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 different 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 = r(1) + r(2). 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 calculations to specific analytic properties of the functions involved. The application of that theory to this new experimental data leaves a rather frustrating situation, where we seem able to do little more than invoke vague concepts, such as flexibility. We conclude that theoretical understanding here lags well behind the power of available computing machinery.