Decomposition of the Schrödinger Equation for Two Identical Particles and a Third Particle of Finite Mass

An angular-momentum decomposition of the Schrödinger equation is extended to the case of two identical particles and a third particle of finite mass. (The case of three unequal-mass particles is treated in an Appendix.) The decomposition is effected with the use of a symmetric choice of Euler angles, and the radial equations are given in two useful forms. The radial equations are shown to yield the Born-Oppenheimer equations for ${\mathrm{H}}_2^{+}$ in the limit that the two identical particles approach infinite mass. Other aspects of this limit are discussed, and general rules which relate the total-angular-momentum states of the three-body system to the molecular states of ${\mathrm{H}}_2^{+}$ are examined.