Note: Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Symbols, units and constants -- Symbols -- Units -- SI Units -- Atomic units -- Molecular units -- Physical constants -- Part I A Modeling Hierarchy for Simulations -- 1 Introduction -- 1.1 What is this book about? -- 1.1.1 Simulation of real systems -- 1.1.2 System limitation -- 1.1.3 Sophistication versus brute force -- 1.2 A modeling hierarchy -- 1.3 Trajectories and distributions -- 1.4 Further reading -- 2 Quantum mechanics: principles and relativistic effects -- 2.1 The wave character of particles -- 2.2 Non-relativistic single free particle -- 2.3 Relativistic energy relations for a free particle -- 2.4 Electrodynamic interactions -- 2.4.1 Homogeneous external magnetic field -- 2.4.2 Electromagnetic plane wave -- 2.5 Fermions, bosons and the parity rule -- Exercises -- 3 From quantum to classical mechanics: when and how -- 3.1 Introduction -- 3.2 From quantum to classical dynamics -- 3.3 Path integral quantum mechanics -- 3.3.1 Feynman's postulate of quantum dynamics -- 3.3.2 Equivalence with the Schrödinger equation -- 3.3.3 The classical limit -- 3.3.4 Evaluation of the path integral -- 3.3.5 Evolution in imaginary time -- 3.3.6 Classical and nearly classical approximations -- 3.3.7 The free particle -- 3.3.8 Non-interacting particles in a harmonic potential -- 3.3.9 Path integral Monte Carlo and molecular dynamics simulation -- 3.4 Quantum hydrodynamics -- 3.4.1 The hydrodynamics approach -- 3.4.2 The classical limit -- 3.5 Quantum corrections to classical behavior -- 3.5.1 Feynman-Hibbs potential -- 3.5.2 The Wigner correction to the free energy -- 3.5.3 Equivalence between Feynman-Hibbs and Wigner corrections -- 3.5.4 Corrections for high-frequency oscillators -- 3.5.5 The fermion-boson exchange correction -- Exercises. , 4 Quantum chemistry: solving the time-independent Schrödinger equation -- 4.1 Introduction -- 4.2 Stationarysolutions of the TDSE -- 4.3 The few-particle problem -- 4.3.1 Shooting methods -- 4.3.2 Expansion on a basis set -- 4.3.3 Variational Monte Carlo methods -- 4.3.4 Relaxation methods -- 4.3.5 Diffusional quantum Monte Carlo methods -- 4.3.6 A practical example -- 4.3.7 Green's function Monte Carlo methods -- 4.3.8 Some applications -- 4.4 The Born-Oppenheimer approximation -- 4.5 The many-electron problem of quantum chemistry -- 4.6 Hartree-Fock methods -- 4.7 Density functional theory -- 4.8 Excited-state quantum mechanics -- 4.9 Approximate quantum methods -- 4.10 Nuclear quantum states -- 5 Dynamics of mixed quantum/classical systems -- 5.1 Introduction -- 5.2 Quantum dynamics in a non-stationary potential -- 5.2.1 Integration on a spatial grid -- 5.2.2 Time-independent basis set -- 5.2.3 Time-dependent basis set -- 5.2.4 The two-level system -- 5.2.5 The multi-level system -- 5.3 Embedding in a classical environment -- 5.3.1 Mean- eld back reaction -- 5.3.2 Forces in the adiabatic limit -- 5.3.3 Surface hopping dynamics -- 5.3.4 Other methods -- Exercises -- 6 Molecular dynamics -- 6.1 Introduction -- 6.2 Boundary conditions of the system -- 6.2.1 Periodic boundary conditions -- 6.2.2 Continuum boundary conditions -- 6.2.3 Restrained-shell boundary conditions -- 6.3 Force field descriptions -- 6.3.1 Ab-Initio molecular dynamics -- 6.3.2 Simple molecular force fields -- 6.3.3 More sophisticated force fields -- 6.3.4 Long-range dispersion interactions -- 6.3.5 Long-range Coulomb interactions -- 6.3.6 Polarizable force fields -- 6.3.7 Choices for polarizability -- 6.3.8 Energies and forces for polarizable models -- 6.3.9 Towards the ideal force field -- 6.3.10 QM/MM approaches -- 6.4 Solving the equations of motion -- 6.4.1 Constraints. , 6.5 Controlling the system -- 6.5.1 Stochastic methods -- 6.5.2 Strong-coupling methods -- 6.5.3 Weak-coupling methods -- 6.5.4 Extended system dynamics -- 6.5.5 Comparison of thermostats -- 6.6 Replica exchange method -- 6.7 Applications of molecular dynamics -- Exercises -- 7 Free energy, entropy and potential of mean force -- 7.1 Introduction -- 7.2 Free energy determination by spatial integration -- 7.3 Thermodynamic potentials and particle insertion -- 7.4 Free energy by perturbation and integration -- 7.5 Free energy and potentials of mean force -- 7.6 Reconstruction of free energy from PMF -- 7.6.1 Harmonic wells -- 7.7 Methods to derive the potential of mean force -- 7.8 Free energy from non-equilibrium processes -- 7.8.1 Proof of Jarzynski's equation -- 7.8.2 Evolution in space only -- 7.8.3 Requirements for validity of Jarzynski's equation -- 7.8.4 Statistical considerations -- 8 Stochastic dynamics: reducing degrees of freedom -- 8.1 Distinguishing relevant degrees