The Theory of Complex Spectra. Part I. Energy Levels

Abstract

A general method of finding the wave function for $\mathrm{LS}$ coupling which is similar to that of Gray and Wills is described. The successive transformations which carry the angular momentum matrices, ${S}^{2}$, ${L}^{2}$ and ${J}^{2}$ to a diagonal form, are determined by writing down these matrices in terms of the unperturbed wave functions and solving the resulting linear equations for the transformation coefficients. This yields the wave functions appropriate for $\mathrm{LS}$ coupling. The method is applied to give the wave functions for all the states of $\mathrm{LS}$ coupling with the smallest value of $|{M}_{J}|$ in the following electronic configurations: ${p}^{2}$, ${d}^{2}$, ${p}^{3}$, ${p}^{2}s$ and ${p}^{3}s$. The matrix of the spin-orbit interaction is calculated with these wave functions and is factored according to $J$ values because ${J}^{2}$ is an integral of the motion. By adding the electrostatic energies as computed by Slater's method to the diagonal elements (the electrostatic energy is known to be a diagonal matrix in $\mathrm{LS}$ coupling), the complete energy matrix is obtained. Setting the determinant of the matrix equal to zero, the secular equation for each $J$ value is found for the above electronic configurations. These equations determine the position of the energy levels in intermediate coupling provided that second order perturbations may be neglected and provided magnetic effects other than the spin-orbit interaction do not contribute appreciably to the Hamiltonian.