Longitudinal instanton fluctuations

The zero mode, which is associated with the longitudinal fluctuations, is now included in (4.10), and when wt>0, the determinants in (4.11) do not suffer from the zero-mode problem. The value klD is simply the rate of tunneling (3.67) in the dynamical one-dimensional barrier V%s) along the instanton trajectory. As for Bt, it incorporates the effect of transverse vibrations around the instanton trajectory. To calculate (4.10), one may employ the apparatus of Chapter 3 designed for one-dimensional tunneling. In particular, now it is possible to make use of (3.69) together with (3.66), which gives... 

The preexponential factor. 4(r) in the rate expression (29.19) consists of a longitudinal component, 4 and a perpendicular component A , representing the contributions of fluctuations that are, respectively, parallel and perpendicular to the instanton path. If in the one-dimensional potential U( (x) the long turmeling action evaluated at = 0 for T = 0 is replaced by the short action evaluated at the zero-point energy hQ(,/2, we can use the approximation, 4 — Q.q/1k for the longitudinal component. The perpendicular component is treated in the adiabatic approximation, which yields = 1 if [/ (x) is replaced by its vibrationally adiabatic counterpart. Using these approximations, we obtain... 

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