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Context. Recent attention has been directed on closed-shell aromatic cations as potential carriers of the diffuse interstellar bands. The spectra of mass-selected, matrix-isolated benzylium, and tropylium cations were recently reported. The visible spectrum of benzylium exhibits a large Franck-Condon (FC) envelope, inconsistent with diffuse interstellar band carriers. Aims. We perform a computational analysis of the experimentally studied benzylium spectrum before extending the methods to a range of larger, closed-shell aromatic cations to determine the potential for this class of systems as diffuse interstellar band carriers. Methods. Density functional theory (DFT), time-dependant ((TD)DFT), and multi-configurational self-consistent field second-order perturbation theory (MRPT2) methods in concert with multidimensional FC analysis is used to model the benzylium spectrum. These methods are extended to larger closed-shell aromatic hydrocarbon cations derived from resonance-stabilized radicals, which are predicted to show strong S0 → Sn transitions in the visible region. The ionization energies of a range of these systems are also calculated by DFT. Results. The simulated benzylium spectrum was found to yield excellent agreement with the experimental spectrum showing an extended progression in a low frequency (510 cm-1) ring distortion mode. The FC progression was found to be significantly quenched in the larger species: 1-indanylium, 1-naphthylmethylium, and fluorenium. Excitation and ionization energies of the closed-shell cations were found to be consistent with diffuse interstellar band carriers, with the former lying in the visible range and the latter straddling the Lyman limit in the 13-14 eV range. Conclusions. Large closed-shell polycyclic aromatic hydrocarbon cations remain viable candidate carriers of the diffuse interstellar bands. [ABSTRACT FROM AUTHOR]