Unraveling the chemical mechanism of atmospheric new particle formation (NPF) has important implications for the broader understanding of the role of aerosols in global climate. We present computational results of the transition states and activation barriers for growth of atmospherically relevant positively charged molecular clusters containing ammonia and sulfuric acid. Sulfuric acid uptake onto the investigated clusters has a small activation free-energy barrier, consistent with nearly collision-limited uptake. Ammonia uptake requires significant reorganization of ions in the preexisting cluster, which yields an activation barrier on the order of 29-53 kJ/mol for the investigated clusters. For this reason, ammonia uptake onto positively charged clusters may be too slow for cluster growth to proceed by the currently accepted mechanism of stepwise addition of sulfuric acid followed by ammonia. The results presented here may have important implications for modeling atmospheric NPF and nanoparticle growth, which typically does not consider an activation barrier along the growth pathway and usually assumes collision-limited molecular uptake.