Compact and accurate integral-transform wave functions. I. The1S1state of the helium-like ions fromH−throughMg10+

Accurate and compact integral-transform wave functions are constructed for the $1{}_{}{}^1{}_{}{}^{}S$ state of the helium-like ions from ${\mathrm{H}}^{-}$ through ${\mathrm{Mg}}^{10+}$. The variational ansatz is of the form $\Psi (r_1, r_2, r_{12})={\left(4\mathrm{}\pi \right)}^{-1\mathrm{}}{\Sigma }_{k=1\mathrm{}}^N{c}_k(1+P_{12})\exp (-{\alpha }_k{r}_1-{\beta }_k{r}_2-{\gamma }_k{r}_{12})$ where the $c_k$ are found by solving the secular equation and the exponents ${\alpha }_k$, ${\beta }_k$, and ${\gamma }_k$ are chosen to be the abscissas of Monte Carlo and number-theoretic quadrature formulas for a variationally optimized parallelotope in $\alpha -\beta -\gamma$ space. A 66-term function of this type for the helium atom yields an energy of -2.903 724363 a.u. as compared with the 1078-term function of Pekeris which yields an energy of -2.903 724376 a.u. In order to test the accuracy of the wave functions a number of properties including $〈r^n〉$ and $〈r_{12}^n〉$ with $n=-2, -1, 1, \dots , 4$, $〈{\overrightarrow{\mathrm{r}}}_1·{\overrightarrow{\mathrm{r}}}_2〉$, $〈{cos\theta }_{12}〉$, $〈\delta \left({\overrightarrow{\mathrm{r}}}_1\right)〉$, and $〈\delta \left({\overrightarrow{\mathrm{r}}}_{12}\right)〉$ are computed and compared with the best available results. The electric dipole polarizability is computed from a simple formula due to Thorhallsson, Fisk, and Fraga. Comments on the limiting accuracy of this formula are made. Electron-nuclear and electron-electron cusp condition tests are made for the functions. Detailed convergence studies are presented for ${\mathrm{H}}^{-}$ and He in the form of a sequence of functions with increasing $N$. The functions are found to be rather accurate and more compact than any other functions available in the literature with the exception of those containing logarithmic terms. Possible refinements to the basis set used are discussed.