A variational method for calculating the critical nuclear charge $Z_C$ 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 heliumlike systems with both infinite and finite nuclear masses. The value of $Z_C=0.9110282\left(3\right)$ 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 $Z_C$ varies from 0.911 030 3(2) for that with a nuclear mass of Ne (the largest heliogenic system considered) to $0.9218024\left(4\right)$ for a system with the nuclear mass of a positron. In all cases the energy varies smoothly as $Z\to 0$. 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 localized near the nucleus at $Z=Z_C$. Additionally, the electron probability distribution is calculated to determine the behavior of the electrons at low $Z$.

• Received 2 March 2015

DOI:https://doi.org/10.1103/PhysRevA.91.042512