Ligand field theory

d-orbitals and f-orbitals as they fall out of the math of the Schrödinger equation
are not in match with the real situation.
Other methods like

• crystal field theory and
• ligand field theory

are used instead.

Usefulness

Tuning colors for gem-gum display technology by choice of

• d-block transition element
• ligands (forced in place via piezomechanosynthesis)
• coordination of those ligands (and background framework, see: Organometallic gemstone-like compounds)
• levels of applied stress/strain (going beyond just electrooptical conversion; possible even active actuation)

And both passive and active colors.

• For the passive side related is: Color emulation
• For the active side related is: mechanooptical conversion

Furthermore:

• Tuning for strong ferromagnetism?
• Accelerating inter system crossing for energy efficient piezomechanosynthesis?
• Having some fun with spins.

How it works

Electron pairing energy ... from electrostatic repulsion due to double-up in an orbital
Smaller orbital => electrons closer together => higher electron pairing energy => high spin more likely

The eight electron d-orbital-block (block … same energy electron states)
due to interaction in the crystal gets split up into two blocks

• the t_2g six electron block (d_xy, d_yz, d_zy)
• the e_g two electron block (d_x²-y², d_z²)

Octahedral coordination: energy of an electron in e_g > energy of an electron in t_2g
Tetrahedral coodination: crystal field splitting energy: vice versa.

Octahedral splitting energy … energy of e_g electron minus energy of t_2g electron
LCOAO-energies … linear combination of atomic orbitals
ESR … electron spin resonance

Weak field ligand and/or smaller metal orbital
Octahedral splitting energy < total electron pairing energy

• e_g block starts filling before any electrons occupy t_2g is fully occupied
• electrons in t_2g block don't get paired up before moving up to the higher energy e_g block (i.e. Hund's rule for all d orbitals)
  

=> high spin complex
oversimplified: red side absorption blue side looking material
more complex but higher signal ESR spectra (spin-spin coupling between multiple unpaired electrons).

Strong field ligand and/or bigger metal orbital
Octahedral splitting energy > total electron pairing energy
– t_2g block (lower energy) gets completely filled first before any electrons occupy e_g
– all electrons in t_2g do get paired up before moving up to the higher energy e_g block
=> low spin complex oversimplified: blue side absorption red side looking material
cleaner but lower signal ESR spectra

Influenced properties:

• electron configuration and thus
• magnetic behavior: diamagnetic (all electrons paired) / paramagnetic (& how many unpaired)
• frequency (sometimes color) of absorbed light

Influencing factors:

• strength of ligand due to chemical identity
  (O,S,halogen, … weaker; C,N,P, … stronger)
• oxidation state
• coordination number and coordination geometry
  tetrahedral => ligands between orbitals => weak field => high spin tendentially
  square planar coordination: d_x²-y² > d_z² > d_xy > (d_yz & d_zx)

spectrochemical series

• For the ligands
• For the metals?

(wiki-TODO: check if there are tables for spectrochemical series that go beyond just randomly deeming lists)

Details

Metal orbitals: – Get reducible representation.
– Decompose reducible representation into irreducible representations
(the ones listed in character table for the relevant pointgroup)
– Construct molecular orbital type diagram for the metal complex of given geometry

Lewis basic ligands (lone pairs facing the metal): – Use projection operator, decompose representation
– factor separate lewis pairs of electrons into symmetries
if there are metal orbitals and ligand orbitals that have the same symmetry => interaction spossible

Linear combination of atomic orbitals (LCOAO): – Mix in phase and out of phase (bonding and antibonding respectively).
How far to shift? The strength of an interaction is inversly proportionsal to
the seperation in energy between the fragments.
=> Farther apart less shift. Noninteracting remnants blocks stay unshifted (nonbonding orbitals)

– Fill up all the low energy bonding orbitals with the ligand electrons.
(The block in the middle energies is dominated by metal d character.)
– Fill up these mid level energy orbitals.
For d⁰ ions nothing changes. For d¹⁰ all is filled up.
Side-note: filling up non-bonding orbitals does not break the bonds (but influences reactivity).
Interesting it gets for the middle fillup levels d⁴ to d⁷ for octahedral coordination.

Side-note:

• Two electrons in HOMO of ligand (lewis base)
• Two holes in LUMO of the metal (lewis acid)

Because its a lewis-acid lewis-base interaction (kinda like an electron deficiency bond)
In the bonding state most of the electron density contribution comes from the ligand little from the metal

• Bonding: big contribution from ligands small contribution from metal
• Antionding: big contribution from the metal small contribution from ligands

Geometries

• octahedral: common
• tetrahedral: less direct ligand approach => less level splitting => tends to be high spin
• planar square: …

• assign symmetry to an orbital … lower case
• assign symmetry to an electronic state … upper case

Frontier atomic orbitals for octahedral coordination:

• _ _ _ _ _ 3d _ 4s _ _ _ 4p … Sc-Zn
• _ _ _ _ _ 4d _ 5s _ _ _ 5p … Y-Cd
• _ _ _ _ _ 5d _ 6s _ _ _ 6p … La-Hg
• _ _ _ _ _ nd _ (n+1)s _ _ _ (n+1)p … generally

lanthanides & actinides: bonding is dominated by electrostatic interactions
f orbitals generally don't spit, don't participate in bonding, stay at about the same energy

Comparison to crystal field theory

Crystal field theory only treats electrostatic point charges and does not take into account covalency of bonds.
Ligand field theory: Combine Crystal field theory with molecular orbital theory

Crystal field theory alone: – relative shifts: center of energy analogous to center of mass – electron pairing energy not taken under consideration?

Related

• Color centers
• Atomic orbitals
• Fun with spins
• Electronic transitions – Inter system crossing

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• Optical effects – see "photonic steampunk" section
• Color emulation – Passive color gemstone display – Active color gemstone display

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See: Organometallic gemstone-like compound – Tuning gemstone color:

• either ultra precisely with Kaehler brackets
• or with fast programmable active pressure adjusting actuation

External links

Wikibook:

• Coordination Chemistry and Crystal Field Theory

Wikipedia:

• Spectrochemical series
• Ligand field theory
• Crystal field theory

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• Jahn–Teller effect

(wiki-TODO: there are some quite good video series on ligand field theory on youtube - add links to those)

Video: Professor Dave Explains – Crystal Field Theory