Explain the difference between Orgel and Tanabe-Sugano diagrams?

Dec 30, 2017

Summary:

• Orgel diagrams show spin-allowed transitions only, not spin-forbidden. They also only show high-spin situations, and mesh together several ${d}^{n}$ cases into one diagram. These are separated out into one $D$ diagram and one $F / P$ diagram.

In general, these are less informative, and more qualitative. I find these harder to read, personally.

• Tanabe Sugano diagrams show both spin-allowed and spin-forbidden transitions. They also show both high-spin and low-spin situations, and are separated out into different diagrams for each ${d}^{n}$ case.

In general, these are way more informative, and more quantitative. I find these easier to read (although they take some brain power to think about), and way more useful.

We go through an example where we examine a transition of a ${t}_{2 g}$ electron to an ${e}_{g}^{\text{*}}$ orbital in "Ti"("H"_2"O")_6.

ORGEL DIAGRAMS (D, F/P)

Orgel diagrams are only for high-spin complexes, showing only the spin-allowed transitions relative to some parent ground state free-ion term ($P , D ,$ or $F$).

These are separated as the $D$ and $F / P$ Orgel diagrams:

General features of the diagram are:

• The horizontal axis is the size of the ligand field splitting energy ${\Delta}_{o}$. Apparently the $o$ is left off for "simplicity". The vertical axis is probably showing energy, but I see no increments...
• They are purely qualitative, and show only the states of highest spin multiplicity $2 S + 1$.
• They combine octahedral and tetrahedral cases together into one diagram for the same parent ground state free-ion term.

Also note that the gerade/ungerade labels are omitted for "simplicity", but apply to octahedral symmetries. This also applies to Tanabe Sugano diagrams.

Also, these are organized as follows:

• For the $D$ Orgel diagram, tetrahedral ${d}^{1} / {d}^{6}$ and octahedral ${d}^{9} / {d}^{4}$ on the left side, and tetrahedral ${d}^{4} / {d}^{9}$ and octahedral ${d}^{6} / {d}^{1}$ on the right side.
• For the $F / P$ Orgel diagram, tetrahedral ${d}^{2} / {d}^{7}$ and octahedral ${d}^{8} / {d}^{3}$ on the left side, and tetrahedral ${d}^{3} / {d}^{8}$ and octahedral ${d}^{7} / {d}^{2}$ on the right side.

If you notice, I purposefully wrote the correlations so that ${d}^{1}$ octahedral complexes correspond to ${d}^{9}$ tetrahedral complexes, for example.

That is described by a so-called "hole formalism" briefly illustrated below:

This will be useful later for Tanabe Sugano diagrams.

TANABE-SUGANO DIAGRAMS

Tanabe-Sugano diagrams show both low-spin and high-spin cases, and also show spin-forbidden transitions. They even show terms that are near to, but not the ground state!

Also, as previously mentioned, there is a hole formalism that says that a ${d}^{n}$ diagram for octahedral complexes applies to ${d}^{10 - n}$ tetrahedral complexes. So these diagrams are recyclable.

Overall, quite a bit more flexible than Orgel diagrams. Here is an example for ${d}^{4}$ octahedral complexes:

General features of the diagram are:

• The ground term (""^5 E_g for high-spin, ""^3 T_(1g) for low spin) is on the horizontal axis which plots ${\Delta}_{o} / B$, and a unitless energy scale $E / B$ relative to the ground state is on the vertical axis, where $B$ is a "Racah parameter" in units of ${\text{cm}}^{- 1}$.
• The vertical divide separates the weak-field and strong-field limits. To its left are the weak-field (high-spin) terms, and to its right are the strong-field (low-spin) terms.
• Solid curves are shown that are labeled by excited-state term symbols, quantitatively correlating with spin-allowed transitions.
• Dashed curves are shown that quantitatively correlate with spin-forbidden transitions.

Unlike Orgel diagrams, these are separated by ${d}^{n}$ configurations, rather than which ground state free-ion term. There are diagrams for ${d}^{2} \to {d}^{8}$ octahedral complexes, or ${d}^{8} \to {d}^{2}$ tetrahedral complexes.

There are many spin-forbidden transitions that we don't see in the Orgel diagram, and the only spin-allowed one is ${\text{^5 E_g -> }}^{5} {T}_{2 g}$ (solid curve to solid curve).

EXAMPLE: bb(""^5 D) FREE-ION GROUND TERM

As an example of a ""^5 D free-ion term for the metal center, take "Ti"("H"_2"O")_6 (we shall see why a ""^5 D). We can use this and compare how we interpret Orgel and Tanabe Sugano diagrams.

(Note that there is a shift from a ""^5 D free-ion term to a ""^5 E_g term when subjected to an octahedral field.)

• This has a $0$ oxidation state on $\text{Ti}$, giving a ${d}^{4}$ octahedral metal complex.
• The $\text{H"_2"O}$ are $\sigma$ donors and really weak $\pi$ donors, and thus are weak-field ligands.
• So, this must be a high-spin complex, and is an applicable choice.

The $d$ orbital configuration was originally:

$\underline{\uparrow \textcolor{w h i t e}{\downarrow}} \text{ "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" } \underline{\textcolor{w h i t e}{\uparrow \downarrow}}$

This ground state has a total orbital angular momentum of:

$\boldsymbol{L} = | {\sum}_{i} {m}_{l} \left(i\right) |$

$= | \left(- 2\right) + \left(- 1\right) + \left(0\right) + \left(1\right) | = \boldsymbol{2}$,

and $L = 2$ corresponds to $D$, just like $l = 2$ corresponds to $d$ orbitals.

It has a total spin angular momentum of:

$\boldsymbol{S} = | {\sum}_{i} {m}_{s} \left(i\right) |$

$= | \left(\frac{1}{2}\right) + \left(\frac{1}{2}\right) + \left(\frac{1}{2}\right) + \left(\frac{1}{2}\right) | = \boldsymbol{2}$,

giving a spin multiplicity of $2 S + 1 = 5$.

Thus, the ground term is color(blue)(""^5 D) and the $D$ Orgel diagram applies.

But remember, this is really a ""^5 E_g term in the octahedral field corresponding to this $d$ orbital splitting diagram (high spin ${d}^{4}$):

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

Based on the $D$ Orgel diagram, we expect that in an octahedral ligand field, it has one spin-allowed transition of the electron marked in $\textcolor{red}{\text{red}}$:

$\overbrace{\text{^5 E_g)^("Corresponds to """^5 D " free-ion term") -> overbrace(""^5 T_(2g))^("Corresponds to T"_2 " on left side of diagram}}$

So the excited state ""^5 T_(2g) is just this:

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

The spin multiplicity of $5$ is implied, because spin-allowed transitions preserve total spin $S$ and thus the spin multiplicity $2 S + 1$ as well.

In the Tanabe Sugano diagram, we look on the left side, the weak-field side, to guarantee a high-spin complex:

And we see ""^5 E on the horizontal axis that can connect vertically to the ""^5 T_2 solid curve. That's the same ${\text{^5 E_g -> }}^{5} {T}_{2 g}$ transition we saw earlier.

Except this time, we can say, for example, at ${\Delta}_{o} = 13 B$ ${\text{cm}}^{- 1}$, the transition requires $E = 13 B$ ${\text{cm}}^{- 1}$ of energy in the weak-field limit (this is where the dashed curve intersects with the solid curve).