Atomic clusters and nanoparticles

Atomic clusters and nanoparticles......Page 1
Preface......Page 2
CONTENTS......Page 6
1......Page 17
1 Introduction......Page 19
2 Subshells, shells and supershells......Page 20
3 The experiment......Page 23
4 Observation of electronic shell structure......Page 24
5 Density functional calculation......Page 28
6 Observation of supershells......Page 31
7 Fission......Page 36
8 Concluding remarks......Page 42
References......Page 43
2......Page 45
1 Introduction......Page 47
2.1 The bulk limit......Page 49
2.2 Calorimetry for free clusters......Page 50
3 Experiment......Page 52
3.1 The source for thermalized cluster ions......Page 54
4 Caloric curves......Page 55
4.1 Melting temperatures......Page 56
4.2 Latent heats......Page 58
4.3 Other experiments measuring thermal properties of free clusters......Page 59
5.1.2 A canonical distribution of initial energies......Page 60
5.1.3 Free clusters in vacuum, a microcanonical ensemble......Page 61
5.2.2 Mapping of the energy on the mass scale......Page 63
5.2.3 Broadening of the mass spectra due to the statistics of evaporation......Page 64
5.3 Canonical or microcanonical data evaluation......Page 65
6.1 Negative heat capacity......Page 66
7 Unsolved problems......Page 68
8 Summary and outlook......Page 69
References......Page 71
3......Page 73
1 Introduction......Page 75
2 Statistical reaction theory......Page 79
2.1 Cluster evaporation rates......Page 82
2.2 Electron emission......Page 85
2.3 Radiative cooling......Page 86
3 Optical properties of small particles......Page 87
3.1 Connections to the bulk......Page 88
3.2 Linear response and short-time behavior......Page 89
3.3 Collective excitations......Page 92
4 Calculating the electron wave function......Page 93
4.1 Time-dependent density functional theory......Page 98
5.1 Alkali metal clusters......Page 100
5.2 Silver clusters......Page 102
6 Carbon structures......Page 105
6.1 Chains......Page 106
6.2 Polyenes......Page 110
6.3 Benzene......Page 111
6.4 $C_{60}$......Page 114
6.5 Carbon nanotubes......Page 115
References......Page 118
4......Page 121
1 Introduction......Page 123
2 Density functional theory......Page 124
2.1 Hohenberg and Kohn theorems......Page 126
2.2 Levy’s constrained search......Page 127
2.3 Kohn–Sham method......Page 128
3 Density matrices and pair correlation functions......Page 129
4 Adiabatic connection or coupling strength integration......Page 131
5 Comparing and constrasting KS-DFT and HF-CI......Page 134
6 Preparing new functionals......Page 138
7 Approximate exchange and correlation functionals......Page 139
7.1 The Local Spin Density Approximation (LSDA)......Page 140
7.2 Gradient Expansion Approximation (GEA)......Page 142
7.3 Generalized Gradient Approximation (GGA)......Page 143
7.4 meta-Generalized Gradient Approximation (meta-GGA)......Page 145
7.5 Hybrid functionals......Page 146
7.6 The Optimized E.ective Potential method (OEP)......Page 147
8 LAP correlation functional......Page 148
9 Solving the Kohn–Sham equations......Page 150
9.1 The Kohn–Sham orbitals......Page 152
9.2 Coulomb potential......Page 154
9.4 Core potential......Page 155
9.6 Functionality......Page 156
10 Applications......Page 157
10.1 Ab initio molecular dynamics for an alanine dipeptide model......Page 158
10.2.1 Vanadium trimer......Page 160
10.2.2 Nickel clusters......Page 161
10.3 The conversion of acetylene to benzene on Fe clusters......Page 165
11 Conclusions......Page 170
References......Page 171
5......Page 177
1 Introduction......Page 180
2.1 Thomas–Fermi approximation......Page 181
2.2 Wigner–Kirkwood expansion......Page 182
2.3 Gradient expansion of density functionals......Page 184
2.4 Density variational method......Page 185
2.5.1 Restricted spherical density variation......Page 189
2.5.2 Unrestricted spherical density variation......Page 193
2.5.3 Liquid drop model for charged spherical metal clusters......Page 194
3 Periodic orbit theory for quantum shell effects......Page 196
3.1 Semiclassical expansion of the Green function......Page 197
3.2 Trace formulae for level density and total energy......Page 198
3.3 Calculation of periodic orbits and their stability......Page 203
3.4 Uniform approximations......Page 206
3.5.1 Supershell structure of spherical alkali clusters......Page 208
3.5.2 Ground-state deformations......Page 210
