The $S$-wave scattering parameters have been calculated for the $e^{+}-\mathrm{H}$ and $e^{+}-\mathrm{He}$ systems below the positronium-formation threshold. The phase shifts were computed using the Harris variational method, while at zero energy the scattering lengths were obtained by the Kohn variational method. For hydrogen, a generalized Hylleraas wave function with three nonlinear parameters was used, giving an upper-bound scattering length $a\le -2.10278\mathrm{}$, agreeing with Schwartz's result $a\le -2.10\mathrm{}$. An extrapolation procedure gives an estimate of $a=-2.1036\mathrm{}\pm 0.0004$. At nonzero energies, agreement with the lower-bound results of Bhatia, Temkin, Drachman and Eiserike, to better than 0.004 rad is achieved. Since the exact atomic-helium wave function is not known, the situation for helium is less clear; various model-target wave functions are examined using a generalized Hylleraas function with two nonlinear parameters. For the exponential model used by Drachman, the scattering is in close agreement with previous results, but the annihilation-rate parameter $Z_{\mathrm{eff}}$ is higher by 15% in poorer agreement with experiment at zero energy.

• Received 21 September 1970

DOI:https://doi.org/10.1103/PhysRevA.3.1335