Foundations — Crystal Field Theory (CFT) — Δ_oct, Δ_tet, splitting diagrams

This page assumes you have seen none of the vocabulary in the parent note. We build every symbol from the ground up, in an order where each piece rests only on pieces already built. Read top to bottom. See the parent: CFT topic note.

1. What is a "metal ion"? (the star of the show)

Picture: a small dense positive ball at the centre. Everything in this topic happens around this ball.

Why the topic needs it: the positive ion is what pulls in ligands, and its leftover outer electrons (the ones we call $d$ electrons, defined in §4) are the ones whose energy we track. No ion → nothing to split.

Reading the notation. ${\text{Fe}}^{2+}$: the number is how many electrons were removed, the $+$ says removed (not added). ${\text{CN}}^{-}$ later will use $-$ meaning one extra electron.

2. What is a "ligand"? (the point negative charge)

Picture — Figure s01: the ion is the accent-red centre dot; each ligand is a small black negative sign placed at a fixed position around it, with an arrow showing the repulsive push it sends inward. That "keep-away" negativity is the only property CFT keeps. Notice in the figure: the ligands are evenly spaced — no orbital is singled out yet; the splitting only appears once we ask which orbital lobe lines up with an arrow.

Why the topic needs it: electrons repel electrons. The ligand's negative charge pushes on the metal's electron clouds. Which clouds get pushed hardest depends only on where the ligands sit — that is the whole mechanism.

3. What is an "orbital"? (a shape, not a path)

Picture: think of a balloon (or two balloons joined at the centre) drawn around the ion. Where the balloon is fat, the electron spends most of its time.

Why the topic needs it: repulsion between a ligand and an electron is strongest where the electron actually is. So the direction the balloon points decides how much the ligand pushes it. This is the single most important idea on the page.

4. The five $d$ orbitals and their shapes (name each balloon)

There are exactly five $d$ orbitals (the five shapes of the $d$ family from §3). Their odd-looking names are just labels for which way the balloons point.

Reading the names. The subscript tells you the axes involved:

• $d_{xy}$ lives in the $xy$-plane, pointing between the $x$ and $y$ axes (at 45°).
• $d_{x^2-y^2}$ points along the $x$ and $y$ axes themselves.
• $d_{z^2}$ points along the $z$ axis (with a small doughnut around the middle).

Picture — Figure s02: the left panel (red lobes) shows the "along-axis" family aligned with the black axis lines; the right panel shows the "between-axis" family sitting in the 45° gaps. Compare the two panels side by side: the only difference is whether a red lobe overlaps a black axis line — that overlap is what a ligand on the axis will attack.

Why the topic needs it: split the five into "point along axes" vs "point between axes" and you have split them into the two energy groups. Everything downstream is bookkeeping on these two families.

5. Energy, and drawing it as height (the vertical axis)

Picture: a vertical arrow labelled "energy". Orbitals are short horizontal lines placed at their height. Electrons are little arrows sitting on those lines.

Why the topic needs it: "splitting" literally means one horizontal line breaks into two lines at different heights. Without the height convention the word "splitting" is meaningless.

6. The barycentre — the "average" reference line (height zero)

Picture — Figure s04: the dashed line is the barycentre at height 0; the lower group ($t_{2g}$) sits below it, the upper group ($e_g$) above it, and the total "amount of line below" exactly balances the "total amount above".

Why the topic needs it: every energy quoted later (like $-0.4\Delta$) is measured from this line. Without it, "$-0.4\Delta$" has no origin.

7. The Greek letter $\Delta$ (delta) — the energy gap

Picture: the arrow spanning the gap between the lower group of lines and the upper group in your energy diagram.

• ${\Delta }_{oct}$ — the gap when 6 ligands sit in an octahedral arrangement.
• ${\Delta }_{tet}$ — the gap in a tetrahedral arrangement (smaller — quantified in §12).

Why the topic needs it: $\Delta$ is the one number CFT computes. Colour, spin, and stability are all read off from it.

