Stochastic Simulations of Clusters: Quantum Methods in Flat and Curved Spaces

STOCHASTIC SIMULATIONS of CLUSTERS: Quantum Methods in Flat and Curved Spaces......Page 3
Contents......Page 6
Preface......Page 14
Acknowledgments......Page 20
Author......Page 22
Part I: Fundamentals......Page 23
1.1 Introduction......Page 25
1.2 What is FORTRAN?......Page 26
1.4 Data Types......Page 27
1.5 The IMPLICIT Statement......Page 28
1.7 Order of Operations......Page 29
1.9 A Typical Chemistry Problem Solved with FORTRAN......Page 30
1.10 Free Format I/O......Page 31
1.11 The FORTRAN Code for the Tertiary Mixtures Problem......Page 32
Exercises......Page 34
1.12.1 Logical Expressions......Page 35
1.12.2 The Logical if Statement......Page 36
1.12.4 The Computed goto Statement......Page 38
1.12.5 The Case Statement (FORTRAN 90)......Page 39
Exercises......Page 40
1.13.1 Counter Controlled do-loop......Page 43
1.14 Intrinsic Functions......Page 44
1.15 User-Defined Functions......Page 45
1.16 Subroutines......Page 48
1.17 Numerical Derivatives......Page 50
1.18 The Extended Trapezoid Rule to Evaluate Integrals......Page 52
Exercises......Page 54
2.1 Introduction......Page 57
2.2 Some Important Variables of Classical Physics......Page 58
2.3 The Lagrangian and the Hamiltonian in Euclidean Spaces......Page 59
Exercises......Page 61
2.4 The Least Action Principle and the Equations of Motion......Page 62
2.5 The Two-Body Problem with Central Potential......Page 64
2.6 Isotropic Potentials and the Two-Body Problem......Page 67
2.7 The Rigid Rotor......Page 68
Exercises......Page 69
2.8 Numerical Integration Methods......Page 71
Exercises......Page 72
2.9 Hamilton’s Equations and Symplectic Integrators......Page 75
Exercises......Page 77
2.10 The Potential Energy Surface......Page 78
Exercises......Page 81
2.11 Dissipative Systems......Page 83
2.12 The Fourier Transform and the Position Autocorrelation Function......Page 84
Exercises......Page 87
3.1 Introduction......Page 91
3.2 Continuous Random Variables and Their Distributions......Page 92
3.3 The Classical Statistical Mechanics of a Single Particle......Page 94
3.4 The Monoatomic Ideal Gas......Page 96
3.5 The Equipartition Theorem......Page 97
Exercises......Page 100
3.6 Basics of Stochastic Computation......Page 101
3.7 Probability Distributions......Page 103
Exercises......Page 107
3.8 Minimizing V by Trial and Error......Page 109
Exercises......Page 111
3.9 The Metropolis Algorithm......Page 112
Exercises......Page 116
3.10 Parallel Tempering......Page 119
3.11 A Random Number Generator......Page 122
Exercises......Page 124
4.1 Introduction......Page 127
4.2 A Few Useful Definitions......Page 129
4.3 Groups......Page 130
4.4 Number Fields......Page 131
4.5 Vector Spaces......Page 132
4.6 Algebras......Page 135
Exercises......Page 138
4.7 The Exponential Mapping of Lie Algebras......Page 143
Exercises......Page 147
4.8 The Determinant of a n X n Matrix and the Levi–Civita Symbol......Page 149
4.9 Scalar Product, Outer Product, and Vector Space Mapping......Page 151
4.10 Rotations in Euclidean Space......Page 152
Exercises......Page 153
4.11 Complex Field Extensions......Page 155
4.12 Dirac Bra–Ket Notation......Page 159
4.13 Eigensystems......Page 161
4.14 The Connection between Diagonalization and Lie Algebras......Page 163
Exercises......Page 165
4.15 Symplectic Lie Algebras and Groups......Page 169
Exercises......Page 171
4.16 Lie Groups as Solutions of Differential Equations......Page 175
Exercises......Page 176
4.17 Split Symplectic Integrators......Page 177
4.18 Supermatrices and Superalgebras......Page 180
Exercises......Page 181
5.1 Introduction......Page 185
5.2 The Failures of Classical Physics......Page 186
5.4 The Heat Capacity of Solids at Low Temperature......Page 187
5.6 Black Body Radiator......Page 188
5.7.1 Planck’s Distribution......Page 190
5.7.2 Wien’s and Stephan–Boltzmann’s Law......Page 191
5.7.4 Einstein’s Equation for the Heat Capacity of Solids......Page 192
5.7.5 Bohr’s Model for Hydrogen......Page 193
