ISBN: 3540429085
TITLE: Atomic clusters and nanoparticles
AUTHOR: Guet, Hobza, Spiegelman, David (Hrsg.)
TOC:

Lecturers xi
Prface xvii
Preface xxi
Contents XXV
Course 1. Experimental Aspects of Metal Clusters
by T.P. Martin 1
Introduction 3
Subshells, shells and supershells 4
The experiment 7
Observation of electronic shell structure 8
Density functional calculation 12
Observation of supershells 15
Fission 20
Concluding remarks 26
Course 2. Melting of Clusters
by H. Haberland 29
1 Introduction 31
2 Cluster calorimetry 33
2.1 The bulk limit 33
2.2 Calorimetry for free clusters 34
3 Experiment 36
3.1 The source for thermalized cluster ions 38
4 Caloric curves 39
4.1 Melting temperatures 40
4.2 Latent heats 42
4.3 Other experiments measuring thermal properties of free clusters 43
5 A closer look at the experiment 44
5.1 Beam preparation 44
5.1.1 Reminder: Canonical versus microcanonical ensemble 44
5.1.2 A canonical distribution of initial energies 44
5.1.3 Free clusters in vacuum, a microcanonical ensemble 45
5.2 Analysis of the fragmentation process 47
5.2.1 Photo-excitation and energy relaxation 47
5.2.2 Mapping of the energy on the mass scale 47
5.2.3 Broadening of the mass spectra due to the statistics of evaporation 48
5.3 Canonical or microcanonical data evaluation 49
6 Results obtained from a closer look 50
6.1 Negative heat capacity 50
6.2 Entropy 52
7 Unsolved problems 52
8 Summary and outlook 53
Course 3. Excitations in Clusters
by G.F. Bertsch 57
1 Introduction 59
2 Statistical reaction theory 63
2.1 Cluster evaporation rates 66
2.2 Electron emission 69
2.3 Radiative cooling 70
3 Optical properties of small particles 71
3.1 Connections to the bulk 72
3.2 Linear response and short-time behavior 73
3.3 Collective excitations 76
4 Calculating the electron wave function 77
4.1 Time-dependent density functional theory 82
5 Linear response of simple metal clusters 84
5.1 Alkali metal clusters 84
5.2 Silver clusters 86
6 Carbon structures 89
6.1 Chains 90
6.2 Polyenes 94
6.3 Benzene 95
6.4 C_60 98
6.5 Carbon nanotubes 99
6.6 Quantized conductance 102
Course 4. Density Functional Theory, Methods, Techniques, and Applications
by S. Chrtien and D.R. Salahub 105
1 Introduction 107
2 Density functional theory 108
2.1 Hohenberg and Kohn theorems 110
2.2 Levy's constrained search 111
2.3 Kahn-Sham method 112
3 Density matrices and pair correlation functions 113
4 Adiabatic connection or coupling strength integration 115
5 Comparing and constrasting KS-DFT and HF-CI 118
6 Preparing new functionals 122
7 Approximate exchange and correlation functionals 123
7.1 The Local Spin Density Approximation (LSDA) 124
7.2 Gradient Expansion Approximation (GEA) 126
7.3 Generalized Gradient Approximation (GGA) 127
7.4 meta-Generalized Gradient Approximation (meta-GGA) 129
7.5 Hybrid functionals 130
7.6 The Optimized Effective potential method (OEP) 131
7.7 Comparison between various approximate functionals 132
8 LAP correlation functional 132
9 Solving the Kohn-Sham equations 134
9.1 The Kohn-Sham orbitals 136
9.2 Coulomb potential 138
9.3 Exchange-correlation potential 139
9.4 Core potential 139
9.5 Other choices and sources of error 140
9.6 Functionality 140
10 Applications 141
10.1 Ab initio molecular dynamics for an alanine dipeptide model 142
10.2 Transition metal clusters: The ecstasy, and the agony 144
10.2.1 Vanadium trimer 144
10.2.2 Nickel clusters 145
10.3 The conversion of acetylene to benzene on Fe clusters 149
11 Conclusions 154
Course 5. Semiclassical Approaches to Mesoscopic systems
by M. Brack 161
1 Introduction 164
2 Extended Thomas-Fermi model for average properties 165
2.1 ThomasFermi approximation 165
2.2 Wigner-Kirkwood expansion 166
2.3 Gradient expansion of density functionals 168
2.4 Density variational method 169
2.5 Applications to metal clusters 173
2.5.1 Restricted spherical density variation 173
2.5.2 Unrestricted spherical density variation 177
2.5.3 Liquid drop model for charged spherical metal clusters 178
3 Periodic orbit theory for quantum shell effects 180
3.1 Semiclassical expansion of the Green function 181
3.2 Trace formulae for level density and total energy 182
3.3 Calculation of periodic orbits and their stability 187
3.4 Uniform approximations 190
3.5 Applications to metal clusters 192
3.5.1 Supershell structure of spherical alkali clusters 192
3.5.2 Ground-state deformations 194
3.6 Applications to two-dimensional electronic systems 195
