A variational method for calculating the critical nuclear charge, Zc, required for the binding of a nucleus to two electrons is reported. The method is very effective and performs well compared to the traditional variational principle for calculating energy. The critical nuclear charge, which corresponds to the minimum charge required for the atomic system to have at least one bound state, has been calculated for helium-like systems both with infinite and finite nuclear masses. The value of $Z_C=$ 0.911 028 2(3) is in very good agreement with recent values in the literature for two-electron atoms with an infinite nuclear mass. When nuclear motion is considered, the value for Zc varies from 0.911 030 3(2) for that with a nuclear mass of Ne (the largest heliogenic system considered) to 0.921 802 4(4) for a system with the nuclear mass of a positron. In all cases the energy varies smoothly as $Z \rightarrow 0$. It is found that for the finite nuclear mass case, in agreement with previous work for the fixed nucleus mass system, the outer electron remains localised near the nucleus at Z = Zc. Additionally, the electron probability distribution is calculated to determine the behaviour of the electrons at low Z.

A method is presented for determining inner and outer one-electron radial density functions for two electron systems by partitioning the fully correlated two-electron radial density function. This is applied to the helium isolectronic series (Z=1 to 10 and 100) and the critical nuclear charge system, which has the minimum charge for which the atomic system has at least one bound state, to separate out the motions of the two electrons in both weakly and strongly correlated systems. It is found that the inner electron experiences an anti-shielding effect due to the perturbation by the other electron which increases with increasing Z. For the weakly bound systems the inner radial density distribution closely resembles that of a hydrogenic atom with the outer radial density distribution becoming very diffuse.