  Based on a true phase space probability distribution function and an ensemble
averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002)]
a non-Markovian quantum Kramers' equation to derive the quantum rate
coefficient for barrier crossing due to thermal activation and tunneling in the
intermediate to strong friction regime. We complement and extend this approach
to weak friction regime to derive quantum Kramers' equation in energy space and
the rate of decay from a metastable well. The theory is valid for arbitrary
temperature and noise correlation. We show that depending on the nature of the
potential there may be a net reduction of the total quantum rate below its
corresponding classical value which is in conformity with earlier observation.
The method is independent of path integral approaches and takes care of quantum
effects to all orders.
