Quantum Reactive Scattering Formulation

X. Wu, R. E. Wyatt, and M. D Mello, Inclusion of the geometric phase in quantum reactive scattering calculations A variational formulation,/. Chem. Phys. 101 2953 (1994). 

ABSTRACT. In this paper, the theoretical basis and the program architecture of some reduced dimensionality quantum reactive scattering computational procedures are illustrated. The aim is to evidence to what extent it is possible to take advantage of parallel and vector performances of modem supercomputers for carrying out extensive calculations of the reactive properties of atom diatom systems. Some efforts have been paid to indicate alternative ways of formulating both the theoretical approaches and the computational codes. Speed-up factors of the suggested solutions have been calculated for some test cases. Results of extensive calculations performed for two prototype reactive systems are presented. 

It should be clear from Fig. 2b (Part I) that either set of mass-scaled Jacobi coordinates alone provides a complete description of the available collinear coordinate space. However, it should be equally clear that while Ra and Va are better suited to describing translational and vibrational motions in the reactant channel, Rc and Tc are more appropriate for a corresponding description of the products. It therefore seems natural to retain both sets of coordinates at once, using each set for convenience as required. Moreover, formulations of quantum reactive scattering based on this idea are quite easy to construct. Indeed a comprehensive account of such a formulation, for the... 

We have derived the ABC formulation of quantum reactive scattering theory, and applied it to the calculation of the initial state selected reaction probability. By exploiting the highly localized nature of forces in reactive scattering, the ABC formulation facilitates the direct calculation of detailed or averaged reaction probabilities while sampling only a finite region of space. 

The final solution to the quantum reactive scattering problem we shall consider is the time-dependent wavepacket method (see Wave Packets). This is one of the simplest ways to solve the Schrddinger equation for a chemical reaction, and it is also rapidly emerging as one of the most powerful. Moreover, it has the great advantage that its numerical implementation is only a small step away from its theoretical formulation, which makes the method especially easy to justify and explain. 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

As will be shown throughout this book, quantum control of molecular dynamics has been applied to a wide variety of processes. Within the framework of chemical applications, control over reactive scattering has dominated. In particular, the two primary chemical processes focused upon are photodissociation, in which a molecule is irradiated and dissociates into various products, and bimolecular reactions, in which two molecules collide to produce new products. In this chapter we formulate fie quantum theory of photodissociation, that is, the light-induced breaking of a chemical bond. In doing so we provide an introduction to concepts essential for the 1 remainder of this book. The quantum theory of bimolecular collisions is also briefly ydiscussed. 

ABSTRACT. Adiabatically adjusting Principal-axis Hyperspherical(APH) coordinates are used in a fully 3-dimensional quantum mechanical formulation of reactive scattering. Exact results will be presented for the following systems ... 

Vector opmtions dominate in irrany quantum mechanical calculatitHis and the advent of computers with a design that greatly enhances the rate at which they can be performed has made it feasible to consider also some very demanding problems, such as die determinatitm of state-to-state cross sections in reactive scattering theo. My report here concerns some aspects of a three-dimensional treatment of atom-diatom collisions based on the Finite Element Method employed in a hyperspheiical coordinate formulation. 

Indeed, it is the presence of the exchange interaction in this formulation of reactive scattering that until recently has stymied this approach. Wolken and Karplus [78] made some early attempts using it, but these were not completely successful. It has ultimately become clear that the most satisfactory way of dealing with exchange is analogous to what quantum chemists do in the Hartree-Fock problem, namely to expand the unknown wavefunctions in a basis set and determine the expansion coefficients via a variational principle. 

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