Molecular orbital studies of polyatomic molecules containing second-row atoms are time-consuming by ab initio techniques. By limiting a calculation to the valence shell electrons, but using a simple correction for interaction between core and valence-electrons, together with the inclusion of most of the two-electron integrals over the valence-electrons, a non-empirical method of reasonable economy and sophistication is possible. The method discussed here has been developed on this basis using Slater-type orbital functions. Calculations have been directed toward obtaining good agreement with available ab initio results as well as relevant experimental data. Some results are presented for molecules containing phosphorus or sulphur.