A gradientless correlation functional is derived within the Kohn-Sham density-functional theory (DFT) based on a spin-polarized pair correlation function of Colle-Salvetti type. The functional involves explicitly only the antiparallel spin correlation while parallel-spin correlation is taken into account indirectly in a manner providing self-interaction-free results. Combined with the local-spin-density (LSD) exchange, and with the nonlocal exchange functional of Becke, the resulting DFT schemes are tested on a set of atoms and small molecules. Compared to experiment, the calculated correlation energies of molecules are on average better than those obtained by other functionals employed in the deMon code [A. St.-Amant and D. R. Salahub, Chem. Phys. Lett. 169, 387 (1990)]: Becke-Perdew (88) and Perdew-Wang (91). For atoms the correlation functional gives slightly worse results than Perdew-Wang (91), but substantially better than the LSD approximation. The corresponding binding energies are on average slightly better than those obtained by the existing gradient-corrected schemes and in excellent agreement with experiment.