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Term symbols (²ˢ⁺¹L) describe the electronic states of multi-electron atoms and ions by specifying the total orbital angular momentum (L), total spin (S), and multiplicity (2S+1). For d-electron configurations, term symbols enumerate all possible electronic states — including ground and excited states — which directly correspond to the energy levels seen in electronic spectra. Deriving term symbols from microstate analysis is the foundation for understanding Tanabe-Sugano diagrams and the full electronic spectrum of any transition metal complex.
Crystal field theory describes d-orbital splitting using a single-electron picture: each d-electron occupies one of the split orbitals. But real multi-electron ions have electron-electron repulsions that create multiple electronic states — a d² ion does not have just one "d²" state but multiple states (³F, ³P, ¹G, ¹D, ¹S) with different energies determined by how the two electrons are arranged. Term symbols label these states, and understanding them is prerequisite to interpreting the full electronic spectra of transition metal complexes through Tanabe-Sugano diagrams.
A term symbol ²ˢ⁺¹L encodes two pieces of information. The orbital part L describes the total orbital angular momentum from all d-electrons coupled together: L = 0 (S term), 1 (P), 2 (D), 3 (F), 4 (G), and so on. The spin multiplicity 2S+1 describes the total spin: singlet (1), doublet (2), triplet (3), quartet (4), etc. Each term represents a distinct electronic state with its own energy, and the number of microstates (individual electron arrangements) within each term is (2S+1)(2L+1). Hund's rules predict the ground-state term: first maximize S, then maximize L for that S, then determine J by L−S (less than half-filled) or L+S (more than half-filled).
Deriving term symbols requires the microstate method. For d², you enumerate all 45 ways to place two electrons in five d-orbitals (with both spatial and spin quantum numbers, respecting Pauli exclusion). These microstates are organized in a table indexed by ML (sum of individual ml values) and MS (sum of individual ms values). The terms are then extracted sequentially: find the largest ML at the largest MS, identify the corresponding ²ˢ⁺¹L term, subtract its microstates, and repeat. The procedure is algorithmic and guarantees that every microstate is assigned to exactly one term.
The hole formalism provides a powerful shortcut: dn and d10−n have identical term symbols. This means you only need to work out terms for d¹ through d⁵; the d⁶ through d⁹ results follow by symmetry. When a crystal field is applied, each free-ion term splits into multiple components labeled by the irreducible representations of the point group — and these split terms become the energy levels plotted in Tanabe-Sugano diagrams. The connection is direct: the term symbols derived here are the y-axis labels at x = 0 (the free-ion limit) of every Tanabe-Sugano diagram.