuracy of the (universal) kinetic energy functional of the ground-state electronic density. Current approximations and future development directions will also be indicated.
REFERENCES
1.
S. A.
Maier
, Plasmonics, Fundamentals and Applications
(Springer
, 2007
).2.
S.
Kawata
, Y.
Inouye
, and P.
Verma
, “Plasmonics for near-field nano-imaging and superlensing
,” Nat. Photonics
3
, 388
(2009
).3.
Modern Plasmonics
, Handbook of Surface Science Vol. 4, edited by A. A.
Maradudin
, J. R.
Sambles
, and W. L.
Barnes
(Elsevier
, Amsterdam
, 2014
).4.
F.
Della Sala
and S.
D’Agostino
, Handbook of Molecular Plasmonics
(CRC Press
, 2013
).5.
H. A.
Atwater
and A.
Polman
, “Plasmonics for improved photovoltaic devices
,” Nat. Mater.
9
, 205
–213
(2010
).6.
A. G.
Brolo
, “Plasmonics for future biosensors
,” Nat. Photonics
6
, 709
–713
(2012
).7.
M. L.
Brongersma
, N. J.
Halas
, and P.
Nordlander
, “Plasmon-induced hot carrier science and technology
,” Nat. Nanotechnol.
10
, 25
(2015
).8.
G. V.
Naik
, V. M.
Shalaev
, and A.
Boltasseva
, “Alternative plasmonic materials: Beyond gold and silver
,” Adv. Mater.
25
, 3264
–3294
(2013
).9.
A. N.
Grigorenko
, M.
Polini
, and K. S.
Novoselov
, “Graphene plasmonics
,” Nat. Photonics
6
, 749
–758
(2012
).10.
A.
Agrawal
, S. H.
Cho
, O.
Zandi
, S.
Ghosh
, R. W.
Johns
, and D. J.
Milliron
, “Localized surface plasmon resonance in semiconductor nanocrystals
,” Chem. Rev.
118
, 3121
–3207
(2018
).11.
X.
Liu
and M. T.
Swihart
, “Heavily-doped colloidal semiconductor and metal oxide nanocrystals: An emerging new class of plasmonic nanomaterials
,” Chem. Soc. Rev.
43
, 3908
–3920
(2014
).12.
U.
Guler
, A.
Boltasseva
, and V. M.
Shalaev
, “Refractory plasmonics
,” Science
344
, 263
–264
(2014
).13.
14.
L.
Novotny
and B.
Hecht
, Principles of Nano-Optics
(Cambridge University Press
, 2012
).15.
M. I.
Mishchenko
, L. D.
Travis
, and A. A.
Lacis
, Scattering, Absorption, and Emission of Light by Small Particles
(Cambridge University Press
, 2002
).16.
W.
Zhu
, R.
Esteban
, A. G.
Borisov
, J. J.
Baumberg
, P.
Nordlander
, H. J.
Lezec
, J.
Aizpurua
, and K. B.
Crozier
, “Quantum mechanical effects in plasmonic structures with subnanometre gaps
,” Nat. Commun.
7
, 11495
(2016
).17.
R.
Esteban
, A. G.
Borisov
, P.
Nordlander
, and J.
Aizpurua
, “Bridging quantum and classical plasmonics with a quantum-corrected model
,” Nat. Commun.
3
, 825
(2012
).18.
Y.
Luo
, A. I.
Fernandez-Dominguez
, A.
Wiener
, S. A.
Maier
, and J. B.
Pendry
, “Surface plasmons and nonlocality: A simple model
,” Phys. Rev. Lett.
111
, 093901
(2013
).19.
W.
Yan
, M.
Wubs
, and N.
Asger Mortensen
, “Projected dipole model for quantum plasmonics
,” Phys. Rev. Lett.
115
, 137403
(2015
).20.
T.
Christensen
, W.
Yan
, A.-P.
Jauho
, M.
Soljačić
, and N. A.
Mortensen
, “Quantum corrections in nanoplasmonics: Shape, scale, and material
,” Phys. Rev. Lett.
118
, 157402
(2017
).21.
Y.
Yang
, D.
Zhu
, W.
Yan
, A.
Agarwal
, M.
Zheng
, J. D.
Joannopoulos
, P.
Lalanne
, T.
Christensen
, K. K.
Berggren
, and M.
Soljačić
, “A general theoretical and experimental framework for nanoscale electromagnetism
,” Nature
576
, 248
–252
(2019
).22.
N. A.
Mortensen
, “Mesoscopic electrodynamics at metal surfaces—From quantum-corrected hydrodynamics to microscopic surface-response formalism
,” Nanophotonics
10
, 2563
–2616
(2021
).23.
X.
Chen
, P.
Liu
, and L.
Jensen
, “Atomistic electrodynamics simulations of plasmonic nanoparticles
,” J. Phys. D: Appl. Phys.
52
, 363002
(2019
).24.
L.
Bonatti
, G.
Gil
, T.
Giovannini
, S.
Corni
, and C.
Cappelli
, “Plasmonic resonances of metal nanoparticles: Atomistic vs continuum approaches
,” Front. Chem.
8
, 340
(2020
).25.
L. L.
Jensen
and L.
Jensen
, “Electrostatic interaction model for the calculation of the polarizability of large noble metal nanoclusters
,” J. Phys. Chem. C
112
, 15697
–15703
(2008
).26.
L. L.
Jensen
and L.
Jensen
, “Atomistic electrodynamics model for optical properties of silver nanoclusters
,” J. Phys. Chem. C
113
, 15182
–15190
(2009
).27.
A.
Mayer
, A. L.
González
, C. M.
Aikens
, and G. C.
Schatz
, “A charge–dipole interaction model for the frequency-dependent polarizability of silver clusters
,” Nanotechnology
20
, 195204
(2009
).28.
S. M.
Morton
and L.
Jensen
, “A discrete interaction model/quantum mechanical method to describe the interaction of metal nanoparticles and molecular absorption
,” J. Chem. Phys.
135
, 134103
(2011
).29.
X.
Chen
, J. E.
Moore
, M.
Zekarias
, and L.
Jensen
, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime
,” Nat. Commun.
6
, 8921
(2015
).30.
J.
Lim
, S.
Kang
, J.
Kim
, W. Y.
Kim
, and S.
Ryu
, “Non-empirical atomistic dipole-interaction-model for quantum plasmon simulation of nanoparticles
,” Sci. Rep.
7
, 15775
(2017
).31.
V. I.
Zakomirnyi
, Z.
Rinkevicius
, G. V.
Baryshnikov
, L. K.
Sørensen
, and H.
Ågren
, “Extended discrete interaction model: Plasmonic excitations of silver nanoparticles
,” J. Phys. Chem. C
123
, 28867
–28880
(2019
).32.
T.
Giovannini
, M.
Rosa
, S.
Corni
, and C.
Cappelli
, “A classical picture of subnanometer junctions: An atomistic Drude approach to nanoplasmonics
,” Nanoscale
11
, 6004
–6015
(2019
).33.
F.
Bloch
, “Bremsvermögen von atomen mit mehreren elektronen
,” Z. Phys.
81
, 363
–376
(1933
).34.
H.
Jensen
, “Eigenschwingungen eines Fermi-gases und anwendung auf die Blochsche bremsformel für schnelle teilchen
,” Z. Phys.
106
, 620
–632
(1937
).35.
R. H.
Ritchie
and R. E.
Wilems
, “Photon-plasmon interaction in a nonuniform electron gas. I
,” Phys. Rev.
178
, 372
–381
(1969
).36.
J.
Heinrichs
, “Hydrodynamic theory of surface-plasmon dispersion
,” Phys. Rev. B
7
, 3487
–3500
(1973
).37.
S. C.
Ying
, “Hydrodynamic response of inhomogeneous metallic systems
,” Nuovo Cimento B
23
, 270
–281
(1974
).38.
A.
Eguiluz
, “Density response function and the dynamic structure factor of thin metal films: Nonlocal effects
,” Phys. Rev. B
19
, 1689
–1705
(1979
).39.