5.8 Modern Quantum Theory and Schrödinger’s Equation......Page 195
5.8.1 de Broglie’s Hypothesis and Heisenberg’s Uncertainty Principle......Page 196
5.8.2 Schrödinger’s Equation......Page 197
5.8.3 The Interpretation of psi: The First Postulate of Quantum Mechanics......Page 200
5.8.4 Expectation Values: The Second Postulate of Quantum Mechanics......Page 201
5.8.5 A Brief List of Operators......Page 202
5.8.6 The Particle in a Box......Page 204
Exercises......Page 205
5.8.7 Atomic Units......Page 208
5.9 Matrix Quantum Mechanics......Page 209
5.10 The Monodimensional Hamiltonian in a Simple Hilbert Space......Page 213
5.11 Numerical Solution Issues in Vector Spaces......Page 215
5.12 The Harmonic Oscillator in Hilbert Space......Page 216
Exercises......Page 219
5.13 A Simple Discrete Variable Representation (DVR)......Page 220
5.14 Accelerating the Convergence of the Simple Discrete Variable Representation (DVR)......Page 224
Exercises......Page 226
5.15 Elements of Sparse Matrix Technology......Page 227
5.16 The Gram–Schmidt Process......Page 228
5.17 The Krylov Space......Page 231
5.18 The Row Representation of a Sparse Matrix......Page 236
Exercises......Page 239
5.19 The Lanczos Algorithm......Page 240
5.20 Orbital Angular Momentum and the Spherical Harmonics......Page 247
5.21 Complete Sets of Commuting Observables......Page 248
5.22 The Addition of Angular Momentum Vectors......Page 252
5.23 Computation of the Vector Coupling Coefficients......Page 255
5.24 Matrix Elements of Anisotropic Potentials in the Angular Momentum Basis......Page 258
5.25 The Physical Rigid Dipole in a Constant Electric Field......Page 260
6.2 The Time-Dependent Schrödinger Equation......Page 265
6.3 Wavepackets, Measurements, and Time Propagation of Wavepackets......Page 267
Exercises......Page 273
6.4 The Time Evolution Operator......Page 276
6.5 The Dyson Series and the Time-Ordered Exponential Representation......Page 277
6.6 The Magnus Expansion......Page 279
6.7 The Trotter Factorization......Page 280
6.8 The time evolution operator Program......Page 281
Exercises......Page 285
6.9 Feynman’s Path Integral......Page 287
6.10 Quantum Monte Carlo......Page 290
Exercises......Page 295
6.11 A Variational Monte Carlo Method for Importance Sampling Diffusion Monte Carlo (IS-DMC)......Page 297
Exercises......Page 302
6.12 Importance Sampling Diffusion Monte Carlo (IS-DMC) with Drift......Page 303
6.13 Green’s Function Diffusion Monte Carlo......Page 304
Exercises......Page 307
7.2 The Harmonic Oscillator......Page 309
7.3 Classical Canonical Average Energy and Heat Capacity......Page 310
7.4 Quantum Canonical Average Energy and Heat Capacity......Page 314
7.5 The Path Integral in Rd......Page 319
Exercises......Page 326
7.6 The Canonical Fourier Path Integral......Page 331
Exercises......Page 337
7.7 The Reweighted Fourier–Wiener Path Integral......Page 338
Exercises......Page 345
Part II: Atomic Clusters......Page 347
8.1 Introduction......Page 349
8.2 Cartesian Coordinates of Atomic Clusters......Page 350
8.3 Rotations and Translations......Page 353
8.4 The Center of Mass......Page 354
8.5 The Inertia Tensor......Page 356
8.6 The Structural Comparison Algorithm......Page 357
Exercises......Page 361
8.7 Gradients and Hessians of Multidimensional Potentials......Page 363
8.8 The Lennard-Jones Potential V (LJ)......Page 366
8.9 The Gradient of V (LJ)......Page 368
Exercises......Page 371
8.10 Brownian Dynamics at 0 K......Page 374
8.11 Basin Hopping......Page 377
8.12 The Genetic Algorithm......Page 384
8.13 The Hessian Matrix......Page 389
8.14 Normal Mode Analysis......Page 395
8.15 Transition States with the Cerjan–Miller Algorithm......Page 399
8.16 Optical Activity......Page 406
Exercises......Page 407
9.1 Introduction......Page 411
9.2 Simulation Techniques: Parallel Tempering Revisited......Page 413
9.3 Thermodynamic Properties of a Cluster with n Atoms......Page 415
9.4 The Program parallel_tempering_r3n.f......Page 417
Exercises......Page 420
