A mixed quantum classical rate theory (MQCLT) is applied to the collinear hydrogen exchange reaction on the LSTH and PK II potential energy surfaces. Classical trajectories are combined with a numerically exact quantum Monte Carlo evaluation of the thermal flux operator to compute the thermal reaction rate. The MQCLT results are compared to quantum transition state theory (QTST) and centroid rate theory computations. The computed rates are found to bound the exact results from above for temperatures ranging from T=200 K to T=1000 K. As in previous studies, the mixed quantum classical theory gives better agreement with numerically exact computations, than the QTST computations, while the added numerical effort is not prohibitive. The MQCLT rate is almost exact at high temperature. At T=200 K it is a factor of 2.8 (2.0) greater than the exact rate on the LSTH (PK II) potential energy surface, a significant improvement over the QTST overestimate of 3.7 (3.4). The mixed quantum classical results are comparable in accuracy to the centroid theory computations, except that the centroid theory is always lower than the exact result while MQCLT is always higher.