Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics

Theory and Computational Methods for Understanding Molecular Spectroscopy and Chemical Reactions

verfasst von: Stavros C. Farantos

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Molecular Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Overview

Abstract

The necessity of treating molecules as nonlinear dynamical systems is explained by referring to both theoretical and computational progress in molecular dynamics as well as experimental advances in reaction dynamics and spectroscopy obtained in the last decades for small and large molecules. The basic concepts of nonlinear mechanics are introduced and time invariant structures in the phase space, such as periodic orbits, tori, normally hyperbolic invariant manifolds and stable/unstable manifolds, are mentioned in association to the molecular phase space. Most importantly, from the very begging it is stated that the hierarchical approach of nonlinear mechanics in understanding dynamical systems by locating invariant phase space structures is the most appropriate in elucidating and predicting molecular behaviour.

Stavros C. Farantos

Chapter 2. The Geometry of Hamiltonian Mechanics

Abstract

In this chapter an introduction to Lagrangian and Hamiltonian mechanics is given. An effort is made to present Hamiltonian theory from the analytical mechanics point of view, which unveils the geometrical characteristics of the theory, such as its symplectic symmetry. The relations among the tangent bundle, cotangent bundle and the mixed tangent-cotangent bundle of the configuration manifold are discussed. The Euler–Lagrange equations and Hamilton’s equations of motion are extracted from the principle of least action. The canonical equations are also formulated by the symplectic \(2-\)form and the symplectic transformations are explained. Poisson brackets, which provide the bridge to pass from classical to quantum mechanics are introduced.

Stavros C. Farantos

Chapter 3. Dynamical Systems

Abstract

The basic theory of dynamical systems is introduced in this chapter. Invariant phase space structures—equilibria, periodic orbits, tori, normally hyperbolic invariant manifolds and stable/unstable manifolds—are defined mainly with graphs produced by numerically solving the equations of motion of 1, 2 and 3 degrees of freedom model Hamiltonian systems. Stability analysis and elementary bifurcations of equilibria and periodic orbits are discussed. The center-saddle, pitchfork, period doubling and complex instability elementary bifurcations encountered in continuation diagrams of equilibria and periodic orbits by varying a parameter in the potential function or the energy of the system are investigated. Methods of analysing non-periodic orbits, regular and chaotic, such as Poincaré surfaces of section, maximal Lyapunov exponent and autocorrelation functions are introduced and explained.

Stavros C. Farantos

Chapter 4. Quantum and Semiclassical Molecular Dynamics

Abstract

The two formulations of classical mechanics, Lagrangian and Hamiltonian, lead to two expressions of quantum mechanics, the path integral and canonical, respectively. After an introduction to the principles of these two theories, subjects related to calculations of molecular dynamics are discussed. Converting Schrödinger equation into quantum Hamilton’s complex equations, the complexification of the classical Hamilton’s equations, and the quantum and classical autocorrelation functions are described. Semiclassical theories based on periodic orbits, tori and the more general approach of initial value representation method are topics of this chapter. Finally, an introduction of how to numerically solve the Schrödinger equation in a Cartesian coordinate system, which results in a simple form molecular Hamiltonian, is provided.

Stavros C. Farantos

Chapter 5. Numerical Methods

Abstract

The theories developed in the previous chapters, classical and quantum mechanical, are put in action by discretizing the corresponding differential equations. The variable order finite difference approximation to the unknown solutions and their derivatives is the preferred method, not only because of their well understood convergence properties and the relatively easy way of their programming, but also, finite differences provide a uniform approach to different type of equations, especially when we work in a Cartesian coordinate system. With respect to Schrödinger equation several grids are examined and comparisons with the more popular pseudospectral methods is made. For the location of periodic orbits the multiple shooting method is developed as it has been thoroughly tested. Finally, computer codes for studying classical nonlinear molecular dynamics and solving the Schrödinger equation are described.

Stavros C. Farantos

Chapter 6. Applications

Abstract

Results from the application of nonlinear mechanics to interpret spectra and dynamics of small and large molecules are presented. Specifically, a vibrational quantum mechanical study for hydrogen hypochlorite with calculated vibrational energy levels up to dissociation are analysed by periodic orbits. A cascade of center-saddle bifurcations of periodic orbits follows the dissociation pathway of the molecule on the ground electronic state. The photodissociation of nitrous oxide on an electronically excited state is investigated by quantum mechanical calculations and nonlinear mechanical analysis. Periodic orbits of alanine dipeptide as well as of the active site of cytochrome c oxidase are employed to understand experimental spectra of such molecular species.

Stavros C. Farantos

Chapter 7. Epilogue

Abstract

In this short chapter a summary is given and it is emphasized that nonlinear mechanics applied to molecular dynamics offer unprecedented details which help to understand the quantum behaviours of the molecules. Some ideas for future developments are put forward.

Stavros C. Farantos

Backmatter

Titel: Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics
verfasst von: Stavros C. Farantos
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-09988-0
Print ISBN: 978-3-319-09987-3
DOI: https://doi.org/10.1007/978-3-319-09988-0