Doubly excited ridge states of atoms.

Abstract

Doubly excited ridge states of atoms are states in which two electrons reach high and comparable excitations. Our goal is to calculate sequences for some of these states by using a wave function that treats the pair of electrons as a single entity in solving the two-electron Schr\"odinger equation in hyperspherical coordinates: R, \ensuremath{\alpha}, ${\mathrm{\ensuremath{\theta}}}_{12}$. For double escape of slow electrons, the so-called Wannier theory predicts that the wave function will be concentrated in the region \ensuremath{\alpha}=\ensuremath{\pi}/4, ${\mathrm{\ensuremath{\theta}}}_{12}$=\ensuremath{\pi} (that is, ${\mathbf{r}}_{1}$=-${\mathbf{r}}_{2}$) and we expect something similar for the ridge states. By expanding the Schr\"odinger equation around these points and retaining the first nontrivial quadratic dependences in \ensuremath{\alpha} and ${\mathrm{\ensuremath{\theta}}}_{12}$, we seek a solution in which the form of the wave function in these two variables is analytically determined as in the Wannier theory. The R dependence of this wave function is then handled numerically and differs from the double-escape solution only in the exponential decay appropriate for bound states. For L=0 a polynomial in R is included, giving the proper number of nodes for the solution. Numerical eigenvalues for autoionizing ridge states are calculated and compared with available results.