Superrotation – A new model to combine internal and overall rotation
Investigators
schmiedt
brackertz
jensen
schlemmer
Description
Extremely floppy molecule exhibit multiple large amplitude vibrations already in the vibrational ground state. This implies that a definition of an equilibrium structure is impossible. The most important large amplitude motion in this context is the internal rotation. By merging this with the overal rotational motion to a fundamentally new five-dimensional rotation, we have found a collective super-rotational motion to describe the low energy dynamics of extremely floppy molecules.
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The most prominent example, protonated methane, CH_{5}^{+}, where its low-energy states has been measured recently, is used as a first test of the new model. The predicted energies and the related symmetry labels from the molecular symmetry group G_{240} compare favourably with the experimental data (CH_{5}^{+}[2]). This very encouraging result puts forward the idea that this model, which at the moment is only of zeroth order, is a first step on the way towards a more thorough understanding of the complex motion in this kind of molecules.
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Methods
Molecular Symmetry
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 CH_{5}^{+}, 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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Recent Results
In customary molecular theory, the most basic assumption is that of a separable Hamiltonian. The different motions, electronic, vibrational, and rotational, take place on different energy scales, which induces a separability of the full molecular Hamiltonian. Intuitively this leads to the well-known ball-and-stick model of molecules, where the nuclei are assumed to be at (almost) fixed positions and the fast electronic motion mediates bonds between them. Vibrations are imagined to only imply small structural deformations, whereas the rotations are approximated as those of a rigid body.
In extremely floppy molecules, multiple large amplitude motions, which deforms the molecule considerably, are present already in the lowest vibrational state. The potential energy surface for this kind of motion is assumed to be flat in the sense that all barriers are small compared to the zero-point vibrational energy. Therefore the different minima on this surface are not well-separated and perturbative treatments of their respective interaction must fail completely. In addition the combination with overall rotational motion is also a formidable task which yet is only attacked by using large computing power.
The most prominent member of this class of molecule is protonated methane, CH_{5}^{+}, where calculations only recently became (at least) comparable to the experimental results of Asvany et al. In CH_{5}^{+} there are 120 equivelent equilibrium geometries which all are separated by very small potential energy surface. The vibrational ground state therefore is described as being highly delocalized and the definition of a single fixed geometry is out of question. Hence, this molecule is a fascinating example of quantum mechanics in general and challenges the molecular theory community since its first detection in the 1950's.
This research project aims to describe the dynamics in the low-energy regime of extremely floppy molecules in a fundamentally new way: We investigate an analytical model which combines the internal and the overall rotations in a collective five-dimensional motion. From the respective five-dimensional symmetry, one finds an appropriate kinetic energy operator and respective energy eigenvalues. Although this energy is described by a single free parameter, a comparison to the experimental results of Asvany et al. are very promising.
The combination of internal and overall rotation is shown in Fig. 1, where the first excited states of internal p=1 and overall J=1 rotations merge to a single fivefold degenerated energy level. In Fig. 2, the comparison of the predicted energy levels and the measured combination differences is shown. The latter are assumed to start at the respective lowest energy level of appropriate symmetry. One obviously sees the good agreement to the theoretical prediction. The dashed energy levels to the side of the central SO(5) symmetric states are Pauli-forbidden, which we can determine by mapping the moleular symmetry group to the continuous SO(5) group and determine the perturbation symmetry of the [n_{1},n_{2}] states.