Two-Electron Atoms. I. Perturbation Study of the Ground State

Abstract

A variational perturbation method used by Hylleraas has been extended to include 70-term basis sets as approximations to the first-order wave function of the two-electron atomic species. An upper limit of -${0.15766625}^{+}$ a.u. has been found for the second-order perturbation energy coefficient. It is estimated that this value is converged to at least two units in the seventh decimal place. A value of the third-order perturbation energy coefficient of 0.00869868 a.u. is calculated from the same variational first-order wave function. A number of expectation values to first-order for certain operators [e.g., $\delta \left({\mathrm{r}}_1\right)$, $\delta \left({\mathrm{r}}_{12}\right)$, ${p_1}^4$, ${r_1}^n$, ${cos\theta }_{12}$, etc.] are computed, and compared, where possible, to known values. A variety of basis sets is studied, including a set that contains negative powers of the metric variables ("Kinoshita" type) and one that contains only positive powers ("Hylleraas" type). A scheme is proposed that uses first and higher order expectation values for the analysis and characterization of approximate wave functions. This scheme, which is not restricted to two electrons, also opens the possibility of a nonenergetic variational procedure for obtaining wave functions.