Electronic Spectra — formula sheet

The whole problem of Part 5 in one sentence: an electron configuration such as t_2g² is not a single state of definite energy — it is a whole family of states that differ in how the electrons are arranged relative to one another, and therefore in how strongly they repel one another.

Microstate: one specific, allowed assignment of every electron to an orbital (a value of m_l) and a spin (a value of m_s), consistent with the Pauli principle. Two microstates that differ by swapping which electron is which are the same microstate — electrons are indistinguishable.

Term symbol: ^2S+1L_J. The letter encodes the total orbital angular momentum L using the sequence L = 0, 1, 2, 3, 4, 5, 6 → S, P, D, F, G, H, I. The left superscript 2S+1 is the spin multiplicity. The right subscript J is the total angular momentum, obtained by coupling L and S; it is usually omitted in ligand-field work.

Hole formalism: a d^n configuration and a d^10−n configuration have the same set of terms with the same degeneracies, because n electrons in ten boxes and n holes in ten boxes are the same combinatorial object. The ordering of those terms is the same too in the free ion, but in a ligand field the splitting pattern inverts, because a hole has the opposite sign of charge to an electron.

The rule: a term of orbital angular momentum L splits in an octahedral field into exactly the same components as a set of 2L+1 orbitals with that same l. So a D term splits like d orbitals, an F term splits like f orbitals, and an S term (L = 0, spherically symmetric) does not split at all.

Orgel diagram: a qualitative plot of the energies of the ligand-field states arising from the ground term and from the excited term of the same multiplicity, against the ligand-field strength. The free ion sits at the centre; Δ increases outwards in both directions, one direction serving the d^n/O_h case and the other the d^10−n/O_h case.

Design decision 1 — divide everything by B. The vertical axis is E/B and the horizontal axis is Δ_o/B. Both axes are therefore dimensionless. This is what makes one diagram serve every d³ complex ever made, whatever its B: complexes with different B values are simply at different points along the same curves.

Design decision 2 — put the ground state on the horizontal axis. All energies are plotted relative to the ground state, so the ground state is the line E/B = 0 by construction. Every vertical distance you measure from the axis to a curve is directly a transition energy. Compare the Orgel diagram, where the ground state slopes and you must measure gaps between two sloping lines.

Design decision 3 — fix C/B. The diagrams are drawn for one chosen ratio C/B, close to the value found for the free ion of that configuration (typically between 4 and 5). This is the only real approximation in the construction, and it affects the spin-forbidden lines much more than the spin-allowed ones.

Laporte selection rule: in a molecule with a centre of symmetry, an electric-dipole transition is allowed only between states of opposite parity — g → u or u → g. Transitions g → g and u → u are forbidden. For a one-electron atomic transition the same rule reads Δl = ±1, so s → p and p → d are allowed while d → d and s → d are not.

Spin selection rule: ΔS = 0. A transition between states of different spin multiplicity is forbidden. The electric-dipole operator does not act on spin at all, so to a first approximation such a transition has no intensity whatever.

Charge-transfer (CT) band: an electronic transition in which an electron moves between an orbital that is mainly ligand-based and one that is mainly metal-based. Both orbitals belong to the same molecule, so this is an intramolecular redox process driven by light. Because the two orbitals have different parity and no spin change is required, CT bands are fully allowed and therefore intense: ε of 10³ to 10⁵.