Onset of Atomic-Subshell Filling in Ordinary and Superheavy Elements

Energy considerations in the central-field approximation to the one-electron problem are used to derive a theoretical expression, namely, $Z_1\mathrm{}\left(l\right)\mathrm{}>~\left(\frac{1}{6}\right) {\mathrm{}(2l+1)\mathrm{}}^3$, for the atomic number $Z_1\mathrm{}\left(l\right)\mathrm{}$ at which a given atomic $l$ subshell first begins to become occupied. The lowest integral values of $Z_1\mathrm{}\left(l\right)\mathrm{}$ satisfying this relation for $l=0, 1, 2$, and 3 are found to be in exact agreement with the corresponding empirical values, i.e., $Z_1=1, 5, 21$, and 58, respectively. In the domain of the superheavy elements ($100\lesssim Z\lesssim 200$) and beyond, this relation gives $Z_1\mathrm{}\left(l\right)=122\mathrm{}$, 222, and 367 for $l=4, 5$, and 6, respectively. The first member of this latter triad is in close accord with a recent estimate of $Z_1\mathrm{}\left(4\right)=125\mathrm{}$, cited by Seaborg. Comparison is made with the work of others in this field. To the writer's knowledge, this is the first calculation, without arbitrary introduction or adjustment of parameters, to yield results in complete accord with experiment.