Electronic Spectra
This is the part that turns a coloured solution into a number. It builds the free-ion term symbols from first principles by counting microstates, follows each term into the ligand field, and then uses Orgel and Tanabe–Sugano diagrams to assign the bands of a real spectrum and extract Δo and the Racah parameter B from them. It also answers the questions students rarely get answered: why Mn(II) salts are almost colourless, why permanganate is intensely purple without having any d electrons to excite, and why a ‘forbidden’ transition happens at all. Two layers on every section: a slow, hand-held beginner path and a research-grade advanced/reference path.
The 10 sections in Part 5
- 1Electron–electron repulsion: the term nobody drew Free below
- 2Microstates and Russell–Saunders terms
- 3Ground terms, Hund’s rules and the full term tables
- 4How each term splits in an octahedral field
- 5Orgel diagrams
- 6Tanabe–Sugano diagrams
- 7Selection rules and the intensity of a band
- 8Charge transfer, Racah B and the nephelauxetic effect
- 9Capstone — extracting Δo and B from a measured spectrum
- 10Where this leads — Part 6
Electron–electron repulsion: the term nobody drew
Free extractSection G.1 of Part 5, reproduced in full from the book — figures and all. No sign-in, no paywall on this section.
Why two electrons in the ‘same’ configuration can have different energies, and how big that difference is compared with Δo.
Take the simplest possible case. A gaseous Ti³⁺ ion has one 3d electron. There are five d orbitals, so there are ten places to put it counting spin — five orbitals × two spin orientations. All ten are the same energy, because in a free spherical ion the five d orbitals are degenerate and nothing distinguishes spin up from spin down. One configuration, one energy. Fine.
Now take gaseous V³⁺, which has two 3d electrons. Again the five orbitals are degenerate, so you might expect one energy again. But the two electrons see each other. Consider two ways of placing them:
- both electrons in the same orbital (say dxy), necessarily with opposite spins. They occupy the same region of space and repel one another strongly.
- the two electrons in different orbitals with parallel spins — say one in dxy and one in dxz. They are in different regions of space and the Pauli principle keeps same-spin electrons apart, so they repel one another weakly.
Same configuration — 3d² in both cases — but different repulsion energies, and the difference is not small. For a first-row M²⁺/M³⁺ ion the spread of energies within one dn configuration is of the order of 10⁴–10⁵ cm⁻¹ — tens of thousands of wavenumbers — which is the same order as Δo itself. That is why neither effect can be treated as a small correction to the other, and why the resulting theory (Tanabe–Sugano) has to handle both at once.
orbital (one-electron) energies ≈ 10⁵ cm⁻¹ ≫ electron–electron repulsion ≈ 10⁴–10⁵ cm⁻¹ ≈ ligand field Δo ≈ 10⁴ cm⁻¹ ≫ spin–orbit coupling ≈ 10²–10³ cm⁻¹
That inequality chain is the justification for the whole procedure that follows. Because repulsion is much larger than spin–orbit coupling for 3d ions, we build states by first coupling all the orbital angular momenta together and all the spins together, and only afterwards let the two couple to each other. That is the Russell–Saunders or LS coupling scheme, and it is a good approximation for the first transition series. (For 4d and especially 5d metals, and for the lanthanides and actinides, the assumption weakens — see the advanced layer.)
The four-step picture
The Hamiltonian, the two coupling schemes, and where the Russell–Saunders approximation fails.
For an n-electron atom or ion in a ligand field the electronic Hamiltonian, in the Born–Oppenheimer and fixed-core approximations, is
The first sum is one-electron and spherically symmetric; it fixes the configuration and contributes nothing to the splitting within a configuration. The second sum is the interelectron repulsion — the term that Parts 3 and 4 silently absorbed into an average. The third is spin–orbit coupling. The fourth is the ligand field. The hierarchy of these last three decides which scheme to use:
| Regime | Ordering | Scheme | Applies to |
|---|---|---|---|
| Russell–Saunders (LS) | repulsion ≫ spin–orbit | couple all l to give L and all s to give S, then L·S gives J | light atoms; the first (3d) transition series — a good approximation |
| Intermediate coupling | repulsion ≈ spin–orbit | diagonalise both together; L and S stop being good quantum numbers | 4d and 5d metals, and the actinides |
| jj coupling | spin–orbit ≫ repulsion | couple each electron’s own l and s to give j, then couple the j | very heavy elements; rarely needed at this level |
| Weak ligand field | repulsion ≫ VLF | free-ion terms first, then split them — the Orgel approach | high-spin first-row complexes with weak-field ligands |
| Strong ligand field | VLF ≫ repulsion | orbital configurations (t2gxegy) first, then apply repulsion | low-spin complexes, cyanides, second- and third-row metals |
It is worth being explicit about what ‘a good approximation’ means for the first row. Russell–Saunders coupling is not exact even for Ti³⁺; L and S are only approximately good quantum numbers because the spin–orbit operator does not commute with L² or S². What justifies it is that ζ3d is a few hundred cm⁻¹ while the term separations are tens of thousands, so the mixing coefficients are small — small enough that term labels remain meaningful, but not zero, which is exactly the loophole that lets ‘spin-forbidden’ bands appear at all (G.7).
Why the repulsion cannot be averaged away
A natural objection: surely the interelectron repulsion is largely a spherically symmetric screening effect, which just shifts the whole configuration? Partly, yes — and that spherically symmetric part is exactly the Racah parameter A, which appears in the energy of every term of a configuration with the same coefficient and therefore cancels out of every transition energy. It is the non-spherical remainder, parameterised by B and C, that splits the configuration into terms. This is why spectroscopists never quote A: it is unmeasurable from a spectrum.
Formally, the repulsion integrals are expanded in Legendre polynomials, giving Slater–Condon parameters F⁰, F², F⁴ (or their scaled versions F0, F2, F4) for a d shell. Racah’s A, B, C are the linear combinations chosen so that the algebra of dn term energies comes out as simply as possible:
The practical consequence is that two numbers, B and C, describe all the interelectron repulsion in a dn free ion, and B alone describes the separation of terms of maximum multiplicity. For the spin-allowed spectra that dominate this part, B is usually the only repulsion parameter you need — C enters only when spin-forbidden transitions are involved.
Read the rest of Part 5
The remaining 9 sections of this part — Microstates and Russell–Saunders terms, Ground terms, Hund’s rules and the full term tables, How each term splits in an octahedral field, Orgel diagrams — and all nine parts of Coordination Chemistry are part of ChemVidya Full Access, along with the other books, 55 Study Notes and 6,000+ practice questions.
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