G.
Barton
, “Some surface effects in the hydrodynamic model of metals
,” Rep. Prog. Phys.
42
, 963
–1016
(1979
).40.
R.
Ruppin
, “Optical properties of a plasma sphere
,” Phys. Rev. Lett.
31
, 1434
–1437
(1973
).41.
B. B.
Dasgupta
and R.
Fuchs
, “Polarizability of a small sphere including nonlocal effects
,” Phys. Rev. B
24
, 554
–561
(1981
).42.
C.
Ciracì
, R. T.
Hill
, J. J.
Mock
, Y.
Urzhumov
, A. I.
Fernández-Domínguez
, S. A.
Maier
, J. B.
Pendry
, A.
Chilkoti
, and D. R.
Smith
, “Probing the ultimate limits of plasmonic enhancement
,” Science
337
, 1072
–1074
(2012
).43.
K. R.
Hiremath
, L.
Zschiedrich
, and F.
Schmidt
, “Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using nédélec finite elements
,” J. Comput. Phys.
231
, 5890
–5896
(2012
).44.
J. M.
McMahon
, S. K.
Gray
, and G. C.
Schatz
, “Nonlocal optical response of metal nanostructures with arbitrary shape
,” Phys. Rev. Lett.
103
, 097403
(2009
).45.
C.
David
and F. J.
García de Abajo
, “Spatial nonlocality in the optical response of metal nanoparticles
,” J. Phys. Chem. C
115
, 19470
–19475
(2011
).46.
G.
Toscano
, S.
Raza
, A.-P.
Jauho
, N. A.
Mortensen
, and M.
Wubs
, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response
,” Opt. Express
20
, 4176
–4188
(2012
).47.
L.
Stella
, P.
Zhang
, F. J.
García-Vidal
, A.
Rubio
, and P.
García-González
, “Performance of nonlocal optics when applied to plasmonic nanostructures
,” J. Phys. Chem. C
117
, 8941
–8949
(2013
).48.
S.
Raza
, N.
Stenger
, S.
Kadkhodazadeh
, S. V.
Fischer
, N.
Kostesha
, A.-P.
Jauho
, A.
Burrows
, M.
Wubs
, and N. A.
Mortensen
, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS
,” Nanophotonics
2
, 131
–138
(2013
).49.
T. V.
Teperik
, P.
Nordlander
, J.
Aizpurua
, and A. G.
Borisov
, “Quantum effects and nonlocality in strongly coupled plasmonic nanowire dimers
,” Opt. Express
21
, 27306
–27325
(2013
).50.
T.
Christensen
, W.
Yan
, S.
Raza
, A.-P.
Jauho
, N. A.
Mortensen
, and M.
Wubs
, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms
,” ACS Nano
8
, 1745
–1758
(2014
).51.
S.
Raza
, S. I.
Bozhevolnyi
, M.
Wubs
, and N.
Asger Mortensen
, “Nonlocal optical response in metallic nanostructures
,” J. Phys.: Condens. Matter
27
, 183204
(2015
).52.
A.
Moradi
, “Quantum nonlocal effects on optical properties of spherical nanoparticles
,” Phys. Plasmas
22
, 022119
(2015
).53.
M.
Kupresak
, X.
Zheng
, G. A. E.
Vandenbosch
, and V. V.
Moshchalkov
, “Comparison of hydrodynamic models for the electromagnetic nonlocal response of nanoparticles
,” Adv. Theory Simul.
1
, 1800076
(2018
).54.
M.
Kupresak
, X.
Zheng
, G. A. E.
Vandenbosch
, and V. V.
Moshchalkov
, “Appropriate nonlocal hydrodynamic models for the characterization of deep-nanometer scale plasmonic scatterers
,” Adv. Theory Simul.
3
, 1900172
(2020
).55.
C.
Ciracì
, J. B.
Pendry
, and D. R.
Smith
, “Hydrodynamic model for plasmonics: A macroscopic approach to a microscopic problem
,” Chem. Phys. Chem.
14
, 1109
–1116
(2013
).56.
J. A.
Scholl
, A. L.
Koh
, and J. A.
Dionne
, “Quantum plasmon resonances of individual metallic nanoparticles
,” Nature
483
, 421
–427
(2012
).57.
R.
Carmina Monreal
, T. J.
Antosiewicz
, and S.
Peter Apell
, “Competition between surface screening and size quantization for surface plasmons in nanoparticles
,” New J. Phys.
15
, 083044
(2013
).58.
G.
Toscano
, J.
Straubel
, A.
Kwiatkowski
, C.
Rockstuhl
, F.
Evers
, H.
Xu
, N.
Asger Mortensen
, and M.
Wubs
, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics
,” Nat. Commun.
6
, 7132
(2015
).59.
C.
Ciracì
and F.
Della Sala
, “Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail
,” Phys. Rev. B
93
, 205405
(2016
).60.
G.
Manfredi
, “How to model quantum plasma
,” Fields Inst. Commun.
46
, 263
(2005
).61.
Complex Plasmas: Scientific Challenges and Technological Opportunities
, Springer Series on Atomic, Optical, and Plasma Physics Vol. 82, edited by M.
Bonitz
, J.
Lopez
, K.
Becker
, and H.
Thomsen
(Springer International Publishing
, Cham
, 2014
).62.
P. K.
Shukla
and B.
Eliasson
, “Novel attractive force between ions in quantum plasmas
,” Phys. Rev. Lett.
108
, 165007
(2012
).63.
D.
Michta
, F.
Graziani
, and M.
Bonitz
, “Quantum hydrodynamics for plasmas - a Thomas-Fermi theory perspective
,” Contrib. Plasma Phys.
55
, 437
–443
(2015
).64.
Z. A.
Moldabekov
, M.
Bonitz
, and T. S.
Ramazanov
, “Theoretical foundations of quantum hydrodynamics for plasmas
,” Phys. Plasmas
25
, 031903
(2018
).65.
M.
Bonitz
, Z. A.
Moldabekov
, and T. S.
Ramazanov
, “Quantum hydrodynamics for plasmas—Quo vadis?
,” Phys. Plasmas
26
, 090601
(2019
).66.
G.
Manfredi
, P.-A.
Hervieux
, and J.
Hurst
, “Fluid descriptions of quantum plasmas
,” Rev. Mod. Plasma Phys.
5
, 7
(2021
).67.
E.
Zaremba
and H. C.
Tso
, “Thomas-Fermi-Dirac-von Weizsäcker hydrodynamics in parabolic wells
,” Phys. Rev. B
49
, 8147
–8162
(1994
).68.
B. P.
van Zyl
and E.
Zaremba
, “Thomas-Fermi-Dirac-von Weizsäcker hydrodynamics in laterally modulated electronic systems
,” Phys. Rev. B
59
, 2079
–2094
(1999
).69.
E.
Zaremba
and H. C.
Tso
, “Hydrodynamics in the Thomas-Fermi-Dirac-von Weizsäcker approximation
,” in Electronic Density Functional Theory
, edited by J.
Dobson
, G.
Vignale
, and M.
Das
(Springer
, Boston
, 1998
), pp. 227
–242
.70.
A.
Banerjee
and M. K.
Harbola
, “Hydrodynamical approach to collective oscillations in metal clusters
,” Phys. Lett. A
372
, 2881
–2886
(2008
).71.
K.
Ding
and C. T.
Chan
, “Plasmonic modes of polygonal rods calculated using a quantum hydrodynamics method
,” Phys. Rev. B
96
, 125134
(2017
).72.
K.
Ding
and C. T.
Chan
, “Optical forces, torques, and force densities calculated at a microscopic level using a self-consistent hydrodynamics method
,” Phys. Rev. B
97
, 155118
(2018
).73.
M.
Khalid
, F.
Della Sala
, and C.
Ciracì
, “Optical properties of plasmonic core-shell nanomatryoshkas: A quantum hydrodynamic analysis
,” Opt. Express
26
, 17322
(2018
).74.