3.6 Applications to two-dimensional electronic systems......Page 211
3.6.1 Conductance oscillations in a circular quantum dot......Page 213
3.6.3 Conductance oscillations in a channel with antidots......Page 216
4 Local-current approximation for linear response......Page 218
4.1 Quantum-mechanical equations of motion......Page 219
4.2 Variational equation for the local current density......Page 221
4.3 Secular equation using a finite basis......Page 223
4.4 Applications to metal clusters......Page 226
4.4.2 Optic response with ionic structure......Page 227
References......Page 231
6......Page 236
1 Introduction......Page 239
2 Basic mechanism: Cooper pair and condensation......Page 241
2.1 Condensed matter perspective: Electron pairs......Page 242
2.2 Nuclear physics perspective: Two nucleons in a shell......Page 244
2.3 Condensation of Cooper’s pairs......Page 245
3 Mean-.eld approach at .nite temperature......Page 246
3.2 Wick theorem......Page 250
3.3.1 Density operator, entropy, average particle number......Page 252
3.3.2 BCS equations......Page 253
3.3.3 Discussion; problems for finite systems......Page 254
3.3.4 Discussion; size of a Cooper pair......Page 255
3.4 Discussion; low temperature BCS properties......Page 256
4.1 Particle number projection......Page 258
4.2 Projected density operator......Page 259
4.3 Expectation values......Page 260
4.4 Projected BCS at $T = 0$, expectation values......Page 261
4.5 Projected BCS at $T = 0$, equations......Page 262
4.6 Projected BCS at $T = 0$, generalized gaps and single particle shifts......Page 263
5.1 General method for constructing stationary principles......Page 265
5.2.1 Characteristic function......Page 266
5.2.2 Transposition of the general procedure......Page 267
5.2.3 General properties......Page 268
6 Variational principle applied to extended BCS......Page 269
6.1 Variational spaces and group properties......Page 270
6.2 Extended BCS functional......Page 271
6.3 Extended BCS equations......Page 272
6.4 Properties of the extended BCS equations......Page 273
6.5 Recovering the BCS solution......Page 274
6.6 Beyond the BCS solution......Page 275
7.1 Particle number projected action......Page 276
7.2 Number projected stationary equations: sketch of the method......Page 277
8.1 Projection and action......Page 278
3.1.1 Duplicated representation......Page 247
3.1.2 Basic operators......Page 248
8.2 Variational equations......Page 280
8.3 Average values and thermodynamic potentials......Page 283
8.4 Small temperatures......Page 284
8.4.2 Odd number systems......Page 285
8.5 Numerical illustration......Page 287
9.1 Number parity projected free energy differences......Page 289
9.2 Nuclear odd–even energy differences......Page 292
10.1 Zero temperature......Page 298
10.2 Finite temperatures......Page 302
11 Conclusions and perspectives......Page 306
References......Page 308
7......Page 311
2 Jellium model and the density functional theory......Page 313
3 Spherical jellium clusters......Page 316
4 Effect of the lattice......Page 319
5 Tight-binding model......Page 322
6 Shape deformation......Page 323
8 Odd–even staggering in metal clusters......Page 329
9 Ab initio electronic structure: Shape and photoabsorption......Page 331
10 Quantum dots: Hund’s rule and spin-density waves......Page 334
11 Deformation in quantum dots......Page 338
12 Localization of electrons in a strong magnetic field......Page 340
References......Page 344
8......Page 348
1 Introduction......Page 350
2.1 Localized electron magnetism......Page 351
2.1.1 Magnetic configurations of atoms: Hund’s rules......Page 352
2.1.2 Magnetic susceptibility of open-shell ions in insulators......Page 354
2.1.3 Interaction between local moments: Heisenberg model......Page 356
2.2 Stoner model of itinerant magnetism......Page 358
2.3 Localized and itinerant aspects of magnetism in solids......Page 360
3 Experiments on magnetic clusters......Page 361
4.1 Model Hamiltonians......Page 365
4.2 Mean-field approximation......Page 367
4.3 Second-moment approximation: Enhancement of local and average spin magnetic moments......Page 369
4.4.1 Free clusters: Surface effects......Page 371
4.4.2 Embedded clusters: Interface effects......Page 374
4.5.1 Relativistic corrections......Page 377
4.5.2 Magnetic anisotropy of small clusters......Page 379
4.5.3 Enhancement of orbital magnetism......Page 382
5.1 The Hubbard model......Page 386
