A relatively simple approximate ab initio method for the evaluation of correlation effects on atomic energy differences is developed. In this method, which is a modification of the semiempirical version of Sinanoglu's non-closed-shell many-electron theory (NCMET), the all-external contributions to the correlation energies are evaluated by a scheme based on the knowledge of second-order pair energies calculated for approximate closed-shell atomic systems. The present method can be applied to a much larger set of atomic systems than the semiempirical NCMET. To test the reliability of the present approach extensive calculations are performed for energy differences involving states of the 1s22sm2pn configurations for various atomic numbers. These calculations could be to a large extent based on previously published values of both the 'charge' contributions to the energies and the relevant pair energies. The results for (i) term splittings within a single configuration for neutral atoms as well as positive and negative ions, (ii) excitation energies, (iii) ionisation potentials and (iv) electron affinities prove that the present approach can be used for reliable predictions of atomic energy differences. Since the method is non-relativistic, some attempts have been made to account for the relativistic corrections. In all cases considered the inclusion of these corrections has further improved the agreement with the experimental results.