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Rotationally highly excited molecules  –  A semi-classical approach to the energy levels and their symmetry


  • schmiedt
  • schlemmer
  • jensen


In the full description of molecular spectra, one is interested not only in the ground states and the few states nearby but also to extend the knowledge to highly excited states. Sine the Hilbert space gets enormously large for high quantum numbers, these states, i.e. their properties like energy and symmetry of the respective wave functions, are difficult to calculate quantum mechanically. In addition one expects some kind of transition to the classical dynamics of that system. In this work we focus on the rotational dynamics of molecules, which may be treated as rigid in a first approximation but can also be distorted by centrifugal distortion terms at higher order of the J quantum operators. ...more


Semi Classical Aproach
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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Recent Results

In Figure 1 we see the good agreement (∼ 0.25%) of the quantum mechanical energy levels for the rigid asymmetric top molecule and the quantized energy values calculated from the rotational energy surface. The two different choices of the quantization axis differ in their accuracy. Above the seperating energy (also called the seperatrix), which is indicated as a dashed line, the kz axis is the better one and below it is the kx axis. This can be explained by looking at the phase space portraits of the respective classical model. For the kx we find closed trajectories whenever the energy smaller than the seperatrix, while there are no closed trajectories otherwise. Indeed, this is observation is in a way inverted when looking at kz (cf. fig. 2).

From this observation, the question arise how to find these closed trajectories for more difficult model Hamiltonians like for non-rigid molecules, where centrifugal distortion plays a prominent role especcially at high J-quantum numbers. The respective Hamiltonians have several restrictions on the order of the Ji operators from symmetry. They can of course be generalized to any power in J, where also time-reversal symmetry must be taken into account. For these higher order Hamiltonians our goal is to find a proper description of the possible closed orbits. They may be injected into the Gutzwiller trace formula (cf. Methods), like it was done for the next to leading order Hamiltonian for the SF6 molecule by J. Robbins et al., to get quantization rules for the different symmetry species of the eigenstates of the respective molecule.

Recently, we published a first proof-of-principle work on the example of sulfur dioxide, a well-described molecule, where we showed the good agreement of semi-classical theory and full quantum calculations also in a few vibrationally excited states.

Figure 1: The energy levels of a hypothetic molecule with rotational constants A=8, B=4, C=3 for a fixed rotational quantum number J = 40. The middle column is determined by diagonalizing the quantum mechanical Hamiltonian for a rigid asymmetric top rotor while the other two columns show the quantized energy values for the two possible choices of the quantization axis.
Figure 2: The phase space portrait for the hypothetic molecule at fixed J quantum number. The trajectories shown differ in their energy. We see closed trajectories in different energy regimes for the two possible choices of the momentum (p= kx or p= kz).


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  • Funding by SFB 956.
  • Hanno Schmiedt is supported by BCGS.
  • This work is done in collaboration with Bergische Universität Wuppertal (P. Jensen), University College London (S.N. Yurchenko), and DESY, Hamburg (A. Yachmenev).