Energy expression many-shell system

The generalization to many-shell systems (McWeeny, 1974) is straightforward. We suppose that there are niK available orbitals (2mj< spin-orbitals) and electrons in shell K, and obtain the average-energy expression (cf. (6.6.7))... 

In fullerene anions C%q, the n electrons outside closed shells occupy /lu triplet electronic states. Jahn-Teller (JT) coupling between these states and 5-fold h-type vibrations has important consequences for many properties of the fullerene anions. It is therefore important to understand the JT effect experienced by these ions from a theoretical point of view. We will study the cases of n = 2 and 4, where the lowest adiabatic potential energy surface is found to consist of a two-dimensional trough in linear coupling. The motion of the system therefore consists of vibrations in three directions across the trough and pseudo-rotations in two directions around the trough. Analytical expressions for states of the system that reflect this motion are obtained and the resultant energies determined. 

It is easy to see that the size of the onions decreases dramatically if the innermost shell is not filled, which amounts to dropping the first term in the series (12). To generalize the theory, one has to takep to be variable and express the total bending energy in terms of the radii of the outermost and innermost onion skins. We refrain from this exercise in view of many other uncertainties. For instance, the nonspherical shapes of the outermost skins, because of their additional bending energies, should cause the phase transition from the onion phase to the planar multilayer system to be discontinuous. 

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