Time-dependent quantum many-body systems: linear response, electronic transport, and reduced density matrices [Elektronische Ressource] / vorgelegt von Heiko Appel
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Time-dependent quantum many-body systems: linear response, electronic transport, and reduced density matrices [Elektronische Ressource] / vorgelegt von Heiko Appel

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Time-Dependent Quantum Many-Body Systems:Linear Response, Electronic Transport,and Reduced Density MatricesDissertationzur Erlangung des akademischen GradesDoctor rerum naturaliumvorgelegt vonHeiko AppelInstitut fu¨r Theoretische PhysikFreie Universit¨at BerlinMai 2007c Heiko Appel, 2007Diese Arbeit wurde im Zeitraum von Oktober 2003 bis Mai 2007 in der Arbeitsgruppe vonProf. E. K. U. Gross am Fachbereich Physik der Freien Universit¨at Berlin erstellt.Die Arbeit ist in elektronischer Form unter folgender URL verfu¨gbar∼http://www.physik.fu berlin.de/ appel/PhDThesis/Eingereicht am: 21. Mai 20071. Gutachter: Prof. Dr. E. K. U. Gross2. Gutachter: Prof. Dr. K. D. SchotteTag der mu¨ndlichen Pru¨fung: 12. Juli 2007AbstractTime-dependent density functional theory (TDDFT) provides a successful approach to cal-culateexcitationenergiesofatomicandmolecularsystems. InpartIofthisworkwepresenta double-pole approximation (DPA) to the response equations of TDDFT. The double-poleapproximation provides an exact description of systems with two strongly coupled exci-tations which are isolated from the rest of the spectrum. In contrast to the traditionalsingle-pole approximation of TDDFT the DPA also yields corrections to the Kohn-Shamoscillator strengths. Several critical pole separations can be identified, e.g. we find thatthe pole coupling can cause transitions to vanish entirely from the optical spectrum.

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Time-Dependent Quantum Many-Body Systems:
Linear Response, Electronic Transport,
and Reduced Density Matrices
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum naturalium
vorgelegt von
Heiko Appel
Institut fu¨r Theoretische Physik
Freie Universit¨at Berlin
Mai 2007c Heiko Appel, 2007
Diese Arbeit wurde im Zeitraum von Oktober 2003 bis Mai 2007 in der Arbeitsgruppe von
Prof. E. K. U. Gross am Fachbereich Physik der Freien Universit¨at Berlin erstellt.
Die Arbeit ist in elektronischer Form unter folgender URL verfu¨gbar
∼http://www.physik.fu berlin.de/ appel/PhDThesis/
Eingereicht am: 21. Mai 2007
1. Gutachter: Prof. Dr. E. K. U. Gross
2. Gutachter: Prof. Dr. K. D. Schotte
Tag der mu¨ndlichen Pru¨fung: 12. Juli 2007Abstract
Time-dependent density functional theory (TDDFT) provides a successful approach to cal-
culateexcitationenergiesofatomicandmolecularsystems. InpartIofthisworkwepresent
a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole
approximation provides an exact description of systems with two strongly coupled exci-
tations which are isolated from the rest of the spectrum. In contrast to the traditional
single-pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham
oscillator strengths. Several critical pole separations can be identified, e.g. we find that
the pole coupling can cause transitions to vanish entirely from the optical spectrum. We
also demonstrate how to invert the double-pole solution which allows us to predict matrix
elements of the exchange-correlation kernelf from experimental input. This can serve asxc
benchmark for the construction of future approximations for the kernel f .xc
Reduced density matrix functional theory (RDMFT) has emerged recently as promising
candidate to treat strongly correlated electronic many-body systems beyond traditional
density functional theory (DFT). The research within RDMFT was so far focussed on the
static theory. In this work we attempt some first steps towards a time-dependent gen-
eralization of RDMFT. In part II we derive equations of motion for natural orbitals and
occupation numbers. Using the equation of motion for the occupation numbers we show
that an adiabatic extension of presently known ground-state functionals of static RDMFT
always leads to occupation numbers which are constant in time. From the stationary con-
ditions of the equations of motion for the N-body correlations (correlated parts of the
N-body matrices) we derive a new class of ground-state functionals which can be used in
static RDMFT. Applications are presented for a one-dimensional model system where the
time-dependent many-body Schrodin¨ ger equation can be propagated numerically. We use
optimal control theory to find optimized laser pulses for transitions in a model for atomic
Helium. From the numerically exact correlated wavefunction we extract the exact time
evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii)
electron-ion scattering.
