The electron correlation cusp: I. Overview and partial wave analysis of the Kais function

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r₁₂^-1 electron-electron interaction. Partial wave analysis (PWA) of the Kais function yields E_L = E + C₁(L + C̄₂) + O(L^-5) where E is the exact energy and E_L the energy of a renormalized finite sum of partial waves omitting all waves with angular momentum ℓ > L. Slight rearrangement of an earlier result by Hill shows that the corresponding full CI energy differs from E_L only by terms of order O(L^-5) with FCI values of C₁ and C₂ identical to PWA values. The dimensionless C̄₂ parameter is weakly dependent upon the size of the physical system. Its value is 0.788 for the Kais function, and 0.893 for the less diffuse helium atom, and approaches C̄₂ → 1 in the limit of an infinitely compact charge distribution. The ℓth energy increment satisfies an approximate virial theorem which becomes exact in the high ℓ limit. This analysis, formulated to facilitate use of the Maple system for symbolic computing, lays the mathematical ground work for subsequent studies of the electron correlation cusp problem. The direction of future papers in this series is outlined.