Biomass recalcitrance, the resistance of cellulosic biomass to degradation, is due in part to the stability of the hydrogen bond network and stacking forces between the polysaccharide chains in cellulose microfibers. The fragment molecular orbital (FMO) method at the correlated Møller-Plesset second order perturbation level of theory was used on a model of the crystalline cellulose Iα core with a total of 144 glucose units. These computations show that the intersheet chain interactions are stronger than the intrasheet chain interactions for the crystalline structure, while they are more similar to each other for a relaxed structure. An FMO chain pair interaction energy decomposition analysis for both the crystal and relaxed structures reveals an intricate interplay between electrostatic, dispersion, charge transfer, and exchange repulsion effects. The role of the primary alcohol groups in stabilizing the interchain hydrogen bond network in the inner sheet of the crystal and relaxed structures of cellulose Iα, where edge effects are absent, was analyzed. The maximum attractive intrasheet interaction is observed for the GT-TG residue pair with one intrasheet hydrogen bond, suggesting that the relative orientation of the residues is as important as the hydrogen bond network in strengthening the interaction between the residues.