Chapter 6: Molecular Symmetry

The presentation begins with symmetry operations and their corresponding elements—identity operations, rotation axes of various orders, mirror planes, inversion centers, and improper rotation axes—which collectively define how molecules can be transformed while maintaining their geometric and electronic properties. These symmetry elements are then systematized into point groups, a classification scheme that allows chemists to categorize molecules based on their complete symmetry profile. Representative examples such as water, ammonia, carbon dioxide, boron trifluoride, and sulfur hexafluoride illustrate how systematic decision trees guide the assignment of molecules to their appropriate point groups. Character tables emerge as the central mathematical tool in group theory, encoding the behavior of orbitals, molecular vibrations, and electronic states under symmetry operations and correlating these behaviors with symmetry species designations and degeneracy levels. The chapter develops practical applications of symmetry analysis, including determining molecular polarity and identifying whether molecules possess or lack chirality based on the presence or absence of improper rotation axes. A substantial section addresses vibrational spectroscopy, demonstrating how symmetry arguments predict which vibrational modes are active in infrared and Raman spectroscopy, employing the exclusion rule for centrosymmetric molecules and providing detailed analysis of complex systems like square-planar complexes and metal carbonyls. Symmetry-adapted linear combinations represent a systematic methodology for constructing molecular orbital diagrams by combining atomic orbitals of matching symmetry character, enabling qualitative bonding predictions for multi-atom systems. The quantitative techniques of representation reduction and projection operators are introduced as methods for decomposing reducible representations into their irreducible components and for generating symmetry-matched orbital combinations. Throughout, the chapter emphasizes how group theory unifies the prediction and interpretation of molecular structure, bonding characteristics, and spectroscopic properties across diverse inorganic systems.