Ligand Field Theory (LFT) and MO description (overview): Visual walkthrough

Before anything, three plain-word definitions we will lean on constantly:

Step 1 — The bare metal: five $d$ -orbitals, all at one energy

WHAT. Take a lone transition-metal ion, e.g. $Co^{3 +}$ , sitting alone. Its five $d$ -orbitals are just five differently-shaped clouds, and — floating free — they all sit at exactly the same energy. We call that state degenerate (equal energy).

WHY start here. We must know the "before" picture to measure any splitting against it. $Δ_{o}$ is a change from this flat baseline, so the baseline has to be drawn first.

PICTURE. Look at the five clouds below. Two of them ( $d_{z^{2}}$ and $d_{x^{2} - y^{2}}$ ) have lobes lying along the $x, y, z$ axes. The other three ( $d_{x y}, d_{y z}, d_{x z}$ ) have lobes lying between the axes, tucked into the diagonals. This "along vs between" difference is the single geometric fact the entire derivation hangs on — remember it.

Step 2 — Bring in 6 ligands along the axes

WHAT. Now surround the metal with 6 ligands placed at the ends of the axes: $+ x, - x, + y, - y, + z, - z$ . This octahedral ( $O_{h}$ ) arrangement is why the splitting is called $Δ_{o}$ ("o" for octahedral). Each ligand offers one σ-donor orbital — a lone pair pointing straight back at the metal along its axis.

WHY these positions. Six ligands on the axes are as far apart from each other as possible (least repulsion between them) — that is why octahedral geometry is so common. And crucially, their lone pairs lie exactly where the axis-pointing $d$ -orbitals live.

PICTURE. Each purple arrow below is a ligand lone pair aimed at the metal. Watch which metal lobes those arrows collide with:

Step 3 — Who overlaps? Axis-pointers win, gap-pointers get nothing

WHAT. Compare the two $d$ -families against the incoming lone pairs.

$d_{z^{2}}, d_{x^{2} - y^{2}}$ (axis-pointers) point straight into the ligand lone pairs → large overlap.
$d_{x y}, d_{y z}, d_{x z}$ (gap-pointers) point into the empty diagonals → the ligand lobes sail right past them → zero overlap.

WHY this matters. Overlap is measured by a number called the overlap integral:

$S = ⟨ d ∣ LGO ⟩$

$S$ — a single number scoring "how much these two clouds share space".
$⟨ ∣ ⟩$ — read as "multiply the two clouds together everywhere and add it all up".
$d$ — a metal $d$ -orbital cloud.
$LGO$ — a Ligand Group Orbital: the 6 ligand lone pairs, added together in a matching pattern (see Step 4).

The rule that runs everything: an MO forms only when $S \neq = 0$ . No overlap → no new bonding/antibonding pair → the orbital is left alone, i.e. non-bonding.

PICTURE. Green = strong overlap (axis-pointers meet lone pairs). Grey = zero overlap (gap-pointers miss entirely).

Recall Why exactly zero, not just "small"?

Question: The $d_{x y}$ lobe and a ligand lone pair aren't infinitely far apart — why is $S$ exactly zero, not merely tiny? ::: Because of symmetry. For every bit of positive overlap on one side of the $d_{x y}$ lobe, there is an equal bit of negative overlap (opposite lobe sign) on the other side. They cancel perfectly, so the sum is exactly $0$ — not an approximation, an exact cancellation.

Step 4 — Package the 6 lone pairs into symmetry-matched groups (LGOs)

WHAT. We don't feed the 6 lone pairs to the metal one by one. We first add them up in patterns that match the metal's orbital shapes. These patterned sums are the Ligand Group Orbitals (LGOs). In $O_{h}$ symmetry, the 6 σ-lone-pairs combine into exactly three symmetry flavours:

$1 LGO a_{1 g} + 2 LGOs e_{g} + 3 LGOs t_{1 u} (= 6 total)$

$a_{1 g}, e_{g}, t_{1 u}$ — symmetry labels: name-tags describing an orbital's shape/symmetry. Two orbitals can only bond if their name-tags match.
The subscript numbers just confirm we still have 6 LGOs, conserving our 6 lone pairs.

WHY bother. Symmetry is the matchmaker: only same-labelled partners bond.