M.
Khalid
and C.
Ciracì
, “Numerical analysis of nonlocal optical response of metallic nanoshells
,” Photonics
6
, 39
(2019
).75.
H. M.
Baghramyan
, F.
Della Sala
, and C.
Ciracì
, “Laplacian-level quantum hydrodynamic theory for plasmonics
,” Phys. Rev. X
11
, 011049
(2021
).76.
Y.-Y.
Zhang
, S.-B.
An
, Y.-H.
Song
, N.
Kang
, Z. L.
Mišković
, and Y.-N.
Wang
, “Plasmon excitation in metal slab by fast point charge: The role of additional boundary conditions in quantum hydrodynamic model
,” Phys. Plasmas
21
, 102114
(2014
).77.
W.
Yan
, “Hydrodynamic theory for quantum plasmonics: Linear-response dynamics of the inhomogeneous electron gas
,” Phys. Rev. B
91
, 115416
(2015
).78.
D. I.
Palade
, “Multiple surface plasmons in an unbounded quantum plasma half-space
,” Phys. Plasmas
23
, 074504
(2016
).79.
Y.
Zhang
, M. X.
Gao
, and B.
Guo
, “Surface plasmon dispersion relation at an interface between thin metal film and dielectric using a quantum hydrodynamic model
,” Opt. Commun.
402
, 326
–329
(2017
).80.
C.
Ciracì
, “Current-dependent potential for nonlocal absorption in quantum hydrodynamic theory
,” Phys. Rev. B
95
, 245434
(2017
).81.
C.
Ciracì
, R.
Jurga
, M.
Khalid
, and F.
Della Sala
, “Plasmonic quantum effects on single-emitter strong coupling
,” Nanophotonics
8
, 1821
–1833
(2019
).82.
D.
de Ceglia
, M.
Scalora
, M. A.
Vincenti
, S.
Campione
, K.
Kelley
, E. L.
Runnerstrom
, J.-P.
Maria
, G. A.
Keeler
, and T. S.
Luk
, “Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms
,” Sci. Rep.
8
, 9335
(2018
).83.
A.
Diaw
and M. S.
Murillo
, “A viscous quantum hydrodynamics model based on dynamic density functional theory
,” Sci. Rep.
7
, 15352
(2017
).84.
A.
Bergara
, J. M.
Pitarke
, and R. H.
Ritchie
, “Nonlinear quantum hydrodynamical model of the electron gas
,” Nucl. Instrum. Methods Phys. Res., Sect. B
115
, 70
–74
(1996
).85.
N.
Crouseilles
, P.-A.
Hervieux
, and G.
Manfredi
, “Quantum hydrodynamic model for the nonlinear electron dynamics in thin metal films
,” Phys. Rev. B
78
, 155412
(2008
).86.
F.
De Luca
and C.
Ciracì
, “Difference-frequency generation in plasmonic nanostructures: A parameter-free hydrodynamic description
,” J. Opt. Soc. Am. B
36
, 1979
–1986
(2019
).87.
M.
Khalid
and C.
Ciracì
, “Enhancing second-harmonic generation with electron spill-out at metallic surfaces
,” Commun. Phys.
3
, 214
(2020
).88.
A.
Domps
, P.-G.
Reinhard
, and E.
Suraud
, “Time-dependent Thomas-Fermi approach for electron dynamics in metal clusters
,” Phys. Rev. Lett.
80
, 5520
(1998
).89.
M. K.
Harbola
, “Differential virial theorem and quantum fluid dynamics
,” Phys. Rev. A
58
, 1779
–1782
(1998
).90.
D.
Neuhauser
, S.
Pistinner
, A.
Coomar
, X.
Zhang
, and G.
Lu
, “Dynamic kinetic energy potential for orbital-free density functional theory
,” J. Chem. Phys.
134
, 144101
(2011
).91.
D. I.
Palade
, “Nonlocal orbital-free kinetic pressure tensors for the Fermi gas
,” Phys. Rev. B
98
, 245401
(2018
).92.
A. J.
White
, O.
Certik
, Y. H.
Ding
, S. X.
Hu
, and L. A.
Collins
, “Time-dependent orbital-free density functional theory for electronic stopping power: Comparison to the mermin-Kohn-Sham theory at high temperatures
,” Phys. Rev. B
98
, 144302
(2018
).93.
Y. H.
Ding
, A. J.
White
, S. X.
Hu
, O.
Certik
, and L. A.
Collins
, “Ab initio studies on the stopping power of warm dense matter with time-dependent orbital-free density functional theory
,” Phys. Rev. Lett.
121
, 145001
(2018
).94.
K.
Jiang
and M.
Pavanello
, “Time-dependent orbital-free density functional theory: Background and Pauli kernel approximations
,” Phys. Rev. B
103
, 245102
(2021
).95.
R. G.
Parr
and W.
Yang
, Density-Functional Theory of Atoms and Molecules
(Oxford University Press
, USA
, 1994
).96.
Y. A.
Wang
and E. A.
Carter
, “Orbital-free kinetic-energy density functional theory
,” in Theoretical Methods in Condensed Phase Chemistry
, edited by S. D.
Schwartz
(Kluwer Academic Publishers
, Dordrecht
, 2002
), Vol. 5, pp. 117
–184
.97.
V.
Gavini
, K.
Bhattacharya
, and M.
Ortiz
, “Quasi-continuum orbital-free density-functional theory: A route to multi-million atom non-periodic DFT calculation
,” J. Mech. Phys. Solids
55
, 697
–718
(2007
).98.
J.
Xia
, C.
Huang
, I.
Shin
, and E. A.
Carter
, “Can orbital-free density functional theory simulate molecules?
,” J. Chem. Phys.
136
, 084102
(2012
).99.
T. A.
Wesolowski
and Y. A.
Wang
, Recent Progress in Orbital-free Density Functional Theory
(World Scientific
, 2013
).100.
W. C.
Witt
, B. G.
del Rio
, J. M.
Dieterich
, and E. A.
Carter
, “Orbital-free density functional theory for materials research
,” J. Mater. Res.
33
, 777
–795
(2018
).101.
P.
Hohenberg
and W.
Kohn
, “Inhomogeneous electron gas
,” Phys. Rev.
136
, B864
–B871
(1964
).102.
M.
Pi
, M.
Barranco
, J.
Nemeth
, C.
Ngô
, and E.
Tomasi
, “Time-dependent Thomas-Fermi approach to nuclear monopole oscillations
,” Phys. Lett. B
166
, 1
–4
(1986
).103.
J.
Wu
, R.
Feng
, and W.
Nörenberg
, “Effective Schrödinger equation for nuclear fluid dynamics
,” Phys. Lett. B
209
, 430
–433
(1988
).104.
T.
Takeuci
and K.
Yabana
, “Numerical scheme for a nonlinear optical response of a metallic nanostructure: Quantum hydrodynamic theory solved by adopting an effective Schrödinger equation
,” Opt. Express
30
, 11572
–11587
(2022
).105.
H.
Xiang
, X.
Zhang
, D.
Neuhauser
, and G.
Lu
, “Size-dependent plasmonic resonances from large-scale quantum simulations
,” J. Phys. Chem. Lett.
5
, 1163
–1169
(2014
).106.
H.
Xiang
, M.
Zhang
, X.
Zhang
, and G.
Lu
, “Understanding quantum plasmonics from time-dependent orbital-free density functional theory
,” J. Phys. Chem. C
120
, 14330
–14336
(2016
).107.
X.
Zhang
, H.
Xiang
, M.
Zhang
, and G.
Lu
, “Plasmonic resonances of nanoparticles from large-scale quantum mechanical simulations
,” Int. J. Mod. Phys. B
31
, 1740003
(2017
).108.
H.
Xiang
, Z.
Wang
, L.
Xu
, X.
Zhang
, and G.
Lu
, “Quantum plasmonics in nanorods: A time-dependent orbital-free density functional theory study with thousands of atoms
,” J. Phys. Chem. C
124
, 945
–951
(2020
).109.
