Preface ........................................................ xi
Symbols, units and constants ................................... xv
Part I A Modeling Hierarchy for Simulations .................... 1
1 Introduction ................................................. 3
1.1 What is this book about? ................................ 3
1.2 A modeling hierarchy .................................... 9
1.3 Trajectories and distributions ......................... 13
1.4 Further reading ........................................ 14
2 Quantum mechanics: principles and relativistic effects ...... 19
2.1 The wave character of particles ........................ 19
2.2 Non-relativistic single free particle .................. 23
2.3 Relativistic energy relations for a free particle ...... 25
2.4 Electrodynamic interactions ............................ 31
2.5 Fermions, bosons and the parity rule ................... 36
3 From quantum to classical mechanics: when and how ........... 39
3.1 Introduction ........................................... 39
3.2 From quantum to classical dynamics ..................... 42
3.3 Path integral quantum mechanics ........................ 44
3.4 Quantum hydrodynamics .................................. 64
3.5 Quantum corrections to classical behavior .............. 70
4 Quantum chemistry: solving the time-independent
Schrödinger equation ........................................ 77
4.1 Introduction ........................................... 77
4.2 Stationary solutions of the TDSE ....................... 78
4.3 The few-particle problem ............................... 79
4.4 The Born-Oppenheimer approximation ..................... 97
4.5 The many-electron problem of quantum chemistry ......... 98
4.6 Hartree-Fock methods ................................... 99
4.7 Density functional theory ............................. 102
4.8 Excited-state quantum mechanics ....................... 105
4.9 Approximate quantum methods ........................... 106
4.10 Nuclear quantum states ................................ 107
5 Dynamics of mixed quantum/classical systems ................ 109
5.1 Introduction .......................................... 109
5.2 Quantum dynamics in a non-stationary potential ........ 114
5.3 Embedding in a classical environment .................. 129
6 Molecular dynamics ......................................... 139
6.1 Introduction .......................................... 139
6.2 Boundary conditions of the system ..................... 140
6.3 Force field descriptions .............................. 149
6.4 Solving the equations of motion ....................... 189
6.5 Controlling the system ................................ 194
6.6 Replica exchange method ............................... 204
6.7 Applications of molecular dynamics .................... 207
7 Free energy, entropy and potential of mean force .......... 211
7.1 Introduction .......................................... 211
7.2 Free energy determination by spatial integration ...... 213
7.3 Thermodynamic potentials and particle insertion ....... 218
7.4 Free energy by perturbation and integration ........... 221
7.5 Free energy and potentials of mean force .............. 227
7.6 Reconstruction of free energy from PMF ................ 231
7.7 Methods to derive the potential of mean force ......... 234
7.8 Free energy from non-equilibrium processes ............ 239
8 Stochastic dynamics: reducing degrees of freedom ........... 249
8.1 Distinguishing relevant degrees of freedom ............ 249
8.2 The generalized Langevin equation ..................... 251
8.3 The potential of mean force ........................... 255
8.4 Superatom approach .................................... 256
8.5 The fluctuation-dissipation theorem ................... 257
8.6 Langevin dynamics ..................................... 263
8.7 Brownian dynamics ..................................... 268
8.8 Probability distributions and Fokker-Planck
equations ............................................. 269
8.9 Smart Monte Carlo methods ............................. 272
8.10 How to obtain the friction tensor ..................... 274
9 Coarse graining from particles to fluid dynamics ........... 279
9.1 Introduction .......................................... 279
9.2 The macroscopic equations of fluid dynamics ........... 281
9.3 Coarse graining in space .............................. 288
9.4 Conclusion ............................................ 295
10 Mesoscopic continuum dynamics .............................. 297
10.1 Introduction .......................................... 297
10.2 Connection to irreversible thermodynamics ............. 298
10.3 The mean field approach to the chemical potential ..... 301
11 Dissipative particle dynamics .............................. 305
11.1 Representing continuum equations by particles ......... 307
11.2 Prescribing fluid parameters .......................... 308
11.3 Numerical solutions ................................... 309
11.4 Applications .......................................... 309
Part II Physical and Theoretical Concepts .................... 313
