Exact Theory Applied to the Lithium Atom

The free complement (FC) theory for solving the scaled Schrödinger equation (SSE) was applied to the Li atom for calculating the exact wave functions, the energies, and the various properties of the ground doublet S and excited P states. The SSE is equivalent to the Schrödinger equation (SE) but does not have the divergence difficulty of the variational equation of the SE. Because the Li atom is a three-electron system, the variational exact FC calculations for solving the SSE are possible using the function <i>g</i> = 1 - exp(-γ<i>r</i>) as the "correct" scaling function of the SSE. The "reasonable" scaling function <i>g</i> = <i>r</i> was also used as comparative calculations. We performed variational calculations to the order eight of the FC theory and could obtain essentially exact solutions of the SSE or SE with the "correct" <i>g</i> function of the FC theory. We report here the values of the exact energy, spin density, electron density, and electron-nuclear and electron-electron cusp values of the doublet S and P states. They agreed very well with the experimental values and the best theoretical values presented by Drake and collaborators. This is a simple example that the exact theory gives the exact solutions.