## Advantages of Tanabe-Sugano Diagrams over Orgel Diagrams

In Orgel diagrams, the magnitude of the splitting energy exerted by the ligands on d orbitals, as a free ion approach a ligand field, is compared to the electron-repulsion energy, which are both sufficient at providing the placement of electrons. However, if the ligand field splitting energy, 10Dq, is greater than the electron-repulsion energy, then Orgel diagrams fail in determining electron placement. In this case, Orgel diagrams are restricted to only high spin complexes.^{[6]}

Tanabe-Sugano diagrams do not have this restriction, and can be applied to situations when 10Dq is significantly greater than electron repulsion. Thus, Tanabe-Sugano diagrams are utilized in determining electron placements for high spin and low spin metal complexes. Like Orgel diagrams, Tanabe-Sugano diagrams has limitation. That is, it has only qualitative significance. Nonetheless, Tanabe-Sugano diagrams are useful in interpreting UV-vis spectra and determining the value of 10Dq.

^{[6]}

## Applications: Tanabe-Sugano Diagrams as a Qualitative Tool

In a centrosymmetric ligand field, such as in octahedral complexes of transition metals, the arrangement of electrons in the d-orbital is not only limited by electron repulsion energy, but it is also related to the splitting of the orbitals due to the ligand field. This leads to many more electron configuration states than is the case for the free ion. The relative energy of the repulsion energy and splitting energy defines the high-spin and low-spin states.Considering both weak and strong ligand fields, a Tanabe-Sugano diagram shows the energy splitting of the spectral terms with the increase of the ligand field strength. It is possible for us to understand how the energy of the different configuration states is distributed at certain ligand strengths. The restriction of the spin selection rule makes it is even easier to predict the possible transitions and their relative intensity. Although they are qualitative, Tanabe-Sugano diagrams are very useful tools for analyzing UV-vis spectra: they are used to assign bands and calculate Dq values for ligand field splitting.

### Examples

#### Manganese(II) Hexahydrate

In the [Mn(H_{2}O)

_{6}]

^{2}metal complex, manganese has an oxidation state of +2, thus it is a d

^{5}ion. H

_{2}O is a weak field ligand (spectrum shown below), and according to the Tanabe-Sugano diagram for d

^{5}ions, the ground state is

^{6}A

_{1}. Note that there is no sextet spin multiplicity in any excited state, hence the transitions from this ground state are expected to be spin-forbidden and the band intensities should be low. From the spectra, only very low intensity bands are observed (low Molar absorptivity (ε) values on y-axis).

#### Cobalt(II) Hexahydrate

#### Another example is [Co(H

_{2}O)

_{6}]

^{2+}.Note that the ligand is the same as the last example. Here the cobalt ion has the oxidation state of +2, and it is a d

^{7}ion. From the high-spin (left) side of the d

^{7}Tanabe-Sugano diagram, the ground state is

^{4}T

_{1}(F), and the spin multiplicity is a quartet. The diagram shows that there are three quartet excited states:

^{4}T

_{2},

^{4}A

_{2}, and

^{4}T

_{1}(P). From the diagram one can predict that there are three spin-allowed transitions. However, the spectra of [Co(H

_{2}O)

_{6}]

^{2+}does not show three distinct peaks that correspond to the three predicted excited states. Instead, the spectrum has a broad peak (spectrum shown below). Based on the T-S diagram, the lowest energy transition is

^{4}T

_{1}to

^{4}T

_{2}, which is seen in the near IR and is not observed in the visible spectrum. The main peak is the energy transition

^{4}T

_{1}(F) to

^{4}T

_{1}(P), and the slightly higher energy transition (the shoulder) is predicted to be

^{4}T

_{1}to

^{4}A

_{2}. The small energy difference leads to the overlap of the two peaks, which explains the broad peak observed in the visible spectrum.

###
Using Tanabe-Sugano diagrams to solve for B and Δ_{O}

For the d^{2}complex [V(H

_{2}O)

_{6}]

^{3+}, two bands are observed with maxima at around 17,500 and 26,000 cm

^{−1}.

^{[citation needed]}The ratio of experimental band energies is E(ν

_{2})/E(ν

_{1}) is 1.49. There are three possible transitions expected, which include: ν

_{1}:

^{3}T

_{1g}→

^{3}T

_{2g}, ν

_{2}:

^{3}T

_{1g}→

^{3}T

_{1g}(P), and ν

_{3}:

^{3}T

_{1g}→

^{3}A

_{2g}. There are three possible transitions, but only two are observed, so the unobserved

transition must be determined.

Δ_{O} / B = |
10 | 20 | 30 | 40 |
---|---|---|---|---|

Height E(ν_{1})/B |
10 | 19 | 28 | 37 |

Height E(ν_{2})/B |
23 | 33 | 42 | 52 |

Height E(ν_{3})/B |
19 | 38 | 56 | 75 |

Ratio E(ν_{3})/E(ν_{1}) |
1.9 | 2.0 | 2.0 | 2.0 |

Ratio E(ν_{2})/E(ν_{1}) |
2.3 | 1.73 | 1.5 | 1.4 |

_{O}/ B. Then find the ratio of these values (E(ν

_{2})/E(ν

_{1}) and E(ν

_{3})/E(ν

_{1})). Note that the ratio

of E(ν

_{3})/E(ν

_{1}) does not contain the calculated ratio for the experimental band energy, so we can determine that the

^{3}T

_{1g}→

^{3}A

_{2g}band is unobserved. Use ratios for E(ν

_{2})/E(ν

_{1}) and the values of Δ

_{O}/ B to

plot a line with E(ν

_{2})/E(ν

_{1}) being the y-values and Δ

_{O}/B being the x-values. Using this line, it is possible to determine the value of Δ

_{O}/ B for the experimental ratio. (Δ

_{O}/ B = 31 for a chart ratio of 1.49 in this example).

Find on the T-S diagram where Δ

_{O}/ B = 31 for

^{3}T

_{1g}→

^{3}T

_{2g}and

^{3}T

_{1g}→

^{3}T

_{1g}(P). For

^{3}T

_{2g}, E(ν

_{1}) / B = 27 and for

^{3}T

_{1g}(P), E(ν

_{2}) / B = 43.

The Racah parameter can be found by calculating B from both E(ν

_{2}) and E(ν

_{1}). For

^{3}T

_{1g}(P), B = 26,000 cm

^{−1}/43 = 604 cm

^{−1}. For

^{3}T

_{2g}, B = 17,500 cm

^{−1}/ 27 = 648 cm

^{−1}. From the average value of the Racah parameter, the ligand field splitting parameter can be found (Δ

_{O}). If Δ

_{O}/ B = 31 and B = 625 cm

^{−1}, then Δ

_{O}= 15,800 cm

^{−1}.

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