Averaged electron densities of the helium-like atoms

Abstract

Different kinds of averaging of the wavefunctions/densities of the two-electron atomic systems are investigated. Using the Pekeris-like method, the ground state wave functions $\Psi$ of the helium-like atoms with nucleus charge $1\le Z\le 5$ are calculated in a few coordinate systems including the hyperspherical coordinates $\left\{R,\alpha ,\theta \right\}$. The wave functions ${\Psi }_{av}(R)$ of the hyperspherical radius $R$ are calculated numerically by averaging $\Psi$ over the hyperspherical angles $\alpha$ and $\theta$. The exact analytic representations for the relative derivatives ${\Psi }_{av}^{\prime }(0)/{\Psi }_{av}(0)$ and ${\Psi }_{av}^{\prime \prime }(0)/{\Psi }_{av}(0)$ are derived. Analytic approximations very close to the actual ${\Psi }_{av}(R)$ are obtained. Using actual wave functions $\Psi$, the one-electron densities $\rho (r)$ are calculated as functions of the electron-nucleus distance $r$. The relevant derivatives ${\rho }^{\prime }(0)/\rho (0)$ and ${\rho }^{\prime \prime }(0)/\rho (0)$ characterizing the behavior of $\rho (r)$ near the nucleus are calculated numerically. Very accurate analytical approximations, representing the actual one-electron density both near the nucleus and far away from it, are derived. All the analytical and numerical results are supplemented with tables and graphs.

29 pages, 7 figures, 3 tables