Averaged electron densities of the helium-like atoms

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 $1\leq Z\leq5$ 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 $\Psi_{av}'(0)/\Psi_{av}(0)$ and $\Psi_{av}''(0)/\Psi_{av}(0)$ 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 $\rho'(0)/\rho(0)$ and $\rho''(0)/\rho(0)$ 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.