We have modeled [111]-cryptand, H+[111]-cryptate, H[111]-cryptate, H-2(2+)[111]-cryptate, and H-2[111]-cryptate at the HF/6-31G (and 6-31G+) level. Ground-state equilibrium geometries of H+c111 and Hc111 belong to the C-3 point group, while the cryptand itself may belong to either C-3 or C-3h Inspection of all equilibrium structures indicated no significant geometric difference between the respective charged and neutral species, although complexation (whether charged or not) leads to significant changes in the N--N and O--O distances. Factors affecting proton hopping between nitrogens are discussed, including barrier height, tunneling, and a possible coupling between the electronic degrees of freedom and migration of the encapsulated proton. We argue that none of these effects are important. Instead, solvent effects probably dominate this process. We have found a barrier height of 17 kcal/mol for internal exchange of the internal proton (or hydrogen), where each saddle point corresponds to a stationary point in C-3 space within the Hartree-Fock description. The related isomerization (C-3 to C-3) involving the uncomplexed cryptand is nearly barrier free. The lowest energy isomer of the Hc111 is clearly Rydberg-like (with a complexed proton), while the lowest energy isomer of H(2)c111 has a more common valence-like electron distribution.