Using the band theory, the nature of the spin exchange interaction between the unpaired electrons within the half-filled band of 1-D organic radical crystals, consisting of galvinoxyl radicals, is investigated theoretically. The various contributions to the Heisenberg effective exchange integral, J eff , the direct, kinetic, and indirect spin exchange, are evaluated quantitatively for the experimentally determined crystal structure. To establish the dependence of J eff on the crystal topology and geometry, the calculations are carried out for several artificial model 1-D crystals of the galvinoxyl radical and of the parent hydrocarbon radical. J eff was also calculated for different interplanar distances of the stacking in the real crystal. † Part 2: Chem. Phys. 1997, 218, 43. ‡ Dedicated to Professor Cornelius Weiss on the occasion of his 65th birthday.