A combination of numerical techniques designed for efficient computations with very large basis sets which enables the spectral analysis of transitional-mode eigenstates in the dissociation of ketene is described. At the heart of our methods is a pseudo-spectral algorithm for the action of the transitional mode Hamiltonian (describing the rocking/bending motion of the dissociating fragments) on a state vector. This allows the multiplication of the state vector by the Hamiltonian matrix without explicitly storing the matrix, thus enabling very large basis sets to be managed. With the radial separation between the fragments frozen, transitional-mode eigenstates at energies close to threshold, where the adiabatic channels are moderately well separated, are readily computed with the Lanczos iterative technique. At higher energies the spectrum rapidly becomes extremely dense and convergence of the Lanczos algorithm is very slow. However, combinations of eigenfunctions with a bandwidth of a few wavenumbers or better can be obtained by shifted inverse iteration. Illustrative results from computations using the Klippenstein–Marcus model interaction potential for ketene are presented.