Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics: Theory and Computational Methods for Understanding Molecular Spectroscopy and Chemical Reactions (SpringerBriefs in Molecular Science)

This Brief Presents Numerical Methods For Describing And Calculating Invariant Phase Space Structures, As Well As Solving The Classical And Quantum Equations Of Motion For Polyatomic Molecules. Examples Covered Include Simple Model Systems To Realistic Cases Of Molecules Spectroscopically Studied. Vibrationally Excited And Reacting Molecules Are Nonlinear Dynamical Systems, And Thus, Nonlinear Mechanics Is The Proper Theory To Elucidate Molecular Dynamics By Investigating Invariant Structures In Phase Space. Intramolecular Energy Transfer, And The Breaking And Forming Of A Chemical Bond Have Now Found A Rigorous Explanation By Studying Phase Space Structures. Introduction And Overview -- The Geometry Of Hamiltonian Mechanics -- Dynamical Systems -- Quantum And Semiclassical Molecular Dynamics -- Numerical Methods -- Applications -- Epilogue -- Appendix. By Stavros C. Farantos. 3 Dynamical Systems3.1 Equilibria and Elementary Bifurcations; 3.1.1 Cubic Potential; 3.1.2 Quartic Potential; 3.2 Periodic Orbits; 3.2.1 Stability Analysis of Equilibria and Periodic Orbits; 3.2.2 Complex Instability; 3.2.3 Existence Theorems for Periodic Orbits; 3.3 Tori and Normally Hyperbolic Invariant Manifolds; 3.3.1 Kolmogorov--Arnold--Moser Theorem; 3.3.2 Normally Hyperbolic Invariant Manifold; 3.4 Poincaré Surfaces of Section; 3.5 Non-periodic Trajectories; 3.5.1 Maximal Lyapunov Exponent; 3.5.2 Autocorrelation Functions; References; 4 Quantum and Semiclassical Molecular Dynamics 4.5.2 The Angular Momentum Projection OperatorsReferences; 5 Numerical Methods; 5.1 Discretizing the Schrödinger Equation; 5.1.1 Variable Order Finite Difference Methods; 5.1.2 Pseudospectral Methods; 5.1.3 Relations Between FD and PS Methods; 5.1.4 Remarks; 5.2 Shooting Methods; 5.2.1 The 2-Point Boundary Value Problem; 5.2.2 The Initial Value Problem; 5.2.3 The Newton-Raphson Iterative Method; 5.2.4 The Underrelaxed Newton-Raphson Method; 5.2.5 The Multiple Shooting Method; 5.2.6 Implementation; 5.3 Computer Software; 5.3.1 GridTDSE; 5.3.2 POMULT; 5.3.3 Normal Forms; 5.3.4 AUTO_DERIV 4.1 Canonical Quantum Mechanics4.1.1 Quantum Hamilton's Equations; 4.1.2 Complexification of Classical Hamilton's Equations; 4.2 Quantum and Classical Autocorrelation Functions and Spectra ; 4.3 Path Integral Quantum Mechanics; 4.4 Semiclassical Approximation; 4.4.1 Gutzwiller's Periodic Orbit Semiclassical Quantization; 4.4.2 The Semiclassical Trace Formula; 4.4.3 Einstein-Brillouin-Keller Quantization Rule for Integrable Systems; 4.4.4 Initial Value Representation Semiclassical Quantization; 4.5 Solving Schrödinger Equation in Cartesian Coordinates ... ; 4.5.1 Computational Methods Preface; Contents; 1 Introduction and Overview; 1.1 Nonlinear Mechanics and Molecular Dynamics; 1.2 Hierarchical Study of Nonlinear Molecular Dynamics; References; 2 The Geometry of Hamiltonian Mechanics; 2.1 Configuration Manifolds and Coordinate Systems; 2.1.1 Cartesian Coordinates; 2.1.2 Curvilinear Coordinates; 2.2 The Topological Map of Lagrangian and Hamiltonian Mechanics; 2.3 The Principle of Least Action; 2.4 Hamiltonian Vector Fields; 2.5 The Canonical Equations Expressed with the Symplectic 2-Form; 2.5.1 Symplectic Transformations; 2.5.2 Poisson Brackets; References Front Matter....Pages i-xi Introduction and Overview....Pages 1-11 The Geometry of Hamiltonian Mechanics....Pages 13-31 Dynamical Systems....Pages 33-54 Quantum and Semiclassical Molecular Dynamics....Pages 55-77 Numerical Methods....Pages 79-101 Applications....Pages 103-121 Epilogue....Pages 123-124 Back Matter....Pages 125-158