Calculations of helium and the negative hydrogen ion are presented using the pseudospectral method. The fundamental analytic properties, including the presence of Kato cusps and logarithmic terms of the solutions to the Schrodinger equation, are explored and their effect on the convergence properties analyzed. We find that by most measures of error the pseudospectral method converges at an exponential rate. With this method, we calculate energies and perturbations due to the finite nuclear mass and relativity and interactions with the electromagnetic field. The value calculated for the absorption oscillator strength of the 1[EXP]1 S [RIGHTWARDS ARROW] 2[EXP]1 P transition is about as accurate as the best in the literature. A general prescription is given for choosing subdomains needed for exponential convergence. With this prescription and the overall general applicability of the method, we conclude pseudospectral methods can be applied to general fewelectron problems.