Molecular Dynamics

This molecular dynamics textbook takes the reader from classical mechanics to quantum mechanics and vice versa, and from few-body systems to many-body systems. It is self-contained, comprehensive, and builds the theory of molecular dynamics from basic principles to applications, allowing the subject to be appreciated by readers from physics, chemistry, and biology backgrounds while maintaining mathematical rigor. The book is enhanced with illustrations, problems and solutions, and suggested reading, making it ideal for undergraduate and graduate courses or self-study. With coverage of recent developments, the book is essential reading for students who explore and characterize phenomena at the atomic level. It is a useful reference for researchers in physics and chemistry, and can act as an entry point for researchers in nanoscience, materials engineering, genetics, and related fields who are seeking a deeper understanding of nature. Preface Prologue Basics of Classical Mechanics Basics of Quantum Mechanics First-Principles Molecular Dynamics Classical Molecular Dynamics Time Evolution Operators Contents Part I Basics of Classical Mechanics 1 Principles of Classical Dynamics 1.1 Newtonian Dynamics 1.2 Space and Time 1.3 Mass 1.4 Energy 1.5 Electric Charge 1.6 Reference System of Coordinates 1.7 Newtonian Time 1.8 Linear Motion 1.9 Angular Motion 1.10 Descriptions Between Inertial Reference Frames References 2 Foundations of Newtonian Dynamics 2.1 First Newton's Law 2.2 Second Newton's Law 2.3 Third Newton's Law 2.4 Reduced Mass of a Two-Particle System 2.5 Time Reversibility 2.6 Angular Momentum and Torque 2.7 Impulse, Work, and Power 2.8 Kinetic and Potential Energies 2.9 Energy Conservation References 3 Many-Particle Systems 3.1 Reference Frame of a Many-Particle System 3.2 Angular Momentum and Torque of a Many-Particle System 3.3 Mechanical Energies of a Many-Particle System 3.4 Transformation of the Energy Components 3.5 Energy Balance Equation 3.6 Statistical and Time Averages of Physical Observables 3.7 Ergodic Hypothesis 3.8 Breaking the Ergodic Hypothesis 3.9 Velocity Distribution Function 3.10 Temperature of a System of Particles 3.11 Temperature Scaling as a Thermostat 3.12 Temperature Fluctuations 3.13 Pressure and Volume 3.14 The Virial and the Equation of State References 4 Mechanical Descriptors 4.1 Caloric Curve 4.2 Interatomic Distance Fluctuations 4.3 Root Mean Square Deviation of Positions 4.4 Orientational Order Parameter 4.5 Probability Distribution Functions 4.6 Correlation Functions 4.7 Properties of Correlation Functions 4.8 Vibrational Spectra from Autocorrelation Functions References 5 Rigid Body 5.1 Angular Momentum of a Rotating System of Particles 5.2 External Torques Acting on a Rotating Body 5.3 Total Energy of a Rotating Rigid Body A.1 Appendix: Matrix Diagonalization 6 Analytical Mechanics 6.1 Action Function 6.2 Principle of Stationary Action 6.3 Classifying Molecular Systems 6.4 Lagrange's Equations of Motion 6.5 Newtonian Equations of Motion from Lagrange Theory 6.6 Non-uniqueness of the Lagrangian 6.7 Invariance of the Lagrange Equations of Motion 6.8 Motion with Constraints 6.9 Hamilton's Function 6.10 Preservation of the Hamiltonian in Time 6.11 Conserved Observables and Symmetries 6.12 Space Homogeneity 6.13 Space Isotropy 6.14 Uniform Passage of Time 6.15 Hamilton's Equations of Motion 6.16 Invariance Under Canonical Transformations 6.17 Time Reversibility in Hamiltonian Theory 6.18 Hamilton-Jacobi Theory 6.19 Illustrating with the Harmonic Oscillator 6.20 Contact Between Quantum and Classical Mechanics 6.21 Poisson's Brackets 6.22 Classical Time Propagator References Part II Basics of Quantum Mechanics 7 Wave-Particle Duality of Matter 7.1 Young's Experiment 7.2 Interference of Waves 7.3 Photo-Electron Experiment 7.4 Compton's Experiment 7.5 Davisson-Germer's Experiment 7.6 de Broglie's Hypothesis 7.7 Bohr's Complementary Principle References 8 Quantization of the Energy 8.1 Planck's Energy Equation 8.2 