K.
Jiang
, X.
Shao
, and M.
Pavanello
, “Nonlocal and nonadiabatic pauli potential for time-dependent orbital-free density functional theory
,” Phys. Rev. B
104
, 235110
(2021
).110.
X.
Shao
, K.
Jiang
, W.
Mi
, A.
Genova
, and M.
Pavanello
, “DFTpy: An efficient and object-oriented platform for orbital-free DFT simulations
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
11
, e1482
(2021
).111.
E.
Runge
and E. K. U.
Gross
, “Density-functional theory for time-dependent systems
,” Phys. Rev. Lett.
52
, 997
–1000
(1984
).112.
C. A.
Ullrich
, Time-Dependent Density-Functional Theory: Concepts and Applications
, Oxford Graduate Texts (Oxford University Press
, Oxford; New York
, 2012
).113.
W.
Kohn
and L. J.
Sham
, “Self-consistent equations including exchange and correlation effects
,” Phys. Rev.
140
, A1133
(1965
).114.
R. M.
Dreizler
and E. K. U.
Gross
, Density Functional Theory
(Springer
, Heidelberg
, 1990
).115.
K.
Yabana
and G. F.
Bertsch
, “Time-dependent local-density approximation in real time
,” Phys. Rev. B
54
, 4484
–4487
(1996
).116.
C.
Lian
, M.
Guan
, S.
Hu
, J.
Zhang
, and S.
Meng
, “Photoexcitation in solids: First-principles quantum simulations by real-time TDDFT
,” Adv. Theory Simul.
1
, 1800055
(2018
).117.
X.
Li
, N.
Govind
, C.
Isborn
, A. E.
DePrince
, and K.
Lopata
, “Real-time time-dependent electronic structure theory
,” Chem. Rev.
120
, 9951
–9993
(2020
).118.
M. E.
Casida
, “Time-dependent density functional response theory for molecules
,” in Recent Advances in Density Functional Methods
, edited by D. P.
Chong
(World Scientific
, Singapore
, 1995
), Vol. 1, pp. 155
–192
.119.
D. A.
Strubbe
, L.
Lehtovaara
, A.
Rubio
, M. A. L.
Marques
, and S. G.
Louie
, “Response functions in TDDFT: Concepts and implementation
,” in Fundamentals of Time-Dependent Density Functional Theory
(Springer
, 2012
), pp. 139
–166
.120.
B.
Walker
, A. M.
Saitta
, R.
Gebauer
, and S.
Baroni
, “Efficient approach to time-dependent density-functional perturbation theory for optical spectroscopy
,” Phys. Rev. Lett.
96
, 113001
(2006
).121.
D.
Rocca
, R.
Gebauer
, Y.
Saad
, and S.
Baroni
, “Turbo charging time-dependent density-functional theory with Lanczos chains
,” J. Chem. Phys.
128
, 154105
(2008
).122.
F.
Wu
, W.
Liu
, Y.
Zhang
, and Z.
Li
, “Linear-scaling time-dependent density functional theory based on the idea of from fragments to molecule
,” J. Chem. Theory Comput.
7
, 3643
–3660
(2011
).123.
T. J.
Zuehlsdorff
, N. D. M.
Hine
, J. S.
Spencer
, N. M.
Harrison
, D. J.
Riley
, and P. D.
Haynes
, “Linear-scaling time-dependent density-functional theory in the linear response formalism
,” J. Chem. Phys.
139
, 064104
(2013
).124.
O.
Baseggio
, G.
Fronzoni
, and M.
Stener
, “A new time dependent density functional algorithm for large systems and plasmons in metal clusters
,” J. Chem. Phys.
143
, 024106
(2015
).125.
J.
Brabec
, L.
Lin
, M.
Shao
, N.
Govind
, C.
Yang
, Y.
Saad
, and E. G.
Ng
, “Efficient algorithms for estimating the absorption spectrum within linear response TDDFT
,” J. Chem. Theory Comput.
11
, 5197
(2015
).126.
Y.
Gao
, D.
Neuhauser
, R.
Baer
, and E.
Rabani
, “Sublinear scaling for time-dependent stochastic density functional theory
,” J. Chem. Phys.
142
, 034106
(2015
).127.
O.
Baseggio
, M.
De Vetta
, G.
Fronzoni
, M.
Stener
, L.
Sementa
, A.
Fortunelli
, and A.
Calzolari
, “Photoabsorption of icosahedral noble metal clusters: An efficient TDDFT approach to large-scale systems
,” J. Phys. Chem. C
120
, 12773
–12782
(2016
).128.
G.
Giannone
and F.
Della Sala
, “Minimal auxiliary basis set for time-dependent density functional theory and comparison with tight-binding approximations: Application to silver nanoparticles
,” J. Chem. Phys.
153
, 084110
(2020
).129.
J. C.
Idrobo
and S. T.
Pantelides
, “Origin of bulklike optical response in noble-metal Ag and Au nanoparticles
,” Phys. Rev. B
82
, 085420
(2010
).130.
C. M.
Aikens
, S.
Li
, and G. C.
Schatz
, “From discrete electronic states to plasmons: TDDFT optical absorption properties of Ag n (n = 10, 20, 35, 56, 84, 120) tetrahedral clusters
,” J. Phys. Chem. C
112
, 11272
–11279
(2008
).131.
J.-O.
Joswig
, L. O.
Tunturivuori
, and R. M.
Nieminen
, “Photoabsorption in sodium clusters on the basis of time-dependent density-functional theory
,” J. Chem. Phys.
128
, 014707
(2008
).132.
H.-C.
Weissker
and C.
Mottet
, “Optical properties of pure and core-shell noble-metal nanoclusters from TDDFT: The influence of the atomic structure
,” Phys. Rev. B
84
, 165443
(2011
).133.
G.-T.
Bae
and C. M.
Aikens
, “Time-dependent density functional theory studies of optical properties of Ag nanoparticles: Octahedra, truncated octahedra, and icosahedra
,” J. Phys. Chem. C
116
, 10356
(2012
).134.
X.
López-Lozano
, H.
Barron
, C.
Mottet
, and H.-C.
Weissker
, “Aspect-ratio- and size-dependent emergence of the surface-plasmon resonance in gold nanorods-an ab initio TDDFT study
,” Phys. Chem. Chem. Phys.
16
, 1820
–1823
(2014
).135.
G.
Barcaro
, L.
Sementa
, A.
Fortunelli
, and M.
Stener
, “Optical properties of silver nanoshells from time-dependent density functional theory calculations
,” J. Phys. Chem. C
118
, 12450
(2014
).136.
M.
Kuisma
, A.
Sakko
, T. P.
Rossi
, A. H.
Larsen
, J.
Enkovaara
, L.
Lehtovaara
, and T. T.
Rantala
, “Localized surface plasmon resonance in silver nanoparticles: Atomistic first-principles time-dependent density-functional theory calculations
,” Phys. Rev. B
91
, 115431
(2015
).137.
H.-C.
Weissker
and X.
López-Lozano
, “Surface plasmons in quantum-sized noble-metal clusters: TDDFT quantum calculations and the classical picture of charge oscillations
,” Phys. Chem. Chem. Phys.
17
, 28379
(2015
).138.
P.
Koval
, F.
Marchesin
, D.
Foerster
, and D.
Sánchez-Portal
, “Optical response of silver clusters and their hollow shells from linear-response TDDFT
,” J. Phys.: Condens. Matter
28
, 214001
(2016
).139.
T. P.
Rossi
, M.
Kuisma
, M. J.
Puska
, R. M.
Nieminen
, and P.
Erhart
, “Kohn–Sham decomposition in real-time time-dependent density-functional theory: An efficient tool for analyzing plasmonic excitations
,” J. Chem. Theory Comput.
13
, 4779
(2017
).140.
R.
Schira
and F.
Rabilloud
, “Localized surface plasmon resonance in free silver nanoclusters Agn, n = 20–147
,” J. Phys. Chem. C
123
, 6205
(2019
).141.