Metal orbital	Its label	Matching ligand LGO?
$s$	$a_{1 g}$	✔ $a_{1 g}$
$p_{x}, p_{y}, p_{z}$	$t_{1 u}$	✔ $t_{1 u}$
$d_{z^{2}}, d_{x^{2} - y^{2}}$	$e_{g}$	✔ $e_{g}$ ← the key match
$d_{x y}, d_{y z}, d_{x z}$	$t_{2 g}$	✘ no σ-LGO has this label

PICTURE. The $e_{g}$ -labelled LGO pattern lines up lobe-for-lobe with the $d_{x^{2} - y^{2}}$ metal orbital — same "petals in the same places". The $t_{2 g}$ metal orbitals have no σ-LGO wearing their name-tag, so they are left partnerless.

Step 5 — Overlap builds the MO diagram: bonding down, antibonding up

WHAT. Take the matched pairs and apply "one in, two out". The metal $e_{g}$ orbitals and the $e_{g}$ LGO merge into:

a bonding $e_{g}$ (pushed down, mostly ligand character), and
an antibonding $e_{g}^{*}$ (pushed up, mostly metal character).

The $t_{2 g}$ metal orbitals, having no partner, stay put in the middle: non-bonding.

WHY up/down and why "mostly metal/ligand". When two orbitals of different starting energy mix, the new bonding orbital sits nearer the lower starter (the ligand lone pair, which starts low) and the antibonding sits nearer the higher starter (the metal $d$ , which starts higher). So $e_{g}^{*}$ inherits mostly metal character. This detail matters: the electrons we track (the metal's own $d$ -electrons) end up living in $t_{2 g}$ (non-bonding) and $e_{g}^{*}$ (antibonding) — the two levels straddling the gap.

PICTURE. Ligand $e_{g}$ starts low, metal $e_{g}$ starts higher; they split into $e_{g}$ (down) and $e_{g}^{*}$ (up). The lonely $t_{2 g}$ line floats untouched between them.

Step 6 — Read off $Δ_{o}$ : it's simply the frontier gap

WHAT. The two levels our $d$ -electrons occupy are $t_{2 g}$ (non-bonding) and $e_{g}^{*}$ (antibonding). The vertical distance between them is the splitting:

$Δ_{o} = E (e_{g}^{*}) - E (t_{2 g})$

$Δ_{o}$ — the octahedral splitting energy (an energy gap, in joules or cm $^{- 1}$ ).
$E (e_{g}^{*})$ — energy of the σ-antibonding level (the metal $d$ pushed up by the ligands).
$E (t_{2 g})$ — energy of the non-bonding level (essentially the original bare-metal $d$ energy).

WHY this is the deep upgrade over CFT. Crystal Field Theory (CFT) also gets a gap between $t_{2 g}$ and $e_{g}$ , but it credits it to electrostatic repulsion from point charges. LFT says: the gap is $e_{g}^{*}$ rising due to covalent σ-overlap. The better the overlap and energy match, the higher $e_{g}^{*}$ climbs, the bigger $Δ_{o}$ . That is a bonding explanation, not a repulsion fudge.

PICTURE. The green double-arrow between $t_{2 g}$ and $e_{g}^{*}$ is $Δ_{o}$ — literally a length you can measure on the diagram.

Step 7 — Turn on π: the $t_{2 g}$ orbitals finally get a partner (edge case)

WHAT. So far the ligands only offered head-on σ lone pairs, and $t_{2 g}$ sat idle. But many ligands also carry π-orbitals — clouds that point sideways, perpendicular to the bond axis. Sideways clouds have $t_{2 g}$ symmetry, so they match the gap-pointing $d$ -orbitals we abandoned in Step 3.

WHY this is a genuinely new case. Pure electrostatics (CFT) can never produce this — it has no notion of a sideways orbital match. This is the effect that lets LFT explain things CFT cannot. There are two sub-cases:

π-donor ( $F^{-}, Cl^{-}, O^{2 -}$ ): has filled, low-energy π orbitals. They donate up into metal $t_{2 g}$ , pushing $t_{2 g}$ UP → since $Δ_{o} = E (e_{g}^{*}) - E (t_{2 g})$ , raising $E (t_{2 g})$ shrinks $Δ_{o}$ . Weak field.
π-acceptor ( $CO, CN^{-}$ ): has empty, high-energy π* orbitals. Metal $t_{2 g}$ donates down into them (back-bonding), pulling $t_{2 g}$ DOWN → lowering $E (t_{2 g})$ grows $Δ_{o}$ . Strong field.

