Precise nonvariational calculations on the helium atom

The Schr\"odinger equation is solved directly for the ground state and excited 2 ¹S state of the helium atom by using a rapidly convergent hyperspherical method which involves no adjustable parameters. The double and triple coalescence points are taken into account analytically. The center-of-mass motion is treated nonperturbatively, and the cases of infinite and finite nuclear masses are considered. The inclusion of 169 hyperspherical functions yields the precision of a few parts in 10⁹ and 10⁸ for the expectation value of the Hamiltonian operator and for all other expectation values, respectively, for the ground state, with only slightly less accuracy for the excited state.