Deriving the octahedral split values $-0.4\Delta$ and $+0.6\Delta$

We now earn the numbers the parent uses. Two facts do all the work:

1. The octahedral split gives a lower group of 3 orbitals ($t_{2g}$, points between → dropped) and an upper group of 2 ($e_g$, points at ligands → raised).
2. The barycentre rule from §6: the five shifts, measured from the average line, must sum to zero.

Let the lower group each drop by $x$ (a positive number, so each sits at $-x$) and the upper group each rise by $y$ (each at $+y$). The gap between them is ${\Delta }_{oct}$: $x+y={\Delta }_{oct}$

Now impose the zero-sum rule — 3 orbitals at $-x$, 2 orbitals at $+y$: $3(-x)+2(+y)=0\Rightarrow 3x=2y$

Substitute $x={\Delta }_{oct}-y$: $3({\Delta }_{oct}-y)=2y\Rightarrow 3{\Delta }_{oct}=5y\Rightarrow y=\frac{3}{5}{\Delta }_{oct},x=\frac{2}{5}{\Delta }_{oct}$

8. Geometry words: octahedral & tetrahedral (where the ligands sit)

Picture — Figure s03: the left panel puts the red ligands on the black axes (octahedral); the right panel puts them in the gaps between axes (tetrahedral). Trace which orbital family from Figure s02 each set of red dots lines up with — that is the whole reason the energy order flips.

Why the topic needs it: the positions of ligands decide which orbitals get pushed. Octahedral ligands sit on axes → they attack the "point-along-axis" orbitals ($d_{x^2-y^2},d_{z^2}$). Tetrahedral ligands sit off-axis → they attack the "between" orbitals. That is why the energy order flips between the two geometries.

9. The group labels: $e_g$, $t_{2g}$, $e$, $t_2$

Why no $g$ in tetrahedral? A tetrahedron has no centre of inversion, so we drop the $g$: the labels become $e$ and $t_2$. (This is the parent's steel-manned mistake — the $g$ is not decorative.)

Why the topic needs it: these are the words the splitting diagram is written in. "$t_{2g}^4{e}_g^2$" is unreadable until you know $t_{2g}$ = 3 lower orbitals and $e_g$ = 2 upper ones.

10. Filling rules: electrons, pairing energy $P$, and spin

Why the topic needs it: every electron entering the split diagram faces a choice — climb the gap $\Delta$ to an empty upper orbital, or pay $P$ to pair in a lower one. Whichever is cheaper wins. That contest ($\Delta$ vs $P$) is exactly high-spin vs low-spin.

11. Reading a configuration like $d^6$ or $t_{2g}^4{e}_g^2$, and CFSE

• $d^6$: six $d$ electrons to place.
• $t_{2g}^4{e}_g^2$: 4 electrons sit in the lower ($t_{2g}$) group, 2 in the upper ($e_g$) group.

Why those exact heights are legal to use here: §7 derived $-0.4\Delta$ and $+0.6\Delta$ purely from the barycentre rule, so they are already measured from the height-zero line — exactly the reference CFSE needs.

Why the topic needs it: CFSE is the numeric payoff — it ranks which complexes are extra stable.

12. Why ${\Delta }_{tet}$ is smaller — the $\frac{4}{9}$ ratio

13. Edge case — square-planar geometry (don't forget it exists)

14. Colour symbols: $h$, $\nu$, $\lambda$, $c$, and the Planck relation

Why the topic needs it: without $E=h\nu$ there is no reason $\Delta$ (an energy) should have anything to do with $\lambda$ (a length). Planck's relation is the bridge.

Prerequisite map

See also how these feed forward into Magnetic Properties — Spin-only Formula μ = √(n(n+2)), Colour & d–d Transitions, Spectrochemical Series & Ligand Strength, Stability of Complexes & CFSE, and contrast with Valence Bond Theory of Complexes and Jahn–Teller Distortion. Naming of the complexes comes from Coordination Compounds — Nomenclature.

Equipment checklist

Cover the right side and answer aloud; reveal to check.

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