Chapter 10: Molecular Symmetry

The approach relies on matrix representations to characterize symmetry operations such as rotations, reflections, and inversions, establishing a rigorous mathematical foundation for analyzing molecular structure. Students learn to identify symmetry elements within molecules and classify them into specific point groups based on their combination of symmetry operations. Character tables emerge as essential reference tools, providing irreducible representations that allow chemists to classify molecular orbitals, vibrational modes, and electronic states according to their symmetry properties. The character of a symmetry operation, defined as the trace of its matrix representative, provides a quantitative measure for determining how different orbitals and states transform under molecular symmetry. A critical application involves using symmetry arguments to predict when integrals vanish, directly enabling assessment of orbital overlap and identification of which atomic orbital combinations can meaningfully contribute to molecular orbital formation. The chapter covers the construction of symmetry-adapted linear combinations of atomic orbitals, which represent the most appropriate basis functions for describing bonding and antibonding interactions within a given molecular point group. Additionally, group theory provides the theoretical foundation for deriving selection rules that govern allowed versus forbidden transitions in spectroscopic experiments, connecting molecular symmetry to observable spectroscopic phenomena. The chapter establishes that degeneracy in molecular energy levels is always intrinsically linked to symmetry, allowing students to predict degenerate states and understand why symmetry breaking occurs under perturbations. Throughout, the emphasis remains on using symmetry as a practical predictive tool rather than as abstract mathematics, empowering students to rapidly assess molecular properties without performing extensive computational calculations.