  An alternative method is introduced to solve a simple two-dimensional models
describing vibrational excitation and dissociation processes during the
electron-molecule collisions. The model works with one electronic and one
nuclear degree of freedom. The two-dimensional $R$-matrix can be constructed
simultaneously on the electronic and nuclear surfaces using all three forms
developed previously for electron-atom and electron-molecule collisions. These
are the eigenchannel $R$-matrix form, inversion technique of Nesbet and
Robicheaux, and the Wigner-Eisenbud-type form using expansion over the poles of
the symmetrized Hamiltonian. The 2D $R$-matrix method is employed to solve a
simple model tailored to describe the dissociative recombination and the
vibrational excitation of H$_2^+$ cation in the singlet ungerade symmetry
$^1\Sigma_u$. These results then serve as a (near-exact) benchmark for the
following calculation in which the $R$-matrix states are replaced by their
Born-Oppenheimer approximations. The accuracy of this approach and its
correction with the first-order nonadiabatic couplings are discussed.
