We report here on how a known method from standard perturbation theory for estimating the energy of the D-line splitting in hydrogen can be modified to effectively approximate this quantity for all of the alkali metals. The approach utilizes a Rayleigh-Schrödinger perturbation theory first order correction to the energy. The perturbing Hamiltonian is the standard relativistically corrected spin-orbit Hamiltonian. From this, one derives an energy difference between the doublet lines that is theoretically appropriate for any one electron atom. This energy difference is written in terms of the Bohr energy. The results are good for hydrogen but, as expected, are significantly off from the experimental values for the multi-electron alkali metals. We show here that by replacing the Bohr energy with a first ionization potential, greatly improved values for the D-line splitting energy can be obtained from the model. However, this approach overestimates the splitting energy for the light alkali metals and underestimates it for the heavier ones. The best result was for Rb where the estimate only varied from the experimentally reported value by 3.2%. Accurate results are finally obtained from the model for all of the alkali metals when the original Bohr energy is adjusted with an appropriate screening constant. Screening constants generated using the Slater scheme however, which yield accurate estimates for ionization potentials, do not give the correct results for the D-line splitting energies. By fitting our results to reported energy splitting values we find that the correct screening constant is a function of the atomic number. We show that this result can be reproduced via a simple model with a relativistic correction for the contraction of the radius of the inner shell electrons.