The spin-forbidden oxidative addition of H-2 to Fe(CO)(4), Fe(PH3)(4), Fe(dpe)(2) and Fe(dmpe)(2) [dpe=H2PCH2CH2PH2, dmpe=(CH3)(2)PCH2CH2P(CH3)(2)] has been investigated by density functional theory using a modified B3PW91 functional. All 16-electron fragments are found to adopt a spin triplet ground state. The H-2 addition involves a spin crossover in the reagents region of configurational space, at a significantly higher energy relative to the triplet dissociation asymptote and, for the case of Fe(CO)(4).H-2, even higher than the singlet dissociation asymptote. After crossing to the singlet surface, the addition proceeds directly to the classical cis-dihydride product. Only for the Fe(CO)(4) was it possible to locate a stable energy minimum for the non-classical tautomer (dihydrogen complex), but the energy difference between this minimum and the tautomerisation transition state inverts when taking into account the zero-point energy correction. The geometries at the crossing points indicate a "side-on" approach of the H-2 molecule to the metal for the Fe(CO)(4), Fe(CO)(2)(PH3)(2) and Fe(PH3)(4) systems. These geometries are more reactants-like for the Fe(CO)(4) system and more product-like for the Fe(PH3)(4) system. The crossing point geometry for the Fe(dpe)(2) system, on the other hand, is nearly C-2-symmetric. The presence of an energy barrier on going from (FeL4)-Fe-3+H-2 to the crossing point is in agreement with the slow observed rates for addition of H-2 to these unsaturated organometallic fragments.