Analytical formulas for the third-order diamagnetic energy of a hydrogen atom

Analytical expressions are presented for the third-order diamagnetic corrections to the energy of nondegenerate hydrogen levels with arbitrary principal quantum number $n$ and the magnetic quantum number $|m|=n-1,n-2,n-3$. The leading term for the third-order energy correction for levels with high $n$ is determined to be $\Delta E^{\mathrm{}\left(3\right)\mathrm{}}\approx \frac{3}{128}{n}^{16}{B}^6$. Together with the well-known first- and second-order corrections $\Delta E^{\mathrm{}\left(1\right)\mathrm{}}\approx \frac{1}{8}{n}^4{B}^2$ and $\Delta E^{\mathrm{}\left(2\right)\mathrm{}}\approx -\frac{1}{32}{n}^{10}{B}^4$ it determines the upper and lower bounds for the level energy in field and also the range of magnetic fields where the first- and second-order perturbation theory terms are valid for calculating the Zeeman energy in hydrogenlike states of atoms.