Interacting electrons: theory and computational approaches
Gespeichert in:
| Beteiligte Personen: | , , |
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| Format: | Buch |
| Sprache: | Englisch |
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Cambridge
Cambridge University Press
2016
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| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028913652&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028913652&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Literaturverzeichnis Seite [750]-805 |
| Umfang: | xxiv, 818 Seiten Illustrationen, Diagramme |
| ISBN: | 9780521871501 0521871506 |
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| 245 | 1 | 0 | |a Interacting electrons |b theory and computational approaches |c Richard M. Martin (University of Illinois Urbana-Champaign), Lucia Reining (École Polytechnique Palaiseau), David M. Ceperley (University of Illinois Urbana-Champaign) |
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Contents
Preface
Acknowledgments
Notation
Part I Interacting electrons: beyond the independent-particle picture
1 The many-electron problem: introduction
Summary
1.1 The electronic structure problem
1.2 Why is this problem hard?
1.3 Why is the independent-electron picture so successful?
1.4 Development of theoretical approaches to the many-body problem
1.5 The many-body problem and computation
1.6 The scope of this book
Select further reading
2 Signatures of electron correlation
Summary
2.1 What is meant by correlation?
2.2 Ground-state and thermodynamic properties
2.3 Magnetism and local moments
2.4 Electron addition and removal: the bandgap problem and more
2.5 Satellites and sidebands
2.6 Particle-hole and collective excitations
2.7 The Kondo effect and heavy fermions
2.8 Mott insulators and metal—insulator transitions
2.9 Lower dimensions: stronger interaction effects
2.10 Wrap-up
3 Concepts and models for interacting electrons
Summary
3.1 The Wigner transition and the homogeneous electron system
3.2 The Mott transition and the Hubbard model
3.3 Magnetism and spin models
page xvii
xix
xx
1
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2
3
5
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10
13
14
15
15
16
17
20
21
26
29
32
33
36
39
40
40
40
43
48
viii
Contents
3.4 Normal metals and Fermi liquid theory 49
3.5 The Kondo effect and the Anderson impurity model 51
3.6 The Luttinger theorem and the Friedel sum rule 54
Select further reading 56
Exercises 56
Part II Foundations of theory for many-body systems
4 Mean fields and auxiliary systems 59
Summary 59
4.1 The Hartree and Hartree-Fock approximations 61
4.2 Weiss mean field and the Curie-Weiss approximation 64
4.3 Density functional theory and the Kohn—Sham auxiliary system 65
4.4 The Kohn—Sham electronic structure 70
4.5 Extensions of the Kohn-Sham approach 72
4.6 Time-dependent density and current density functional theory 76
4.7 Symmetry breaking in mean-field approximations and beyond 78
4.8 Wrap-up 80
Select further reading 80
Exercises 81
5 Correlation functions 84
Summary 84
5.1 Expectation values and correlation functions 85
5.2 Static one-electron properties 86
5.3 Static two-particle correlations: density correlations and the structure
factor 90
5.4 Dynamic correlation functions 93
5.5 Response functions 98
5.6 The one-particle Green’s function 105
5.7 Useful quantities derived from the one-particle Green’s function 111
5.8 Two-particle Green’s functions 116
Select further reading 120
Exercises 120
6 Many-body wavefunctions 122
Summary 122
6.1 Properties of the many-body wavefunction 123
6.2 Boundary conditions 124
6.3 The ground-state wavefunction of insulators 126
