The effect of the nuclear charge on any given orbital electron is reduced by the screening effect of the other electrons in the atom. Values of sigma 2, the Sommerfeld constant which is a measure of this screening, can be obtained from the difference between energy levels belonging to a spin doublet. New values of sigma 2(Z), with probable errors, have been calculated for the L2L3, M2M3, M4M5, N2N3, N4N5, N6,N7, O2O2, and O4O5 doublets over the entire range of atomic numbers. Z. The method used is shown to be superior to the usually employed Sommerfeld's approximation. A study of sigma 2 against Z yields several interesting conclusions, the most important one being that the best calculated values for the binding energies of electrons in the transuranium elements give too large a separation between the members of the L shell spin doublet. Several anomalous energy level spacings are pointed out and new values for the L shell spin doublet separation for the transuranium elements are given.