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A set of approximations (“EPCE-F2s”) is given for estimating the correlation energies of molecules. The approximations start from Sinanoglu's pair correlations and his many-electron theory for ground states (MET). For application of this method, “effective pair correlation energies” must be known. In the present paper three sets of one-center effective pair correlation energies are obtained from the atomic pair correlation energies. A weighted average method is introduced for handling the non-transferable atomic correlation energies. For the corresponding two-center effective pair correlation energies several empirical expressions are given. Then using three sets of one-center effective pair correlation energies and for the two-center ones, the correlation energies of some diatomic hydrides, N2 and CH4 molecules are estimated. The results agree quite well with each other, with experimental values, and with those obtained by other methods.