Experiment and Theory in Computations of the He Atom Ground State

Extensive variational computations are reported for the ground state energy of the non-relativistic two-electron atom. Several di erent sets of basis functions were systematically explored, starting with the original scheme of Hylleraas. The most rapid convergence is found with a combination of negative powers and a logarithm of the coordinate s = r1+r2. At N = 3091 terms we pass the previous best calculation (Korobov's 25 decimal accuracy with N = 5200 terms) and we stop at N = 10257 with E = -2.90372, 43770, 34119, 59831, 11592, 45194, 40444, ... Previous mathematical analysis sought to link the convergence rate of such cal-culations to speci c analytic properties of the functions involved. The application ofthat theory to this new experimental data leaves a rather frustrating situation, wherewe seem able to do little more than invoke vague concepts, such as "flexibility." Weconclude that theoretical understanding here lags well behind the power of availablecomputing machinery