The　size-selected　copper　single-cluster　catalysts(SCCs)　play　a　crucial　role　in　catalyzing　of　CO2　hydrogenation.　However,　nature　of　size　effect　and　hydrogenation　mechanism　is　still　unclear.　Here,　Density　functional　theory　(DFT)　calculations　were　performed　to　investigate　activation　and　reaction　mechanism　of　CO2　hydrogenation　to　methanol　over　copper　single-clusters　catalyst　anchored　on　alpha-Al2O3(0001).　Our　results　show　that　Cun(n　=　1-9)/　Al2O3　catalysts　significantly　promote　activation　of　CO2　in　a　bent　configuration　due　to　the　synergistic　interaction　associated　with　Cun-Al2O3(0001)　interfacial　site.　Bader　charge　analysis　suggested　electron　transfer　occurring　from　Cun-Al2O3(0001)　interface　to　CO2,　leading　to　negatively　charged　CO2.　We　find　descriptors　of　deviation　of　CO2　bonds　length　and　bonds　angle,　D-band　center　of　Cun(n　=　1-9)　cluster　that　can　well　descript　for　determining　activation　degree　of　CO2.　Especially,　the　Cu5/alpha-Al2O3(0001)　has　the　best　activity　for　CO2　hydrogenation　among　them.　Furthermore,　reaction　mechanism　of　CO2　hydrogenation　at　Cu5/Al2O3　interface　have　been　explored.　Methanol　synthesis　takes　formate　route　and　direct　C-O　bond　cleavage　route　competitive　with　rate-dominant　step　of　HCOO*　and　HO*　hydrogenation,　respectively.　While　CO-hydrogenation　produced　by　RWGS　route　is　not　feasible　because　of　high　activation　barrier　of　formed　CO　intermediate(1.95　eV).　This　study　will　provide　insight　into　designing　and　application　of　ultrahigh-performance　Cu-based　single-cluster　catalysts　for　CO2　hydrogenation.