6.4 Correlation in two-electron systems 129
6.5 Trial function local energy, Feynman-Kac formula, and wavefunction
quality 131
6.6 The pair product or Slater—Jastrow wavefunction 134
Contents
ix
6.7 Beyond Slater determinants 139
Exercises 141
7 Particles and quasi-particles 144
Summary 144
7.1 Dynamical equations and Green’s functions for coupled systems 145
7.2 The self-energy and the Dyson equation 148
7.3 Illustration: a single state coupled to a continuum 151
7.4 Interacting systems: the self-energy and spectral function 152
7.5 Quasi-particles 157
7.6 Quasi-particle equations 161
7.7 Separating different contributions to a Dyson equation 163
7.8 Wrap-up 165
Select further reading 166
Exercises 166
8 Functionals in many-particle physics 169
Summary 169
8.1 Density functional theory and the Hartree-Fock approximation 171
8.2 Functionals of the Green’s function G and self-energy £ 174
8.3 Functionals of the screened interaction W 179
8.4 Generating functionals 182
8.5 Conservation laws and conserving approximations 187
8.6 Wrap-up 190
Select further reading 190
Exercises 191
Part III Many-body Green’s function methods
9 Many-body perturbation theory: expansion in the interaction 193
Summary 193
9.1 The Coulomb interaction and perturbation theory 194
9.2 Connecting the interacting and non-interacting systems 199
9.3 Telling the story of particles: diagrams 202
9.4 Making the story easier: two theorems 206
9.5 Dyson equation for the one-particle Green’s function, and the
self-energy 212
9.6 Diagrammatic expansion at non-vanishing temperature 213
9.7 Self-consistent perturbation theory: from bare to dressed building
blocks 215
9.8 The Luttinger-Ward functional 217
9.9 Wrap-up 219
Select further reading 219
Exercises 220
X
Contents
10 Many-body perturbation theory via functional derivatives 222
Summary 222
10.1 The equation of motion 223
10.2 The functional derivative approach 226
10.3 Dyson equations 228
10.4 Conservation laws 231
10.5 A starting point for approximations 234
10.6 Wrap-up 242
Select further reading 243
Exercises 243
11 The RPA and the GW approximation for the self-energy 245
Summary 245
11.1 Hedin’s equations 246
11.2 Neglecting vertex corrections in the polarizability: the RPA 251
11.3 Neglecting vertex corrections in the self-energy: the GW approximation 253
11.4 Link between the GWA and static mean-field approaches 260
11.5 Ground-state properties from the GWA 262
11.6 The GWA in the homogeneous electron gas 265
11.7 The GWA in small model systems 272
11.8 Wrap-up 277
Select further reading 278
Exercises 278
12 GWA calculations in practice 280
Summary 280
12.1 The task: a summary 281
12.2 Frequently used approximations 283
12.3 Core and valence 289
12.4 Different levels of self-consistency 292
12.5 Frequency integrations 298
12.6 GW calculations in a basis 302
12.7 Scaling and convergence 306
12.8 Wrap-up 308
Select further reading 309
Exercises 310
13 GWA calculations: illustrative results 311
Summary 311
13.1 From the HEG to a real semiconductor: silicon as a prototype system 312
13.2 Materials properties in the GWA: an overview 319
13.3 Energy levels in finite and low-dimensional systems 326
13.4 Transition metals and their oxides 329
13.5 GW results for the ground state 337
Contents
xi
13.6 A comment on temperature 341
13.7 Wrap-up 343
Select further reading 343
Exercises 344
14 RPA and beyond: the Bethe-Salpeter equation 345
Summary 345
14.1 The two-particle correlation function and measurable quantities 346
14.2 The two-particle correlation function: basic relations 348
14.3 The RPA: what can it yield? 350
14.4 Beyond the RPA: spin and frequency structure of the BSE 353
