A formula to compute the microcanonical volume of reactive initial conditions in transition state theory

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.