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 $\delta$ 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 $\rho \left({\theta }_{12}\right)$ for states corresponding to shells $n_1=n_2=2,3,4\mathrm{}$. 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 $\rho \left({\theta }_{12}\right)$, which reveal both collective rotations and vibrations and even the influence of centrifugal distortion. A minimal basis ($l_{max}=n-1\mathrm{}$) is inadequate to represent angular correlation in the higher-energy states of each shell. The repulsive Gaussian potential, which is finite at ${\theta }_{12}=0\mathrm{}$, 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.