  The perfect quenching of spin tunneling that has previously been discussed in
terms of interfering instantons, and has recently been observed in the magnetic
molecule \Fe8, is treated using a discrete phase integral (or
Wentzel-Kramers-Brillouin) method. The simplest model Hamiltonian for the
phenomenon leads to a Schr\"odinger equation that is a five-term recursion
relation. This recursion relation is reflection-symmetric when the magnetic
field applied to the molecule is along the hard magnetic axis. A completely
general Herring formula for the tunnel splittings for all reflection-symmetric
five-term recursion relations is obtained. Using connection formulas for a new
type of turning point that may be described as lying "under the barrier", and
which underlies the oscillations in the splitting as a function of magnetic
field, this Herring formula is transformed into two other formulas that express
the splittings in terms of a small number of action and action-like integrals.
These latter formulas appear to be generally valid, even for problems where the
recursion contains more than five terms. The results for the model Hamiltonian
are compared with experiment, numerics, previous instanton based approaches,
and the limiting case of no magnetic field.
