Double-bound equivalent of the three-body Coulomb double-continuum wave function

In cross-section calculations of electron or photon double ionization processes of two-electron atoms, it is desirable to have a symmetric description of initial and final states. In this contribution, we search the doubly bound analog, for S states, of the well-known and widely used three-body Coulomb (C3) double-continuum wave function. This is performed with two alternative approaches: through an analytic continuation of the C3 continuum and through a "C3 approach" of the Hylleraas equation for S bound states. The double-bound analog consists of the product of two Coulomb bound states (one for each electron-nucleus interaction) multiplied by a Coulomb distortion factor which describes the electron-electron correlation. Our result differs from Pluvinage's wave function which is commonly misbelieved to be the bound counterpart of the C3 double continuum. With a rigorous treatment of the analytic continuation of the distortion factor, the double-bound equivalent of the C3 model is also found. Though the purpose is not to obtain good two-electron bound states, the found bound wave functions are tested, in the case of helium, through various local and mean quantities which probe different regions of the configuration space, in particular those close to the two-particle coalescence points.