Quantum states of two particles on concentric spheres

The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities ρ(θ₁₂) exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.