Question

Explain the difference between Orgel and Tanabe-Sugano diagrams?

Answer 1

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_(2g)` electron to an `e_g^"*"` orbital in `"Ti"("H"_2"O")_6`.

For this answer, you should recall or read up on:

• Spectroscopic Selection Rules
• Crystal field theory
• High and low spin complexes
• Term symbols
• Character tables of octahedral (`O_h`) complexes

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DISCLAIMER: LONG ANSWER!

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 `2S+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 `"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 -> d^8` octahedral complexes, or `d^8 -> 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 `""^5 E_g -> ""^5 T_(2g)` (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 `"Ti"`, giving a `d^4` octahedral metal complex.
• The `"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:

`ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(color(white)(uarr darr))`

This ground state has a total orbital angular momentum of:

`bbL = |sum_i m_l(i)|`

`= |(-2) + (-1) + (0) + (1)| = bb2`,

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

It has a total spin angular momentum of:

`bbS = |sum_i m_s(i)|`

`= |(1/2) + (1/2) + (1/2) + (1/2)| = bb2`,

giving a spin multiplicity of `2S + 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 `color(red)("red")`:

`overbrace(""^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 `2S+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 `""^5 E_g -> ""^5 T_(2g)` transition we saw earlier.

Except this time, we can say, for example, at `Delta_o = 13B` `"cm"^(-1)`, the transition requires `E = 13B` `"cm"^(-1)` of energy in the weak-field limit (this is where the dashed curve intersects with the solid curve).