Finite-element-method expectation values for correlated two-electron wave functions

The Schrödinger equation for the ground state of correlated two-electron atoms is treated by an accurate finite-element method (FEM) yielding energy eigenvalues of -2.903 724 377 021 a.u. for the helium atom and -0.527 751 016 532 a.u. for the hydrogen ion ${\mathrm{H}}^{\mathrm{-}}$. By means of an adaptive multilevel grid refinement the FEM energy eigenvalue is improved to a precision of 1×${10}^{\mathrm{-}11}$ a.u., which is comparable to results obtained with sophisticated global basis sets. The local and overall precision of the FEM wave function approximation is studied and discussed. Benchmark values for the expectation values 〈${\mathit{r}}^2$〉, 〈r〉, 〈1/r〉, and 〈1/${\mathit{r}}_{12}$〉 are presented.