Configuration Interaction in Complex Spectra

The electrostatic interaction is calculated between the terms in Russell-Saunders coupling of the electron configurations $n{d}^2{n}^{\prime }s$, $n{d}^3$, $\mathrm{nd}{n}^{\prime }{s}^2$ and $n{d}^2{n}^{\prime \prime }s$. The first order eigenfunctions, for the terms of these configurations in the $\mathrm{LS}{M}_L{M}_S$ scheme with $M_L=0\mathrm{}$, $M_S=\frac{1}{2}$ are given in terms of the zero order functions of Slater. The matrix elements between the quartets of $d^{2}s$ and $d^3$ are found to vanish. The sums over closed shells in non-diagonal matrix elements of electrostatic interaction are found to vanish except when the two electrons not in closed shells have the same value of $l$. This sum is evaluated in general and for the configurations $n{d}^2{n}^{\prime }s-n{d}^2{n}^{\prime \prime }s$. The matrix elements of magnetic spin orbit interaction are given between the initial states without configuration interaction. The electrostatic energies and magnetic splittings are then calculated for the terms of $n{d}^2{n}^{\prime }s$, $n{d}^3$ and $\mathrm{nd}{n}^{\prime }{s}^2$; and compared with the values observed in Ti II and Zr II. Zr II agrees particularly well, especially if the observed ${}_{}{}^2{}_{}{}^{}D$ terms are rearranged.