The cumulative reaction probability as eigenvalue problem

It is shown that the cumulative reaction probability for a chemical reaction can be expressed (absolutely rigorously) as N(E)=summation kpk(E), where {pk} are the eigenvalues of a certain Hermitian matrix (or operator). The eigenvalues {pk} all lie between 0 and 1 and thus have the interpretation as probabilities, eigenreaction probabilities which may be thought of as the rigorous generalization of the transmission coefficients for the various states of the activated complex in transition state theory. The eigenreaction probabilities {pk} can be determined by diagonalizing a matrix that is directly available from the Hamiltonian matrix itself. It is also shown how a very efficient iterative method can be used to determine the eigenreaction probabilities for problems that are too large for a direct diagonalization to be possible. The number of iterations required is much smaller than that of previous methods, approximately the number of eigenreaction probabilities that are significantly different from zero. All of these new ideas are illustrated by application to three model problems---transmission through a one-dimensional (Eckart potential) barrier, the collinear H+H2→H2+H reaction, and the three-dimensional version of this reaction for total angular momentum J=0