We have used density functional theory, both within the local density (LDA) and generalized gradient (GGA) approximations, to study the structure, energetics, and vibrational properties of zeolite offretite in the presence of different monovalent cations (H+, Na+, M+, and Cu+). We find that the spatial locations of the most favorable cation-binding sites are similar for the different cations, being related to the minima of the electrostatic potential. However, the relative stability of the sites does depend on the nature of the counterion, as well as on the Al/Si ratio and on the mutual interactions between cations. At low Al/Si ratios, the preferred sire for H+ is in the channel, where it is accessible for reaction with incoming molecules. For both Na+ and Cu+, the most stable site is within the 6-fold ring of the gmelinite cage, but for Na+, two other sites are present within a few tenths of a kilocalorie/mole from the lowest site (small site selectivity). For K+, two sites, one inside the cancrinite cage and the other near the 8-fold ring of the gmelinite cage, are very close in energy, consistent with the X-ray experiments on natural hydrated and dehydrated offretites. Dynamical simulations have been carried out for H- and Na-offretite. The vibrational spectrum of the framework agrees well with the available experiment. OH stretching frequencies calculated for a number of different H+ locations show that more "open" positions, e.g., in the channel, have higher frequencies, in agreement with experiment.