Molecules consisting entirely of nitrogen have been studied extensively for their potential as high energy density materials (HEDM). However, many such molecules are too unstable to serve as practical energy sources. This has prompted many studies of molecules that are mostly nitrogen but which incorporate heteroatoms into the structure to provide additional stability. In the current study, cages of three-coordinate nitrogen are viewed as candidates for stabilization by insertion of oxygen atoms into the nitrogen framework. Cages of N-12, N-14, and N-16 with four-membered rings are studied because four-membered rings have been previously shown to be a destabilizing influence. Insertion of oxygen atoms, which converts N-N bonds to N-O-N bonding groups, relieves ring strain and can potentially result in stable molecules. These molecules are studied by theoretical calculations, using Hartree-Fock and Moller-Plesset (MP3 and MP4) theories, to determine the dissociation energies of the molecules. The primary result of the study is that stable molecules can result from oxygen insertion but that oxygen-oxygen proximity destabilizes the insertion products.