R.
Sinha-Roy
, P.
García-González
, and H.-C.
Weissker
, “How metallic are noble-metal clusters? Static screening and polarizability in quantum-sized silver and gold nanoparticles
,” Nanoscale
12
, 4452
(2020
).142.
G.
Piccini
, R. W. A.
Havenith
, R.
Broer
, and M.
Stener
, “Gold nanowires: A time-dependent density functional assessment of plasmonic behavior
,” J. Phys. Chem. C
117
, 17196
(2013
).143.
S.
Malola
, L.
Lehtovaara
, J.
Enkovaara
, and H.
Häkkinen
, “Birth of the localized surface plasmon resonance in monolayer-protected gold nanoclusters
,” ACS Nano
7
, 10263
(2013
).144.
K.
Iida
, M.
Noda
, K.
Ishimura
, and K.
Nobusada
, “First-principles computational visualization of localized surface plasmon resonance in gold nanoclusters
,” J. Phys. Chem. A
118
, 11317
(2014
).145.
R. W.
Burgess
and V. J.
Keast
, “TDDFT study of the optical absorption spectra of bare gold clusters
,” J. Phys. Chem. C
118
, 3194
(2014
).146.
K.
Iida
, M.
Noda
, and K.
Nobusada
, “Interface electronic properties between a gold core and thiolate ligands: Effects on an optical absorption spectrum in Au133(SPh-tBu)52
,” J. Phys. Chem. C
120
, 2753
(2016
).147.
A.
Varas
, P.
García-González
, J.
Feist
, F. J.
García-Vidal
, and A.
Rubio
, “Quantum plasmonics: From jellium models to ab initio calculations
,” Nanophotonics
5
, 409
–426
(2016
).148.
S.
Hernandez
, Y.
Xia
, V.
Vlček
, R.
Boutelle
, R.
Baer
, E.
Rabani
, and D.
Neuhauser
, “First-principles spectra of au nanoparticles: From quantum to classical absorption
,” Mol. Phys.
116
, 2506
(2018
).149.
G.-T.
Bae
and C. M.
Aikens
, “TDDFT and CIS studies of optical properties of dimers of silver tetrahedra
,” J. Phys. Chem. A
116
, 8260
–8269
(2012
).150.
P.
Zhang
, J.
Feist
, A.
Rubio
, P.
García-González
, and F. J.
García-Vidal
, “Ab initio nanoplasmonics: The impact of atomic structure
,” Phys. Rev. B
90
, 161407
(2014
).151.
A.
Varas
, P.
García-González
, F. J.
García-Vidal
, and A.
Rubio
, “Anisotropy effects on the plasmonic response of nanoparticle dimers
,” J. Phys. Chem. Lett.
6
, 1891
(2015
).152.
M.
Barbry
, P.
Koval
, F.
Marchesin
, R.
Esteban
, A. G.
Borisov
, J.
Aizpurua
, and D.
Sánchez-Portal
, “Atomistic near-field nanoplasmonics: Reaching atomic-scale resolution in nanooptics
,” Nano Lett.
15
, 3410
–3419
(2015
).153.
M.
Urbieta
, M.
Barbry
, Y.
Zhang
, P.
Koval
, D.
Sánchez-Portal
, N.
Zabala
, and J.
Aizpurua
, “Atomic-scale lightning rod effect in plasmonic picocavities: A classical view to a quantum effect
,” ACS Nano
12
, 585
–595
(2018
).154.
P.
Liu
, D. V.
Chulhai
, and L.
Jensen
, “Atomistic characterization of plasmonic dimers in the quantum size regime
,” J. Phys. Chem. C
123
, 13900
–13907
(2019
).155.
F.
Della Sala
, M.
Pezzolla
, S.
D’Agostino
, and E.
Fabiano
, “Ab initio plasmonics of externally doped silicon nanocrystals
,” ACS Photonics
6
, 1474
–1484
(2019
).156.
E.
Selenius
, S.
Malola
, M.
Kuisma
, and H.
Häkkinen
, “Charge transfer plasmons in dimeric electron clusters
,” J. Phys. Chem. C
124
, 12645
–12654
(2020
).157.
W.
Ekardt
, “Size-dependent photoabsorption and photoemission of small metal particles
,” Phys. Rev. B
31
, 6360
–6370
(1985
).158.
D. E.
Beck
, “Self-consistent calculation of the eigenfrequencies for the electronic excitations in small jellium spheres
,” Phys. Rev. B
35
, 7325
–7333
(1987
).159.
G.
Bertsch
, “An RPA program for jellium spheres
,” Comput. Phys. Commun.
60
, 247
–255
(1990
).160.
M.
Brack
, “The physics of simple metal clusters: Self-consistent jellium model and semiclassical approaches
,” Rev. Mod. Phys.
65
, 677
–732
(1993
).161.
B.
Palpant
, B.
Prével
, J.
Lermé
, E.
Cottancin
, M.
Pellarin
, M.
Treilleux
, A.
Perez
, J. L.
Vialle
, and M.
Broyer
, “Optical properties of gold clusters in the size range 2–4 nm
,” Phys. Rev. B
57
, 1963
–1970
(1998
).162.
E.
Townsend
and G. W.
Bryant
, “Plasmonic properties of metallic nanoparticles: The effects of size quantization
,” Nano Lett.
12
, 429
–434
(2012
).163.
M.
Zapata Herrera
, J.
Aizpurua
, A. K.
Kazansky
, and A. G.
Borisov
, “Plasmon response and electron dynamics in charged metallic nanoparticles
,” Langmuir
32
, 2829
–2840
(2016
).164.
J.
Zuloaga
, E.
Prodan
, and P.
Nordlander
, “Quantum description of the plasmon resonances of a nanoparticle dimer
,” Nano Lett.
9
, 887
–891
(2009
).165.
D. C.
Marinica
, A. K.
Kazansky
, P.
Nordlander
, J.
Aizpurua
, and A. G.
Borisov
, “Quantum plasmonics: Nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer
,” Nano Lett.
12
, 1333
–1339
(2012
).166.
E.
Prodan
and P.
Nordlander
, “Electronic structure and polarizability of metallic nanoshells
,” Chem. Phys. Lett.
352
, 140
–146
(2002
).167.
C.
Yannouleas
, “Microscopic description of the surface dipole plasmon in large clusters
,” Phys. Rev. B
58
, 6748
–6751
(1998
).168.
R.
van Leeuwen
, “Mapping from densities to potentials in time-dependent density-functional theory
,” Phys. Rev. Lett.
82
, 3863
–3866
(1999
).169.
R.
Van Leeuwen
, “Key concepts in time-dependent density-functional theory
,” Int. J. Mod. Phys. B
15
, 1969
–2023
(2001
).170.
I. V.
Tokatly
, “Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws
,” Phys. Rev. B
71
, 165104
(2005
).171.
W.
Tarantino
and C. A.
Ullrich
, “A reformulation of time-dependent Kohn–Sham theory in terms of the second time derivative of the density
,” J. Chem. Phys.
154
, 204112
(2021
).172.
X.
Andrade
, S.
Botti
, M. A. L.
Marques
, and A.
Rubio
, “Time-dependent density functional theory scheme for efficient calculations of dynamic (hyper)polarizabilities
,” J. Chem. Phys.
126
, 184106
(2007
).173.
F.
Hofmann
, I.
Schelter
, and S.
Kümmel
, “Linear response time-dependent density functional theory without unoccupied states: The Kohn-Sham-Sternheimer scheme revisited
,” J. Chem. Phys.
149
, 024105
(2018
).174.
F.
Furche
and K.
Burke
, “Time-dependent density functional theory in quantum chemistry
,” Annu. Rep. Comput. Chem.
1
, 19
–30
(2005
).175.
G.
Onida
, L.
Reining
, and A.
Rubio
, “Electronic excitations: Density-functional versus many-body green’s-function approaches
,” Rev. Mod. Phys.
74
, 601
–659
(2002
).176.
A.