14.5 The Bethe-Salpeter equation in the GW approximation 355
14.6 A two-body Schrodinger equation 357
14.7 Importance and analysis of electron-hole interaction effects 361
14.8 Bethe-Salpeter calculations in practice 368
14.9 Applications 372
14.10 Extensions 379
14.11 Linear response using Green’s functions or density functionals 382
14.12 Wrap-up 385
Select further reading 387
Exercises 387
15 Beyond the GW approximation 389
Summary 389
15.1 The need to go beyond GW: analysis and observations 391
15.2 Iterating Hedin’s equations 393
15.3 Effects of vertex corrections 394
15.4 The T-matrix and related approximations 399
15.5 Beyond the T-matrix approximation: combining channels 402
15.6 T-matrix and related approaches in practice 406
15.7 Cumulants in electron spectroscopy 410
15.8 Use of exact constraints 415
15.9 Retrospective and outlook 417
Select further reading 418
Exercises 419
16 Dynamical mean-field theory 421
Summary 421
16.1 Auxiliary systems and embedding in Green’s function methods 423
16.2 Overview of DMFT 425
16.3 Expansion around an atomic limit: low energy scales and strong
temperature dependence 429
16.4 Background for mean-field theories and auxiliary systems 431
16.5 Dynamical mean-field equations 435
xii
Contents
16.6 Self-energy functional and variational equations 441
16.7 Static properties and density matrix embedding 442
16.8 Single-site DMFA in a two-site model 444
16.9 The Mott transition in infinite dimensions 445
16.10 Hybridized bands and consequences for the Mott transition 450
16.11 Interacting bands and spin transitions 451
16.12 Wrap-up 453
Select further reading 454
Exercises 454
17 Beyond the single-site approximation in DMFT 457
Summary 457
17.1 Supercells and clusters 45 8
17.2 Cellular DMFA 460
17.3 Dynamic cluster approximation 463
17.4 Variational cluster and nested cluster approximations 466
17.5 Extended DMFT and auxiliary bosons 467
17.6 Results for Hubbard models in one, two, and three dimensions 470
17.7 Wrap-up 475
Select further reading 476
Exercises 477
18 Solvers for embedded systems 479
Summary 479
18.1 The problem(s) to be solved 480
18.2 Exact diagonalization and related methods 481
18.3 Path-integral formulation in terms of the action 483
18.4 Auxiliary-field methods and the Hirsch—Fye algorithm 485
18.5 CTQMC: expansion in the interaction 487
18.6 CTQMC: expansion in the hybridization 491
18.7 Dynamical interactions in CTQMC 496
18.8 Other methods 498
18.9 Wrap-up 499
Select further reading 500
Exercises 500
19 Characteristic hamiltonians for solids with d and/ states 502
Summary 502
19.1 Transition elements: atomic-like behavior and local moments 503
19.2 Hamiltonian in a localized basis: crystal fields, bands, Mott-Hubbard
vs. charge transfer 507
19.3 Effective interaction hamiltonian 512
19.4 Identification of localized orbitals 513
Contents xiii
19.5 Combining DMFT and DFT 515
19.6 Static mean-field approximations: DFT+U, etc. 522
19.7 Wrap-up 525
Select further reading 525
Exercises 526
20 Examples of calculations for solids with d and/ states 527
Summary 527
20.1 Kondo effect in realistic multi-orbital problems 528
20.2 Lanthanides - magnetism, volume collapse, heavy fermions, mixed
valence, etc. 529
20.3 Actinides - transition from band to localized 536
20.4 Transition metals - local moments and ferromagnetism: Fe and Ni 537
20.5 Transition metal oxides: overview 540
20.6 Vanadium compounds and metal—insulator transitions 541
20.7 NiO - charge-transfer insulator, antiferromagnetism, and doping 543
20.8 MnO - metal-insulator and spin transitions 547
20.9 Wrap-up 549
Select further reading 551
Exercises 551
21 Combining Green’s functions approaches: an outlook 553
Summary 553
