Cumulative reaction probability via transition state wave packets

A new time-dependent approach to the cumulative reaction probability, N(E), has been developed based on the famous formulation given by Miller and co-workers @J. Chem. Phys. 79, 4889 ~1983!#, N(E)5@(2p)/2# tr@d(E2H)Fd(E2H)F# . Taking advantage of the fact that the flux operator has only two nonzero eigenvalues, we evaluate the trace efficiently in a direct product basis of the first flux operator eigenstates and the Hamiltonian eigenstates on the dividing surface ~internal states!. Because the microcanonical density operator, d(E2H), will eliminate contributions to N(E) from an internal state with the energy much higher than the total energy E , we can minimize the number of internal states required by choosing a dividing surface with the lowest density of internal states. If the dividing surface is located in an asymptotic region, one just needs to include all the open channels, i.e., with internal energy lower than the total energy. Utilizing the Fourier transform for d(E2H), we can obtain the information for all the energies desired by propagating these wave packets once. Thus the present approach will be much more efficient than the initial state selected wave packet ~ISSWP! approach to N(E) for systems with many rotation degrees of freedom because the density of states in asymptotic region for such systems is much higher than that in the transition state region. With the present method one can also calculate the cumulative reaction probability from an initial state ~or to a final state! by locating the second flux operator in the corresponding asymptotic region. This provides an alternative to the ISSWP approach which may be more efficient if the reaction probabilities from a large number of initial states are desired. The method is applied to the 3D H 1 H2 ~even rotation! reaction for J50 by locating the first dividing surface in the transition state region. The demonstration also shows an aspect less than ideal; the contribution to N(E) from a wave packet may be slightly larger than 1 or slightly smaller than 0, making it improper to interpret the contribution as a probability. © 1996 American Institute of Physics. @S0021-9606~96!01216-2#