Distance dependence of tunneling in dissipative systems

We investigate the distance dependence of the rate of tunneling between two sites a distance r apart interacting with a thermal bath of phonons characterized by a density of states of the form g()d-1. We show that in such systems, the correction to the tunneling rate is of the form exp[-(r/r0)3-d] where r0 is a characteristic length. Ohmic dissipation corresponds to d=1 and a Gaussian correction exp[-(r/r0)2] to the tunneling rate arises as predicted by one of us [P. Phillips, J. Chem. Phys. 84, 976 (1986)].