Finite-element calculations for theSstates of helium

The finite-element method provides a convenient and accurate procedure for the calculation of the expectation values of quantum observables. We calculated energies, wave functions, and expectation values of ${\mathit{r}}_1^{\mathit{n}}$ for n=-1, 1, and 2, and of πδ(${\mathbf{r}}_1$) for the singlet n ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ and triplet n ${}_{}{}^3{}_{}{}^{}\mathrm{S}$ states (n=1,2,3,4) of helium. In contrast to the standard methods with globally defined basis functions, the accuracy of the expectation values of physical observables is comparable to the accuracy of the eigenvalues. The results are reported here and compared with those of Baker et al. 2 [Relativistic, Quantum Electrodynamic, and Weak Interaction Effects in Atoms, edited by Walter Johnson, Peter Mohr, and Joseph Sucher, AIP Conf. Proc. No. 189 (AIP, New York, 1989); Phys. Rev. A 41, 1247 (1990)], Drake [Nucl. Instrum. Methods Phys. Res. B 31, 7 (1988)], Pekeris [Phys. Rev. 115, 1216 (1959)], Accad et al. [Phys. Rev. A 4, 516 (1971)], and Haftel and Mandelzweig [Phys. Rev. A 38, 5995 (1988)].