Motivated by recent experimental results on the reactivity of the diatomic Group 13 subhalogenides GaCl and InCl in solid Ar matrices, the mechanisms for spontaneous and photoactivated reactions of these species are studied herein by means of quantum chemical methods as applied to their reactions with H₂. The experimental results show that reaction with H₂ in a solid Ar matrix does not occur spontaneously, but requires photoactivation, the trivalent derivatives H₂MCl (M = Ga or In) being the detectable products. Furthermore, these results indicate that the photoactivated reactions proceed in a concerted fashion. Quantum chemical calculations are employed to explain the observed reactivities and provide quantitative estimates for the barriers to reaction. Calculations were performed for several electronic states and multiplicities of the GaCl/H₂ and InCl/H₂ systems likely to be of relevance and enable a detailed overall analysis of the reaction mechanisms to be made. The reactions of Group 13 subhalogenides with other molecules are anticipated to follow the same pattern and, thus, the results reported herein should be of relevance to the reactivity of Group 13 subhalogenides in general.