Simulations and diagnostics of high-energy-density plasmas and warm dense matter rely on models of material response properties, both static and dynamic (frequency-dependent). Here, we systematically investigate variations in dynamic electron–ion collision frequencies in warm dense matter using data from a self-consistent-field average-atom model. We show that including the full quantum density of states, strong collisions, and inelastic collisions lead to significant changes in . These changes result in red shifts and broadening of the plasmon peak in the dynamic structure factor, an effect observable in x-ray Thomson scattering spectra, and modify stopping powers around the Bragg peak. These changes improve the agreement of computationally efficient average-atom models with first-principles time-dependent density functional theory in warm dense aluminum, carbon, and deuterium.
REFERENCES
1.
B.
Hammel
,
S.
Haan
,
D.
Clark
,
M.
Edwards
,
S.
Langer
,
M.
Marinak
,
M.
Patel
,
J.
Salmonson
, and
H.
Scott
, “
High-mode Rayleigh-Taylor growth in NIF ignition capsules
,” High Energy Density Phys.
6
, 171
–178
(2010
).2.
S. H.
Glenzer
and
R.
Redmer
, “
X-ray Thomson scattering in high energy density plasmas
,” Rev. Mod. Phys.
81
, 1625
–1663
(2009
).3.
A. B.
Zylstra
,
J. A.
Frenje
,
P. E.
Grabowski
,
C. K.
Li
,
G. W.
Collins
,
P.
Fitzsimmons
,
S.
Glenzer
,
F.
Graziani
,
S. B.
Hansen
,
S. X.
Hu
,
M. G.
Johnson
,
P.
Keiter
,
H.
Reynolds
,
J. R.
Rygg
,
F. H.
Séguin
, and
R. D.
Petrasso
, “
Measurement of charged-particle stopping in warm dense plasma
,” Phys. Rev. Lett.
114
, 215002
(2015
).4.
S.
Malko
,
W.
Cayzac
,
V.
Ospina-Bohorquez
,
K.
Bhutwala
,
M.
Bailly-Grandvaux
,
C.
McGuffey
,
R.
Fedosejevs
,
X.
Vaisseau
,
A.
Tauschwitz
,
J.
Apiñaniz
et al, “
Proton stopping measurements at low velocity in warm dense carbon
,” Nat. Commun.
13
, 1
–12
(2022
).5.
J.
Lindl
, “
Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain
,” Phys. Plasmas
2
, 3933
–4024
(1995
).6.
O. A.
Hurricane
,
D. A.
Callahan
,
D. T.
Casey
,
E. L.
Dewald
,
T. R.
Dittrich
,
T.
Döppner
,
S.
Haan
,
D. E.
Hinkel
,
L. F.
Berzak Hopkins
,
O.
Jones
,
A. L.
Kritcher
,
S.
Le Pape
,
T.
Ma
,
A. G.
MacPhee
,
J. L.
Milovich
,
J.
Moody
,
A.
Pak
,
H.-S.
Park
,
P. K.
Patel
,
J. E.
Ralph
,
H. F.
Robey
,
J. S.
Ross
,
J. D.
Salmonson
,
B. K.
Spears
,
P. T.
Springer
,
R.
Tommasini
,
F.
Albert
,
L. R.
Benedetti
,
R.
Bionta
,
E.
Bond
,
D. K.
Bradley
,
J.
Caggiano
,
P. M.
Celliers
,
C.
Cerjan
,
J. A.
Church
,
R.
Dylla-Spears
,
D.
Edgell
,
M. J.
Edwards
,
D.
Fittinghoff
,
M. A.
Barrios Garcia
,
A.
Hamza
,
R.
Hatarik
,
H.
Herrmann
,
M.
Hohenberger
,
D.
Hoover
,
J. L.
Kline
,
G.
Kyrala
,
B.
Kozioziemski
,
G.
Grim
,
J. E.
Field
,
J.
Frenje
,
N.
Izumi
,
M.
Gatu Johnson
,
S. F.
Khan
,
J.
Knauer
,
T.
Kohut
,
O.
Landen
,
F.
Merrill
,
P.
Michel
,
A.
Moore
,
S. R.
Nagel
,
A.
Nikroo
,
T.
Parham
,
R. R.
Rygg
,
D.
Sayre
,
M.
Schneider
,
D.
Shaughnessy
,
D.
Strozzi
,
R. P. J.
Town
,
D.
Turnbull
,
P.