21.1 Taking advantage of different Green’s function methods 555
21.2 Partitioning the system 557
21.3 Combining different levels of diagrammatic approaches 559
21.4 Combining Green’s function methods: GW and DMFT 561
21.5 Dynamical interactions and constrained RPA 568
21.6 Consequences of dynamical interactions 570
21.7 Diagrammatic extensions: dynamical vertex approximation and dual
fermions 571
21.8 Wrap-up 574
Select further reading 574
Exercises 575
Part IV Stochastic methods
22 Introduction to stochastic methods 577
Summary 577
22.1 Simulations 578
22.2 Random walks and Markov chains 579
22.3 The Metropolis Monte Carlo method 581
22.4 Computing error bars 583
xiv
Contents
22.5 The “heat bath” algorithm 586
22.6 Remarks 587
Select further reading 588
Exercises 588
23 Variational Monte Carlo 590
Summary 590
23.1 Details of the variational Monte Carlo method 592
23.2 Optimizing trial wavefunctions 596
23.3 The momentum distribution and single-particle density matrix 598
23.4 Non-local pseudopotentials 599
23.5 Finite-size effects 601
23.6 VMC for lattice models 603
23.7 Excitations and orthogonality 603
23.8 Strengths and weaknesses of VMC 606
Select further reading 607
Exercises 608
24 Projector quantum Monte Carlo 609
Summary 609
24.1 Types and properties of projectors 610
24.2 The diffusion Monte Carlo method 612
24.3 Exact fermion methods: the sign or phase problem 621
24.4 The fixed-node and fixed-phase methods 623
24.5 Mixed estimators, exact estimators, and the overlap 628
24.6 Non-local pseudopotentials in PMC 630
24.7 Projector auxiliary-field quantum Monte Carlo methods 632
24.8 Applications of projector MC 636
24.9 The pluses and minuses of projector MC 639
Select further reading 642
Exercises 642
25 Path-integral Monte Carlo 644
Summary 644
25.1 The path-integral representation 645
25.2 Exchange of localized electrons 650
25.3 Quantum statistics and PIMC 652
25.4 Ground-state path integrals (GSPI) 659
25.5 Finite-temperature QMC for the Hubbard model 662
25.6 Estimating real-time correlation functions 665
25.7 Correlation-function QMC for excitations 669
Select further reading 672
Exercises 673
26 Concluding remarks
674
Contents
XV
PartV Appendices
Appendix A Second quantization 677
Summary 677
A. 1 First quantization 677
A. 2 Second quantization 678
Select further reading 682
Appendix B Pictures 683
Summary 683
B. 1 Schrodinger picture 684
B.2 Heisenberg picture 684
B. 3 Interaction picture 686
Select further reading 689
Exercises 689
Appendix C Green’s functions: general properties 690
Summary 690
C. 1 Green’s functions for differential equations 690
C.2 Fourier transforms and spectral representations 691
C.3 Frequency integrals 693
C.4 From many-body to few-body Green’s functions 695
C. 5 The thermodynamic limit 696
Select further reading 697
Exercises 697
Appendix D Matsubara formulation for Green’s functions for T ^ 0 699
Summary 699
D. l Green’s functions at T ^ 0: Matsubara frequencies 699
D.2 Analytic properties in the complex-frequency plane 702
D.3 Illustration of the structure of G°(i(on) and G°(r) 705
D.4 The grand potential £2 707
D. 5 Transformation to real frequencies 709
Select further reading 709
Exercises 709
Appendix E Time ordering, contours, and non-equilibrium 710
Summary 710
E. l The task 710
E.2 The contour interpretation 710
E. 3 Contours for all purposes 712
Select further reading 714
Appendix F Hedin’s equations in a basis 715
Summary 715
F. 1 Generalization of Hedin’s equations 715
XVI
Contents
F. 2 Hedin’s equations in a basis 717
Select further reading 717
Appendix G Unique solutions in Green’s function theory 719
Summary 719
G. l Which G°? Boundary conditions in time 719
G.2 Which G? Self-consistent Dyson equations 720
