Ab initio calculations at the MP2/6-311++G(d,p) and MP2/aug-cc-pVDZ//MP2/aug-cc-pVTZ levels have been performed for the following complexes: H2OH+••• HBeH, H2OH+••• HBeBeH, H2OH+••• HBeF, ClOH+••• HBeH, Cl2OH+••• HBeH, and Cl2OH+••• HBeF. For all dimers considered, extremely short H••• H intermolecular contacts (1.0-1.3 &ANGS;) were obtained. These are the shortest intermolecular distances which have ever been reported, with binding energies within the range of 13.7-24.3 kcal/mol (MP2/aug-cc-pVDZ//MP2/aug-cc-pVTZ level). The interaction energies of the complexes analyzed were also extrapolated to the complete basis set (CBS) limit. To explain the nature of such strong interactions, the Bader theory was applied, and the characteristics of the bond critical points (BCPs) were analyzed. It was pointed out that for the major part of the H••• H contacts considered here the Laplacian of the electron density at H••• H BCP is negative indicating the partly covalent nature of such a connection. The term "covalent character of the hydrogen bond" used sometimes in recent studies is discussed. An analysis of the interaction energy components for dihydrogen bonded systems considered indicates that in contrast to conventional hydrogen bonded systems the attractive electrostatic term is outweighed by the repulsive exchange energy term and that the higher order delocalization energy term is the most important attractive term.