1.1: Symmetry Elements and Operations Consider the symmetry properties of an object (e.g. atoms of a molecule, set of orbitals, vibrations). The collection of objects is commonly referred to as a basis set: (1) classify objects of the basis set into symmetry operations, (2) symmetry operations form a group group mathematically defined and (3) manipulated by group theory 1.2: Operator Properties and Mathematical Groups 1.3: Irreducible Representations and Character Tables Similarity transformations yield irreducible representations, Γi, which lead to the useful tool in group theory – the character table. The general strategy for determining Γi is as follows: A, B and C are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). 1.4: Molecular Point Groups 1 The symmetry properties of molecules (i.e. the atoms of a molecule form a basis set) are described by point groups, since all the symmetry elements in a molecule will intersect at a common point, which is not shifted by any of the symmetry operations. There are also symmetry groups, called space groups, which contain operators involving translational motion. 1.5: Molecular Point Groups 2 The D point groups are distiguished from C point groups by the presence of rotation axes that are perpindicular to the principal axis of rotation. 1.6: LCAO and Hückel Theory 1 (Eigenfunctions) A common approximation employed in the construction of molecular orbitals (MOs) is the linear combination of atomic orbitals (LCAOs). In the LCAO method, the molecular orbitals are expanded in an atomic orbital basis. 1.7: Hückel Theory 2 (Eigenvalues) 1.8: N-dimensional cyclic systems This lecture will provide a derivation of the LCAO eigenfunctions and eigenvalues of N total number of orbitals in a cyclic arrangement. 1.9: Band Theory in Solids The LCAO method for cyclic systems provides a convenient starting point for the development of the electronic structure of solids. 1.10: General electronic considerations of metal-ligand complexes 1.11: Frontier molecular orbitals of σ-donor, π-donor and π-acceptor ligands 1.12: Octahedral ML₆ Sigma Complexes 1.13: Octahedral ML₆ π complexes 1.14: Angular Overlap Method and M-L Diatomics