Part III of this work considers time-dependent quantum transport within TDDFT. We
present an algorithm for the calculation of extended eigenstates of single-particle Hamil-
tonians which is especially tailored to a finite-difference discretization of the Schrodin¨ ger
equation. We consider the propagation of finite mesoscopic systems and demonstrate the
limitationsofsuchanapproach. Toovercometheshortcomingsofadescriptionofquantum
transport in terms of finite systems we develop a time-propagation scheme for extended
stateswhichutilizesamixedbasisrepresentation. Ourdiscretizationschemeallowstotreat
central device and lead regions on the same footing thus preventing artificial reflections at
grid boundaries.Contents
List of Figures v
List of Tables vii
Abbreviations ix
Notation xi
1 Introduction 1
I Double-Pole Approximation in Time-Dependent Density Functional
Theory 5
2 Foundations of Time-Dependent Density Functional Theory 7
2.1 Runge-Gross Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Time-Dependent Kohn-Sham Scheme. . . . . . . . . . . . . . . . . . . . . . 10
2.3 Linear-Response Formulation of TDDFT . . . . . . . . . . . . . . . . . . . . 12
3 Excitation Energies in Time-Dependent Density Functional Theory 15
3.1 Single-Pole Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Double-Pole Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Exact Solution of Casida’s Equations . . . . . . . . . . . . . . . . . . 18
3.2.2 Model Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 Magic Positions in the Spectra . . . . . . . . . . . . . . . . . . . . . 23
3.2.4 Strength of the Interaction . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.5 Frequency-Dependent Kohn-Sham Oscillator Strengths. . . . . . . . 26
3.2.6 Inversion of the Double-Pole Solution . . . . . . . . . . . . . . . . . 26
3.3 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
II Time-Dependent Natural Orbitals and Reduced Density Matrices 29
4 Static Reduced Density Matrix Functional Theory 31
4.1 Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Direct Minimization of the Total Energy in Terms of γ . . . . . . . . . . . 352
4.3 Direct Minimization of the Total Energy in Terms of γ . . . . . . . . . . . 371ii Contents
5 Time-Dependent Reduced Density Matrix Functional Theory 41
5.1 The BBGKY Hierarchy of Reduced Density Matrices . . . . . . . . . . . . . 42
5.1.1 Formulation in Terms of N-Body Matrices. . . . . . . . . . . . . . . 43
5.1.2 Formu in Terms of N-Body Correlations . . . . . . . . . . . . 45
5.1.3 Orbital Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Time-Dependent Natural Spin Orbitals and Occupation Numbers . . . . . . 49
5.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.2 Cluster Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.3 Phase Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.4 Time-Dependent Hartree-Fock Limit . . . . . . . . . . . . . . . . . . 55
5.2.5 Adiabatic Extension of Ground-State Functionals . . . . . . . . . . . 56
5.3 Obtaining Static Functionals from TDRDMFT . . . . . . . . . . . . . . . . 58
5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4.1 Correlation Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4.2 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4.3 Model System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.4 Atoms in Strong Laser Fields . . . . . . . . . . . . . . . . . . . . . . 71
5.4.5 Electron-Ion Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
III ElectronicTransportwithinTime-DependentDensityFunctionalTheory 83
6 Ab-Initio Approaches to Electronic Transport 85
6.1 Real-Space Algorithm for Scattering States . . . . . . . . . . . . . . . . . . 91
7 Ab-Initio Methods for Time-Dependent Electronic Transport 97
7.1 Comparison of Approaches for the Solution of the TDKS Equations . . . . 97
7.2 Quantum Transport in Finite Systems . . . . . . . . . . . . . . . . . . . . . 103
7.3 Propagation with a Hybrid Basis . . . . . . . . . . . . . . . . . . . . . . . . 112
7.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
IV Appendix 121
A Matrix Formulation of the TDDFT Response Equations 123
B Domain Parallelization 127
B.1 Parallelization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 Technical Aspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.3 Application to Cs @C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298 60
Bibliography 131
Deutsche Kurzfassung 145
Publications 147
Acknowledgements 149Contents iii
Index 151