9.5 The Variational Ground State Energy......Page 421
9.6 Diffusion Monte Carlo (DMC) of Atomic Clusters......Page 423
9.7 Path Integral Simulations of Ar7......Page 425
Exercises......Page 429
9.8 Characterization Techniques: The Lindemann Index......Page 430
9.9 Characterization Techniques: Bond Orientational Parameters......Page 433
9.10 Characterization Techniques: Structural Comparison......Page 437
Exercises......Page 442
9.11 Appendix A: parallel tempering r3n......Page 444
9.12 Appendix B: gfis dmc......Page 448
9.13 Appendix C: rewfpi......Page 451
Part III: Methods in Curved Spaces......Page 457
10.1 Introduction......Page 459
10.2 Coordinate Changes, Einstein’s Sum Convention, and the Metric Tensor......Page 460
10.3 Contravariant Tensors......Page 461
10.5 Tensors of Higher Ranks......Page 462
10.6 The Metric Tensor of a Space......Page 463
10.6.2 Example: The One-Sphere (S1)......Page 466
10.7 Integration on Manifolds......Page 467
10.8 Stereographic Projections......Page 469
Exercises......Page 472
10.9 Dynamics in Manifolds......Page 474
10.10 The Hessian Metric......Page 475
10.11 The Christofell Connections and the Geodesic Equations......Page 476
10.12 The Laplace–Beltrami Operator......Page 477
10.13 The Riemann–Cartan Curvature Scalar......Page 478
Exercises......Page 480
10.14 The Two-Body Problem Revisited......Page 483
10.15 Stereographic Projections for the Two-Body Problem......Page 486
10.16 The Rigid Rotor and the Infinitely Stiff Spring Constant Limit......Page 488
10.17 Relative Coordinates for the Three-Body Problem......Page 490
10.18 The Rigid Body Problem and the Body Fixed Frame......Page 492
Exercises......Page 496
10.19 Stereographic Projections for the Ellipsoid of Inertia......Page 497
10.20 The Spherical Top......Page 500
10.21 The Riemann Curvature Scalar for a Spherical Top......Page 503
10.22 Coefficients and the Curvature for Spherical Tops with Stereographic Projection Coordinates (SPCs)......Page 506
10.23 The Riemann Curvature Scalar for a Symmetric Nonspherical Top......Page 507
10.24 A Split Operator for Symplectic Integrators in Curved Manifolds......Page 509
10.25 The Verlet Algorithm for Manifolds......Page 513
11.1 Introduction......Page 517
11.2 The Invariance of the Phase Space Volume......Page 522
11.2.1 Soft and Hard Constraints and Parallel Tempering......Page 523
11.3 Variational Ground States......Page 526
11.4 Diffusion Monte Carlo (DMC) in Manifolds......Page 527
11.5 The Path Integral in Space-Like Curved Manifolds......Page 529
11.6 The Virial Estimator for the Total Energy......Page 532
11.7 Angular Momentum Theory Solution for a Particle in S2......Page 534
11.8 Variational Ground State for S2......Page 537
11.9 Diffusion Monte Carlo (DMC) in S2......Page 541
11.10 Stereographic Projection Path Integral in S2......Page 542
11.11 Higher Dimensional Tops......Page 547
Exercises......Page 556
11.12 The Free Particle in a Ring......Page 560
11.13 The Particle in a Ring Subject to Smooth Potentials......Page 565
Exercises......Page 575
Part IV: Applications to Molecular Systems......Page 579
12.1 Introduction......Page 581
12.2 The Stockmayer Model......Page 583
12.3 The Map for…......Page 587
12.4 The Gradient of the Lennard-Jones Dipole-Dipole (LJDD) Potential......Page 589
12.5 Beyond the Stockmayer Model for Rigid Linear Tops......Page 595
12.6 The Hessian Metric Tensor on…......Page 599
12.7 Reweighted Random Series Action for Clusters of Linear Rigid Tops......Page 605
12.9 Clusters of Rigid Nonlinear Tops......Page 608
12.10 Coordinate Transformations for…......Page 609
12.11 The Hessian Metric Tensor for…......Page 616
12.12 Local Energy and Action for…......Page 618
12.13 Concluding Remarks......Page 622
Symplectic Integrators......Page 623
Discrete Variable Representation......Page 627
Variational Monte Carlo......Page 637
Quantum Monte Carlo......Page 638
Path Integral......Page 657
Lennard-Jones Systems......Page 682
Angular Momentum Theory......Page 689
Water TIP4P Clusters......Page 690