Spectra of Two Electron Systems

Following the method of Güttinger and Pauli four angular momentum vectors are treated in matrix mechanics. Analytic formulas are obtained for the matrix components of the spin-orbit interaction of two electrons in $\mathrm{LS}$ coupling. Since the electrostatic energy in $\mathrm{LS}$ coupling is a diagonal matrix whose elements may be calculated by Slater's method, this gives the complete energy matrix. Then for any two electron configuration one easily obtains the secular equations whose roots give the energy levels in intermediate coupling. In this manner the secular equations are worked out for the configurations $p^2$, $p·p$, $d^2$, $d·d$, and $d·p$.