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Course description:
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Sets, relations, and functions. Foundations of group theory. Subgroups, group homomorphisms. Isometries in the 3D space, linear transformations: rotation, inversion, reflection in a plane. General systematics of linear transformations in 3D space: rotations and inversion-rotations or rotations and improper rotations. Point symmetry of 3D objects (polyhedra, mole-cules), and point-symmetry groups. Symmetry operations and symmetry elements; classes of symmetry operations. Inernational (Hermann-Mauguin) and Schoenflies notation for point groups. A systematics of point groups (including the 32 crystallographic ones). Complex vector spaces, linear transformations and their matrix representation. The theory of group representations. Irreducible representations, the Grand Orthogonality Theorem and its multiple consequences. Representation characters and the character tables for the irreducible representations of point groups. Methods for converting a reducible representation into a sum of the irreducible ones. Group theory and the solutions of the Schroedinger equation. The use of symmetry for the block-diagonalization of the Hamiltonian matrix; finding the symmetry orbitals in the LCAO MO calculations (example - the HMO model). The simple product of representations. Finding the symmetry-allowed dipole electronic transitions. Molecular vibrations and the symmetry classification of the normal modes. Symmetry and chemical reactions. The Woodward-Hoffmann rules.
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