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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 $Ψ$ of the helium-like atoms with nucleus charge $1\leq Z\leq5$ are calculated in a few coordinate systems including the hyperspherical coordinates $\left\{R,α,θ\right\}$. The wave functions $Ψ_{av}(R)$ of the hyperspherical radius $R$ are calculated numerically by averaging $Ψ$ over the hyperspherical angles $α$ and $θ$. The exact analytic representations for the relative derivatives $Ψ_{av}'(0)/Ψ_{av}(0)$ and $Ψ_{av}''(0)/Ψ_{av}(0)$ are derived. Analytic approximations very close to the actual $Ψ_{av}(R)$ are obtained. Using actual wave functions $Ψ$, the one-electron densities $ρ(r)$ are calculated as functions of the electron-nucleus distance $r$. The relevant derivatives $ρ'(0)/ρ(0)$ and $ρ''(0)/ρ(0)$ characterizing the behavior of $ρ(r)$ 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.