This chapter extends quantum mechanics beyond hydrogen to polyelectronic atoms, where electron–electron repulsion prevents exact analytical solutions. To make progress, the orbital approximation is introduced, allowing electrons to be treated as occupying hydrogen-like orbitals while electron–electron interactions are handled phenomenologically. Building on this approximation, the chapter develops electronic configurations using the Aufbau principle, Pauli Exclusion Principle, and Hund’s rules, and introduces the formal machinery needed to understand atomic spectra: angular momentum coupling, term symbols, microstates, and spin–orbit interactions. These tools culminate in practical methods—such as the Landé Interval Rule and Deslandres tables—that allow complex atomic spectra to be interpreted and assigned systematically.
The Hamiltonian of a many-electron atom includes kinetic energy terms for each electron, attractive electron–nucleus interactions, and repulsive electron–electron interactions. The electron–electron repulsion term makes the Schrödinger equation non-separable, turning the problem into a three-body (or many-body) problem that cannot be solved analytically. This fundamental difficulty motivates the approximations used throughout the chapter.
The orbital approximation neglects explicit electron–electron repulsion in the wavefunction and treats each electron as moving independently in an effective potential. This allows the Hamiltonian to be separated into one-electron terms and enables the use of hydrogen-like orbitals. It is this approximation that underlies electronic configurations and orbital diagrams in chemistry.
The Aufbau principle describes how electrons “build up” atomic structure as nuclear charge increases, filling orbitals in order of increasing energy. In polyelectronic atoms, subshell energy ordering differs from hydrogen and can change with atomic number, explaining features such as the filling of 4s before 3d and the order of electron removal upon ionization. Electronic configurations summarize these occupancies compactly.
Shells are defined by the principal quantum number n, subshells by n and l, and orbitals by n, l, and ml. Each orbital can hold two electrons with opposite spin projections ms = ±1/2. This nomenclature provides the framework for orbital diagrams and later angular momentum coupling schemes. Orbital diagrams graphically represent electronic configurations, showing orbitals as boxes (or lines) and electrons as arrows. They make spin pairing, unpaired electrons, and magnetic properties such as paramagnetism and diamagnetism immediately apparent. Orbital diagrams are a practical bridge between abstract quantum numbers and observable properties.
Total angular momentum is descibed according the Russell-Saunders (L-S) coupling. Microstates are distinct combinations of ml and ms values for all electrons consistent with the Pauli Exclusion Principle. Enumerating microstates allows all possible term symbols for a given configuration to be identified systematically. The microstate method ensures that no allowed states are missed and no forbidden states are included. Spin–orbit coupling arises from the interaction between an electron’s spin and its orbital motion, splitting term states into multiple J levels. This fine structure explains the multiplet patterns observed in high-resolution atomic spectra. The magnitude and ordering of these splittings depend on both electronic configuration and whether subshells are less than or more than half filled. Hund's Rules are also discussed as to how they are used to determine the lowest energy state in a Russell-Saunders manifold.
Electrons are fermions, requiring the total wavefunction to be antisymmetric under exchange of identical particles. This constraint explains why certain combinations of spin and orbital symmetry are allowed while others are forbidden. Exchange symmetry provides a deeper foundation for Hund’s rules and term-symbol restrictions.
Atomic spectra can be interpreted using term symbols and selection rules derived from Russell–Saunders coupling. The Landé Interval Rule predicts systematic splittings within a term manifold, while Deslandres tables organize spectral lines to reveal constant energy differences. Together, these tools allow complex spectra to be assigned and spin–orbit coupling constants to be extracted.