THE CUMULATIVE REACTION PROBABILITY AS EIGENVALUE PROBLEM

It is shown that the cumulative reaction probability for a chemical re action can be expressed (absolutely rigorously) as N(E) = SIGMA(k)p(k) (E), where {p(k)} are the eigenvalues of a certain Hermitian matrix (o r operator). The eigenvalues {P(k)} all lie between 0 and 1 and thus h ave the interpretation as probabilities, eigenreaction probabilities w hich may be thought of as the rigorous generalization of the transmiss ion coefficients for the various states of the activated complex in tr ansition state theory. The eigenreaction probabilities {p(k)} can be d etermined by diagonalizing a matrix that is directly available from th e Hamiltonian matrix itself. It is also shown how a very efficient ite rative method can be used to determine the eigenreaction probabilities for problems that are too large for a direct diagonalization to be po ssible. The number of iterations required is much smaller than that of previous methods, approximately the number of eigenreaction probabili ties that are significantly different from zero. All of these new idea s are illustrated by application to three model problems-transmission through a one-dimensional (Eckart potential) barrier, the collinear H + H-2 --> H-2 + H reaction, and the three-dimensional version of this reaction for total angular momentum J = 0.