Volegov
,
A.
Wan
,
K.
Widmann
,
C.
Wilde
, and
C.
Yeamans
, “
Inertially confined fusion plasmas dominated by alpha-particle self-heating
,” Nat. Phys.
12
, 800
–806
(2016
).7.
A.
Zylstra
and
O.
Hurricane
, “
On alpha-particle transport in inertial fusion
,” Phys. Plasmas
26
, 062701
(2019
).8.
H.
Milchberg
,
R.
Freeman
,
S.
Davey
, and
R.
More
, “
Resistivity of a simple metal from room temperature to 106 K
,” Phys. Rev. Lett.
61
, 2364
(1988
).9.
A. W.
DeSilva
and
A. D.
Rakhel
, “
Progress in measurements of the electrical conductivity of metal plasmas
,” Contrib. Plasma Phys.
45
, 236
–242
(2005
).10.
P.
Sperling
,
E. J.
Gamboa
,
H. J.
Lee
,
H. K.
Chung
,
E.
Galtier
,
Y.
Omarbakiyeva
,
H.
Reinholz
,
G.
Röpke
,
U.
Zastrau
,
J.
Hastings
,
L. B.
Fletcher
, and
S. H.
Glenzer
, “
Free-electron x-ray laser measurements of collisional-damped plasmons in isochorically heated warm dense matter
,” Phys. Rev. Lett.
115
, 115001
(2015
).11.
J.
Frenje
,
P.
Grabowski
,
C.
Li
,
F.
Séguin
,
A.
Zylstra
,
M. G.
Johnson
,
R.
Petrasso
,
V. Y.
Glebov
, and
T.
Sangster
, “
Measurements of ion stopping around the bragg peak in high-energy-density plasmas
,” Phys. Rev. Lett.
115
, 205001
(2015
).12.
H.
Brysk
, “
Electron-ion equilibration in a partially degenerate plasma
,” Plasma Phys.
16
, 927
–932
(1974
).13.
P.
Hohenberg
and
W.
Kohn
, “
Inhomogeneous electron gas
,” Phys. Rev.
136
, B864
(1964
).14.
W.
Kohn
and
L. J.
Sham
, “
Self-consistent equations including exchange and correlation effects
,” Phys. Rev.
140
, A1133
(1965
).15.
M.
Desjarlais
,
J.
Kress
, and
L.
Collins
, “
Electrical conductivity for warm, dense aluminum plasmas and liquids
,” Phys. Rev. E
66
, 025401
(2002
).16.
M.
Schörner
,
B. B.
Witte
,
A. D.
Baczewski
,
A.
Cangi
, and
R.
Redmer
, “
Ab initio study of shock-compressed copper
,” Phys. Rev. B
106
, 054304
(2022
).17.
E.
Runge
and
E. K.
Gross
, “
Density-functional theory for time-dependent systems
,” Phys. Rev. Lett.
52
, 997
(1984
).18.
M. A.
Marques
,
C. A.
Ullrich
,
F.
Nogueira
,
A.
Rubio
,
K.
Burke
, and
E. K.
Gross
, Time-Dependent Density Functional Theory
(
Springer Science and Business Media
, 2006
), Vol.
706
.19.
C. A.
Ullrich
, Time-Dependent Density-Functional Theory: Concepts and Applications
(
OUP
Oxford
, 2011
).20.
A.
Kononov
,
C.-W.
Lee
,
T. P.
dos Santos
,
B.
Robinson
,
Y.
Yao
,
Y.
Yao
,
X.
Andrade
,
A. D.
Baczewski
,
E.
Constantinescu
,
A. A.
Correa
et al, “
Electron dynamics in extended systems within real-time time-dependent density-functional theory
,” MRS Commun.
12
, 1002
–1014
(2022
).21.
A. A.
Correa
,
J.
Kohanoff
,
E.
Artacho
,
D.
Sánchez-Portal
, and
A.
Caro
, “
Nonadiabatic forces in ion-solid interactions: The initial stages of radiation damage
,” Phys. Rev. Lett.
108
, 213201
(2012
).22.
A.
Schleife
,
Y.
Kanai
, and
A. A.
Correa
, “
Accurate atomistic first-principles calculations of electronic stopping
,” Phys. Rev. B
91
, 014306
(2015
).23.
A. A.
Shukri
,
F.
Bruneval
, and
L.
Reining
, “
Ab initio electronic stopping power of protons in bulk materials
,” Phys. Rev. B
93
, 035128
(2016
).24.
