The thermochemical stability of hydrogen rings is studied using ab initio SCF-CI methods. The rings include H 3 +, H 4, H 5 +, H 5 -, H 6, H 7 +, and H 8. Optimized geometries for the D nh rings are reported at theoretical levels from SCF to multireference CI. Vibrational frequencies are reported at the optimized geometries, in order to establish whether the ring geometries are absolute minima, relative minima, or saddle points on the potential surface. Only H 3 + is an absolute minimum. Three relatively stable systems are H 5 -, H 6, and H 7 +; in general, low-energy ring systems contain 4n + 2 electrons. High-energy systems contain 4n electrons, including H 4, H 5 +, and both the cubic and octagonal forms of H 8. Existence or absence of local minima depends strongly on the type of correlated products; e.g., for square planar H 4 the lowest singlet state is unstable because it dissociates along a rectangular or trapezoidal path to ground state H 2 products, whereas the next two singlets are stable with respect to both these vibrational modes because they dissociate to excited products. Contrary to previous reports in the literature, the lowest state of square planar H 4 is a triplet state, so there is no violation of Hund's rule. The possible stabilization of the hydrogen rings by formation of inorganic complexes is briefly discussed.