Second-order perturbative calculation of hydrogenic Zeeman levels

The degenerate perturbative treatment of the hydrogenic Zeeman problem is put on a more rigorous basis and is extended to higher levels belonging to the manifolds of $M=n-\left|m\right|\mathrm{}\le 4$ basis states. The $16n=4$ levels are calculated and the large-$n$ behavior of the basis states is discussed.