12 Fourier transforms ......................................... 315
12.1 Definitions and properties ............................ 315
12.2 Convolution and autocorrelation ....................... 316
12.3 Operators ............................................. 317
12.4 Uncertainty relations ................................. 318
12.5 Examples of functions and transforms .................. 320
12.6 Discrete Fourier transforms ........................... 323
12.7 Fast Fourier transforms ............................... 324
12.8 Autocorrelation and spectral density from FFT ......... 325
12.9 Multidimensional Fourier transforms ................... 331
13 Electromagnetism ........................................... 335
13.1 Maxwell's equation for vacuum ......................... 335
13.2 Maxwell's equation for polarizable matter ............. 336
13.3 Integrated form of Maxwell's equations ................ 337
13.4 Potentials ............................................ 337
13.5 Waves ................................................. 338
13.6 Energies .............................................. 339
13.7 Quasi-stationary electrostatics ....................... 340
13.8 Multipole expansion ................................... 353
13.9 Potentials and fields in non-periodic systems ......... 362
13.10 Potentials and fields in periodic systems of
charges ............................................... 362
14 Vectors, operators and vector spaces ....................... 379
14.1 Introduction .......................................... 379
14.2 Definitions ........................................... 380
14.3 Hilbert spaces of wave functions ...................... 381
14.4 Operators in Hilbert space ............................ 382
14.5 Transformations of the basis set ...................... 384
14.6 Exponential operators and matrices .................... 385
14.7 Equations of motion ................................... 390
14.8 The density matrix .................................... 392
15 Lagrangian and Hamiltonian mechanics ....................... 397
15.1 Introduction .......................................... 397
15.2 Lagrangian mechanics .................................. 398
15.3 Hamiltonian mechanics ................................. 399
15.4 Cyclic coordinates .................................... 400
15.5 Coordinate transformations ............................ 401
15.6 Translation and rotation .............................. 403
15.7 Rigid body motion ..................................... 405
15.8 Holonomic constraints ................................. 417
16 Review of thermodynamics ................................... 423
16.1 Introduction and history .............................. 423
16.2 Definitions ........................................... 425
16.3 Thermodynamic equilibrium relations ................... 429
16.4 The second law ........................................ 432
16.5 Phase behavior ........................................ 433
16.6 Activities and standard states ........................ 435
16.7 Reaction equilibria ................................... 437
16.8 Colligative properties ................................ 441
16.9 Tabulated thermodynamic quantities .................... 443
16.10 Thermodynamics of irreversible processes ............. 444
17 Review of statistical mechanics ............................ 453
17.1 Introduction .......................................... 453
17.2 Ensembles and the postulates of statistical
mechanics ............................................. 454
17.3 Identification of thermodynamical variables ........... 457
17.4 Other ensembles ....................................... 459
17.5 Fermi-Dirac, Bose-Einstein and Boltzmann statistics ... 463
17.6 The classical approximation ........................... 472
17.7 Pressure and virial ................................... 479
17.8 Liouville equations in phase space .................... 492
17.9 Canonical distribution functions ...................... 497
17.10 The generalized equipartition theorem ................ 502
18 Linear response theory ..................................... 505
18.1 Introduction .......................................... 505
18.2 Linear response relations ............................. 506
18.3 Relation to time correlation functions ................ 511
18.4 The Einstein relation ................................. 518
18.5 Non-equilibrium molecular dynamics .................... 519
19 Splines for everything ..................................... 523
19.1 Introduction .......................................... 523
19.2 Cubic splines through points .......................... 526
19.3 Fitting splines ....................................... 530
19.4 Fitting distribution functions ........................ 536
19.5 Splines for tabulation ................................ 539
19.6 Algorithms for spline interpolation ................... 542
19.7 B-splines ............................................. 548
References .................................................... 557
Index ......................................................... 587
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