In this paper, high-level ab initio calculations by using multi-configuration self-consistent field method with atomic effective core potential, polarization potential, and uncontracted Gaussian basis function, are performed to compute the potential energy curves of a total of 36 low-lying ∧-S states with ∑_g,u, Π_g,u, △_g,u symmetries of Na₂⁺ cation associated with the lowest 9 dissociation limits Na (3s, 3p, 4s, 3d, 4p, 5s, 4d, 4f, 5p)+Na⁺. On the basis of the potential energy curves, the spectroscopic constants (T_e, R_e, ω_e, ω_eχ_e, B_e, α _e, D_e) of the bound states are determined, which are in good agreement with the existing available experimental and theoretical values. Our results indicate that 5²∑_g⁺-7²∑_g⁺, 3²∑_u⁺-7²∑_u⁺, 2²Π_g, 4²Π_g, 1²△_u and 2²△_u states are repulsive, which supports Berriche's results, and we report 10 electron states for the first time, that is, 8²∑_{g, u}⁺-9²∑_{g, u}⁺, 5²Π_{g, u}-7²Π_{g, u} and 3²△_{g, u}. The vibrational-rotational spectroscopic constants and lowest vibrational-rotational energy levels (ν=0-20) of the bound states are also presented. Moreover, in order to illustrate the strong state interactions of adjacent states with same symmetry, the information about the avoided crossing points is shown in detail. Finally, the transition dipole moments from a few low-lying excited states (1²Π_u-3²Π_u) to the ground state X²∑_g⁺ are computed. Therefore, it is expected that our computational results in the present calculations are significant for the molecular spectroscopy, ion-atom interaction and molecular cold collision fields.