The direct evaluation of Eq. (6.20) by numerical methods was performed to generate the theoretical function used in fitting the temperature dependence of the Mu hop rate in terms of three parameters; the tunnelling bandwidth , and dimensionless coupling parameters and . Since the phonon spectrum includes an arbitrary normalization that is folded into the coupling parameters, the overall normalization is arbitrary. However, the ratio of the coupling parameters tells us their relative strength, and determines the course of the temperature dependence.

Figure 6.29(a) shows an example of the integrand
of Eq. (6.20),
a function of time.
The point to be made here is that the high temperature
expansion of the general theory is clearly not appropriate for all
values of the coupling constants. The structure in this function away
from *t*=0 is due to phonons in the middle of the phonon
spectrum; the highest frequencies contribute most to the region
near *t*=0.

Numerical evaluation of the integrals is made
difficult due to the oscillatory nature of the integrand, which
contains terms in while ranges from zero to, in principle, infinity.
However, the existence of an upper cut-off in the phonon
spectrum at the Debye frequency allows one to carry
out the integration to a time *t*
such that an accurate estimate of the integral in the limit
that can be made.
(The program that performs this calculation is
discussed in Apppendix B, complete with the code.)
Fitting the data with hop rates calculated in this model
gives the parameter values
and .The resulting curve is also shown by the solid line
in figure Fig. 6.30.
The values of the parameters imply that the suppression of tunnelling
due to the polaron self-trapping is overwhelmingly
large in comparison to the benefit gained from barrier height
fluctuations.
A fit with the model of Flynn and Stoneham, with the
hop rate given by Eq. (5.40)
in which
the barrier fluctuations do not enter, gives a similar
fit with parameter values = 2.50(5) and
activation energy *E* = 0.0508(2) eV, also essentially indistinguishable
from the full one-phonon theory.