of freedom -- 8.2 The generalized Langevin equation -- 8.3 The potential of mean force -- 8.4 Superatom approach -- 8.5 The fluctuation-dissipation theorem -- 8.6 Langevin dynamics -- 8.6.1 Langevin dynamics in generalized coordinates -- 8.6.2 Markovian Langevin dynamics -- 8.7 Brownian dynamics -- 8.8 Probability distributions and Fokker-Planck equations -- 8.8.1 General Fokker-Planck equations -- 8.8.2 Application to generalized Langevin dynamics -- 8.8.3 Application to Brownian dynamics -- 8.9 Smart Monte Carlo methods -- 8.10 How to obtain the friction tensor -- 8.10.1 Solute molecules in a solvent -- 8.10.2 Friction from simulation -- Exercises -- 9 Coarse graining from particles to .uid dynamics -- 9.1 Introduction -- 9.2 The macroscopic equations of fluid dynamics -- 9.2.1 Conservation of mass -- 9.2.2 The equation of motion -- 9.2.3 Conservation of linear momentum. , 9.2.4 The stress tensor and the Navier-Stokes equation -- 9.2.5 The equation of state -- 9.2.6 Heat conduction and the conservation of energy -- 9.3 Coarse graining in space -- 9.3.1 Definitions -- 9.3.2 Stress tensor and pressure -- 9.3.3 Conservation of mass -- 9.3.4 Conservation of momentum -- 9.3.5 The equation of motion -- 9.4 Conclusion -- Exercises -- 10 Mesoscopic continuum dynamics -- 10.1 Introduction -- 10.2 Connection to irreversible thermodynamics -- 10.3 The mean field approach to the chemical potential -- 11 Dissipative particle dynamics -- 11.1 Representing continuum equations by particles -- 11.2 Prescribing fluid parameters -- 11.3 Numerical solutions -- 11.4 Applications -- Part II Physical and Theoretical Concepts -- 12 Fourier transforms -- 12.1 Definitions and properties -- 12.2 Convolution and autocorrelation -- 12.3 Operators -- 12.4 Uncertainty relations -- 12.5 Examples of functions and transforms -- 12.5.1 Square pulse -- 12.5.2 Triangular pulse -- 12.5.3 Gaussian function -- 12.6 Discrete Fourier transforms -- 12.7 Fast Fourier transforms -- 12.8 Autocorrelation and spectral density from FFT -- 12.9 Multidimensional Fourier transforms -- Exercises -- 13 Electromagnetism -- 13.1 Maxwell's equation for vacuum -- 13.2 Maxwell's equation for polarizable matter -- 13.3 Integrated form of Maxwell's equations -- 13.4 Potentials -- 13.5 Waves -- 13.6 Energies -- 13.7 Quasi-stationary electrostatics -- 13.7.1 The Poisson and Poisson-Boltzmann equations -- 13.7.2 Charge in a medium -- 13.7.3 Dipole in a medium -- 13.7.4 Charge distribution in a medium -- 13.7.5 The generalized Born solvation model -- 13.8 Multipole expansion -- 13.8.1 Expansion of the potential -- 13.8.2 Expansion of the source terms -- 13.9 Potentials and fields in non-periodic systems -- 13.10 Potentials and fields in periodic systems of charges. , 13.10.1 Short-range contribution -- 13.10.2 Long-range contribution -- 13.10.3 Gaussian spread function -- 13.10.4 Cubic spread function -- 13.10.5 Net dipolar energy -- 13.10.6 Particle-mesh methods -- 13.10.7 Potentials and fields in periodic systems of charges and dipoles -- Exercises -- 14 Vectors, operators and vector spaces -- 14.1 Introduction -- 14.2 Definitions -- 14.3 Hilbert spaces of wave functions -- 14.4 Operators in Hilbert space -- 14.5 Transformations of the basis set -- 14.6 Exponential operators and matrices -- 14.6.1 Example of a degenerate case -- 14.7 Equations of motion -- 14.7.1 Equations of motion for the wave function and its representation -- 14.7.2 Equation of motion for observables -- 14.8 The density matrix -- 14.8.1 The ensemble-averaged density matrix -- 14.8.2 The density matrix in coordinate representation -- 15 Lagrangian and Hamiltonian mechanics -- 15.1 Introduction -- 15.2 Lagrangian mechanics -- 15.3 Hamiltonian mechanics -- 15.4 Cyclic coordinates -- 15.5 Coordinate transformations -- 15.6 Translation and rotation -- 15.6.1 Translation -- 15.6.2 Rotation -- 15.7 Rigid body motion -- 15.7.1 Description in terms of angular velocities -- 15.7.2 Unit vectors -- 15.7.3 Euler angles -- 15.7.4 Quaternions -- 15.8 Holonomic constraints -- 15.8.1 Generalized coordinates -- 15.8.2 Coordinate resetting -- 15.8.3 Projection methods -- 16 Review of thermodynamics -- 16.1 Introduction and history -- 16.2 Definitions -- 16.2.1 Partial molar quantities -- 16.3 Thermodynamic equilibrium relations -- 16.3.1 Relations between partial differentials -- 16.4 The second law -- 16.5 Phase behavior -- 16.6 Activities and standard states -- 16.6.1 Virial expansion -- 16.7 Reaction equilibria -- 16.7.1 Proton transfer reactions -- 16.7.2 Electron transfer reactions -- 16.8 Colligative properties -- 16.9 Tabulated thermodynamic quantities. , 16.10 Thermodynamics of irreversible processes.