Nodal structures of intrashell states of three-valence-electron atoms

The nodal structures of intrashell states of a three-valence-electron atom are investigated. We expand the wave function in the body frame of the three electrons and show that the effect of symmetry such as rotation, space inversion, and permutation is to impose nodal surfaces on the components of the wave function. The equilibrium configuration is deduced to be a coplanar equilateral triangle with the core at the center and the three electrons at the vertices. Three basic modes of internal oscillations have been identified and the relative energy levels of the atom are interpreted in terms of the degrees of excitations of these basic modes. Calculations are carried out using a model of three electrons moving on the surface of a sphere. Results from this r-frozen model, as well as the limited data in the literature, are used to illustrate conclusions drawn only from the symmetry consideration alone. We thus claim that the relative energy levels of intrashell states of a three-electron atom is determined less by the detailed dynamics and interactions, but more by the inherent quantum symmetry. copyright 1997 The American Physical Society