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## Über dieses Buch

The amazing growth of computational resources has made possible the modeling of complex chemical processes. To develop these models one needs to proceed from rigorous theoretical methods to approximate ones by exploiting the potential of innovative architectural features of modern concurrent processors. This book reviews some of the most advanced theoretical approaches in the field of molecular reaction dynamics in order to cope as rigorously as possible with the complexity of real systems.

## Inhaltsverzeichnis

### Computational Reaction and Molecular Dynamics: from Simple Systems and Rigorous Methods to Large Systems and Approximate Methods

Abstract
The problem of a priori calculations of the efficiency parameters of chemical reactions is discussed in view of illustrating the logical percourse that has to be followed to design related computational procedures for a realistic modeling of chemical applications. The role played by innovative architectural trends of modern computing and by the impressive development of networking and hypermedia are also considered.
Antonio Laganà, Antonio Riganelli

### Fitting Potential Energy Surfaces

Abstract
This paper reviews several methods for determining analytical representations of potential energy surfaces for small molecule systems which are involved in unimolecular dissociation and/or bimolecular reaction. These methods may be categorized as “global” or “local” and they may involve either “fitting” or “interpolation” of ab initio data. The methods may be applied either directly to the full potential energy surface, or to individual terms in a many-body expansion of the full potential surface. In addition, most of the methods may be applied to the description of multiple coupled potential energy surfaces (typically using a diabatic representation) as well as to Born-Oppenheimer surfaces. Included in the global methods are least squares fitting and spline, Morse-spline, rotated Morse-spline, and reproducing kernel Hilbert space interpolation methods. The local methods include Shepard and moving least squares interpolation. Examples of the application of these methods to several triatomic reactive surfaces are discussed.
George C. Schatz

### Multivalued Potential Energy Surfaces for Dynamics Studies

Abstract
After reviewing basic concepts and the hierarchy of approximations to the solution of the many-body problem, we concentrate on the calculation and modelling of potential energy surfaces for dynamics studies with emphasis on the double many-body expansion method for multivalued global functions. A simple scheme to give spectroscopic accuracy to such functions is also highlighted. The focus is on methodological aspects although the case of H3 is examined in detail. The complications in the computational treatment of nuclear dynamics in the adiabatic state basis through the geometric phase effect are also briefly addressed.
António J. C. Varandas

### Distributed Approximating Functionals: a Robust, New Approach to Computational Chemistry and Physics

Abstract
Distributed approximating functionals (DAFs) result from a general approach to describing a class of functions (the “DAF-class”) by expanding them at a point x’ in the neighborhood of x in terms of a complete basis set constructed using the point x as the origin of coordinates:
$$f\left( {x'} \right) = {\sum\nolimits_n {{a_n}\left( x \right)\phi } _n}\left( {x' - x} \right).$$
One may use any convenient complete set {øn} for expanding f (x’)with x’ near x and furthermore different bases can be used for different neighborhoods. This results in an extremely general basis set approximation. In fact, we use the truncated (on the summation index n) approximation above only at the single pointx’x. The specific form of the expansion coefficients, an (x), is determined by a variational optimization. We shall give a detailed derivation in the lecture, and show how a particularly useful approximation to the a n (x) can be obtained. In the course of the discussion, we shall obtain the resulting DAFs. We also show the relationship of DAFs to “two-parameter Delta sequences”. The theory will be illustrated in terms of the DAF that results from choosing Hermite polynomials as the expansion basis. The properties of these DAFs will be explicated by considering their structure both in coordinate space and in Fourier space. Of greatest importance is the so-called “well-tempered” property of the DAFs. DAFs have been widely applied to solving partial differential equations (with particular emphasis on quantum scattering), as well as the more general problem of constructing approximations to functions based on a finite, discrete sampling. The original derivation of a DAF was for the quantum mechanical free propagator. General, non-product sampling in multidimensional systems can be employed, including Monte Carlo and number theoretic methods. Finally, DAFs have an intimate connection to wavelets. In addition to quantum mechanics, example potential applications (some of which have been realized) include solving nonlinear partial differential equations in situations where instabilities are encountered by other methods, signal denoising, signal enhancement, signal inversion, data compression (both lossless and lossy), imaging, pattern recognition and characterization, medical imaging, teleradiology, target acquisition, periodic and non-periodic extensions of functions in 1D, 2D, 3D, 4D,..., filling in gaps in data (including noisy experimental data), imposition of general boundary conditions onto experimental or computational data, etc.
David K. Hoffman, Donald J. Kouri