Blackbody Radiation Experiment 8.3 Rayleigh-Jeans Law 8.4 Wien's Displacement Law 8.5 Ultraviolet Catastrophe 8.6 Planck's Law 8.7 Franck-Hertz Experiment 8.8 Heisenberg's Uncertainty Principle A.1 Appendix: Planck's Radiation Intensity Law A.1.1 Average Energy A.1.2 Density of States A.1.3 Energy per Volume per Frequency A.1.4 Irradiance A.1.5 Maximum Intensity A.1.6 Time Interval of Temperature Cooling Reference 9 Quantization of the Angular Momentum 9.1 Orbital Angular Momentum and Spin 9.2 Characterizing a Particle with Spin 9.3 Stern-Gerlach Experiment 9.4 Wave-Particle Duality and Spin of a Particle 9.5 Fermions and Bosons 9.6 Pauli's Exclusion Principle and Hund's Rule A.1 Appendix: Magnetic Moment A.1.1 Electric Current in a Circular Loop A.1.2 Magnetic g Factor A.1.3 Magnetic Energy and Magnetic Work A.1.4 Zeeman Effect A.1.5 Electron Spin A.1.6 Paschen-Back Effect A.1.7 Applications of the Spin Resonance Technique B.1 Appendix: Spin Functions and Operators References 10 Postulates of Quantum Mechanics 10.1 Reformulating the Conceptual World 10.2 Postulates of Quantum Mechanics 10.2.1 First Postulate 10.2.2 Second Postulate 10.2.3 Third Postulate 10.2.4 Fourth Postulate 10.2.5 Fifth Postulate 10.2.6 Sixth Postulate 10.3 Time Reversibility in Quantum Mechanics 10.4 Stationary States 10.5 Superposition Principle of Quantum States 10.6 Bohr's Correspondence Principle 10.7 Selection Rules 10.8 Pauli's Principle in the Electronic Wave Function 10.9 Wave Function of the Electrons in a Molecule 10.10 Variational Principle of the Energy A.1 Appendix: Proposing the Wave Equation for Matter Waves B.1 Appendix: Expansion of a Determinantal Wave Function References Part III First-Principles Molecular Dynamics 11 Dynamics of Electrons and Nuclei 11.1 The Electronic and Nuclear Dynamics Are Coupled 11.2 Molecular Hamiltonian 11.3 Approximating the Total Wave Function 11.4 Time-Dependent Self-Consistent Field Equations References 12 Classical Limit of the Nuclear Motion 12.1 Polar Form of the Nuclear Wave Equation 12.2 Continuity and Hamilton-Jacobi Equations 12.3 Conditions to Describe the Nuclear Particles Classically 12.4 Simplification of the Nuclear Potential 12.5 Parameterizing the Potential Function 12.6 Total Energy of the Molecular System 12.7 Establishing the Accuracy of Atomic Forces 12.8 Diffusion from the Continuity Equation 12.9 Diffusion Equation and Particle Flux 12.10 Expansion of the Electronic Wave Equation 12.11 Expansion of the Newtonian Equation of the Nuclei A.1 Appendix: Bohm's Quantum Potential References Part IV Classical Molecular Dynamics 13 Classical Molecular Dynamics 13.1 Model Interaction Potentials 13.2 Forcefields 13.3 Atom Types 13.4 United Atom 13.5 Bond Elongation and Compression 13.6 Combination Rules 13.7 Bond Angle Vibration 13.8 Plane Bending 13.9 Angle Inversion 13.10 Torsional Motion 13.11 Electrostatic Interaction 13.12 van der Waals Forces 13.13 Interaction Potential Functions of Water 13.14 Atom Polarization 13.15 External Fields and Potentials 13.16 Parameterization of Forcefields 13.17 Model Potentials of Non-biological Systems 13.18 Sutton-Chen Potential Function 13.19 Gupta Potential Function 13.20 Tersoff Potential Function A.1 Appendix: Harmonic Model of the Dispersion Energy References 14 Extended Systems 14.1 Fixed and Flexible Boundaries 14.2 Periodic Boundary Conditions 14.3 The PBC System Is an Open System 14.4 Electrostatics in the PBC Approach 14.5 Ewald Sum Approach 14.6 Using the Poisson Equation 14.7 Short-Range Interactions 14.8 Electrostatic Self-Interactions 14.9 Long-Range Interactions 14.10 Ewald Electrostatic Energy 14.11 Smooth Particle Mesh Ewald Approach 14.12 Shifted Potentials and Forces References Part V Time Evolution Operators 15 Integrating the Equations of Motion 15.1 Liouville Operator as a Time Propagator 15.2 Discretizing the Time Propagator 15.3 Evolving Positions and Momenta 15.4 Simplified Time Integrators 15.5 Leapfrog Algorithm 15.6 Verlet Algorithm 15.7 Bond Constraints References Index