G. 3 Convergence of perturbation expansions and consequences 721
Select further reading 722
Exercises 722
Appendix H Properties of functionals 724
Summary 724
H. l Functionals and functional equations 724
H.2 Legendre transformations and invertibility 725
H.3 Examples of functionals for the total energy in Kohn-Sham DFT
calculations 726
H.4 Free-energy functionals for spin systems and proof of invertibility 727
H. 5 Extension to quantum spins and density functional theory 729
Select further reading 730
Exercises 730
Appendix I Auxiliary systems and constrained search 731
Summary 731
LI Auxiliary system to reproduce selected quantities 731
I. 2 Constrained search with an interacting auxiliary system 732
Exercises 734
Appendix J Derivation of the Luttinger theorem 735
Summary 735
Select further reading 737
Exercises 738
Appendix K Gutzwiller and Hubbard approaches 739
Summary 739
K. 1 Gutzwiller approach in terms of the wavefunction 740
K.2 Hubbard approach in terms of the Green’s function 742
K.3 Two scenarios for the Mott transition 747
Select further reading 748
Exercises 748
References 750
Index 806
Recent progress in the theory and computation of electronic structure is bringing
an unprecedented level of capability for research. Many-body methods are becoming
essential tools vital for quantitative calculations and understanding materials phenomena
in physics, chemistry, materials science, and other fields. This book provides a unified
exposition of the most-used tools: many-body perturbation theory, dynamical mean-field
theory, and quantum Monte Carlo simulations.
Each topic is introduced with a less technical overview for a broad readership, followed by
in-depth descriptions and mathematical formulation. Practical guidelines, illustrations, and
exercises are chosen to enable readers to appreciate the complementary approaches, their
relationships, and the advantages and disadvantages of each method. This book is designed
for graduate students and researchers who want to use and understand these advanced
computational tools, get a broad overview, and acquire a basis for participating in new
developments.
“This excellent book written by three world-leading experts provides dear explanations
and profound insights into the theory of computational methods for interacting electrons.”
Richard Needs, University of Cambridge
“This timely and excellent book is essential reading for anyone interested in modern materials
theory; a crucial resource for graduate students, postdocs and established researchers.”
Andrew Millis, Columbia University
Richard M. Martin is Emeritus Professorat the University of Illinois at Urbana-Champaign and
Consulting Professor at Stanford University. He has made extensive contributions to the field
of modern electronic structure methods and the theory of interacting electron systems and he
is the author of the companion book, Electronic Structure: Basic Theory and Methods.
Lucia Reining is CNRS Senior Researcher at École Polytechnique Palaiseau and founding
member of the European Theoretical Spectroscopy Facility. Her work covers many-body
perturbation theory and time-dependent density functional theory. She is a recipient of the
CNRS silver medal and fellow of the American Physical Society.
David M. Ceperley is a Founders and Blue Waters Professor at the University of Illinois at
Urbana-Champaign where he has pioneered the quantum Monte Carlo method, including
the development of variational, diffusion, and path-integral Monte Carlo. He is a member of
the US National Academy of Sciences and recipient of the Rahman Prize for Computational
Physics of the APS and the Feenberg Medal for Many-Body Physics.
