Ground State of the Helium Atom

Abstract

Following a recent attempt to analyze the rate of convergence of Rayleigh-Ritz variational calculations on the ground state of helium, we were led to re-investigate the usefulness of inserting fractional powers of the variables into the conventional Hylleraas series. The results have been very successful: With a 164-term trial function containing half-powers of the variable $s={r}_{1}+{r}_{2}$, we have matched the best eigenvalue obtained by Pekeris, who used 1078 terms of the conventional type. Our extrapolated value for the nonrelativistic eigenvalue is -2.9037243771 a.u., with an estimated uncertainty of about 1/${10}^{10}$.