The popular approach of softening the Coulomb singularity in models with reduced dimensionality leads to uncontrollable discrepancies with numerical results obtained using exact potentials. This is demonstrated for ionization and dissociation probabilities as well as for high-order harmonic generation spectra even of the simple hydrogen molecular ion in aligned geometry. Soft-core models perform better in full dimensions, thereby losing one of their advantages, namely the gain in computation time that is achieved by going to reduced dimensionality. However, the much simpler numerical implementation of soft-core potentials carries over to full dimensions. Moreover, a generalization of the soft-core potential given in previous work [M. Lein, P. P. Corso, J. P. Marangos, and P. L. Knight, Phys. Rev. A 67, 023819 (2003)] is proposed, leading to a tremendous improvement toward the exact results, which is very promising for future calculations without cylindrical symmetry.