We present a new method based on set oriented computations for the calculation of reaction rates in chemical systems. The method is demonstrated with the Rydberg atom, an example for which traditional Transition State Theory fails. Coupled with dynamical systems theory, the set oriented approach provides a global description of the dynamics. The main idea of the method is as follows. We construct a box covering of a Poincaré section under consideration, use the Poincaré first return time for the identification of those regions relevant for transport and then we apply an adaptation of recently developed techniques for the computation of transport rates ([12], [27]). The reaction rates in chemical systems are of great interest in chemistry, especially for realistic three and higher dimensional systems. Our approach is applied to the Rydberg atom in crossed electric and magnetic fields. Our methods are complementary to, but in common problems considered, agree with, the results of [14]. For the Rydberg atom, we consider the half and full scattering problems in both the 2- and the 3-degree of freedom systems. The ionization of such atoms is a system on which many experiments have been done and it serves to illustrate the elegance of our method.