### Time Dependent Quantum Approaches to Chemical Reactivity

Abstract
Time dependent approaches to quantum reactive scattering are becoming increasingly popular. Here we give a description of basic equations and technical aspects of numerical implementations. More in detail we discuss the following arguments:
• wavepacket propagation. Solving the time-dependent Schrodinger equation.
• Expansion of the propagator.
• Energy resolution and energy range of a wavepacket.
• The wavefunction and the wavepacket.
• The Hamiltonian.
• The radial kinetic energy terms.
• The angular kinetic energy terms.
• Preparation of the initial wavepacket.
• Analysis of the product states. Calculation of S matrix.
• Reactant and Product coordinate system.
• Calculation for J > 0, centrifugal coupling and helicity decoupling.
• Angular basis functions and angula grids.
• Cross sections and reaction probabilities.
• Fourier transforms: Absorbing the wavepacket at grid edge.
• Formsof Complex absorbing potential.
• J-shifting and capture models for estimating cross sections.
• New Developments and examples.
Gabriel G. Balint-Kurti

### Quantum Reactive Scattering for Three Particle Systems using Hyperspherical Coordinates

Abstract
Quantum reactive scattering or rearrangement processes continue to be of considerable theoretical and experimental interest. A review of our Adiabatically-adjusting, Principal Axes Hyperspherical (APH) coordinates method will be presented. Basis functions defined as a product of Wigner rotation matrices [5] times surfaces functions will be defined. We will briefly describe four methods for obtaining accurate surface functions and give relative merits of each. The surface functions methods are:
• Finite Element Method (FEM)
• Discrete Variable Representation (DVR)
• Analytic Basis set Method (ABM)
• Distributed Approximating Functions (DAF)
Our choice of product basis functions giverisetoasetof second-order differential outline the solution tothese equations and present a new method equations. We will for smoothly transforming from hyperspherical to Jacobi coordinates outside the rear-rangement region. This new procedure eliminates the two dimensional projection used demonstrate that Quantum resonances are important forsomere-previously. We will arrangement processes. We also argue that it is important for theoreticians to properly the experimental parameters for a complete comparison with experiment.
Gregory A. Parker, Stefano Crocchianti, Mark Keil

### Approximate Time Independent Methods for Polyatomic Reactions

Abstract
In this chapter I review approximation methods to describe the quantum reactive scattering of polyatomic molecules. These methods are known generically as “reduced dimensionality” approximations. I will review several versions of this method, and focus on the so-called J-shifting approximation and very recent developments of it. These new developments are reviewed for the OH + H2reaction, where apparent discrepancies between coupled-states and standard J-shifting rate constants are resolved. I also present new expressions in the spirit of J-shifting for reactions that proceed via complex formation.
Joel M. Bowman

### Quantum-Classical Methods

Abstract
Theoretical foundations of Quantum-classical theories to incorporate quantum effects in molecular dynamics calculations are discussed to give a more solid ground for their applications to many degrees of freedom study of chemical reactions.
Gert D. Billing

### Direct Calculation of Reaction Rates

Abstract
Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. Thus, quantum effects, e.g. tunneling and zero point energy, are correctly included. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and their relation to transition state theory, describes computational schemes to obtain accurate rate constants, presents applications, and discusses approximations.
Uwe Manthe

### CAS-SCF and MM-VB Dynamics: Applications to Organic Photochemistry

Abstract
The increased speed of modern computers makes trajectory computations where the gradient is evaluated on the fly a realistic possibility. For excited state processes one must deal with the non-adiabatic event where the trajectory passes from the excited state to the ground state. For this we have implemented (in CAS-SCF and in our MM-VB method) a surface hop type algorithm as well as a method where the trajectory propagates on a mixed state. Investigations have been performed for several problems relevant to photochemistry. The dynamics results show interesting chemical effects associated with tipped versus sloped conical intersections and effects associated with dynamically locked transient species.
Michael A. Robb, Marco Garavelli

