  We compare the coupled channels procedure to the semiclassical approach to
describe two-body emission processes, in particular $\alpha$-decay, from
deformed nuclei within the propagator method. We express the scattering
amplitudes in terms of a propagator matrix, describing the effect of the
deformed field, multiplied by the ratio between internal wave function
components and irregular Coulomb waves. In the spherical case the propagator
becomes diagonal and scattering amplitudes acquire the well-known form. We
describe a more rigorous formulation of the 3D semiclassical approach,
corresponding to deformed potentials, which leads to the exact results and we
also compare them with the much simpler expressions given by the Angular
Wentzel-Krames-Brillouin (AWKB) and Linearized WKB (LWKB) with its
approximation, known as Fr\"oman WKB (FWKB) method. We will show that LWKB
approach is closer than AWKB to the exact coupled-channels formalism. An
analysis of alpha-emission from ground states of even-even nuclei evidences the
important role played by deformation upon the channel decay widths.
