Abstract
As in many fields of physics and chemistry, symmetry plays a major role in the discussion of inter- and intra-molecular dynamics. Since we will use fundamental concepts of symmetry throughout the whole present work, this first chapter is intended to introduce some of the basic concepts. We start with a short summary of the symmetry groups of the molecular Hamiltonian. Subsequently, we discuss the general representation theory of the groups we use in this work. In contrast to these two group theory based sections, we introduce the main dynamical models of molecular physics in Sect. 2.2. The connection of these so-called zero-order models and representation theory is done in Sect. 2.3 where we also include a short outlook on the upcoming parts.
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Notes
- 1.
For a more rigorous definition of the group S \(_n\), see Sect. 2.1.2.
- 2.
An English translation was published in [21].
- 3.
The elements of the Lie algebra are abstractly defined as operators. By choosing a suitable basis they correspond to matrices and we therefore use these terms equivalently.
- 4.
We refer here only to the exchange symmetry of identical nuclei since we do not deal with the electronic motion in any particular sense in the following chapters. However, the definitions of permutations, symmetric groups, etc., are indeed fully general.
- 5.
We have subtracted three degrees of freedom for the translational motion of the center of mass.
- 6.
Notice that we set \(\hbar =1\) for the moment.
- 7.
The principal axis of rotation for an oblate top is the molecule-fixed axis corresponding to the moment of inertia \(I_3\). For the prolate case, it is the axis of \(I_1\).
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Schmiedt, H. (2017). Basic Concepts. In: Molecular Symmetry, Super-Rotation, and Semiclassical Motion. Springer Series on Atomic, Optical, and Plasma Physics, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-66071-4_2
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DOI: https://doi.org/10.1007/978-3-319-66071-4_2
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