Zangwill
and P.
Soven
, “Density-functional approach to local-field effects in finite systems: Photoabsorption in the rare gases
,” Phys. Rev. A
21
, 1561
–1572
(1980
).177.
G.
Giuliani
and G.
Vignale
, Quantum Theory of the Electron Liquid
(Cambridge University Press
, 2005
).178.
O.
Keller
, “Local fields in the electrodynamics of mesoscopic media
,” Phys. Rep.
268
, 85
–262
(1996
).179.
O. S.
Jenkins
and K. L. C.
Hunt
, “Non-local dielectric functions on the nanoscale: Electronic polarization and fluctuations
,” J. Mol. Struct.: THEOCHEM
633
, 145
–155
(2003
).180.
E.
Madelung
, “Quantum theory in hydrodynamical form
,” Z. Phys.
40
, 322
(1927
).181.
C. F.
von Weizsäcker
, “Zur theorie der kernmassen
,” Z. Phys.
96
, 431
–458
(1935
).182.
L. J.
Bartolotti
, “The hydrodynamic formulation of time-dependent Kohn-Sham orbital density functional theory
,” J. Phys. Chem.
90
, 5518
–5523
(1986
).183.
R.
Bauernschmitt
, M.
Häser
, O.
Treutler
, and R.
Ahlrichs
, “Calculation of excitation energies within time-dependent density functional theory using auxiliary basis set expansions
,” Chem. Phys. Lett.
264
, 573
(1997
).184.
L.
Brey
, N. F.
Johnson
, and B. I.
Halperin
, “Optical and magneto-optical absorption in parabolic quantum wells
,” Phys. Rev. B
40
, 10647
–10649
(1989
).185.
J. F.
Dobson
, “Harmonic-potential theorem: Implications for approximate many-body theories
,” Phys. Rev. Lett.
73
, 2244
–2247
(1994
).186.
G.
Vignale
, “Center of mass and relative motion in time dependent density functional theory
,” Phys. Rev. Lett.
74
, 3233
–3236
(1995
).187.
K.
Hagino
, “Anharmonicity of the dipole resonance of metal clusters
,” Phys. Rev. B
60
, R2197
–R2199
(1999
).188.
L. G.
Gerchikov
, C.
Guet
, and A. N.
Ipatov
, “Multiple plasmons and anharmonic effects in small metallic clusters
,” Phys. Rev. A
66
, 053202
(2002
).189.
G.
Weick
, G.-L.
Ingold
, R. A.
Jalabert
, and D.
Weinmann
, “Surface plasmon in metallic nanoparticles: Renormalization effects due to electron-hole excitations
,” Phys. Rev. B
74
, 165421
(2006
).190.
J.
Lermé
, H.
Baida
, C.
Bonnet
, M.
Broyer
, E.
Cottancin
, A.
Crut
, P.
Maioli
, N.
Del Fatti
, F.
Vallée
, and M.
Pellarin
, “Size dependence of the surface plasmon resonance damping in metal nanospheres
,” J. Phys. Chem. Lett.
1
, 2922
–2928
(2010
).191.
J.
Lermé
, “Size evolution of the surface plasmon resonance damping in silver nanoparticles: Confinement and dielectric effects
,” J. Phys. Chem. C
115
, 14098
–14110
(2011
).192.
U.
Kreibig
and M.
Vollmer
, Optical Properties of Metal Clusters
(Springer Berlin
, Heidelberg
, 1995
).193.
A. J.
Bennett
, “Influence of the electron charge distribution on surface-plasmon dispersion
,” Phys. Rev. B
1
, 203
–207
(1970
).194.
K.-D.
Tsuei
, E. W.
Plummer
, A.
Liebsch
, K.
Kempa
, and P.
Bakshi
, “Multipole plasmon modes at a metal surface
,” Phys. Rev. Lett.
64
, 44
–47
(1990
).195.
K.-D.
Tsuei
, E. W.
Plummer
, A.
Liebsch
, E.
Pehlke
, K.
Kempa
, and P.
Bakshi
, “The normal modes at the surface of simple metals
,” Surf. Sci.
247
, 302
–326
(1991
).196.
A.
Liebsch
, “Surface-plasmon dispersion and size dependence of Mie resonance: Silver versus simple metals
,” Phys. Rev. B
48
, 11317
–11328
(1993
).197.
A.
Liebsch
, Electronic Excitations at Metal Surfaces
(Springer
, 1997
).198.
W.
Yang
, P. W.
Ayers
, and Q.
Wu
, “Potential functionals: Dual to density functionals and solution to the -representability problem
,” Phys. Rev. Lett.
92
, 146404
(2004
).199.
A.
Cangi
, D.
Lee
, P.
Elliott
, K.
Burke
, and E. K. U.
Gross
, “Electronic structure via potential functional approximations
,” Phys. Rev. Lett.
106
, 236404
(2011
).200.
B.
Goodman
and A.
Sjölander
, “Application of the third moment to the electric and magnetic response function
,” Phys. Rev. B
8
, 200
–214
(1973
).201.
A.
Griffin
and J.
Harris
, “Sum rules for a bounded electron gas
,” Can. J. Phys.
54
, 1396
–1408
(1976
).202.
A. A.
Lushnikov
and A. J.
Simonov
, “Surface plasmons in small metal particles
,” Z. Phys.
270
, 17
–24
(1974
).203.
G.
Mukhopadhyay
and S.
Lundqvist
, “Density oscillations and density response in systems with nonuniform electron density
,” Nuovo Cimento B
27
, 1
–18
(1975
).204.
G.
Mukhopadhyay
and S.
Lundqvist
, “Collective motion in heavy atoms
,” J. Phys. B: At. Mol. Phys.
12
, 1297
–1304
(1979
).205.
V. V.
Kresin
, “Collective resonances and response properties of electrons in metal clusters
,” Phys. Rep.
220
, 1
–52
(1992
).206.
E.
Prodan
and P.
Nordlander
, “Structural tunability of the plasmon resonances in metallic nanoshells
,” Nano Lett.
3
, 543
–547
(2003
).207.
M.
Ichikawa
, “Theory of localized plasmons for metal nanostructures in random-phase approximation
,” J. Phys. Soc. Jpn.
80
, 044606
(2011
).208.
Y.
Pavlyukh
, J.
Berakdar
, and K.
Köksal
, “Fast computations of the dielectric response of systems with spherical or axial symmetry
,” Phys. Rev. B
85
, 195418
(2012
).209.
J.
Höller
, E.
Krotscheck
, and É.
Suraud
, “On the response function of simple metal clusters
,” Eur. Phys. J. D
69
, 141
(2015
).210.
R.
Ruppin
, “Optical properties of a metal sphere with a diffuse surface
,” J. Opt. Soc. Am.
66
, 449
–453
(1976
).211.
S. V.
Fomichev
and W.
Becker
, “Linear and nonlinear light scattering and absorption in free-electron nanoclusters with diffuse surface: General considerations and linear response
,” Phys. Rev. A
81
, 063201
(2010
).212.
V. V.
Karasiev
and S. B.
Trickey
, “Issues and challenges in orbital-free density functional calculations
,” Comput. Phys. Commun.
183
, 2519
–2527
(2012
).213.
F.
Tran
and T. A.
Wesolowski
, “Semilocal approximation for the kinetic energy
,” in Recent Advances in Computational Chemistry
, edited by T. A.
Wesolowski
and Y. A.
Wang
(World Scientific
, Singapore
, 2013
), Vol. 6, pp. 429
–442
.214.
A.
Lembarki
and H.
Chermette
, “Obtaining a gradient-corrected kinetic-energy functional from the Perdew-Wang exchange functional
,” Phys. Rev. A
50
, 5328
–5331
(1994
).215.
Y. A.
Wang
, N.
Govind
, and E. A.
Carter
, “Orbital-free kinetic-energy functionals for the nearly free electron gas
,” Phys. Rev. B
58
, 13465
(1998
).216.
J. P.
Perdew
and L. A.
