Ground State of the Helium Atom

Abstract

The energy level of the ground state of atomic helium is restudied in detail. The nonrelativistic part is treated by the conventional Ritz variation technique. However, the trial functions used are more general than the ordinary Hylleraas-type functions since they contain negative power terms in addition to the positive power terms. Linear combinations of up to 39 terms are employed in the numerical computation. The best approximation for the ground state energy of the nonrelativistic Schrödinger equation obtained so far is -2.9037225 atomic units. The precision of the variation result is estimated by the evaluation of lower bounds to the ground state energy. Our final result is about twelve times more accurate than the best published value. On the basis of these calculations, it is conjectured that the actual nonrelativistic energy will be lower than our best value by not more than 0.0000012 atomic unit. The accuracy of various approximate eigenfunctions is also estimated. It is thus found that the total contribution to our 39-parameter function from all the excited states will be of the order of 0.1%. Mass polarization and relativistic corrections are evaluated with various trial functions including our best ones. They seem to converge to certain limits with reasonable speed. We therefore believe that the mass polarization and relativistic corrections to the ionization potential of the He ground state will be very close to -${4.78}_6$ ${\mathrm{cm}}^{-1\mathrm{}}$ and -${0.57}_0$ ${\mathrm{cm}}^{-1\mathrm{}}$, respectively. With these corrections, and also the Lamb-shift correction for the ground state of the He atom (-1.23 ${\mathrm{cm}}^{-1\mathrm{}}$), the theoretical ionization potential becomes 198310.38 ${\mathrm{cm}}^{-1\mathrm{}}$ which is in a very good agreement with the best observed value 198310.5±1 ${\mathrm{cm}}^{-1\mathrm{}}$.