The transition state theory of chemical reactions rests on the assumption that motion along a reaction path is separable from motion in directions transverse to it and that the latter are adiabatic, i.e., that their quantum numbers are preserved as the reaction proceeds. This assumption requires that, in calculating quantum mechanical tunneling factors, the zero-point energy of the transverse motions be added to the potential energy along the reaction path to furnish an effective vibrationally adiabatic barrier which should be used in such calculations. All tunneling calculations reported so far have neglected such zero-point energy corrections or used unrealistic approximations for them, and most have also replaced the reaction barrier by approximate analytical fits for which transmission probabilities could be determined analytically. We have performed accurate quantum mechanical calculations by numerical techniques for the collinear exchange reactions of H + H2 and D + D2 and reached the following conclusions. (1) The results of transmission probability and tunneling factor calculations from Eckart potential fits to the potential energy barrier lead to substantial systematic errors, especially at low temperatures, and therefore should not be used. Further, correcting Shavitt's calculations to eliminate his numerical approximations brings the model he used into better agreement with experiment. (2) The results of calculations ignoring the zero-point energy of the transverse motion and its variation along the reaction path are dramatically different from the ones including it. Since the latter are in accord with the adiabatic derivation of transition state theory and the former are not, agreement of the latter with gas-phase experiments must be considered the result of fortuitous cancellations of errors and should not be construed as support for the assumption behind the theory.