High-precision calculation of the geometric quantities of two-electron atoms based on the Hylleraas configuration-interaction basis

The expectation values of radial geometric quantities $\ensuremath{\langle}{r}_{l}\ensuremath{\rangle}, \ensuremath{\langle}{r}_{g}\ensuremath{\rangle}, \ensuremath{\langle}r\ensuremath{\rangle}$, and $\ensuremath{\langle}{r}_{12}\ensuremath{\rangle}$ and angular ones $\ensuremath{\langle}{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$ and $\ensuremath{\langle}cos{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$ are investigated in detail for two-electron systems. Although these quantities can be calculated straightforwardly in the framework of Hartree-Fock or configuration-interaction methods, their computations based on explicitly correlated type wave functions are nontrivial tasks. In this work we employ the Hylleraas configuration-interaction basis functions to produce accurate system energies and wave functions for the He atom and He-like ions. Computational methods are developed to accurately and efficiently calculate the geometric quantities, especially for inner and outer electron radii, i.e., $\ensuremath{\langle}{r}_{l}\ensuremath{\rangle}$ and $\ensuremath{\langle}{r}_{g}\ensuremath{\rangle}$, and the average interelectronic angle $\ensuremath{\langle}{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$. Compared to previous Hartree-Fock and multiconfiguration Hartree-Fock predictions and the configuration-interaction calculations based on Slater-type orbitals, our present work improve significantly the accuracy of all geometric quantities for He atom in the ground and singly excited states. The application of the present method to He-like ions with different nuclear charge reveals asymptotic power laws for both radial and angular quantities as they approach to the high-$Z$ limit. Further extensions of the present work to other systems are discussed.