The phenalenyl radical (1) is a prototype of the hydrocarbon radical. Recently, the single crystal of 2,5,8-tri-tert-butylphenalenyl (2) was isolated and showed that the two phenalenyl radicals form a staggered dimeric pair, giving rise to strong antiferromagnetic interactions. The origin of the antiferromagnetic interactions and the nature of the chemical bond for the dimeric pair are challenging issues for chemists. First, spin-polarized hybrid DFT (Becke's half and half LYP (UB2LYP)) and CASSCF calculations were performed for 2 and its simplified model, the staggered-stacking phenalenyl radical dimeric pair (3a), to elucidate the origin of the strong antiferromagnetic coupling and the characteristics of the chemical bond. The calculated results showed that a SOMO-SOMO overlap effect was responsible for the strong antiferromagnetic interactions and weak or intermediate covalent bonding between phenalenyl radicals. The tert-butyl groups introduced at three beta-positions hardly affected the magnetic coupling, mainly causing steric hindrances in the crystalline state. Next, to obtain insight into ferromagnetic stacking, we investigated the stacking effect of staggered (3a)- and eclipsed (3b)-stacking phenalenyl radical dimeric pairs with a change of the SOMO-SOMO overlap on the basis of the extended McConnell model. We found that the stacking mode of the dimeric pair with both a small SOMO-SOMO overlap and a ferromagnetic spin polarization effect provided a ferromagnetic coupling.