The mononuclear and binuclear mechanisms of Ag(I)-catalyzed azide-alkyne cycloaddition (AgAAC) reaction of benzyl azide with phenyl alkyne in gas phase and the presence of toluene as a solvent for formation of the 1,4-disubstituted 1,2,3-triazoles have been investigated by DFT calculations using MN12-L and MN15-L functionals with both basis sets Def2-TZVP including a pseudopotential for Ag and Def2-SVP for the mean elements. The effect of the PPh3, 2-(diphenylphosphaneyl)-N,N-diisopropylbenzamide, and 2-(di-tert-butylphosphaneyl)-N,N-diiso-propylbenzamide (L1, L2, and L3) as ligands on the catalytic cycle has been studied in both mononuclear and binuclear pathways. DFT calculations with MN12-L and MN15-L functionals show that the AgAAC reaction favors the mononuclear pathway by comparison with the uncatalyzed reaction for the formation of 1,4-regioisomer. On the other hand, in the AgAAC reaction, the comparison between the mononuclear and binuclear pathways indicates that the free energy barrier of the mononuclear paths is higher than that of the binuclear cases for L1 ligand, and vice versa for L2 and L3 ligands.