Correlation of two particles on a sphere

We consider the correlation of two particles on a sphere interacting via various repulsive forces: a Coulomb repulsion, a Gaussian, and a δ function. This "rigid-bender" picture with Coulomb repulsion provides a schematic model of intrashell angular correlation in doubly excited two-electron atoms. We examine energy levels and reduced densities ρ(θ₁₂) for states corresponding to shells n₁=n₂=2,3,4. Energy levels for fully converged states fall into remarkable rovibrator patterns, as found in the corresponding three-dimensional case by Kellman and Herrick. The rovibrator nature of the states is directly reflected in plots of ρ(θ₁₂), which reveal both collective rotations and vibrations and even the influence of centrifugal distortion. A minimal basis (l_max=n-1) is inadequate to represent angular correlation in the higher-energy states of each shell. The repulsive Gaussian potential, which is finite at θ₁₂=0, shows the onset of independent-particle shell-model behavior in higher-energy states. Comparison of the Coulomb case with the delta-function potential exhibits the degree to which kinematics governs wave-function character.