R. J.
Magyar
,
L.
Shulenburger
, and
A. D.
Baczewski
, “
Stopping of deuterium in warm dense deuterium from Ehrenfest time-dependent density functional theory
,” Contrib. Plasma Phys.
56
, 459
–466
(2016
).25.
A.
Kononov
and
A.
Schleife
, “
Pre-equilibrium stopping and charge capture in proton-irradiated aluminum sheets
,” Phys. Rev. B
102
, 165401
(2020
).26.
A.
Kononov
and
A.
Schleife
, “
Anomalous stopping and charge transfer in proton-irradiated graphene
,” Nano Lett.
21
, 4816
–4822
(2021
).27.
A. D.
Baczewski
,
L.
Shulenburger
,
M. P.
Desjarlais
,
S. B.
Hansen
, and
R. J.
Magyar
, “
X-ray Thomson scattering in warm dense matter without the Chihara decomposition
,” Phys. Rev. Lett.
116
, 115004
(2016
).28.
K.
Ramakrishna
,
A.
Cangi
,
T.
Dornheim
,
A.
Baczewski
, and
J.
Vorberger
, “
First-principles modeling of plasmons in aluminum under ambient and extreme conditions
,” Phys. Rev. B
103
, 125118
(2021
).29.
A. D.
Baczewski
,
T.
Hentschel
,
A.
Kononov
, and
S. B.
Hansen
, “
Predictions of bound-bound transition signatures in x-ray Thomson scattering
,” arXiv:2109.09576.30.
D. A.
Liberman
, “
Self-consistent field model for condensed matter
,” Phys. Rev. B
20
, 4981
–4989
(1979
).31.
B.
Wilson
,
V.
Sonnad
,
P.
Sterne
, and
W.
Isaacs
, “
Purgatorio—A new implementation of the Inferno algorithm
,” J. Quant. Spectrosc. Radiat. Transfer
99
, 658
–679
(2006
).32.
C. E.
Starrett
and
D.
Saumon
, “
Electronic and ionic structures of warm and hot dense matter
,” Phys. Rev. E
87
, 013104
(2013
).33.
T. J.
Callow
,
S. B.
Hansen
,
E.
Kraisler
, and
A.
Cangi
, “
First-principles derivation and properties of density-functional average-atom models
,” Phys. Rev. Res.
4
, 023055
(2022
).34.
J. M.
Ziman
, “
A theory of the electrical properties of liquid metals. I. The monovalent metals
,” Philos. Mag. A
6
, 1013
–1034
(1961
).35.
D.
Burrill
,
D.
Feinblum
,
M.
Charest
, and
C.
Starrett
, “
Comparison of electron transport calculations in warm dense matter using the Ziman formula
,” High Energy Density Phys.
19
, 1
–10
(2016
).36.
N.
Wetta
and
J.-C.
Pain
, “
Consistent approach for electrical resistivity within ziman's theory from solid state to hot dense plasma: Application to aluminum
,” Phys. Rev. E
102
, 053209
(2020
).37.
G.
Gregori
,
S. H.
Glenzer
,
W.
Rozmus
,
R. W.
Lee
, and
O. L.
Landen
, “
Theoretical model of x-ray scattering as a dense matter probe
,” Phys. Rev. E
67
, 026412
(2003
).38.
W. R.
Johnson
,
J.
Nilsen
, and
K. T.
Cheng
, “
Thomson scattering in the average-atom approximation
,” Phys. Rev. E
86
, 036410
(2012
).39.
P.
Wang
,
T. M.
Mehlhorn
, and
J. J.
MacFarlane
, “
A unified self-consistent model for calculating ion stopping power in ICF plasma
,” Phys. Plasmas
5
, 2977
–2987
(1998
).40.
G.
Faussurier
,
C.
Blancard
,
P.
Cossé
, and
P.
Renaudin
, “
Equation of state, transport coefficients, and stopping power of dense plasmas from the average-atom model self-consistent approach for astrophysical and laboratory plasmas
,” Phys. Plasmas
17
, 052707
(2010
).41.
M. D.
Barriga-Carrasco
, “
Target electron collision effects on energy loss straggling of protons in an electron gas at any degeneracy
,” Phys. Plasmas
15
, 033103
(2008
).42.
M. D.
Barriga-Carrasco
, “
Dynamical local field corrections on energy loss in plasmas of all degeneracies
,” Phys. Rev. E
79
, 027401
(2009
).43.
