Decomposition of the two-electron-atom eigenvalue problem

Following Bhatia and Temkin [Rev. Mod. Phys. 36, 1050 (1964)] we decompose the Hamiltonian of a two-electron atom (or ion) with a fixed nucleus in terms of Euler angles, and thereby reduce the energy-eigenvalue problem to a set of coupled equations involving only three lengths, the distance of the electrons from each other and the distances from the nucleus. However, our equations differ from those of Bhatia and Temkin since we use a different expansion of the wave function. When the total orbital-angular-momentum quantum number L is zero or one our equations are the same as those derived by Hylleraas [Z. Phys. 48, 469 (1928)] and Breit [Phys. Rev. 35, 569 (1930)]. We give the transformation relating the generalized Hylleraas-Breit equations to the equations of Bhatia and Temkin. Our derivation is facilitated by a special factorization of one-particle angular-momentum operators