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 ν ( ω ) 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 ν ( ω ). 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.

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.
T.
Hentschel
(2023). “,”
Zenodo.
https://doi.org/10.5281/zenodo.7812518
You do not currently have access to this content.