3.6.1 Conductance oscillations in a circular quantum dot 197
3.6.2 Integer quantum Hall effect in the two-dimensional electron gas 200
3.6.3 Conductance oscillations in a channel with antidots 200
4 Local-current approximation for linear response 202
4.1 Quantum-mechanical equations of motion 203
4.2 Variational equation for the local current density 205
4.3 Secular equation using a finite basis 207
4.4 Applications to metal clusters 210
4.4.1 Optic response in the jellium model 211
4.4.2 Optic response with ionic structure 211
Course 6. Pairing Correlations in Finite Fermionic systems
by H. Flocard 221
1 Introduction 225
2 Basic mechanism: Cooper pair and condensation 227
2.1 Condensed matter perspective: Electron pairs 228
2.2 Nuclear physics perspective: Two nucleons in a shell 230
2.3 Condensation of Cooper's pairs 231
3 Mean-field approach at finite temperature 232
3.1 Family of basic operators 233
3.1.1 Duplicated representation 233
3.1.2 Basic operators 234
3.1.3 BCS coefficients; quasi-particles 235
3.2 Wick theorem 236
3.3 BCS finite temperature equations 238
3.3.1 Density operator, entropy, average particle number 238
3.3.2 BCS equations 239
3.3.3 Discussion; problems for finite systems 240
3.3.4 Discussion; size of a Cooper pair 241
3.4 Discussion; low temperature BCS properties 242
4 First attempt at particle number restoration 244
4.1 Particle number projection 244
4.2 Projected density operator 245
4.3 Expectation values 246
4.4 Projected BCS at T = 0, expectation values 247
4.5 Projected BCS at T = 0, equations 248
4.6 Projected BCS at T = 0, generalized gaps and single particle shifts 249
5 Stationary variational principle for thermodynamics 251
5.1 General method for constructing stationary principles 251
5.2 Stationary action 252
5.2.1 Characteristic function 252
5.2.2 Transposition of the general procedure 253
5.2.3 General properties 254
6 Variational principle applied to extended BCS 255
6.1 Variational spaces and group properties 256
6.2 Extended BCS functional 257
6.3 Extended BCS equations 258
6.4 Properties of the extended BCS equations 259
6.5 Recovering the BCS solution 260
6.6 Beyond the BCS solution 261
7 Particle number projection at finite temperature 262
7.1 Particle number projected action 262
7.2 Number projected stationary equations: sketch of the method 263
8 Number parity projected BCS at finite temperature 264
8.1 Projection and action 264
8.2 Variational equations 266
8.3 Average values and thermodynamic potentials 269
8.4 Small temperatures 270
8.4.1 Even number systems 270
8.4.2 Odd number systems 271
8.5 Numerical illustration 273
9 Odd-even effects 275
9.1 Number parity projected free energy differences 275
9.2 Nuclear odd-even energy differences 278
10 Extensions to very small systems 284
10.1 Zero temperature 284
10.2 Finite temperatures 288
11 Conclusions and perspectives 292
Course 7. Models of Metal Clusters and Quantum Dots
by M. Manninen 297
1 Introduction 299
2 Jellium model and the density functional theory 299
3 Spherical jellium clusters 302
4 Effect of the lattice 305
5 Tight-binding model 308
6 Shape deformation 309
7 Tetrahedral and triangular shapes 315
8 Odd-even staggering in metal clusters 315
9 Ab initio electronic structure: Shape and photoabsorption 317
10 Quantum dots: Hund's rule and spin-density waves 320
11 Deformation in quantum dots 324
12 Localization of electrons in a strong magnetic field 326
13 Conclusions 330
Course 8. Theory of Cluster Magnetism
by G.M. Pastor 335
1 Introduction 337
2 Background on atomic and solid-state properties 338
2.1 Localized electron magnetism 338
2.1.1 Magnetic configurations of atoms: Hund's rules 339
2.1.2 Magnetic susceptibility of openshell ions in insulators 341
2.1.3 Interaction between local moments: Heisenberg model 343
2.2 Stoner model of itinerant magnetism 345
2.3 Localized and itinerant aspects of magnetism in solids 347
3 Experiments on magnetic clusters 348
4 Ground-state magnetic properties of transition-metal clusters 352
4.1 Model Hamiltonians 352
4.2 Mean-field approximation 354
4.3 Second-moment approximation 356
4.4 Spin magnetic moments and magnetic order 358
4.4.1 Free clusters: Surface effects 358
4.4.2 Embedded clusters: Interface effects 361
4.5 Magnetic anisotropy and orbital magnetism 364
4.5.1 Relativistic corrections 364
4.5.2 Magnetic anisotropy of small clusters 366
4.5.3 Enhancement of orbital magnetism 369
5 Electron-correlation effects on cluster magnetism 373
5.1 The Hubbard model 373
