Rotationally highly excited molecules – A semi-classical approach to the energy levels and their symmetry
Investigators
schmiedt
schlemmer
jensen
Description
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.
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The semi-classical description for large J was brought to the molecule community in the late 70's and early 80's mainly by S. Collwell, N. Handy and W. Miller and W. Harter and C. Patterson. Somehow paralell to their description in terms of a Sommerfeld or WKB- quantization procedure, another approach came into play by using tools from the study of quantum chaos (J. Robbins et al.).
It was shown that one can predict the energy levels for the SF_{6} molecule with high accuracy. Robbins et al. also found the right symmetry for the eigenstates for the individual energy levels. We want to generalize their result to other symmetry groups and we hope to find a general quantization scheme to calculate energies and symmetries of levels at high angular momentum quantum numbers. In addition to this more general aim, we want to implement the semi-classical method to the "TROVE" program by Yurchenko et al., which we want tu use to calculate the vibrational influence on the rotational problem. With this, we hope to be able to calculate the energy levels for high rotational quantum numbers much faster than by full quantum calculations.
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Methods
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" (J_{x}, J_{y},J_{z})^{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.
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 k_{z} axis is the better one and below it is the k_{x} axis. This can be explained by looking at the phase space portraits of the respective classical model. For the k_{x} 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 k_{z} (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 J_{i} 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 SF_{6} 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.
This work is done in collaboration with Bergische Universität Wuppertal (P. Jensen), University College London (S.N. Yurchenko), and DESY, Hamburg (A. Yachmenev).