Effective Hamiltonian models and unimolecular decomposition

Partitioning Hilbert space into two subspaces by using orthogonal projection operators yields compact forms for effective Hamiltonians for each of the subspaces. When one (the Q space) contains molecular bound states and the other (the P space) contains dissociative continua, a simple form for the non-Hermitian Q-space effective Hamiltonian, H-eff, can be obtained, subject to reasonable approximations. Namely, H-eff = H-0 - ih Gamma/2, where H-0 is Hermitian, and the width operator h Gamma accounts for couplings of the Q-space levels to the P-space continua. The P/Q partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. Inputting into this formalism ideas from random matrix theory in order to model independent open channels yields the random matrix H-eff model. Despite numerous efforts, this model has failed to model satisfactorily the statistical transition-state theory of unimolecular decomposition (hereafter referred to as TST) in the regime of overlapping resonances, where nearly all such reactions occur. All statistical models of unimolecular decomposition are premised on rapid intramolecular vibrational redistribution (IVR) for a given set of good quantum numbers. The phase space thus accessed results in a threshold reaction rate of 1/h rho, and for K independent open channels, the rate is K/h rho. This reaction rate corresponds to a resonance width of K/2 pi rho, and when K increases, the resonances (which are rho(-1) apart) overlap. In this regime, the random matrix H-eff model fails because it does not introduce independent open channels. To illustrate the source of the problem, an analysis is carried out of a simple model that is obviously and manifestly inconsistent with TST. This model is solved exactly, and it is then put in the form of the random matrix H-eff model, illustrating the one-to-one correspondence. This reveals the deficiencies of the latter. In manipulating this model into the form H-0 - ih Gamma/2, it becomes clear that the independent open channels in the random matrix H-eff model are inconsistent with TST. Rather, this model is one of gateway states (i.e., bound states that are coupled to their respective continua as well as to a manifold of zero-order bound states, none of which are coupled directly to the continua). Despite the fact that the effective Hamiltonian method is, by itself, beyond reproach, the random matrix H-eff model is flawed as a model of unimolecular decomposition in several respects, most notably, bifurcations of the distributions of resonance widths in the regime of overlapping resonances.