Separation of Angles in the Two-Electron Problem

Neglecting the spin, two electrons are described in quantum mechanics by means of a wave equation in six variables. It is shown that well-known relations between angular momentum operators make it possible to determine the dependence of the wave function on three variables. The problem is thus reduced from six to three dimensions. For a state with an assigned "orbital" angular momentum $l$, say an $S$, $P$, $D$ state the dependence of the wave function on three Euler angles is determined by the value of $l$. The wave function is a linear combination of products of distance and angle functions, the latter depending only on the three Euler angles. The angle functions are well-known solutions of the wave equation for a symmetrical top. The distance functions satisfy wave equations in three variables $r_1$, $r_2$, $r_{12}$ or $r_1$, $r_2$, $\theta$. The case of $P$ terms is worked out in detail. Equations (10), (25) apply to two electrons having the same asimuthal quantum number. Equations (18), (20), (24) describe all the other cases, for instance $S$ and $P$ electrons combining to give ${}_{}{}^1{}_{}{}^{}P$ and ${}_{}{}^3{}_{}{}^{}P$. Triplets are described by (18) and singlets by (20).