5.2 Geometry optimization in graph space 374
5.3 Ground-state structure and total spin 375
5.4 Comparison with non-tollinear Hartree-Fock 378
6 Finite-temperature magnetic properties of clusters 384
6.1 Spin-fluctuation theory of cluster magnetism 385
6.2 Environment dependence of spin fluctuation energies 388
6.3 Role of electron correlations and structural fluctuations 391
7 Conclusion 396
Course 9. Electron Scattering on Metal Clusters and Fullerenes
by A. V. Solov'yov 401
1 Introduction 403
2 Jellium model: Cluster electron wave functions 405
3 Diffraction of fast electrons on clusters: Theory and experiment 407
4 Elements of many-body theory 409
5 Inelastic scattering of fast electrons on metal clusters 412
6 Plasmon resonance approximation: Diffraction phenomena, comparison with experiment and RPAE 415
7 Surface and volume plasmon excitations in the formation of the electron energy loss spectrum 421
8 Polarization effects in low-energy electron cluster collision and the photon emission process 425
9 How electron excitations in a cluster relax 429
10 Concluding remarks 432
Course 10. Energy Landscapes
by D.J. Wales 437
1 Introduction 439
1.1 Levinthal's paradox 440
1.2 "Strong" and "fragile" liquids 443
2 The Born-Oppenheimer approximation 446
2.1 Normal modes 447
2.1.1 Orthogonal transformations 447
2.1.2 The normal mode transformation 449
3 Describing the potential energy landscape 451
3.1 Introduction 451
4 Stationary points and pathways 453
4.1 Zero Hessian eigenvalues 454
4.2 Classification of stationary points 456
4.3 Pathways 457
4.4 Properties of steepest-descent pathways 458
4.4.1 Uniqueness 458
4.4.2 Steepest-descent paths from a transition state 458
4.4.3 Principal directions 461
4.4.4 Birth and death of symmetry elements 462
4.5 Classification of rearrangements 465
4.6 The McIver-Stanton rules 467
4.7 Coordinate transformations 468
4.7.1 "Mass-weighted" steepest-descent paths 471
4.7.2 Sylvester's law of inertia 472
4.8 Branch points 474
5 Tunnelling 477
5.1 Tunnelling in (HF)_2 480
5.2 Tunnelling in (H_2O)_3 480
6 Global thermodynamics 481
6.1 The superposition approximation 481
6.2 Sample incompleteness 485
6.3 Thermodynamics and cluster simulation 486
6.4 Example: Isomerisation dynamics of LJ_7 491
7 Finite size phase transitions 493
7.1 Stability and van der Waals loops 494
8 Global optimisation 499
8.1 Basin-hopping global optimisation 500
Course 11. Confinement Technique for Simulating Finite Many-Body systems
by S.F. Chekmarev 509
1 Introduction 511
2 Key points and advantages of the confinement simulations: General remarks 517
3 Methods for generating phase trajectories 519
3.1 Conventional molecular dynamics 519
3.2 Stochastic molecular dynamics 520
4 Identification of atomic structures 521
4.1 Quenching procedure 521
4.2 Characterization of a minimum 522
5 Confinement procedures 523
5.1 Reversal of the trajectory at the boundary of the basin. Microcanonical ensemble 523
5.2 Initiating the trajectory at the point of the last quenching within the basin. Microcanonical and canonical ensembles 530
6 Confinement to a selected catchment area. Some applications 533
6.1 Fractional caloric curves and densities of states of the isomers 533
6.2 Rates of the transitions between catchment basins. Estimation of the rate of a complex transition by successive confinement 537
6.3 Creating a Subsystem of a complex system. Self-diffusion in the Subsystem of permutational isomers 539
7 Complex study of a system by successive confinement 541
7.1 Surveying a potential energy surface. Strategies 542
7.1.1 Strategies to survey a surface 542
7.1.2 A taboo search strategy. Fermi-like distribution over the minima 542
7.2 Kinetics 551
7.3 Equilibrium properties 553
7.4 Study of the alanine tetrapeptide 554
8 Concluding remarks 560
Course 12. Molecular Clusters: potential Energy and Free Energy Surfaces. Quantum Chemical ab initio and Computer Simulation Studies
by P. Hobza
1 Introduction 567
1.1 The hierarchy of interactions between elementary particles, atoms and molecules 567
1.2 The origin and phenomenological description of vdW interactions 568
2 Calculation of interaction energy 570
3 Vibrational frequencies 573
4 Potential energy surface 574
5 Free energy surface 576
6 Applications 577
6.1 Benzene. Ar_n clusters 577
6.2 Aromatic system dimers and oligomers 578
6.3 Nucleic acid-base pairs 580
Seminars by participants 585
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