Z. A.
Moldabekov
,
T.
Dornheim
,
M.
Bonitz
, and
T. S.
Ramazanov
, “
Ion energy-loss characteristics and friction in a free-electron gas at warm dense matter and nonideal dense plasma conditions
,” Phys. Rev. E
101
, 053203
(2020
).44.
A. N.
Souza
,
D. J.
Perkins
,
C. E.
Starrett
,
D.
Saumon
, and
S. B.
Hansen
, “
Predictions of x-ray scattering spectra for warm dense matter
,” Phys. Rev. E
89
, 023108
(2014
).45.
M. W. C.
Dharma-wardana
, “
Dynamic conductivity and plasmon profile of aluminum in the ultra-fast-matter regime
,” Phys. Rev. E
93
, 063205
(2016
).46.
N. D.
Mermin
, “
Lindhard dielectric function in the relaxation-time approximation
,” Phys. Rev. B
1
, 2362
–2363
(1970
).47.
J.
Lindhard
and
A.
Winther
, “
Stopping power of electron gas and equipartition rule
,” Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd.
34
, 1
–22
(1964
).48.
N. R.
Arista
and
W.
Brandt
, “
Energy loss and straggling of charged particles in plasmas of all degeneracies
,” Phys. Rev. A
23
, 1898
–1905
(1981
).49.
G.
Maynard
and
C.
Deutsch
, “
Energy loss and straggling of ions with any velocity in dense plasmas at any temperature
,” Phys. Rev. A
26
, 665
–668
(1982
).50.
S.
Skupsky
, “
Energy loss of ions moving through high-density matter
,” Phys. Rev. A
16
, 727
–731
(1977
).51.
N. R.
Arista
and
A. R.
Piriz
, “
Energy loss of fast particles in confined atomic systems at very high temperatures
,” Phys. Rev. A
35
, 3450
–3453
(1987
).52.
H.
Reinholz
,
R.
Redmer
,
G.
Röpke
, and
A.
Wierling
, “
Long-wavelength limit of the dynamical local-field factor and dynamical conductivity of a two-component plasma
,” Phys. Rev. E
62
, 5648
–5666
(2000
).53.
R.
Thiele
,
R.
Redmer
,
H.
Reinholz
, and
G.
Röpke
, “
Using the Gould–DeWitt scheme to approximate the dynamic collision frequency in a dense electron gas
,” J. Phys. A: Math. Gen.
39
, 4365
–4368
(2006
).54.
G. A.
Rinker
, “
Electrical conductivity of a strongly coupled plasma
,” Phys. Rev. B
31
, 4207
–4219
(1985
).55.
P.
Sterne
,
S.
Hansen
,
B.
Wilson
, and
W.
Isaacs
, “
Equation of state, occupation probabilities and conductivities in the average atom purgatorio code
,” High Energy Density Phys.
3
, 278
–282
(2007
).56.
D. M.
Ceperley
and
B. J.
Alder
, “
Ground state of the electron gas by a stochastic method
,” Phys. Rev. Lett.
45
, 566
–569
(1980
).57.
M. S.
Murillo
,
J.
Weisheit
,
S. B.
Hansen
, and
M.
Dharma-Wardana
, “
Partial ionization in dense plasmas: Comparisons among average-atom density functional models
,” Phys. Rev. E
87
, 063113
(2013
).58.
C. E.
Starrett
and
D.
Saumon
, “
Fully variational average atom model with ion-ion correlations
,” Phys. Rev. E
85
, 026403
(2012
).59.
R.
Thiele
,
T.
Bornath
,
C.
Fortmann
,
A.
Höll
,
R.
Redmer
,
H.
Reinholz
,
G.
Röpke
,
A.
Wierling
,
S. H.
Glenzer
, and
G.
Gregori
, “
Plasmon resonance in warm dense matter
,” Phys. Rev. E
78
, 026411
(2008
).60.
G.
Faussurier
and
C.
Blancard
, “
Electron-ion collision-frequency for x-ray Thomson scattering in dense plasmas
,” Phys. Plasmas
23
, 012703
(2016
).61.
G. A.
Rinker
, “
Systematic calculations of plasma transport coefficients for the periodic table
,” Phys. Rev. A
37
, 1284
–1297
(1988
).62.
D. J.
Griffiths
, Introduction to Quantum Mechanics
, 2nd ed.
