Potential energy surfaces are evaluated for H atom and H-2 elimination in the gas phase reaction of a Y atom with acetylene, C2H2. Coupled-cluster calculations are performed with extrapolations to the complete basis set limit and zero point energy, core correlation, and spin-orbit corrections. The resulting surfaces reveal that the lowest energy reaction channel leads to H-2 elimination, consistent with the YC2 + H-2 products observed in crossed molecular beam experiments. This reaction proceeds in three steps: (i) YC2H2 adduct formation, (ii) C-H insertion, and (iii) 1,3-elimination of H-2. A higher energy reaction channel leads from the C-H insertion intermediate to the H atom elimination products YC2H + H. Our calculations predict product asymptotes of -7.1 kcal/mol for YC2 + H-2 and 15.0 kcal/mol for YC2H + H, energies that differ considerably from those (-18.5 +/- 4.3 and 21.5 +/- 2.0 kcal/mol, respectively) determined in the beam experiments. Natural bond orbital methods are used to determine how the metal atom influences the redistribution of electrons during the reaction steps. The open-shell metal atom and its partially occupied valence 4d shell usually promote homolytic cleavage and formation of bonds as the reaction proceeds. Our best estimates of the bond dissociation energies D-0(Y-C-2), D-0(Y-C2H), and D-0(H-YC2H) are 147.7, 115.8, and 57.1 kcal/mol, respectively.