Understanding the molecular mechanism of hemoglobin cooperativity remains an enduring challenge. Protein forces that control ligand affinity are not directly accessible by experiment. We demonstrate that computational quantum mechanics/molecular mechanics methods can provide reasonable values of ligand binding energies in Hb, and of their dependence on allostery. About 40% of the binding energy differences between the relaxed state and tense state quaternary structures result from strain induced in the heme and its ligands, especially in one of the pyrrole rings. The proximal histidine also contributes significantly, in particular, in the alpha-chains. The remaining energy difference resides in protein contacts, involving residues responsible for locking the quaternary changes. In the alpha-chains, the most important contacts involve the FG corner, at the "hinge" region of the alpha(1)beta(2) quaternary interface. The energy differences are spread more evenly among the beta-chain residues, suggesting greater flexibility for the beta- than for the alpha-chains along the quaternary transition. Despite this chain differentiation, the chains contribute equally to the relaxed substitute state energy difference. Thus, nature has evolved a symmetric response to the quaternary structure change, which is a requirement for maximum cooperativity, via different mechanisms for the two kinds of chains.