Upper and Lower Bounds to Two and Three-Electron Systems

Upper and lower bounds for Li('+) in non-relativistic approximation with Coulomb interaction only are calculated. We use the familiar Rayleigh-Ritz method to calculate upper bounds, and the method of intermediate problems to calculate lower bounds. Emphasis is given to the calculation of lower bounds. In particular, we have constructed our lower bounding operators using a complete set of basis functions, and we have included a positive semi-definite term that recovers some of what was lost due to truncation of the base operator. This allows us to raise the essential spectrum of the base operator for the lithium atom, which in turn paves the way for the calculation of good lower bounds for three electron atomic systems. We also present, and discuss at some length, the numerically stable algorithm we developed for the computation of two particle matrix elements.