Chapter 9 extends quantum mechanics from atoms to molecules, focusing on the electronic structure and spectroscopy of diatomic molecules. The chapter develops the conceptual and mathematical tools needed to describe molecular bonding, electronic states, angular momentum coupling, and spectral structure. Special emphasis is placed on the construction and interpretation of diatomic molecular term symbols, which encode symmetry, spin, and angular momentum information in a compact form.
The chapter introduces multiple complementary approaches—including the united atom method, the separated atom method, and the molecular orbital method— and shows how these methods are used together to determine allowed electronic states and identify molecular ground states.
The chapter begins by formulating the molecular Hamiltonian and introducing the Born–Oppenheimer approximation, which separates fast electronic motion from slower nuclear motion. This approximation enables the construction of potential energy curves that describe molecular bonding, vibration, and dissociation.
Molecular orbitals are introduced using the linear combination of atomic orbitals (LCAO) approach. Bonding and antibonding orbitals of \( \sigma \), \( \pi \), and higher symmetry are constructed and interpreted in terms of overlap, nodal structure, and energy.
Electronic configurations for diatomic molecules are built by filling molecular orbitals according to the Pauli principle and Hund’s rule. From these configurations, bond order and magnetic properties (diamagnetism versus paramagnetism) are predicted, explaining key observations such as the paramagnetism of \( \mathrm{O_2} \).
The chapter develops Hund’s coupling cases (a) and (b) for diatomic molecules, describing how electronic orbital angular momentum, electron spin, and molecular rotation combine to form total angular momentum. These coupling schemes provide the foundation for understanding molecular fine structure and rotational level patterns.
Diatomic term symbols are introduced as the molecular analogue of atomic term symbols. The meaning of \( \Lambda \), spin multiplicity, reflection symmetry (\( \Sigma^{+}/\Sigma^{-} \)), and inversion symmetry (\( g/u \)) is developed in detail.
Two symmetry-based methods for enumerating diatomic term symbols are presented. The united atom method correlates molecular states with atomic term symbols at short internuclear distances, while the separated atom method combines atomic states at large separations. The Wigner–Witmer rules are used to assign reflection symmetry in both limits.
The molecular orbital method is shown to provide the most reliable determination of diatomic term symbols by using the actual electronic configuration of the molecule. By considering only partially filled molecular orbitals, the correct ground state and symmetry can be identified unambiguously.
Herzberg diagrams are introduced as a powerful tool for predicting rotational branch structure and selection rules in electronic transitions. These diagrams connect molecular symmetry with observed P-, Q-, and R-branch patterns in rovibronic spectra.
Electronic transitions with resolved vibrational structure are analyzed using the Franck–Condon principle. Franck–Condon factors explain the relative intensities of vibronic bands and connect molecular structure to observed spectral progressions.