### Ab-Initio MD Calculations on Dynamics and Reactivity in Confined and Disordered Systems

Abstract
The subject of this lecture is an application of the Car-Parrinello method to a computational investigation of the properties of zeolites. The use of advanced computational methods has become, in fact, an important surrogate of the experiment for several inaccessible information.
Aldo Gamba

### Parallel Paradigms for Scientific Computing

Abstract
In this paper paradigms and tools for innovative parallel software environments will be discussed according to the user requirements of heterogeneus multidisciplinary applications, performance portability, rapid prototyping and software reuse, integration and interoperability of standard tools. The various issues will be demonstrated with reference to the PQE2000 project and its programming environment SkIE (Skeleton-based Integrated Environment). The coordination language SkIE-cl allows the designers to express, in a primitive and structured way, efficient combinations of data parallelism and task parallelism, with the goal of achieving fast development time and efficiency in irregular/dynamic computations. Modules developed by standard languages and tools are encapsulated in SkIE-cl structures to form the global application. A performance model associated to the coordination language allows the static and dynamic tools to introduce a large amount of global optimizations without the direct interventation of the programmer.
Marco Vanneschi

### Networking and Hypermedia in Chemistry

Abstract
The impact of networking and hypermedia on chemical research and education is discussed by outlining the main techonological features and giving some examples.
Osvaldo Gervasi

### Tutorial on Fitting of Potential Energy Surfaces

Abstract
In this paper we present a tutorial on fitting of potential energy surfaces that was given at the I European Computational Chemistry School on Molecular and Reaction Dynamics. The tutorial consists in four exercises that involve the use of a computer running under UNIX operating system, programs and subroutines for assembling and analyzing potential energy surfaces, numerical methods for fitting and graphical tools for the visualization of results. This paper explains the background of the exercises and guides the reader to the succesful completion of the exercises.
The first exercise is devoted to fitting of potential energy surfaces (PES) by inter-polation methods. Thesecond exercise explains the useof the LEPS function to fit potential energy surfaces. In the third exercise the many-body expansion method is three-dimensional PES fora non-symmetric three-atom reaction. The applied tofita last exercise illustrates the useof bond order coordinatesto represent and develop potential energy functions. It also explains theconcept of many-process expansion.
Guillermo Ochoa de Aspuru, María Luz Hernández

### Time-Dependent Techniques

Abstract
Practical aspects of quantum time-dependent calculations of atom-diatom reactive probabilities are discussed. The tutorial describes an application to the collinear H + H2 reaction, to its deuterium isotopic variant and to its zero total angular momentum three dimensional version.
Marlies Hankel, Valentina Piermarini

### The exact Computation of Reactive Cross Sections for Atom-Diatom Systems. The Hyperspherical Propagative Approach

Abstract
Several aspects involved in the theoretical formulation and the practical calculation of reactive cross sections, for atom-diatom systems, are reviewed and discussed, focusing on the time-independent hyperspherical-propagative approach. The general trends of the formalism, that allows a complete scattering calculation for a reactive process, is presented first. Then, the discussion is divided into four additional parts. The first one briefly discusses the coordinate and reference frame problem, showing how the democratic version of the hyperspherical coordinates provides a solution for the cumbersome transition between rearrangement channels. The second part deals with the details of the systematic expansion of the nuclear wavefunction, on a conveniently chosen basis set for the internal (angular) coordinates, performed at fixed values of the remaining (scattering or hyperradial) coordinate. The third part deals with the approach to solving the resulting set of coupled second-order differential equations, the propagative method, which obtains (the logarithmic derivative transform of) the hyperradial solution for a number of increasing discrete values of the hyperradial independent variable. Finally, the fourth part discusses the extraction of the asymptotic information and the calculation for reaction probabilities and integral cross sections. Some examples illustrate each stage into which the calculation process is divided.
Alessandro Bolloni, Fermin Huarte-Larrañaga, Xavier Gimenez

### A Quasiclassical Trajectory Study of Atom Diatom Reactions

Abstract
The tutorial tackles the problem of calculating scattering properties of atom diatomreactions and analyzes the accuracyofintegratedtrajectories,their graphical representations and the agreement between measured and calculated quantities by making reference to the Li + HF reaction.
Ernesto Garcia

