Ligand Field Theory (LFT) and MO description (overview): Foundations

Before you can read the parent note, you need to earn every symbol it throws at you. Below, each item is: plain words → the picture → why the topic needs it. Read top to bottom; every rung stands on the one below it.

1. An orbital — the shape a lone electron lives in

Picture it. Think of a fuzzy balloon of "where the electron probably is." An $s$ -orbital is a round ball. A $p$ -orbital is a dumbbell (two lobes). The ones we care about most are $d$ -orbitals — they have four lobes (a cloverleaf) or a dumbbell with a doughnut.

Why the topic needs it. The entire subject is about what happens to the metal's five $d$ -orbitals when ligands approach. If "orbital" means nothing to you, nothing else can.

Figure — Ligand Field Theory (LFT) and MO description (overview)

2. The five $d$ -orbitals and where their lobes point

There are exactly five $d$ -orbitals. Their names tell you which direction their lobes stick out:

Name	What the lobes do
$d_{x y}$	four lobes lying between the $x$ and $y$ axes
$d_{y z}$	four lobes between the $y$ and $z$ axes
$d_{x z}$	four lobes between the $x$ and $z$ axes
$d_{x^{2} - y^{2}}$	four lobes pointing straight along the $x$ and $y$ axes
$d_{z^{2}}$	a dumbbell along the $z$ axis with a ring around the middle

3. Ligand — the friend the metal holds hands with

Picture it. A ligand is a hand reaching toward the metal, carrying a lone pair of electrons like a gift.

Why the topic needs it. LFT is entirely about the metal–ligand relationship: whose orbitals point where, and how they mix.

4. Octahedral geometry and the $O_{h}$ label

Picture it. Put the metal at the centre of a room. One ligand on each wall, one on the ceiling, one on the floor. That's octahedral. (Connect the six ligand positions and you get an eight-faced solid — an octahedron.)

Why the topic needs it. Because the six ligands sit exactly on the axes, the $d_{z^{2}}$ and $d_{x^{2} - y^{2}}$ lobes (which point along axes) crash into them, while $d_{x y}$ etc. (which point between) slip past. That's why "along vs between" from §2 matters here specifically.

Picture it. Two flashlight beams. Aimed at each other → beams overlap strongly. Aimed at right angles → they never cross, overlap is zero.

Why the topic needs it. A chemical bond only forms when overlap is nonzero. So:

$d_{z^{2}}, d_{x^{2} - y^{2}}$ (aim at ligands) → big overlap → they bond.
$d_{x y}, d_{y z}, d_{x z}$ (aim between) → zero σ-overlap → they don't (in the σ-only story).

That single fact is why the $d$ -orbitals split at all.

6. Molecular Orbitals — bonding, antibonding, non-bonding

When two orbitals overlap, they don't just sit there — they merge into new shared orbitals called Molecular Orbitals (MOs). See Molecular Orbital Theory for the full machinery. Two orbitals in give exactly two MOs out:

Picture it. Two identical waves. Crest-on-crest → bigger wave (bonding). Crest-on-trough → they flatten (antibonding). A wave with no partner → stays exactly as it was (non-bonding).

Why the topic needs it. LFT is MO theory applied to complexes. The famous energy gap is literally the distance between two MO levels: a non-bonding set and an antibonding set.

7. Symmetry labels: $a_{1 g}$ , $e_{g}$ , $t_{1 u}$ , $t_{2 g}$

These strange labels are just tags telling you how an orbital behaves under the octahedron's symmetry. You do not need the group theory to read the parent note; you only need the matching rule.

Picture it. Think of labels as coloured wristbands at a dance. You may only partner someone wearing your colour. A metal $e_{g}$ orbital can only pair with a ligand combination also wearing the $e_{g}$ band.

Why the topic needs it. The parent note keeps saying " $e_{g}$ interacts, $t_{2 g}$ doesn't." That's just: the ligands' σ-donor combination wears an $e_{g}$ band (found a partner) but there's no σ-donor combination wearing a $t_{2 g}$ band — so $t_{2 g}$ sits out the σ-dance and stays non-bonding.

8. Ligand Group Orbitals (LGOs)

Picture it. Six people humming. Instead of six voices, listen for the chords they form together — one bundle where all six hum in phase ( $a_{1 g}$ ), others with mixed phases ( $e_{g}$ , $t_{1 u}$ ). Each chord is an LGO.

Why the topic needs it. MOs form between the metal orbital and the matching LGO chord, not between the metal and one lone ligand. The metal $e_{g}$ finds an $e_{g}$ LGO chord to marry — producing bonding $e_{g}$ and antibonding $e_{g}^{*}$ .

9. σ-bonds vs π-bonds — head-on vs sideways

Picture it. σ = two people shaking hands directly. π = two people high-fiving with hands passing side-by-side.

Why the topic needs it. The magic of LFT over Crystal Field Theory (CFT) is π: sideways-pointing ligand orbitals wear the $t_{2 g}$ wristband — the very band the metal $t_{2 g}$ wears! So once π enters, $t_{2 g}$ finally gets a partner and moves up or down. See Back-bonding and π-Acceptor Ligands.

A cloud-shaped region where one electron is likely to be found.

Name the two

d

-orbitals whose lobes point along the axes.

d_{z^{2}}

and

d_{x^{2} - y^{2}}

(the

e_{g}

pair).

Name the three

d

-orbitals whose lobes point between the axes.

d_{x y}, d_{y z}, d_{x z}

(the

t_{2 g}

trio).

Where do the 6 ligands sit in an octahedral (

O_{h}

) complex?

Along

\pm x, \pm y, \pm z

— one on each axis end.

When is orbital overlap

S

large, and when is it zero?

Large when lobes point at each other; zero when they miss (e.g.

t_{2 g}

with σ-ligands).

What three kinds of MO can form from overlap?

Bonding (lower), antibonding (higher, starred), and non-bonding (unchanged, no partner).

What does a symmetry label like

e_{g}

t_{2 g}

do for you?

It is a wristband — only orbitals with the same label may combine into an MO.

What are LGOs?

Symmetric bundles ("chords") of the six ligand donor orbitals, spanning

a_{1 g} + e_{g} + t_{1 u}

Difference between σ and π overlap?

σ is head-on along the bond; π is sideways above/below the bond.

Write

Δ_{o}

as an MO gap.

Δ_{o} = E (e_{g}^{*}) - E (t_{2 g})

When does an electron avoid pairing and jump up instead?

When

Δ_{o} < P

(the gap is smaller than the pairing penalty).

Relate

Δ_{o}

to absorbed light.

Δ_{o} = h ν = h c / λ

— bigger gap, shorter absorbed wavelength.

Ready? Then head back to the parent overview — every symbol there is now yours.

Foundations — Ligand Field Theory (LFT) and MO description (overview)

1. An orbital — the shape a lone electron lives in

2. The five $d$ -orbitals and where their lobes point

3. Ligand — the friend the metal holds hands with

4. Octahedral geometry and the $O_{h}$ label

6. Molecular Orbitals — bonding, antibonding, non-bonding

7. Symmetry labels: $a_{1 g}$ , $e_{g}$ , $t_{1 u}$ , $t_{2 g}$

8. Ligand Group Orbitals (LGOs)

9. σ-bonds vs π-bonds — head-on vs sideways

10. The splitting gap $Δ_{o}$

11. Pairing energy $P$ , and the light equation

Prerequisite map

Equipment checklist