Published on *I. Physikalisches Institut* (https://www.astro.uni-koeln.de)

To find the electric potential of an rf multipole trap we have to find a solution for the ''Laplace'' equation:
ΔΦ=0
The movement of a particle in a rapidly oscillating field can described by an effective trapping potential. It is calculated by
taking the time-average over one period of the fast oscillation rf field .
This effective potential can be expressed as:
Ueffective = qΦdc+ q²⁄(4mΩ²)·(∇Φrf)²
with Φdc the static part of the potential, Φrf the rf part, Ω the rf frequenz and m the mass of the particle.

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The usual rotation group, SO(3), is conveniently used to describe the symmetry of the rotational wave functions of molecules. Internal rotation, on the other hand, is described by SO(2) symmetry since the rotation axis is fixed with respect to the rigid framework. Combining these two motions leads to the formulation in a five-dimensional rotational symmetry group SO(5). Applied to the example of CH5+, this group is capable of describing the permutation symmetry elements of the molecular symmetry group in terms of five-dimensional rotations. We showed that an "equivalent rotation" treatment is actually impossible in three dimensions implying a conventional description of the molecular wave functions in terms of rotational wave functions to fail.

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Useful descriptions of the intermediate regime between the well-understood quantum mechanical "world'' and the classical limit include the use of semi-classical calculations, which we use to determine the rotational energy spectra of different molecules at high J-quantum numbers. Basis of the semiclassical approach is the rotational energy surface (RES, cf. Fig. 1) which is found by writing the quantum mechanical Hamiltonian in terms of a "classical vector" (Jx, Jy,Jz)T using two angles and the fixed length |J|.
To get to quantization conditions, one can use the so-called WKB- or Sommerfeld quantization rules, which were first applied to standard problems like the quantum harmonic oscillator. Here we can use them analogously to find conditions for the quantization of the energy levels.
An alternative approach makes use of results of quantum chaos theory. This apporach is completely coordinate free and involves only geometrical and topological features of the classical dynamics, which makes it more useful than the heuristic picture of the earlier works. In that approach one uses the Gutzwiller trace formula for the description of the density of states, where the poles signals the energy levels. Going one step further, we can use the symmetry projected Green function in the derivation of the density of states and hence find different quantization conditions for the various representations of the molecular symmetry group.

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**Links:**

[1] https://www.astro.uni-koeln.de/node/784?id=boundary_element_method

[2] https://www.astro.uni-koeln.de/node/784?id=molecular_symmetry

[3] https://www.astro.uni-koeln.de/node/784?id=semi_classical_aproach