### Quantum-Classical Methods: a Quantum-Classical Approach to Diatom-Diatom Reactive Scattering and VV Energy Transfer

Abstract
A semiclassical method to study both reactive scattering and vibrational energy transfer in diatom-diatom collisions is illustrated. The vibrations of the two reagents are treated quantum mechanically by means of an exact solution of the time dependent Schrödinger equation, while translational and rotational motions are treated classically. An effective semiclassical Hamiltonian approach is used to couple quantum and classical degrees of freedom. As an example application of the method to the reaction H2 + CN is shown.
Cecilia Coletti, Gert D. Billing

### Resonance Characterization for the Ne + H 2 + → NeH+ + H System: Time Delays and Argand Diagrams

Abstract.
Accurate three-dimensional quantum mechanical calculations on the Ne + H 2 + → NeH+ + H reaction evidence the importance of quantum effects in the reactivity pattern of this system. In this work, particular attention has been paid to the low energy resonant structure. Analysis of the resonances will be carried out by presenting the integral cross section, reaction probability, opacity function, time delay and Argand diagram plots.
Fermín Huarte-Larrañaga

### Quasiclassical Trajectory Study of the O(1D) + H2O → 2OH, H + HO2 Reactions

Abstract
The dynamics of the O(1D) + H2O → OH,H + HO2 reactions has been studied using the quasiclassical trajectory (QCT) method on a pseudotriatomic (O - H - (OH)) analytical representation of the ground potential energy surface (PES), where an OH bond of the H2O molecule has been treated as an atom of 17 a.m.u.. The OH + OH and H + HO2 reaction channels show a very different behaviour, although the H2O2 (hydrogen peroxide) deep minimum plays a very important role in the dynamics (insertion mechanism) of both channels.
Ramon Sayós, Carolina Oliva, Miguel González

### A Quantum Model Hamiltonian to Study X + YCZ3 ↔ XY + CZ3 Reactions

Abstract
Currently our work is devoted to the study of polyatomic reactions using time independent (TI) quantum methods. In particular, we are working on reactions of the type: {fx286-1} where the CZ3 group has C3 v symmetry. Even for the much simpler case of four-atom reactions, the implementation of an exact TI calculation represents a formidable task and demands a lot of computing time. Therefore, any hopeful approach to a reaction like (1.1) must be based on some sort of approximation. Our approximation was to assume that CZ3 maintains its symmetry during the reaction. This assumption is based on the fact that CZ3 does retain its symmetry when the reaction proceeds through the minimum energy path. Using this approximation, a model Hamiltonian can be obtained from first principles.
Juliana Palma, David C. Clary

### Femtodynamics of Double Proton Transfer in 7-azaindole Dimer

Abstract
The development of new spectroscopic techniques (femtospectroscopy) has made possible the experimental study of a new range of reactions in chemistry. Attention has been focused on the double proton transfer induced tautomerization of 7-azaindole dimer. This reaction can provide an insight into a similar process in DNA base pairs which could explain some mutations in different biological systems. The geometries of the species involved in the reaction are showed in Fig. 1.1.
Carmen M. Redondo, David C. Clary

### Ab Initio Calculations on the Harpooning Reaction in Ba • • • FCH3

Abstract
In ab initio calculations the energetics and the equilibrium geometries of the ground and excited electronic states of the barium-fluoromethane complex are calculated. Furthermore the nonadiabatic coupling between two neighbouring electronic states is determined and discussed. This coupling should be responsible for the experimentally observed charge transfer reaction on the subpicosecond time scale.

### Quantum Dynamics of Gas-Phase S N 2 Reactions

Abstract
A quantum approach to gas-phase S N 2 reactions is presented. Within a time-independent formalism a reduced dimensionality model is developed which takes rotations of reactants and products into account. A local symmetry approximation for the CH3 group is introduced.
Stefan Schmatz, David C. Clary

### A QCT Study of the F + CH4→ HF + CH3 Reaction Dynamics under Thermal Conditions

Summary
Quasiclassical trajectory (QCT) calculations under thermal conditions have been carried out in order to study the F + CH4reaction dynamics on an analytical pseudotriatomic (F-H-(CH3)) representation of the ground PES derived from ab initio calculations. Both scalar and two-vector properties have been analysed to get a deep insight into the reaction dynamics.
Miguel González, Diego Troya, Ramon Sayós, Judith Millán

### Backmatter

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