It is still difficult to obtain a precise structural description of the transition between the deoxy T-state and oxy R-state conformations of human hemoglobin, despite a large number of experimental studies. We used molecular dynamics with the Path Exploration with Distance Constraints (PEDC) method to provide new insights into the allosteric mechanism at the atomic level, by simulating the T-to-R transition. The T-state molecule in the absence of ligands was seen to have a natural propensity for dimer rotation, which nevertheless would be hampered by steric hindrance in the "joint" region. The binding of a ligand to the alpha subunit would prevent such hindrance due to the coupling between this region and the alpha proximal histidine, and thus facilitate completion of the dimer rotation. Near the end of this quaternary transition, the "switch" region adopts the R conformation, resulting in a shift of the beta proximal histidine. This leads to a sliding of the beta-heme, the effect of which is to open the beta-heme's distal side, increasing the accessibility of the Fe atom and thereby the affinity of the protein. Our simulations are globally consistent with the Perutz strereochemical mechanism.