We propose a new long-range correction scheme that combines generalized-gradient-approximation (GGA) exchange functionals in density-functional theory (DFT) with the ab initio Hartree-Fock exchange integral by using the standard error function. To develop this scheme, we suggest a new technique that constructs an approximate first-order density matrix that corresponds to a GGA exchange functional. The calculated results of the long-range correction scheme are found to support a previous argument that the lack of the long-range interactions in conventional exchange functionals may be responsible for the underestimation of 4s-3d interconfigurational energies of the first-row transition metals and for the overestimation of the longitudinal polarizabilities of pi -conjugated polyenes in DFT calculations.