The Modified Nilsson Potential for Ellipsoidal Shapes

In particular, a modified Nilsson Hamiltonian appropriate for metal clusters [35, 36] is given by 

Uo in Eq. (27) is a dimensionless parameter, which for occupied states may depend on the effective principal quantum number n = n ri2- - n 3 associated with the major 

The stretched is not a properly defined angular-momentum operator, but has the advantageous property that it does not mix deformed states that correspond to spherical major shells with different principal quantum numbers n = n ri2- - (see the 

The subtraction of the term (/ = az(/i + 3)/2, where ) denotes the expectation value taken over the /ith-major shell in spherical symmetry, guarantees that the average separation between major oscillator shells is not affected as a result of the lifting of the degeneracy. 

The oscillator frequencies can be related to the principal semiaxes a, b and c (see, Eq. (20)) via the volume-conservation constraint and the requirement that the surface of the cluster is an equipotential one, namely 

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