The enzyme nitrogenase naturally hydrogenates N2 to NH3, achieved through the accumulation of H atoms on FeMo-co, the Fe7MoS9C(homocitrate) cluster that is the catalytically active site. Four intermediates, E1H1, E2H2, E3H3, and E4H4, carry these hydrogen atoms. I report density functional calculations of the numerous possibilities for the geometric and electronic structures of these poly-hydrogenated forms of FeMo-co. This survey involves more than 100 structures, including those with bound H2, and assesses their relative energies and most likely electronic states. Twelve locations for bound H atoms in the active domain of FeMo-co, including Fe–H–Fe and Fe–H–S bridges, are studied. A significant result is that transverse Fe–H–Fe bridges (transverse to the pseudo-threefold axis of FeMo-co and shared with triply-bridging S) are not possible geometrically unless the S is hydrogenated to become doubly-bridging. The favourable Fe–H–Fe bridges are shared with doubly-bridging S. ENDOR data for an E4H4 intermediate trapped at low temperature, and interpretations in terms of the geometrical and electronic structure of E4H4, are assessed in conjunction with the calculated possibilities. The results reported here yield a set of 24 principles for the mechanistically significant coordination chemistry of H and H2 on FeMo-co, in the stages prior to N2 binding.