# Screening factors - why can we use them?


The expression is semi-empirical, and intended to reproduce the famous Balmer formula for the lines of hydrogen: $$\Delta E\sim R_\infty\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\, .$$ The Balmer formula can be obtained using solutions to the Schrödinger equation for the hydrogen atom and generally for single-electron atoms with $Z$ protons, where energy levels $$E_n\sim R_\infty \frac{Z^2}{n^2}\, .$$ Phenomenologically, electrons in the outer shells do not see the full nuclear charge $Z$, but net charge smaller than $Z$ since the inner electrons "screen" or partially cancel the charge of the nucleus, creating a smaller net electric potential (still assumed to be central in nature if the screening is from electrons in closed atomic shells). This is the origin of the $Z-\sigma_n$ factor.
Since the $\sigma_n$ depends mostly on inner electrons (provided their orbitals are more or less completely inside the outer electron orbitals), this quantity does not depend very much on $Z$.