Angular correlation of two electrons on a sphere

As an approach to the electronic structure of the doubly excited and other highly correlated states of atoms, we consider the angular correlation of two electrons on a sphere of radius R. The Hamiltonian is scaled by a factor $R^2$ and the parameter R is reinterpreted as the strength of electronic repulsion. The scaled Hamiltonian $H_S$ is the sum of an analytically solvable Hamiltonian $H_0$+$V_M$ and a perturbation Δ$H_1$+Δ$H_2$ which becomes negligible as R→∞. The spectrum of $H_0$+$V_M$ therefore approaches the spectrum of $H_S$ in the asymptotic limit. The eigenfunctions of $H_0$+$V_M$ show a transition from weak correlation at small R to strong correlation and ‘‘molecular’’ motion at large R.