Radius of convergence of the 1/Z expansion for the ground state of a two-electron atom

An estimation of the radius of convergence of the 1/Z expansion (Z is the charge of the nucleus) for the ground state of the two-electron atom is obtained. The calculation is based on an idea that, with certain conditions being satisfied, the radius of convergence of the 1/Z series can be estimated with good precision if one constructs the function λ(f), inverse to the function f(λ)=[E(λ)-E₀]/E₁ (E is energy, λ=1/Z, while E₀ and E₁ are the first two coefficients of the perturbation expansion of the energy). We find numerically that the nearest singularity to f=0 in the complex f plane of the inverse function λ(f) is at the point f=0.8 corresponding to the threshold point E=-0.5. a.u. We find also that the series for the inverse function λ(f) converges at this point. We discuss the nature of the singularity of the inverse function λ(f). The value for the radius of convergence of the 1/Z expansion of the ground state of a He-like ion obtained is R_λ=1.097 660 79, which we think to be the most accurate value presently available.