  The ionization by photon or electron impact of the inner (2a1) and outer
(1t2) valence orbitals of the CH4 molecule is investigated theoretically. In
spite of a number of approximations, including a monocentric approach and a
rather simple distorting molecular potential, the calculated cross sections are
overall similar to those of other theoretical methods, and in reasonable
agreement with experimental data. The originality of the present approach
stands in the way we evaluate the transition matrix elements. The key
ingredient of the calculation scheme is that the continuum radial wave function
of the ejected electron is represented by a finite sum of complex Gaussian type
orbitals. This numerically expensive optimization task is then largely
compensated by rather simple and rapid analytical calculations of the necessary
integrals, and thus all ionization observables, including cross section angular
distributions. The proposed and implemented Gaussian approach is proved to be
numerically very reliable in all considered kinematical situations with ejected
electrons up to 2.7 a.u.. The analytical formulation of the scheme is provided
here for bound molecular states described by monocentric Slater type orbitals;
however, Gaussian type orbitals may be equally be employed. In combination with
complex Gaussian functions for the continuum states, an all Gaussian approach
with multicentric bound states can be envisaged.
