Chapter 8: Atomic Structure and Spectra

The solution to the Schrödinger equation yields atomic orbitals expressed as products of radial and angular components, where the angular portion emerges from spherical harmonic functions. Three quantum numbers arise naturally from this solution: the principal quantum number n defines electron shells, the orbital angular momentum quantum number l designates subshells, and the magnetic quantum number ml specifies orbital orientation. These quantum numbers connect directly to the Rydberg constant, which predicts the energy levels of hydrogenic systems containing a single electron. Extending this framework to atoms with multiple electrons requires approximations because the Schrödinger equation becomes analytically intractable. The orbital approximation treats each electron as occupying an independent orbital within an effective potential created by the nucleus and other electrons. Two fundamental principles govern electron arrangement in these orbitals: the Pauli exclusion principle restricts each orbital to at most two electrons with opposite spins, and the building-up principle dictates that electrons fill orbitals in order of increasing energy. The self-consistent field procedure iteratively solves the Schrödinger equation by adjusting orbital energies until convergence occurs, accounting for electron-electron repulsion effects. The resulting electronic configurations explain periodic trends including ionization energy variations, electron affinity patterns, and systematic changes in atomic radius across periods and groups. The chapter concludes by connecting atomic electronic structure to observable spectroscopy through term symbols, which encode the total orbital angular momentum L and total spin angular momentum S of an atom. Spin-orbit coupling causes energy splitting of states with identical L and S values, creating fine structure in atomic spectra. Selection rules derived from quantum mechanics determine which transitions between electronic states are allowed, enabling prediction and interpretation of experimental spectroscopic data from both single-electron and multi-electron systems.