ISBN 978-0-521-87150-1 |
| any_adam_object | 1 |
| author | Martin, Richard M. 1942- Reining, Lucia 1961- Ceperley, David M. 1949- |
| author_GND | (DE-588)131743732 (DE-588)1104865327 (DE-588)110486634X |
| author_facet | Martin, Richard M. 1942- Reining, Lucia 1961- Ceperley, David M. 1949- |
| author_role | aut aut aut |
| author_sort | Martin, Richard M. 1942- |
| author_variant | r m m rm rmm l r lr d m c dm dmc |
| building | Verbundindex |
| bvnumber | BV043497179 |
| classification_rvk | UM 1200 UP 3600 |
| classification_tum | PHY 026f |
| ctrlnum | (OCoLC)952552406 (DE-599)BVBBV043497179 |
| dewey-full | 539.7/2112 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 539 - Modern physics |
| dewey-raw | 539.7/2112 |
| dewey-search | 539.7/2112 |
| dewey-sort | 3539.7 42112 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Book |
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| id | DE-604.BV043497179 |
| illustrated | Illustrated |
| indexdate | 2025-02-03T19:00:18Z |
| institution | BVB |
| isbn | 9780521871501 0521871506 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028913652 |
| oclc_num | 952552406 |
| open_access_boolean | |
| owner | DE-11 DE-29T DE-20 DE-703 DE-188 DE-634 DE-384 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-355 DE-BY-UBR |
| owner_facet | DE-11 DE-29T DE-20 DE-703 DE-188 DE-634 DE-384 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-355 DE-BY-UBR |
| physical | xxiv, 818 Seiten Illustrationen, Diagramme |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Cambridge University Press |
| record_format | marc |
| spellingShingle | Martin, Richard M. 1942- Reining, Lucia 1961- Ceperley, David M. 1949- Interacting electrons theory and computational approaches Vielelektronenproblem (DE-588)4188249-0 gnd Dynamische Molekularfeldtheorie (DE-588)1060518759 gnd Vielteilchensystem (DE-588)4063491-7 gnd Elektronenwechselwirkung (DE-588)4508857-3 gnd Computersimulation (DE-588)4148259-1 gnd Green-Funktion (DE-588)4158123-4 gnd Elektronenstruktur (DE-588)4129531-6 gnd Mathematisches Modell (DE-588)4114528-8 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd Vielkörperproblem (DE-588)4078900-7 gnd |
| subject_GND | (DE-588)4188249-0 (DE-588)1060518759 (DE-588)4063491-7 (DE-588)4508857-3 (DE-588)4148259-1 (DE-588)4158123-4 (DE-588)4129531-6 (DE-588)4114528-8 (DE-588)4240945-7 (DE-588)4078900-7 |
| title | Interacting electrons theory and computational approaches |
| title_auth | Interacting electrons theory and computational approaches |
| title_exact_search | Interacting electrons theory and computational approaches |
| title_full | Interacting electrons theory and computational approaches Richard M. Martin (University of Illinois Urbana-Champaign), Lucia Reining (École Polytechnique Palaiseau), David M. Ceperley (University of Illinois Urbana-Champaign) |
| title_fullStr | Interacting electrons theory and computational approaches Richard M. Martin (University of Illinois Urbana-Champaign), Lucia Reining (École Polytechnique Palaiseau), David M. Ceperley (University of Illinois Urbana-Champaign) |
| title_full_unstemmed | Interacting electrons theory and computational approaches Richard M. Martin (University of Illinois Urbana-Champaign), Lucia Reining (École Polytechnique Palaiseau), David M. Ceperley (University of Illinois Urbana-Champaign) |
| title_short | Interacting electrons |
| title_sort | interacting electrons theory and computational approaches |
| title_sub | theory and computational approaches |
| topic | Vielelektronenproblem (DE-588)4188249-0 gnd Dynamische Molekularfeldtheorie (DE-588)1060518759 gnd Vielteilchensystem (DE-588)4063491-7 gnd Elektronenwechselwirkung (DE-588)4508857-3 gnd Computersimulation (DE-588)4148259-1 gnd Green-Funktion (DE-588)4158123-4 gnd Elektronenstruktur (DE-588)4129531-6 gnd Mathematisches Modell (DE-588)4114528-8 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd Vielkörperproblem (DE-588)4078900-7 gnd |
| topic_facet | Vielelektronenproblem Dynamische Molekularfeldtheorie Vielteilchensystem Elektronenwechselwirkung Computersimulation Green-Funktion Elektronenstruktur Mathematisches Modell Monte-Carlo-Simulation Vielkörperproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028913652&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028913652&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT martinrichardm interactingelectronstheoryandcomputationalapproaches AT reininglucia interactingelectronstheoryandcomputationalapproaches AT ceperleydavidm interactingelectronstheoryandcomputationalapproaches |
Inhaltsverzeichnis
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Teilbibliothek Physik
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