Quantum chemical calculations using ab initio methods at the CCSD(T) level and density functional theory have been carried out for the title molecules. The electronic structures of the molecules were analyzed with a variety of charge and energy decomposition methods. The equilibrium geometries of the M2O2 rhombic clusters exhibit very short distances between the transannular metal atoms M = Be, Mg, Ca. The calculated distances are close to standard values between double and triple bonds, but there are no chemical M-M bonds. The metal atoms M carry large positive partial charges, which are even bigger than in diatomic MO. The valence electrons of M are essentially shifted toward oxygen in M2O2, which makes it possible that there is practically no electronic charge in the region between the metal atoms. The bond dissociation energies for fragmentation of M2O2 into two metal oxides MO are very large. The metal-oxide bonds in the rhombic clusters are shorter and stronger than in diatomic MO. A detailed analysis of the electronic structure suggests that there is no significant direct M M interaction in the M2O2 rhombic clusters, albeit weak three-center M-O-M bonding.