Identification and visualization of the collective normal modes of intrashell triply excited states of atoms

We identify the elementary normal modes of the collective motion of three electrons in intrashell triply excited states of atoms. By modelling the atom as consisting of three electrons on the surface of a sphere with the nucleus at the centre, we analysed the calculated wavefunctions in the body-fixed frame. Using contour surfaces in appropriate coordinates, the nodal surfaces and the normal modes of the collective motion of the three electrons can be visualized. Three elementary normal modes and their excited modes are identified, thus allowing the new classification of intrashell triply excited states. (Some figures in this article are in colour only in the electronic version; see www.iop.org) Since the beginning of quantum mechanics, the basic conceptual framework for describing the structure of a many-electron atom is the independent electron model. In the past few decades, it has been well established that the independent electron model fails to describe multiply excited states. For the simpler doubly excited states, alternative equivalent descriptions have been found [1–4] and a new set of approximate quantum numbers have been obtained. In all of these descriptions, the motion of the two electrons is treated together and the correlation of the two electrons is understood as analogous to the bending vibrations and the symmetric or antisymmetric stretch motion of a floppy linear triatomic molecule. It is fair to say that the basic properties of correlations in doubly excited states are now well understood. As a natural extension, the next major step is to understand the correlated motion of triply excited states of an atom. Experimentally, triply excited states have been explored recently at synchrotron radiation facilities [5, 6] and in experiments involving collisions of multiply charged ions with atoms, molecules, and surfaces [7]. Calculations based on the R-matrix method [8] or the general configuration mixing method [9–11] have been able to give a very accurate description of individual triply excited states. However, the nature of how the three electrons are correlated is still not understood and there is no global knowledge of these states. Unless the basic normal modes of the collective motion of the three electrons are delineated, classification of triply excited states in terms of a new set of quantum numbers is not possible. Various aspects of electron correlations in selected triply excited states have been examined in the past, by partial display of the calculated wavefunctions [12, 13]. In our recent works [14, 15] we have solved the three-electron atomic Hamiltonian in hyperspherical coordinates 0953-4075/01/040105+07$30.00 © 2001 IOP Publishing Ltd Printed in the UK L105 L106 Letter to the Editor and examined the correlations of triply excited states of Li. Since these triply excited states lie above both the singly and the doubly excited states, the analysis has been limited only to the lowest eight 2l2l′2l′′ intrashell states. We have found that these eight states can be separated into three groups, and each group is characterized by a certain normal mode of a symmetric top. To elucidate other possible normal modes, triply excited states such as those in the higher 3l3l′3l′′ manifold have to be examined. A full hyperspherical calculation of these states would, however, be too tedious and too complicated since these states lie further above the 2lnl′ml′′ (n,m 2) triply excited states. We have thus studied the correlation of 3l3l′3l′′ triply excited states of a model atom where the three electrons are confined to the surface of a sphere, with the nucleus at the centre. For the 2l2l′2l′′ states, the density distributions obtained using this model are similar to those obtained from the full solution using hyperpsherical coordinates when the three electrons are at the same distance from the nucleus [15]. The model Hamiltonian (in atomic units) we used is