Colophon 155iv ContentsList of Figures
3.1 Schematic illustration of a 3-level model as considered in the double-pole
approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Interacting and Kohn-Sham spectra as function of frequency. . . . . . . . . 22
3.3 Excitation energies and oscillator strengths as function of frequency. . . . . 22
3.4 Interacting and Kohn-Sham spectra at critical values ω and ω . . . . . . . 23d c
3.5 In and Kohn-Sham spectra at the critical value ω . . . . . . . . . . 24e
3.6 Spectra for stronger coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7 The scaled coupling angle θ/π as function of frequency. . . . . . . . . . . . 25
3.8 Linear frequency dependence of Kohn-Sham oscillator strengths. . . . . . . 26
5.1 Natural orbitals of the reduced one-body density matrix of the ground state
and the three lowest excited states of Helium. . . . . . . . . . . . . . . . . . 68
5.2 Real-space representations of reduced one-body density matrices of corre-
lated Helium at different interaction strengths λ. . . . . . . . . . . . . . . . 70
5.3 Optimal laser pulse (t), correlation entropy s(t), and the two largest oc-
cupation numbers n (t) during the transition from the ground state to thek
first excited state of Helium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Same as Fig. 5.3, but with λ=0.6. . . . . . . . . . . . . . . . . . . . . . . . 72
25.5 Timeevolutionoftheelectrondensityγ (x,x;t)andorbitaldensities|ϕ (x;t)|1 j
for different interaction strengths λ. . . . . . . . . . . . . . . . . . . . . . . 75
− +5.6 Correlation entropy and occupation numbers of e – He scattering for
different interaction strengths λ. . . . . . . . . . . . . . . . . . . . . . . . . 77
− +5.7 Space-time plot of the electron density for e – He scattering. . . . . . . . 77
− +5.8 Correlation entropy and occupation numbers of e – He scattering for a
second parameter set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
0 − +5.9 Real-part of the reduced one-body density matrix γ (x,x;t) for e – He1
scattering at different points in time. . . . . . . . . . . . . . . . . . . . . . . 79
5.10 Same as Fig. 5.9, but for an interaction strength of λ=2.0. . . . . . . . . . 80
6.1 Average CMOS structure size of commercially available microprocessors at
their market launch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Atomic force microscope tomographs of a field-programmable nanowire in-
terconnect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Uniform finite-difference mesh spanning a generic transport geometry with
FCC unit cells for the lead materials. . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Scattering state with energy E =0.2 a.u. for an attractive potential well. . 95
6.5 Reflection and transmission coefficients as function of energy. . . . . . . . . 96
7.1 Top view on the external potentials for the 2D model in the x-y plane. . . . 104
7.2 Example for the definition of integration areas. . . . . . . . . . . . . . . . . 104vi List of Figures
7.3 Electron density for transport through a quantum stadium. . . . . . . . . . 106
7.4 SameasFig.7.3,butshownisthelongitudinalcomponentoftheKohn-Sham
current density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.5 Reservoir depletion for dot and stadium reservoirs. . . . . . . . . . . . . . . 107
7.6 Impurity scattering with non-interacting electrons. . . . . . . . . . . . . . . 108
7.7 Impurity scattering in the TD-Hartree approximation. . . . . . . . . . . . . 108
7.8 Integratedchargedensitiesandlongitudinalcurrentdensitiesforchargeflow
through a quantum ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.9 Electron density for charge flow through a quantum ring. . . . . . . . . . . 111
7.10 Same as Fig. 7.9, but shown is the transversal component of the current
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.11 Snapshots for the time evolution of a Gaussian wave packet in the mixed-
basis representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.12 Same as Fig.7.11, buthere the Gaussianwave packet has aninitialmomen-
tum of 1 a.u. pointing to the right. . . . . . . . . . . . . . . . . . . . . . . 116
7.13 Time evolution of a single lead mode in the hybrid scheme. . . . . . . . . . 117
B.1 Parallelization modes for DFT/TDDFT codes. . . . . . . . . . . . . . . . . . 127
B.2 Ghost points in a domain parallelization. . . . . . . . . . . . . . . . . . . . 128
B.3 Measured speedups for a domain-parallel calculation of Cs @C . . . . . . 1308 60