Modern Methods for Multidimensional Dynamics Computations in Chemistry

The following sections are included:

• CONTENTS

• PREFACE

This paper describes computational methods that are commonly used to describe bimolecular reaction dynamics at the state-to-state level for reactions involving four or more atoms. Only purely quantum and quasiclassical methods are considered, with the bulk of the discussion pertaining to classical methods for defining state-resolved .processes in polyatomic molecules and to reduced dimensionality quantum coupled-channel methods based on hyperspherical coordinates. For each of these methods, we give a detailed description of the theory and involved, and numerical procedures that are used in making applications. The methods are illustrated through an application to the reaction CN + H₂ − HCN + H, including comparisons of classical and quantum thermal rate constants, product state distributions, and average product quantum numbers.

A number of methods have been developed to introduce quantum mechanical effects into an otherwise classical mechanical molecular dynamics framework. These methods can be applied to processes involving electronic transitions among multiple potential energy surfaces as well as to processes involving tunneling and quantization of atomic motion on a single potential energy surface. A major challenge for any such mixed quantum-classical approach is feedback; the quantum and classical degrees of freedom must respond to each other properly and self-consistently. This paper examines a number of mixed quantum-classical strategies, including mean-field (Ehrenfest) and surface-hopping, with particular attention to the issue of feedback. Each of the methods discussed has deficiencies, and none displays universal superiority. Directions for further progress are suggested. Nevertheless, the mixed quantumclassical approach has greatly extended the range of chemical processes that can be addressed accurately and practically by molecular dynamics simulation.

Theoretical methods for modeling the interactions of molecules with surfaces are described. Reactive gas-metal collisions are examined, with a focus on quantum close-coupling and pseudospectral approaches. Applications to dissociative adsorption and Eley-Rideal reactions are discussed. Methods for coupling the scattering or reacting molecule to the thermal fluctuations of the lattice are examined, with an emphasis on fully quantum treatments. Both weak and strong coupling cases, and some recent reduced density matrix formulations are reviewed.

Molecular dynamics computer simulation techniques for studying the structure and dynamics at liquid interfaces are described. The neat liquid/vapor, liquid/liquid and liquid/solid systems, as well as solvation and chemical reactions at these interfacial regions are considered. The focus is on methodology that is unique to the interfacial systems.

The use of direct dynamics to simulate polyatomic reactive systems is reviewed. In this computational method, classical trajectories are integrated ‘on the fly’ using an electronic structure theory to provide the potential energy and its derivatives. Previous studies are used to critique and compare the Car-Parrinello and Born-Oppenheimer approaches for performing direct dynamics. Relative merits of integrating trajectories in the cartesian or instantaneous normal mode framework are considered as well as the use of the Hellmann-Feynman force versus the analytic gradient. The applicability of density functional, ab initio, and semiempirical electronic structure theories to direct dynamics is discussed. The chapter concludes with a synopsis of our direct dynamics simulations for H₂CO → H₂ + CO dissociation, and the intramolecular and unimolecular dynamics of Cl⁻⋯CH₃Br and trimethylene.

The intramolecular dynamics of a polyatomic molecule is an intrinsically multidimensional problem of great complexity, especially when the molecule is highly excited. Understanding the mode-mode dynamics requires recourse to novel methods which are designed to bring out important features of the dynamics such as resonances and chaos/order transitions. In this chapter, we demonstrate the use of such a method, Frequency Analysis, by applying it to the mode-mode dynamics of the formaldehyde molecule.

Quantum dynamics in multi-dimensional systems is described in the context of a generalized Langevin equation approach to the computation of time correlation functions. Four imaging procedures based on the principle of using short-time derivatives are presented; namely, a Taylor image, a trigonometric image, a time-packet image, and a maximum-entropy image. The Taylor and trigonometric images simulate the time correlation function directly with the trigonometric image being especially useful for discrete state systems. The time-packet and maximum-entropy images make full use of the GLE structure to model a bath spectral density. Applications to a quartic oscillator and a harmonic oscillator coupled to an ohmic bath are shown to demonstrate the essential features of the imaging procedures.

Methods for time-dependent quantum-mechanical simulations of large polyatomic systems are described and analyzed, and applications to processes in clusters are presented. Two basic methods are discussed: (i) The Time-Dependent Self-Consistent Field (TDSCF) approximation and its generalizations; (ii) The Classically-Based Separable Potential (CSP) approach, and its extensions. At their simplest respective levels, both methods assume separability of the different modes of the system, and describe each mode as moving in a mean field due to the other modes. In TDSCF the effective single-mode potentials are computed quantummechanically, in CSP they are obtained from a classical MD simulation that precedes the quantum calculation. For both methods, improvements are available that correct for correlations between different modes, resulting in approaches of good accuracy. Both the TDSCF and the CSP approach make possible calculations for systems of size and complexity that could hitherto not be treated. Algorithmic and computational aspects of the two methods are examined, and it is found that in typical molecular applications the methods are nearly identical in accuracy, but CSP is greatly superior computationally, especially in applicability to much larger systems than TDSCF. Applications of the methods are presented for: (1) Photoexcitation dynamics and spectroscopy of atomic and molecular impurities in large clusters, e.g., for I₂(Ar)_n. (2) Collision dynamics, energy transfer and state-to-state transitions in atom-cluster scattering, e.g., He + (Ar)_n, Ar + (H₂O)_n. Future research directions are suggested in the light of the algorithmic aspects and the applications.

