Practical recursive solution of degenerate Rayleigh-Schrödinger perturbation theory and application to high-order calculations of the Zeeman effect in hydrogen

With the aim of high-order calculations, a new recursive solution for the degenerate Rayleigh-Schrödinger perturbation-theory wave function and energy has been derived. The final formulas, ${\chi }_{\sigma }^{\mathrm{}\left(N\right)\mathrm{}}=R^{(-\sigma )}{\Sigma }_{k=0\mathrm{}}^{N-1\mathrm{}}{\overline{\mathcal{H}}}_{\sigma +1\mathrm{}}^{(\sigma +1+k)}{\chi }^{\mathrm{}(N-1-k)\mathrm{}}$ $E^{(N+\sigma )}=〈0\left|{\mathcal{H}}_{\sigma +1\mathrm{}}^{(N+\sigma )}\right|0\mathrm{}〉+〈0\left|{\Sigma }_{k=0\mathrm{}}^{N-2\mathrm{}}{\mathcal{H}}_{\sigma +1\mathrm{}}^{(\sigma +1+k)}\right|{\chi }^{\mathrm{}(N-1-k)\mathrm{}}〉$ which involve new Hamiltonian-related operators ${\mathcal{H}}_{\sigma }^{(\sigma +k)}$ and ${\overline{\mathcal{H}}}_{\sigma }^{(\sigma +k)}$, strongly resemble the standard nondegenerate recursive formulas. As an illustration, the perturbed energy coefficients for the $3s-3d_0$ states of hydrogen in the Zeeman effect have been calculated recursively through 87th order in the square of the magnetic field. Our treatment is compared with that of Hirschfelder and Certain [J. Chem. Phys. 60, 1118 (1974)], and some relative advantages of each are pointed out.