C₂N is an ordered two-dimensional carbon nitride with a high density (1.7 × 10¹⁴ cm^-2) of 3.1 Å-sized nanopores, making it promising for high-flux gas sieving for energy-efficient He and H2 purification. Herein, we discuss the accurate calculation of potential energy surfaces for He, H₂, N₂, and CO₂ across C₂N nanopores, to characterize the gas-sieving potential of C₂N. We compare the potential energy surface derived from density-functional theory calculations using five commonly used van der Waals (vdW) approximations. While all five functionals point that the C₂N nanopore yields He/N₂ and H₂/N₂ selectivities over 1000, adsorption energies and energy barriers vary remarkably depending on the approximation chosen. To make progress, we compare the calculations against the results from the adiabatic connection fluctuation dissipation theory, with random-phase approximation, known to be accurate in capturing vdW interactions. The comparison indicates that the interaction energy is less accurate with vdW density functional theory. On the other hand, more empirical corrections work reasonably well, a finding that we also confirm for another carbon nitride lattice, poly(triazine imide). Overall, we recommend these for screening carbon nitride materials for gas separation, but also comparing functionals with higher-order approaches when dealing with different materials.