Theoretical methods and investigations of physical and chemical processes occurring in matrices at cryogenic temperatures are described. Methods for treating phenomenological rate data are discussed with emphasis on inhomogeneous systems that exhibit a distribution of diffusion coefficients within the matrix environment. Previous investigations of mobility and diffusion are reviewed. Methods employing energy decay plots and power spectrum line splittings to obtain energy transfer rate coefficients between an embedded subtrate and the phonon modes of the matrix are described. The Chapter examines the methods that have proven to be the most useful in the theoretical study of matrix-isolated chemical reactions. Applications to isomerization reactions, unimolecular decomposition and bimolecular reactions are given and discussed.

In this review, we will begin with a perspective on the use and usefulness of the application of classical molecular dynamics to complex macromolecular systems. This will be followed by sections that describe issues in the development/application of molecular dynamics to such systems. Sections will include: 2. Force Fields; 3. Boundary Conditions and Long Range Electrostatics; 4. Molecular Dynamics Integrators and Temperature and Pressure Coupling Approaches; 5. Conformational Sampling; and 6. Summary.

Molecular dynamics simulations are ideally suited for studying the properties of aqueous solutions of biological molecules, which have been difficult to probe experimentally. A series of such simulations of carbohydrates in recent years makes possible a general survey of solvation for this class of molecules. The incorporation of solvation effects into theoretical models of sugars was found to be necessary to produce realistic results. The relative rigidity of the pyranose ring forms of the sugars allows a mapping of the anisotropic structuring of water around carbohydrate solutes. The detailed structuring imposed on the solvent was found to be a function of each solute molecular structure and to have significant effects on the properties of the solutions.

Computer simulation and modeling have become powerful tools for understanding the structure and properties of a broad range of materials. Simulations, both atomistic and continuum, are playing an increasingly prominent role in materials science, chemistry, engineering, and molecular biology. Beyond the relatively conventional studies of point and planar defects or simply providing more detailed and better understanding of experimental data, modern computational chemistry provides tools for studying complex processes involving fracture, fatigue, machining and for examining and extending analytic theories. In this chapter we present an introduction to a number of computational methods that can be used to examine the structure and properties of materials (ranging from small molecules to polymers). The methods are discussed in the context of molecular-based materials (systems that are bound primarily by covalent bonding) but can also be used for condensed matter (van der Waals), ionic, and metallic systems.

Methods for performing molecular simulations of detonation, a chemical reaction wave that propagates at supersonic velocities, are described. Also, a review of the potential energy functions used to describe the exothermal chemical reactions needed to sustain the detonation wave and a description of various simulation approaches are provided. Suggestions for improving the models used in detonation simulation are given.

The present article is a tutorial on the use of Monte Carlo methods in chemistry. It is intended to convey a feel for the character and utility of these methods to the non-expert. Various classical and quantum Monte Carlo methods are illustrated by applying them to a relatively simple model problem. In keeping with the tutorial character, sample programs are included to provide a starting point for the benefit of first-time users.

Recently developed microcanonical Monte Carlo sampling methods are described and applications to the evaluation of classical transition-state theory rate constant expressions are presented. These include unimolecular and bimolecular gas-phase reaction rates and a method for computing nonadiabatic unimolecular reaction rates. Monte Carlo methods for estimating thermal desorption, diffusion and tunnelling rates in solids and on surfaces are also discussed. The methods described provide a bridge between simple models and full molecular dynamics simulations, and are of particular use for the study of reaction rates in large molecular systems where quantum mechanical techniques cannot be easily applied.

Variational transition state theory (VTST) with multidimensional tunneling (MT) contributions provides a practical way to include zero-point energies and semiclassical estimates of barrier penetration in reaction rate computations for polyatomic reactions. In this chapter we compare rate coefficients calculated by variational transition state theory (VTST) with optimized multidimensional tunneling (OMT) transmission coefficients to accurate quantal calculations for 74 three-body systems for which accurate quantal rate coefficients are available. These comparisons include both one-dimensional and three-dimensional treatments and both thermal and state-selected reactions. Rate coefficient data are given at a standard set of temperatures in the range 200-2400 K. Rate coefficients are calculated by conventional transition state theory (TST) and improved canonical variational transition state theory (ICVT). Transmission coefficients to account for tunneling are calculated using the small-curvature-tunneling (SCT) approximation, the large-curvature-tunneling (LCT) approximation, the microcanonical optimized multidimensional tunneling (μOMT) approximation, and the least-action-tunneling (LAT) approximation. It is shown that ICVT/μOMT and ICVT/LAT rate coefficients are generally in excellent agreement with the accurate quantal data.

A semiclassical approach for treating tunneling in multidimensional systems is described. The method incorporates tunneling calculations into standard classical trajectory simulations and thus allows for full-dimensional treatment of tunneling. The accuracy of the method is discussed and applications to molecular collisions and unimolecular isomerization reactions are reviewed.