The exact quantum expression for the thermal rate of reaction is the trace of a product of two operators. It may therefore be written exactly as a phase space integral over the Wigner phase space representations of the two operators. The two are a projection operator onto the product's space, which is difficult to compute, and the symmetrized thermal flux operator, which can be computed using Monte Carlo methods. A quantum transition state theory was presented recently, in which the exact projection operator was replaced by its parabolic barrier limit. Alternatively, the exact projection operator may be replaced by its classical limit. Both approximations give thermodynamic estimates for the quantum rates. In this paper, we derive a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit. The correction terms are then used to evaluate the leading order corrections to the rate estimates based on the parabolic barrier or classical limits of the projection operator. The expansion is applied to a symmetric and an asymmetric Eckart barrier. The first two terms in the expansion give excellent results for temperatures above the crossover between quantum tunneling and thermal activation. For deep tunneling and asymmetric systems, the use of variational transition state theory, the classical limit, and perturbation theory leads to significant improvement in the estimate of the tunneling rate. Multidimensional extensions are presented and discussed.