Darboux functions and perturbative theory for the ground-state energy of two-electron atoms

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Abstract
The power series of the lowest eigenvalue epsilon ( lambda ) of the helium isoelectronic sequence, lambda being the coupling constant for the interaction between the two electrons, is summed by the ansatz of a simple analytic function of class D (Darboux). Stationarity criteria are used in constructing such a function, and the basic information needed is only a few terms of the perturbative power series. The ground-state energy is evaluated with great accuracy without using any free parameter. The Darboux method is used to give an estimate of the asymptotic coefficients of the perturbative expansion. The contrast between different empirical procedures and the self-consistency of the present one is remarkable.