The mechanism of the cyclic reaction N2O(1∑) + CO(1∑+) → N2(1∑ g + ) + CO2(1∑ g + ) catalyzed by Re+ has been investigated on quintet and septet potential energy surfaces (PES). The reactions were studied by the B3LYP density functional method and the CCSD(T) theory. The calculated results of different PES show that the reaction proceeds in a two-step manner and spin crossing between different PES occurs. The involving crossing points (CPs) between the quintet and septet PES have been discussed by means of the intrinsic reaction coordinate approach. And the O-atom affinities testified that Re+ can capture O from N2O and transfer O atom to CO in the two spin state, which are thermodynamically allowed. Furthermore, the spin–orbit coupling (SOC) is calculated between electronic states of different multiplicities at the CPs. For CP1 and CP2, the computed SOC constants are 8.34 and 10.09 cm−1, respectively, obtained by using one-electron spin–orbit Hamiltonian in GAMESS. Therefore, the intersystem crossing at CP1 and CP2 occurs with a little probability because of the small SOC involved.