Constantin
, “Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy
,” Phys. Rev. B
75
, 155109
(2007
).217.
V. V.
Karasiev
, R. S.
Jones
, S. B.
Trickey
, and F. E.
Harris
, “Properties of constraint-based single-point approximate kinetic energy functionals
,” Phys. Rev. B
80
, 245120
(2009
).218.
S.
Laricchia
, L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Laplacian-level kinetic energy approximations based on the fourth-order gradient expansion: Global assessment and application to the subsystem formulation of density functional theory
,” J. Chem. Theory Comput.
10
, 164
–179
(2014
).219.
S.
Laricchia
, E.
Fabiano
, L. A.
Constantin
, and F.
Della Sala
, “Generalized gradient approximations of the noninteracting kinetic energy from the semiclassical atom theory: Rationalization of the accuracy of the frozen density embedding theory for nonbonded interactions
,” J. Chem. Theory Comput.
7
, 2439
–2451
(2011
).220.
L. A.
Constantin
, E.
Fabiano
, S.
Laricchia
, and F.
Della Sala
, “Semiclassical neutral atom as a reference system in density functional theory
,” Phys. Rev. Lett.
106
, 186406
(2011
).221.
I.
Shin
and E. A.
Carter
, “Enhanced von Weizsäcker Wang-Govind-Carter kinetic energy density functional for semiconductors
,” J. Chem. Phys.
140
, 18A531
(2014
).222.
A. C.
Cancio
, D.
Stewart
, and A.
Kuna
, “Visualization and analysis of the Kohn-Sham kinetic energy density and its orbital-free description in molecules
,” J. Chem. Phys.
144
, 084107
(2016
).223.
L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Modified fourth-order kinetic energy gradient expansion with Hartree potential-dependent coefficients
,” J. Chem. Theory Comput.
13
, 4228
–4239
(2017
).224.
J.
Seino
, R.
Kageyama
, M.
Fujinami
, Y.
Ikabata
, and H.
Nakai
, “Semi-local machine-learned kinetic energy density functional with third-order gradients of electron density
,” J. Chem. Phys.
148
, 241705
(2018
).225.
P.
Golub
and S.
Manzhos
, “Kinetic energy densities based on the fourth order gradient expansion: Performance in different classes of materials and improvement via machine learning
,” Phys. Chem. Chem. Phys.
21
, 378
(2018
).226.
L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Nonlocal kinetic energy functional from the jellium-with-gap model: Applications to orbital-free density functional theory
,” Phys. Rev. B
97
, 205137
(2018
).227.
L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Semilocal Pauli–Gaussian kinetic functionals for orbital-free density functional theory calculations of solids
,” J. Phys. Chem. Lett.
9
, 4385
–4390
(2018
).228.
L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Performance of semilocal kinetic energy functionals for orbital-free density functional theory
,” J. Chem. Theory Comput.
15
, 3044
–3055
(2019
).229.
F.
Sarcinella
, E.
Fabiano
, L. A.
Constantin
, and F.
Della Sala
, “Nonlocal kinetic energy functionals in real space using a Yukawa-potential kernel: Properties, linear response, and model functionals
,” Phys. Rev. B
103
, 155127
(2021
).230.
F.
Imoto
, M.
Imada
, and A.
Oshiyama
, “Order-n orbital-free density-functional calculations with machine learning of functional derivatives for semiconductors and metals
,” Phys. Rev. Res.
3
, 033198
(2021
).231.
L. H.
Thomas
, “The calculation of atomic fields
,” Math. Proc. Cambridge Philos. Soc.
23
, 542
–548
(1927
).232.
E.
Fermi
, “Un metodo statistico per la determinazione di alcune prioprietà dell’atomo
,” Rend. Accad. Naz. Lincei
6
, 602
–607
(1927
).233.
G. K.-L.
Chan
, A. J.
Cohen
, and N. C.
Handy
, “Thomas–Fermi–Dirac–von Weizsäcker models in finite systems
,” J. Chem. Phys.
114
, 631
–638
(2001
).234.
L. A.
Espinosa Leal
, A.
Karpenko
, M. A.
Caro
, and O.
Lopez-Acevedo
, “Optimizing a parametrized Thomas–Fermi–Dirac–Weizsacker density functional for atoms
,” Phys. Chem. Chem. Phys.
17
, 31463
–31471
(2015
).235.
B. M.
Deb
and S. K.
Ghosh
, “New method for the direct calculation of electron density in many-electron systems. I. Application to closed-shell atoms
,” Int. J. Quantum Chem.
23
, 1
–26
(1983
).236.
M.
Levy
and H.
Ou-Yang
, “Exact properties of the Pauli potential for the square root of the electron density and the kinetic energy functional
,” Phys. Rev. A
38
, 625
(1988
).237.
L.-W.
Wang
and M. P.
Teter
, “Kinetic-energy functional of the electron density
,” Phys. Rev. B
45
, 13196
(1992
).238.
F.
Perrot
, “Hydrogen-hydrogen interaction in an electron gas
,” J. Phys.: Condens. Matter
6
, 431
(1994
).239.
E.
Smargiassi
and P. A.
Madden
, “Orbital-free kinetic-energy functionals for first-principles molecular dynamics
,” Phys. Rev. B
49
, 5220
(1994
).240.
D.
García-Aldea
and J. E.
Alvarellos
, “Fully nonlocal kinetic energy density functionals: A proposal and a general assessment for atomic systems
,” J. Chem. Phys.
129
, 074103
(2008
).241.
C.
Huang
and E. A.
Carter
, “Nonlocal orbital-free kinetic energy density functional for semiconductors
,” Phys. Rev. B
81
, 045206
(2010
).242.
W.
Mi
, A.
Genova
, and M.
Pavanello
, “Nonlocal kinetic energy functionals by functional integration
,” J. Chem. Phys.
148
, 184107
(2018
).243.
W.
Mi
and M.
Pavanello
, “Orbital-free density functional theory correctly models quantum dots when asymptotics, nonlocality, and nonhomogeneity are accounted for
,” Phys. Rev. B
100
, 041105
(2019
).244.
Q.
Xu
, J.
Lv
, Y.
Wang
, and Y.
Ma
, “Nonlocal kinetic energy density functionals for isolated systems obtained via local density approximation kernels
,” Phys. Rev. B
101
, 045110
(2020
).245.
C.
Herring
, “Explicit estimation of ground-state kinetic energies from electron densities
,” Phys. Rev. A
34
, 2614
–2631
(1986
).246.
N.
Choly
and E.
Kaxiras
, “Kinetic energy density functionals for non-periodic systems
,” Solid State Commun.
121
, 281
–286
(2002
).247.
C. J.
García-Cervera
., “An efficient real space method for orbital-free density-functional theory
,” Commun. Comput. Phys.
2
, 334
–357
(2007
), see https://global-sci.org/intro/article_detail/cicp/7909.html.248.
E.
Fabiano
, F.
Sarcinella
, L. A.
Constantin
, and F.
Della Sala
, “Kinetic energy density functionals based on a generalized screened Coulomb potential: Linear response and future perspectives
,” Computation
10
, 30
(2022
).249.
A.
Schild
and E. K. U.
Gross
, “Exact single-electron approach to the dynamics of molecules in strong laser fields
,” Phys. Rev. Lett.
118
, 163202
(2017
).250.
R.
Singh
and B. M.
Deb
, “Developments in excited-state density functional theory
,” Phys. Rep.
311
, 47
–94
(1999
).251.
B. M.
Deb
and S. K.
Ghosh
, “Schrödinger fluid dynamics of many-electron systems in a time-dependent density-functional framework
,” J. Chem. Phys.
77
, 342
–348
(1982
).252.
L. J.
Bartolotti
, “Time-dependent Kohn-Sham density-functional theory
,” Phys. Rev. A
26
, 2243
–2244
(1982
).253.
S. K.
Ghosh
and B. M.
Deb
, “Densities, density-functionals and electron fluids
,” Phys. Rep.
92
, 1
–44
(1982
).254.
