The exchange energy between two multielectron atomic systems is shown to be a product of an angular momentum factor and the energy of the triplet-singlet splitting of a single pair of electrons. The angular momentum factor accounts for the coupling of the angular momentum of the valence electrons and was first given by Duman and Smirnov [Opt. Spectrosc. (USSR) 29, 229 (1970)]. Here it is rederived and in the cases of interactions between hydrogen, rare gas, alkali and alkaline earth atomic systems the new corrected expressions are shown to reduce to a simple physical model. The angular momentum factors are listed for all these interacting systems. The important factors in the analytic expression for the distance dependent asymptotic exchange energies are also given for all the homonuclear alkali, alkaline earth, and rare gas dimers.