PICTURE. Left panel: filled π-donor pushes $t_{2 g}$ up, arrow closing the gap. Right panel: empty π*-acceptor pulls $t_{2 g}$ down, arrow opening the gap.

Step 8 — Filling electrons: the big-vs-small $Δ_{o}$ payoff (all cases)

WHAT. The size of $Δ_{o}$ decides how the $d$ -electrons fill $t_{2 g}$ and $e_{g}^{*}$ . There is a competition between two energies:

$Δ_{o}$ — cost of promoting an electron up to $e_{g}^{*}$ .
$P$ — the pairing energy: cost of forcing two electrons into the same orbital (they repel).

The electron takes the cheaper option:

$if Δ_{o} > P \Rightarrow pair up (low-spin); if Δ_{o} < P \Rightarrow spread out (high-spin)$

WHY this closes the loop. This is the whole reason chemists care about $Δ_{o}$ . For a $d^{6}$ ion like $Co^{3 +}$ :

$[Co(NH_{3})_{6}]^{3 +}$ : $NH_{3}$ is σ-donor with essentially no π → clean big $Δ_{o} > P$ → low-spin $t_{2 g}^{6} e_{g}^{* 0}$ , zero unpaired electrons.
$[CoF_{6}]^{3 -}$ : $F^{-}$ is a π-donor → $t_{2 g}$ pushed up → $Δ_{o} < P$ → high-spin $t_{2 g}^{4} e_{g}^{* 2}$ , four unpaired electrons.

PICTURE. Same $d^{6}$ count, two fillings, decided purely by whether the green gap beats $P$ .

The one-picture summary

Everything above compressed into a single frame: bare degenerate $d$ (Step 1) → σ-overlap raises $e_{g}^{*}$ and defines $Δ_{o}$ (Steps 5–6) → π-donors close it / π-acceptors open it (Step 7) → the gap decides high- vs low-spin and colour (Step 8).

Recall Feynman retelling — the whole walkthrough in plain words

Picture a metal atom's five "rooms" (the $d$ -orbitals), all on the same floor. Two rooms have windows facing straight down the corridors (axis-pointers); three have windows facing the corners between corridors (gap-pointers).

Now six friends (ligands) walk up the six corridors and knock on the metal's door with their hands out (lone pairs). The two rooms whose windows face the corridors get a strong handshake — that handshake lifts those rooms up to a new higher floor, and we star them ( $e_{g}^{*}$ ). The three corner-facing rooms get no handshake at all, so they stay on the original floor ( $t_{2 g}$ ).

The height between the untouched middle rooms and the lifted rooms is $Δ_{o}$ — that's the whole splitting.

Then a twist: some friends also reach sideways. If their hands are full and they push (π-donors like $F^{-}$ ), they lift the corner rooms up, shrinking the gap. If their hands are empty and they pull (π-acceptors like CO, "back-bonding"), they drag the corner rooms down, widening the gap. That is why boring neutral CO out-splits a charged fluoride — it opens the gap from both sides.

Finally, the electrons choose: if the gap is tall (bigger than the cost of crowding, $P$ ), they pair up and stay low (low-spin); if it's short, they spread out into the high rooms (high-spin). And when an electron hops the gap by swallowing a photon, that's the colour we see.

Recall Rebuild it yourself

One-line definition of $Δ_{o}$ in MO terms? ::: $Δ_{o} = E (e_{g}^{*}) - E (t_{2 g})$ . Why is $t_{2 g}$ non-bonding in a σ-only complex? ::: Its lobes point between the axes, so overlap $S$ with the axis-directed σ-LGOs is exactly zero (symmetry cancellation). A π-donor does what to $Δ_{o}$ ? ::: Raises $t_{2 g}$ → $Δ_{o}$ shrinks (weak field). A π-acceptor does what to $Δ_{o}$ ? ::: Lowers $t_{2 g}$ via back-bonding → $Δ_{o}$ grows (strong field). $d^{6}$ with $Δ_{o} > P$ gives which config? ::: Low-spin $t_{2 g}^{6} e_{g}^{* 0}$ , zero unpaired electrons.