(
Pearson Prentice Hall
, 2004
), Chap. 11.63.
J.
Sakurai
and
J. J.
Napolitano
, Modern Quantum Mechanics
, 2nd ed.
(
Pearson
, 2010
), Chap. 6.64.
C. E.
Starrett
,
R.
Perriot
,
N. R.
Shaffer
,
T.
Nelson
,
L. A.
Collins
, and
C.
Ticknor
, “
Tabular electrical conductivity for aluminium
,” Contrib. Plasma Phys.
60
, e201900123
(2020
).65.
C.
Fortmann
,
A.
Wierling
, and
G.
Röpke
, “
Influence of local-field corrections on Thomson scattering in collision-dominated two-component plasmas
,” Phys. Rev. E
81
, 026405
(2010
).66.
D. A.
Baiko
,
A. D.
Kaminker
,
A. Y.
Potekhin
, and
D. G.
Yakovlev
, “
Ion structure factors and electron transport in dense Coulomb plasmas
,” Phys. Rev. Lett.
81
, 5556
–5559
(1998
).67.
W.
Johnson
,
C.
Guet
, and
G.
Bertsch
, “
Optical properties of plasmas based on an average-atom model
,” J. Quant. Spectrosc. Radiat. Transfer
99
, 327
–340
(2006
).68.
B. B. L.
Witte
,
L. B.
Fletcher
,
E.
Galtier
,
E.
Gamboa
,
H. J.
Lee
,
U.
Zastrau
,
R.
Redmer
,
S. H.
Glenzer
, and
P.
Sperling
, “
Warm dense matter demonstrating non-drude conductivity from observations of nonlinear plasmon damping
,” Phys. Rev. Lett.
118
, 225001
(2017
).69.
J.
Chihara
, “
Difference in x-ray scattering between metallic and non-metallic liquids due to conduction electrons
,” J. Phys. F: Met. Phys.
17
, 295
(1987
).70.
J.
Chihara
, “
Interaction of photons with plasmas and liquid metals—Photoabsorption and scattering
,” J. Phys.: Condens. Matter
12
, 231
–247
(2000
).71.
G. D.
Mahan
, Many-Particle Physics
, 2nd ed
. (
Springer
, 1990
), Chap. 5.72.
N. R.
Arista
and
W.
Brandt
, “
Dielectric response of quantum plasmas in thermal equilibrium
,” Phys. Rev. A
29
, 1471
–1480
(1984
).73.
J.
Lindhard
, “
On the properties of a gas of charged particles
,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd.
28
(8
), 1
–58
(1954
).74.
J.
Hubbard
, “
The description of collective motions in terms of many-body perturbation theory. II. the correlation energy of a free-electron gas
,” Proc. R. Soc. London, Ser. A
243
, 336
–352
(1958
).75.
K. S.
Singwi
,
M. P.
Tosi
,
R. H.
Land
, and
A.
Sjölander
, “
Electron correlations at metallic densities
,” Phys. Rev.
176
, 589
–599
(1968
).76.
T.
Dornheim
,
S.
Groth
, and
M.
Bonitz
, “
The uniform electron gas at warm dense matter conditions
,” Phys. Rep.
744
, 1
–86
(2018
).77.
T.
Dornheim
,
S.
Groth
,
J.
Vorberger
, and
M.
Bonitz
, “
Ab initio path integral Monte Carlo results for the dynamic structure factor of correlated electrons: From the electron liquid to warm dense matter
,” Phys. Rev. Lett.
121
, 255001
(2018
).78.
T.
Dornheim
,
J.
Vorberger
,
S.
Groth
,
N.
Hoffmann
,
Z. A.
Moldabekov
, and
M.
Bonitz
, “
The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation
,” J. Chem. Phys.
151
, 194104
(2019
).79.
P.
Hamann
,
T.
Dornheim
,
J.
Vorberger
,
Z. A.
Moldabekov
, and
M.
Bonitz
, “
Dynamic properties of the warm dense electron gas based on abinitio path integral Monte Carlo simulations
,” Phys. Rev. B
102
, 125150
(2020
).80.
T.
Dornheim
,
A.
Cangi
,
K.
Ramakrishna
,
M.
Böhme
,
S.
Tanaka
, and
J.
Vorberger
, “
Effective static approximation: A fast and reliable tool for warm-dense matter theory
,” Phys. Rev. Lett.
125
, 235001
(2020
).81.
T.
Dornheim
,
Z. A.
Moldabekov
,
K.