M. T.
Michalewicz
, “Hydrodynamic model of collective electronic oscillations in spherical geometries
,” in Topics in Condensed Matter Physics
(Nova Science Publishers
, 1995
), pp. 143
–167
.255.
A.
Banerjee
and M. K.
Harbola
, “Hydrodynamic approach to time-dependent density functional theory; Response properties of metal clusters
,” J. Chem. Phys.
113
, 5614
–5623
(2000
).256.
J. R.
Maack
, N. A.
Mortensen
, and M.
Wubs
, “Size-dependent nonlocal effects in plasmonic semiconductor particles
,” Europhys. Lett.
119
, 17003
(2017
).257.
F.
De Luca
, M.
Ortolani
, and C.
Ciracì
, “Free electron nonlinearities in heavily doped semiconductors plasmonics
,” Phys. Rev. B
103
, 115305
(2021
).258.
J. F.
Dobson
and H. M.
Le
, “High-frequency hydrodynamics and Thomas–Fermi theory
,” J. Mol. Struct.: THEOCHEM
501-502
, 327
–338
(2000
).259.
M.
Thiele
and S.
Kümmel
, “Hydrodynamic perspective on memory in time-dependent density-functional theory
,” Phys. Rev. A
79
, 052503
(2009
).260.
X.
Shao
, W.
Mi
, and M.
Pavanello
, “Efficient DFT solver for nanoscale simulations and beyond
,” J. Phys. Chem. Lett.
12
, 4134
–4139
(2021
).261.
K.
Ding
and C. T.
Chan
, “An eigenvalue approach to quantum plasmonics based on a self-consistent hydrodynamics method
,” J. Phys.: Condens. Matter
30
, 084007
(2018
).262.
S. A.
Chin
and E.
Krotscheck
, “Structure and collective excitations of 4He clusters
,” Phys. Rev. B
45
, 852
(1992
).263.
P.
Halevi
, “Hydrodynamic model for the degenerate free-electron gas: Generalization to arbitrary frequencies
,” Phys. Rev. B
51
, 7497
–7499
(1995
).264.
I.
Tokatly
and O.
Pankratov
, “Hydrodynamic theory of an electron gas
,” Phys. Rev. B
60
, 15550
–15553
(1999
).265.
J. M.
Pitarke
, V. M.
Silkin
, E. V.
Chulkov
, and P. M.
Echenique
, “Theory of surface plasmons and surface-plasmon polaritons
,” Rep. Prog. Phys.
70
, 1
–87
(2006
).266.
R.
Slavchov
and R.
Tsekov
, “Quantum hydrodynamics of electron gases
,” J. Chem. Phys.
132
, 084505
(2010
).267.
M.
Akbari-Moghanjoughi
, “Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited
,” Phys. Plasmas
22
, 022103
(2015
).268.
C. E.
Patrick
and K. S.
Thygesen
, “Adiabatic-connection fluctuation-dissipation DFT for the structural properties of solids—The renormalized ALDA and electron gas kernels
,” J. Chem. Phys.
143
, 102802
(2015
).269.
A.
Ruzsinszky
, N. K.
Nepal
, J. M.
Pitarke
, and J. P.
Perdew
, “Constraint-based wave vector and frequency dependent exchange-correlation kernel of the uniform electron gas
,” Phys. Rev. B
101
, 245135
(2020
).270.
A. D.
Kaplan
, N. K.
Nepal
, A.
Ruzsinszky
, P.
Ballone
, and J. P.
Perdew
, “First-principles wave-vector- and frequency-dependent exchange-correlation kernel for jellium at all densities
,” Phys. Rev. B
105
, 035123
(2022
).271.
J.
Crowell
and R. H.
Ritchie
, “Radiative decay of Coulomb-stimulated plasmons in spheres
,” Phys. Rev.
172
, 436
–440
(1968
).272.
S.
Raza
, G.
Toscano
, A.-P.
Jauho
, M.
Wubs
, and N. A.
Mortensen
, “Unusual resonances in nanoplasmonic structures due to nonlocal response
,” Phys. Rev. B
84
, 121412
(2011
).273.
A.
Eguiluz
, S. C.
Ying
, and J. J.
Quinn
, “Influence of the electron density profile on surface plasmons in a hydrodynamic model
,” Phys. Rev. B
11
, 2118
(1975
).274.
A.
Eguiluz
and J. J.
Quinn
, “Hydrodynamic model for surface plasmons in metals and degenerate semiconductors
,” Phys. Rev. B
14
, 1347
–1361
(1976
).275.
C.
Schwartz
and W. L.
Schaich
, “Hydrodynamic models of surface plasmons
,” Phys. Rev. B
26
, 7008
–7011
(1982
).276.
C.
David
and F. J.
García de Abajo
, “Surface plasmon dependence on the electron density profile at metal surfaces
,” ACS Nano
8
, 9558
–9566
(2014
).277.
A.
Kawabata
and R.
Kubo
, “Electronic properties of fine metallic particles. II. Plasma resonance absorption
,” J. Phys. Soc. Jpn.
21
, 1765
–1772
(1966
).278.
M.
Barma
and V.
Subrahmanyam
, “Optical absorption in small metal particles
,” J. Phys.: Condens. Matter
1
, 7681
–7688
(1989
).279.
C.
Yannouleas
and R. A.
Broglia
, “Landau damping and wall dissipation in large metal clusters
,” Ann. Phys.
217
, 105
–141
(1992
).280.
H.
Kurasawa
, K.
Yabana
, and T.
Suzuki
, “Damping width of the Mie plasmon
,” Phys. Rev. B
56
, R10063
–R10066
(1997
).281.
R. A.
Molina
, D.
Weinmann
, and R. A.
Jalabert
, “Oscillatory size dependence of the surface plasmon linewidth in metallic nanoparticles
,” Phys. Rev. B
65
, 155427
(2002
).282.
G.
Weick
, R. A.
Molina
, D.
Weinmann
, and R. A.
Jalabert
, “Lifetime of the first and second collective excitations in metallic nanoparticles
,” Phys. Rev. B
72
, 115410
(2005
).283.
X.
Li
, D.
Xiao
, and Z.
Zhang
, “Landau damping of quantum plasmons in metal nanostructures
,” New J. Phys.
15
, 023011
(2013
).284.
T. V.
Shahbazyan
, “Landau damping of surface plasmons in metal nanostructures
,” Phys. Rev. B
94
, 235431
(2016
).285.
J.
Khurgin
, W.-Y.
Tsai
, D. P.
Tsai
, and G.
Sun
, “Landau damping and limit to field confinement and enhancement in plasmonic dimers
,” ACS Photonics
4
, 2871
–2880
(2017
).286.
X.
You
, S.
Ramakrishna
, and T.
Seideman
, “Origin of plasmon lineshape and enhanced hot electron generation in metal nanoparticles
,” J. Phys. Chem. Lett.
9
, 141
–145
(2018
).287.
X.
Li
, H.
Fang
, X.
Weng
, L.
Zhang
, X.
Dou
, A.
Yang
, and X.
Yuan
, “Electronic spill-out induced spectral broadening in quantum hydrodynamic nanoplasmonics
,” Opt. Express
23
, 29738
–29745
(2015
).288.
N. A.
Mortensen
, S.
Raza
, M.
Wubs
, T.
Søndergaard
, and S. I.
Bozhevolnyi
, “A generalized non-local optical response theory for plasmonic nanostructures
,” Nat. Commun.
5
, 3809
(2014
).289.
P. J.
Feibelman
, “Surface electromagnetic fields
,” Prog. Surf. Sci.
12
, 287
–407
(1982
).290.
A.
Liebsch
and W. L.
Schaich
, “Influence of a polarizable medium on the nonlocal optical response of a metal surface
,” Phys. Rev. B
52
, 14219
–14234
(1995
).291.
R.
van Leeuwen
and E. J.
Baerends
, “Exchange-correlation potential with correct asymptotic behavior
,” Phys. Rev. A
49
, 2421
–2431
(1994
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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