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The spectra for H5+ and D5+ are extended to cover the region between 4830 and 7300 cm−1. These spectra are obtained using mass-selected photodissociation spectroscopy. To understand the nature of the states that are accessed by the transitions in this and prior studies, we develop a four-dimensional model Hamiltonian. This Hamiltonian is expressed in terms of the two outer H2 stretches, the displacement of the shared proton from the center of mass of these two H2 groups, and the distance between the H2 groups. This choice is motivated by the large oscillator strength associated with the shared proton stretch and the fact that the spectral regions that have been probed correspond to zero, one, and two quanta of excitation in the H2 stretches. This model is analyzed using an adiabatic separation of the H2 stretches from the other two vibrations and includes the non-adiabatic couplings between H2 stretch states with the same number of quanta of excitation in the H2 stretches. Based on the analysis of the energies and wave functions obtained from this model, we find that when there are one or more quanta of excitation in the H2 stretches the states come in pairs that reflect tunneling doublets. The states accessed by the transitions in the spectrum with the largest intensity are assigned to the members of the doublets with requisite symmetry that are localized on the lowest-energy adiabat for a given level of H2 excitation.