Ramakrishna
,
P.
Tolias
,
A. D.
Baczewski
,
D.
Kraus
,
T. R.
Preston
,
D. A.
Chapman
,
M. P.
Böhme
,
T.
Döppner
et al, “
Electronic density response of warm dense matter
,” arXiv:2212.08326 (2022
).82.
A.
Wierling
, “
Dynamic local field corrections for two-component plasmas at intermediate coupling
,” J. Phys. A: Math. Theor.
42
, 214051
(2009
).83.
C. P.
Race
,
D. R.
Mason
,
M. W.
Finnis
,
W. M. C.
Foulkes
,
A. P.
Horsfield
, and
A. P.
Sutton
, “
The treatment of electronic excitations in atomistic models of radiation damage in metals
,” Rep. Prog. Phys.
73
, 116501
(2010
).84.
W. D.
Wilson
,
L. G.
Haggmark
, and
J. P.
Biersack
, “
Calculations of nuclear stopping, ranges, and straggling in the low-energy region
,” Phys. Rev. B
15
, 2458
–2468
(1977
).85.
C. F.
Clauser
and
N. R.
Arista
, “
Stopping power of dense plasmas: The collisional method and limitations of the dielectric formalism
,” Phys. Rev. E
97
, 023202
(2018
).86.
X.
Andrade
,
S.
Hamel
, and
A. A.
Correa
, “
Negative differential conductivity in liquid aluminum from real-time quantum simulations
,” Eur. Phys. J. B
91
, 229
(2018
).87.
A.
Kononov
,
T.
Hentschel
,
S.
Hansen
, and
A.
Baczewski
, “
Trajectory sampling and finite-size effects in first-principles stopping calculations
,” (unpublished).88.
H.
Bethe
, “
Zur theorie des durchgangs schneller korpuskularstrahlen durch materie
,” Ann. Phys.
397
, 325
–400
(1930
).89.
G.
Kresse
and
J.
Furthmüller
, “
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
,” Phys. Rev. B
54
, 11169
(1996
).90.
G.
Kresse
and
J.
Furthmüller
, “
Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
,” Comput. Mater. Sci.
6
, 15
–50
(1996
).91.
G.
Kresse
and
D.
Joubert
, “
From ultrasoft pseudopotentials to the projector augmented-wave method
,” Phys. Rev. B
59
, 1758
(1999
).92.
P. E.
Blöchl
, “
Projector augmented-wave method
,” Phys. Rev. B
50
, 17953
(1994
).93.
N. D.
Mermin
, “
Thermal properties of the inhomogeneous electron gas
,” Phys. Rev.
137
, A1441
(1965
).94.
A. E.
Mattsson
,
P. A.
Schultz
,
M. P.
Desjarlais
,
T. R.
Mattsson
, and
K.
Leung
, “
Designing meaningful density functional theory calculations in materials science—A primer
,” Modell. Simul. Mater. Sci. Eng.
13
, R1
(2004
).95.
A.
Zangwill
and
P.
Soven
, “
Resonant photoemission in barium and cerium
,” Phys. Rev. Lett.
45
, 204
–207
(1980
).96.
A.
Zangwill
and
P.
Soven
, “
Resonant two-electron excitation in copper
,” Phys. Rev. B
24
, 4121
–4127
(1980
).97.
X.
Qian
,
J.
Li
,
X.
Lin
, and
S.
Yip
, “
Time-dependent density functional theory with ultrasoft pseudopotentials: Real-time electron propagation across a molecular junction
,” Phys. Rev. B
73
, 035408
(2006
).98.
Y.
Saad
and
M. H.
Schultz
, “
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
,” SIAM J. Sci. Stat. Comput.
7
, 856
–869
(1986
).99.
A.
Ojanperä
,
V.
Havu
,
L.
Lehtovaara
, and
M.
Puska
, “
Nonadiabatic Ehrenfest molecular dynamics within the projector augmented-wave method
,” J. Chem. Phys.
136
, 144103
(2012
).100.
A.
Sakko
,
A.
Rubio
,
M.
Hakala
, and
K.
Hämäläinen
, “
Time-dependent density functional approach for the calculation of inelastic x-ray scattering spectra of molecules
,” J. Chem. Phys.
133
, 174111
(2010
).101.
A.
Baldereschi
, “
Mean-value point in the Brillouin zone
,” Phys. Rev. B
7
, 5212
–5215
(1973
).102.
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