Abstract Internal rotation in symmetric top molecules offers an excellent opportunity to investigate large amplitude motion in a relatively simple intramolecular environment. Due to specific symmetry characteristics of a symmetric top molecular frame, the internal rotation degree of freedom is in the zeroth order approximation separable from the small amplitude vibrations and the overall rotation, thus enabling to characterize the vibrational-torsional-rotational energy structure with a relatively simple Hamiltonian. Lessons from symmetric internal rotor studies may be applied to more complex systems, such as asymmetric internal rotors and macromolecules. This thesis deals with internal rotation in CH3SiH3, CH3SiD3, CH3CF3 which have become a prototype of symmetric internal rotors. The thesis presents high resolution vibration-torsion-rotation spectra and detailed analysis of these molecules. Particular attention is focused on torsion-mediated interactions, such as Coriolis-type interactions and Fermi-type interactions, coupling the internal rotation and the small amplitude vibrational motion. The studies show that the expansion of the data to the small amplitude vibrations and inclusion of the torsion-mediated interactions play a crucial role in order to obtain an appropriate characterization of the vibrational-torsional-rotational energy level structure and physically meaningful molecular parameters.

Abstract This thesis deals with the rotation-vibration theory and high resolution infrared spectroscopy of semirigid C3 molecules. Semirigid molecules form a class of molecules which are strongly bound with one well defined structure, and without low frequency internal motions. The theory, as well as the experimental studies of semirigid molecules are of special importance in the field of rotation-vibration spectroscopy. They provide a good starting point for interpreting and analyzing the spectra of practically all types of molecules. In this work, the theory is reviewed fromthe standpoint of one particular molecule, 13CH3I, which is a prolate symmetric top with C3 symmetry. The origin and the properties of the rotation-vibration Hamiltonian are discussed in detail. Molecular symmetry plays an important role in these studies. The expansion of the Hamiltonian for nuclear motion in powers of the vibrational operators converges rapidly as numerical examples thoughout the treatment indicate. The molecule is thus a good subject for the perturbation calculations, also reviewed here in detail. 13CH3I can be considered as a model example of semirigid molecules. From the spectroscopic point of view, this thesis is a study of the six fundamental bands of 13CH3I. The rotational analysis of the vibrational ground state is first given. Special attention is paid to obtaining the axial rotational constants which are problematic for symmetric top molecules. The relatively high energy level density of 13CH3I leads to several resonances. The fundamental bands, especially the higher ones, must therefore be treated as parts of band systems. Care is paid to properly take into account the effects of the near-lying vibrational levels on the constants of the fundamentals. Certain ambiguities in the rotation-vibration Hamiltonian of 13CH3I are also discussed.

The problem of the quadrupole interaction occurring in a vibrating-rotating C₃v symmetric top molecule has been studied in detail. The quadrupole interaction has been treated as another perturbation term to a general frequency expression accounting for the vibrating-rotating interaction of the molecule so that a complete frequency formula is obtained for both interactions, and from which hyperfine spectral components are predicted and measured. The hyperfine transitions in the ground, and v₈=1 and v₈=2 excited vibrational states of the ¹³C isotopes of methyl cyanide have been investigated in the frequency range 17-72 GHz, primarily in the low J transitions (0≤J≤3). The study of the ground state of isotope i3CH3i3CN, and the v₈=1, v₈=2 excited vibrational states for all the isotopes have been conducted here for the first time. A substantial perturbation has been discovered and discussed at the ΔJ=3→4 transitions within the Kl=1 sets in the v₈=1 mode for isotopes ¹³CH₃CN and CH₃¹³CN. A total of 716 hyperfine transitions have been assigned from measurements, only 7 of which have been measured previously. A total of 84 molecular constants have been reported; 70 of these constants are derived for the first time from microwave data.

<p>In case of the PDE's the concept of solving by separation of variables</p><p>has a well defined meaning. One seeks a solution in a form of a</p><p>product or sum and tries to build the general solution out of these</p><p>particular solutions. There are also known systems of second order</p><p>ODE's describing potential motions and certain rigid bodies that are</p><p>considered to be separable. However, in those cases, the concept of</p><p>separation of variables is more elusive; no general definition is</p><p>given.</p><p>In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.</p>

<p>In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustrated</p><p>In the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops.</p><p>The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions.</p> / Report code: LiU-TEK-LIC-2004:35.

In case of the PDE's the concept of solving by separation of variables has a well defined meaning. One seeks a solution in a form of a product or sum and tries to build the general solution out of these particular solutions. There are also known systems of second order ODE's describing potential motions and certain rigid bodies that are considered to be separable. However, in those cases, the concept of separation of variables is more elusive; no general definition is given. In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustrated In the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops. The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions. / <p>Report code: LiU-TEK-LIC-2004:35.</p>