Parity violation in molecular magnetic resonance properties

The Standard Model of particle physics consists of the quantum electrodynamics (QED) and the weak and strong nuclear interactions. The QED is the basis for molecular properties, and thus it defines much of the world we see. The weak nuclear interaction is responsible for decays of nuclei, among other things, and in principle, it should also effects at the molecular scale. The strong nuclear interaction is hidden in interactions inside nuclei. From the high-energy and atomic experiments it is known that the weak interaction does not conserve parity. Consequently, the weak interaction and specifically the exchange of the Z^0 boson between a nucleon and an electron induces small energy shifts of different sign for mirror image molecules. This in turn will make the other enantiomer of a molecule energetically favorable than the other and also shifts the spectral lines of the mirror image pair of molecules into different directions creating a split. Parity violation (PV) in molecules, however, has not been observed. The topic of this thesis is how the weak interaction affects certain molecular magnetic properties, namely certain parameters of nuclear magnetic resonance (NMR) and electron spin resonance (ESR) spectroscopies. The thesis consists of numerical estimates of NMR and ESR spectral parameters and investigations of the effects of different aspects of quantum chemical computations to them. PV contributions to the NMR shielding and spin-spin coupling constants are investigated from the computational point of view. All the aspects of quantum chemical electronic structure computations are found to be very important, which makes accurate computations challenging. Effects of molecular geometry are also investigated using a model system of polysilyene chains. PV contribution to the NMR shielding constant is found to saturate after the chain reaches a certain length, but the effects of local geometry can be large. Rigorous vibrational averaging is also performed for a relatively small and rigid molecule. Vibrational corrections to the PV contribution are found to be only a couple of per cents. PV contributions to the ESR g-tensor are also evaluated using a series of molecules. Unfortunately, all the estimates are below the experimental limits, but PV in some of the heavier molecules comes close to the present day experimental resolution.

Tannakian Formalism Applied to the Category of Mixed Hodge Structures

Many problems in analysis have been solved using the theory of Hodge structures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. Category of mixed Hodge structures over the field of real or complex numbers is a rigid abelian tensor category, and in fact, a neutral Tannakian category. Therefore it is equivalent to the category of representations of an affine group scheme. The direct sums of pure Hodge structures of different weights over real or complex numbers can be realized as a representation of the torus group, whose complex points is the Cartesian product of two punctured complex planes. Mixed Hodge structures turn out to consist of information of a direct sum of pure Hodge structures of different weights and a nilpotent automorphism. Therefore mixed Hodge structures correspond to the representations of certain semidirect product of a nilpotent group and the torus group acting on it.

Hyperfine effects in the nuclear magnetic resonance of paramagnetic molecules

Paramagnetic, or open-shell, systems are often encountered in the context of metalloproteins, and they are also an essential part of molecular magnets. Nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for chemical structure elucidation, but for paramagnetic molecules it is substantially more complicated than in the diamagnetic case. Before the present work, the theory of NMR of paramagnetic molecules was limited to spin-1/2 systems and it did not include relativistic corrections to the hyperfine effects. It also was not systematically expandable. --- The theory was first expanded by including hyperfine contributions up to the fourth power in the fine structure constant α. It was then reformulated and its scope widened to allow any spin state in any spatial symmetry. This involved including zero-field splitting effects. In both stages the theory was implemented into a separate analysis program. The different levels of theory were tested by demonstrative density functional calculations on molecules selected to showcase the relative strength of new NMR shielding terms. The theory was also tested in a joint experimental and computational effort to confirm assignment of 11 B signals. The new terms were found to be significant and comparable with the terms in the earlier levels of theory. The leading-order magnetic-field dependence of shielding in paramagnetic systems was formulated. The theory is now systematically expandable, allowing for higher-order field dependence and relativistic contributions. The prevailing experimental view of pseudocontact shift was found to be significantly incomplete, as it only includes specific geometric dependence, which is not present in most of the new terms introduced here. The computational uncertainty in density functional calculations of the Fermi contact hyperfine constant and zero-field splitting tensor sets a limit for quantitative prediction of paramagnetic shielding for now.