  The problem of Kramers' turnover is a central issue of dynamical theory of
reaction rate. Since its classical solution in the Markovian limit in
mid-eighties by Melnikov and Meshkov, the problem has been addressed by a
number of groups in the last decade both in classical non-Markovian and quantum
mechanical context. Based on a coherent state representation of noise operators
and a positive definite Wigner canonical thermal distribution function we have
recently developed a c-number quantum Langevin equation [Barik \textit{et al},
J. Chem. Phys. {\bf 119}, 680 (2003); Banerjee \textit{et al}, Phys. Rev. E
{\bf 65}, 021109 (2002)]. We implement this scheme within Pollak's well known
normal mode description to calculate the quantum transmission coefficient over
an arbitrary range of friction, noise correlation and temperature. The theory
generalizes the quantum correction to Grote-Hynes factor in the rate expression
down to vacuum limit which reduces to well known high temperature quantum
correction, \textit{i.e.}, the Wolynes term for quantum transmission and
reflection for the barrier in the appropriate limit and also considers the
quantum corrections due to nonlinearity of the system potential order by order
which contributes to energy loss and dispersion due to coupling between
unstable and stable normal modes near the barrier top and is valid for both
above and below the activated tunneling regime. Our results have been compared
with those obtained earlier for a model potential and found to be good
agreement.
