Chapter
2.4.2 Electromagnetic plane wave
2.5 Fermions, bosons and the parity rule
3 From quantum to classical mechanics: when and how
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
3.3.8 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
4 Quantum chemistry: solving the time-independent Schrödinger equation
4.2 Stationarysolutions of the TDSE
4.3 The few-particle problem
4.3.2 Expansion on a basis set
4.3.3 Variational Monte Carlo methods
4.3.5 Diffusional quantum Monte Carlo methods
4.3.6 A practical example
4.3.7 Green’s function Monte Carlo methods
4.4 The Born–Oppenheimer approximation
4.5 The many-electron problem of quantum chemistry
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
5.2 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-field back reaction
5.3.2 Forces in the adiabatic limit
5.3.3 Surface hopping dynamics
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
6.4 Solving the equations of motion
6.5 Controlling the system
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
7 Free energy, entropy and potential of mean force
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
7.7 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
8.5 The fluctuation–dissipation theorem
8.6.1 Langevin dynamics in generalized coordinates
8.6.2 Markovian Langevin 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
9 Coarse graining from particles to .uid dynamics
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
9.3.2 Stress tensor and pressure
9.3.3 Conservation of mass
9.3.4 Conservation of momentum
9.3.5 The equation of motion
10 Mesoscopic continuum dynamics
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
Part II Physical and Theoretical Concepts
12.1 Definitions and properties
12.2 Convolution and autocorrelation
12.4 Uncertainty relations
12.5 Examples of functions and transforms
12.6 Discrete Fourier transforms
12.7 Fast Fourier transforms
12.8 Autocorrelation and spectral density from FFT
12.9 Multidimensional Fourier transforms
13.1 Maxwell’s equation for vacuum
13.2 Maxwell’s equation for polarizable matter
13.3 Integrated form of Maxwell’s equations
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
13.8.1 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
14 Vectors, operators and vector spaces
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
14.7.1 Equations of motion for the wave function and its representation
14.7.2 Equation of motion for observables
14.8.1 The ensemble-averaged density matrix
14.8.2 The density matrix in coordinate representation
15 Lagrangian and Hamiltonian mechanics
15.2 Lagrangian mechanics
15.3 Hamiltonian mechanics
15.5 Coordinate transformations
15.6 Translation and rotation
15.7.1 Description in terms of angular velocities
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
16.2.1 Partial molar quantities
16.3 Thermodynamic equilibrium relations
16.3.1 Relations between partial differentials
16.6 Activities and standard states
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
16.10.1 Irreversible entropy production
16.10.2 Chemical reactions
16.10.3 Phenomenological and Onsager relations
16.10.4 Stationary states
17 Review of statistical mechanics
17.2 Ensembles and the postulates of statistical mechanics
17.2.1 Conditional maximization of H
17.3 Identification of thermodynamical variables
17.3.1 Temperature and entropy
17.3.2 Free energy and other thermodynamic variables
17.4.1 Ensemble and size dependency
17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics
17.5.1 Canonical partition function as trace of matrix
17.5.2 Ideal gas: FD and BE distributions
17.5.3 The Boltzmann limit
17.6 The classical approximation
17.7.1 The mechanical pressure and its localization
17.7.2 The statistical mechanical pressure
17.8 Liouville equations in phase space
17.9 Canonical distribution functions
17.9.1 Canonical distribution in cartesian coordinates
17.9.2 Canonical distribution in generalized coordinates
17.9.3 Metric tensor effects from constraints
17.10 The generalized equipartition theorem
18 Linear response theory
18.2 Linear response relations
The Kramers–Kronig relations
18.3 Relation to time correlation functions
18.3.1 Dielectric properties
18.4 The Einstein relation
18.5 Non-equilibrium molecular dynamics
18.5.3 Thermal conductivity
19 Splines for everything
19.2 Cubic splines through points
19.4 Fitting distribution functions
19.5 Splines for tabulation
19.6 Algorithms for spline interpolation
Simulating the Physical World
：Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics
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