For a general spin dimer that has different numbers of unpaired electrons at the two spin sites, we examined the relationship between the spin exchange parameter J and the energy difference between the broken-symmetry and highest-spin states of the spin dimer on the basis of the spin-Hamiltonian (H) over cap=-J (S) over cap (1).(S) over cap (2). The resulting relationship was shown to be a generalization of Noodleman's formulas derived from the broken-symmetry method within density functional theory. We proposed a new method of calculating spin exchange parameters J within density functional theory on the basis of Slater's transition state concept. The intrachain J values of magnetic solids A(2)MnF(5) (A=Rb, Cs, NH4, Na, Li) were calculated by using the "transition-state" and Noodleman's methods. The J values calculated by the transition-state method are in much better quantitative agreement with experiment. (C) 2001 American Institute of Physics.