5.2 Geometry optimization in graph space......Page 387
5.3 Ground-state structure and total spin......Page 388
5.4 Comparison with non-collinear Hartree–Fock......Page 391
6 Finite-temperature magnetic properties of clusters......Page 397
6.1 Spin-fluctuation theory of cluster magnetism......Page 398
6.2 Environment dependence of spin fluctuation energies......Page 401
6.3 Role of electron correlations and structural fluctuations......Page 404
7 Conclusion......Page 409
References......Page 410
9......Page 414
1 Introduction......Page 416
2 Jellium model: Cluster electron wave functions......Page 418
3 Diffraction of fast electrons on clusters: Theory and experiment......Page 420
4 Elements of many-body theory......Page 422
5 Inelastic scattering of fast electrons on metal clusters......Page 425
6 Plasmon resonance approximation: Diffraction phenomena, comparison with experiment and RPAE......Page 428
7 Surface and volume plasmon excitations in the formation of the electron energy loss spectrum......Page 434
8 Polarization effects in low-energy electron cluster collision and the photon emission process......Page 438
9 How electron excitations in a cluster relax......Page 442
References......Page 445
10......Page 449
1 Introduction......Page 451
1.1 Levinthal’s paradox......Page 452
1.2 “Strong” and “fragile” liquids......Page 455
2 The Born–Oppenheimer approximation......Page 458
2.1.1 Orthogonal transformations......Page 459
2.1.2 The normal mode transformation......Page 461
3.1 Introduction......Page 463
4 Stationary points and pathways......Page 465
4.1 Zero Hessian eigenvalues......Page 466
4.2 Classification of stationary points......Page 468
4.3 Pathways......Page 469
4.4.2 Steepest-descent paths from a transition state......Page 470
4.4.3 Principal directions......Page 473
4.4.4 Birth and death of symmetry elements......Page 474
4.5 Classification of rearrangements......Page 477
4.6 The McIver–Stanton rules......Page 479
4.7 Coordinate transformations......Page 480
4.7.1 “Mass-weighted” steepest-descent paths......Page 483
4.7.2 Sylvester’s law of inertia......Page 484
4.8 Branch points......Page 486
5 Tunnelling......Page 489
5.2 Tunnelling in $(H_{2}O)_3$......Page 492
6.1 The superposition approximation......Page 493
6.2 Sample incompleteness......Page 497
6.3 Thermodynamics and cluster simulation......Page 498
6.4 Example: Isomerisation dynamics of $LJ_7$......Page 503
7 Finite size phase transitions......Page 505
7.1 Stability and van der Waals loops......Page 506
8 Global optimisation......Page 511
8.1 Basin-hopping global optimisation......Page 512
References......Page 514
11......Page 520
1 Introduction......Page 522
2 Key points and advantages of the confinement simulations: General remarks......Page 528
3.1 Conventional molecular dynamics......Page 530
3.2 Stochastic molecular dynamics......Page 531
4.1 Quenching procedure......Page 532
4.2 Characterization of a minimum......Page 533
5.1 Reversal of the trajectory at the boundary of the basin. Microcanonical ensemble......Page 534
5.2 Initiating the trajectory at the point of the last quenching within the basin. Microcanonical and canonical ensembles......Page 541
6.1 Fractional caloric curves and densities of states of the isomers [51, 52]......Page 544
6.2 Rates of the transitions between catchment basins. Estimation of the rate of a complex transition by successive confinement [50, 52]......Page 548
6.3 Creating a subsystem of a complex system. Self-di.usion in the subsystem of permutational isomers [52, 63]......Page 550
7 Complex study of a system by successive confinement......Page 552
7.1.2 A taboo search strategy. Fermi-like distribution over the minima......Page 553
7.2 Kinetics......Page 562
7.3 Equilibrium properties......Page 564
7.4 Study of the alanine tetrapeptide......Page 565
8 Concluding remarks......Page 571
References......Page 572
12......Page 575
1.1 The hierarchy of interactions between elementary particles, atoms and molecules......Page 577
1.2 The origin and phenomenological description of vdW interactions......Page 578
2 Calculation of interaction energy......Page 580
3 Vibrational frequencies......Page 583
4 Potential energy surface......Page 584
5 Free energy surface......Page 586
6.1 Benzene$...Ar_n$ clusters......Page 587
6.2 Aromatic system dimers and oligomers......Page 588
6.3 Nucleic acid–base pairs......Page 590
References......Page 592