  The ground electronic, vibrational and rotational state of the OH molecule is
currently of interest as it can be manipulated by electric and magnetic fields
for experimental studies in ultracold chemistry and quantum degeneracy. Based
on our recent exact solution of the corresponding effective Stark-Zeeman
Hamiltonian, we present an analytical study of the crossings and avoided
crossings in the spectrum. These features are relevant to non-adiabatic
transitions, conical intersections and Berry phases. Specifically, for an
avoided crossing employed in the evaporative cooling of OH, we compare our
exact results to those derived earlier from perturbation theory.
