# What are high and low spin complexes in crystal field theory?

Nov 23, 2017

It just categorizes, qualitatively, how the metal $d$ orbitals are filled in crystal field theory after they are split by what the theory proposes are the ligand-induced electron repulsions.

The usual Hund's rule and Aufbau Principle apply.

Examples of low-spin ${d}^{6}$ complexes are ["Cr"("CN")_6]^(3-) and "Cr"("CO")_6, and examples of high-spin ${d}^{6}$ complexes are ${\left[{\text{CrCl}}_{6}\right]}^{3 -}$ and "Cr"("H"_2"O")_6.

I assume you know the basic facets of crystal field theory:

1. Ligands come in, and their important orbitals interact with the metal $d$ orbitals.
2. Electrons repel electrons to destabilize certain metal $d$ orbitals. In an octahedral field, these are known as the ${e}_{g}^{\text{*}}$ orbitals.
3. Electrons are attracted to the electropositive metal center to stabilize certain metal $d$ orbitals. In an octahedral field, these are known as the ${t}_{2 g}$ orbitals.

The crystal field splitting energy is called ${\Delta}_{o}$ in an octahedral field for simplicity, and the resultant $d$ orbital splitting is:

uarrE" "color(white)({(" "" "color(black)(ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "e_g^"*")),(color(black)(Delta_o)),(" "color(black)(ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "t_(2g))):})

Orbitals close in energy simultaneously fill more easily and vice versa. And so, depending on the magnitude of ${\Delta}_{o}$, there are two cases. Take a ${d}^{6}$ configuration as an example...

• When ${\Delta}_{o}$ is large, the complex is likely low-spin:

uarrE" "color(white)({(" "" "color(black)(ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "e_g^"*")),(),(),(),(),(color(black)(Delta_o)),(),(),(),(),(" "color(black)(ul(uarr darr)" "ul(uarr darr)" "ul(uarr darr)" "t_(2g))):})

• When ${\Delta}_{o}$ is small, the complex is likely high-spin:

uarrE" "color(white)({(" "" "color(black)(ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "e_g^"*")),(),(color(black)(Delta_o)),(),(" "color(black)(ul(uarr darr)" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "t_(2g))):})

Of course, I am exaggerating the energy scale, but hopefully that brings the point across.

• High spin complexes half-fill the lower energy $\boldsymbol{d}$ orbitals first, and THEN move up to the higher energy $d$ orbitals to half-fill those next, before pairing starts occurring, BECAUSE those orbitals are so similar in energy to the lower energy orbitals.
• Low spin complexes fill the lower energy orbitals completely first, before moving on to the higher energy orbitals, BECAUSE those orbitals are so much higher in energy.