Two Electrons in a Coulomb Potential. Double-Continuum Wave Functions and Threshold Law for Electron-Atom Ionization

Abstract

The Schrödinger equation for two electrons in a Coulomb field is studied in the critical region where both electrons have near-zero kinetic energies. The main feature of this problem is that the mutual screening between the two electrons determines and is determined by the partition of the available energy between them. This energy-dependent screening can be taken into account to yield a complex potential in the radial variable $R={(r_1^2+r_2^2)}^{\frac{1}{2}}$ of the six-dimensional configuration space of the two electrons. Solutions of this equation are obtained and are shown to correspond to the classical orbits given in an early paper by Wannier. A possible way is indicated of using these wave functions to establish the Wannier threshold law which, for ionization of neutral atoms, is $\sigma \propto E^{1.127}$. Finally, the interplay between the total energy and the Coulomb potential is discussed both for this problem and for the case of one electron in the field of a nucleus.