This document discusses using Orgel and Tanabe-Sugano diagrams to calculate crystal field splitting energies (10 Dq) and Racah parameters (B) for various d-block metal complexes. It provides examples of calculating these values for d1, d2, d7, and d9 systems using observed absorption bands and relationships between the transitions. The values obtained are then used to determine 10 Dq, B, and the nephelauxetic ratio β for complexes such as [Ti(H2O)6]3+, [V(H2O)6]3+, and [Co(H2O)6]2+.

1. Calculations using
Orgel and Tanabe-Sugano Diagrams
V.SANTHANAM
Department of Chemistry
SCSVMV

2. Konig’s Method for Dq and B values when ground state is T

3. Konig’s Method for Dq and B values when ground state is A

4. d1 -system

5. ▪ Metal complexes with d1configuration do not have any inter electronic
repulsion .
▪ The single electron resides in the t2g orbital ground state.
▪ When t2g orbital set holds the single electron, six microstates
will have 2T2g state energy of -4 Dq and when the electron is promoted
to the eg orbital, the four microstates will have 2Eg state energy of +6
Dq.
▪ The only parameter to be calculated is the magnitude of crystal field
splitting energy (10 Dq).
▪ The single absorption band in a UV-vis experiment is exactly what
we are looking for.
▪ The energy of the transition 2T2g → 2Eg gives the value of Δ directly .
▪ The electronically degenerate t2g
1 configuration will undergo Jahn-Teller
distortion and it produces a shoulder in the peak.

6. ▪ Consider the example of [Ti(H2O)6]3+
▪ Calculation of B: No need to calculate the Racah parameter since
there is only one electron.
▪ Calculation of Δo: The purple color of the complex ion [Ti(H2O)6]3+
is due to a broad absorption band at 20300 cm-1 arising from 2T2g →
2Eg transition.
▪ Hence, 10 Dq for this complex is 20300 cm-1.
▪ Calculation of β: No need to calculate the nephelauxetic ratio, since
only one electron is there.

7. d9 system
▪ In d9 octahedral metal complexes, the ground state filling of
electrons (t2g
6 eg
3) has only four microstates that have 2Eg
energy state with -6Dq.
▪ When the electron from t2g is promoted to the eg orbital set
,the new configuration will have six microstates that have
2T2g energy state with +4 Dq.
▪ This could also be described as a positive "hole" that moves
from the eg to the t2g orbital set.

8. d9 system
▪ The sign of Dq is opposite that for d1, with a 2Eg ground state and a
2T2g excited state.
▪ Like the d1 case, the only parameter that is needed to be calculated in
d9 complexes is the magnitude of crystal field splitting energy (10 Dq)
▪ The single absorption band in a UV-vis experiment is exactly what we
are looking for. Hence, the energy of the transition 2Eg → 2T2g gives
the value of Δ directly.

9. ▪ Consider the example of [Cu(H2O)6]2+
▪ Calculation of B: No need to calculate the Racah parameter
▪ Calculation of Δo: In the UV-visible spectra of [Cu(H2O)6]2+the
broad band at 12000 cm-1
▪ This is due to spin allowed 2Eg → 2T2g transition; and hence,
10 Dq for this complex is 12000 cm-1.
▪ Calculation of β: No need to calculate the nephelauxetic ratio.

10. d2- system

11. ▪ Metal complexes with d2-configuration have 3F ground state
term symbol in the absence of any crystal field.
▪ However, when six ligands approach in octahedral coordination,
the ground state term symbol becomes 3T1g and remains as
such in weak as well as in strong ligand fields.
▪ The Orgel and Tanabe-Sugano diagram for d2-configuration can
be used to estimate the value of crystal field splitting energy
for these transition metal complexes.
Example:
[V(H2O)6]3+
V3+ contains two d electrons

12. Calculation of B
▪ From the Orgel diagram, it can be clearly seen that the ground state
for d2-octahedral complexes is 3T1g and there are three main transitions
before the crossover point.
▪ Moreover, it is worthy to note down that the order of second and
third transitions is reversed after the crossover point and only two
bands will be observed at or near the crossover point.
▪ As the magnitude of the crystal field splitting energy increases,
the 3T1g(F) and 3T1g(P) states repel each other more and more with a
magnitude of x energy value.

13. ▪ It is known that
ν1 = 8 Dq+x ------- (1)
ν2 = 18 Dq+x ------- (2)
ν3 = 15 B + 6Dq + 2x --------(3)
(2) – (1) = (ν2- ν1) = 10Dq --------(4)
(2) + (1) = (ν2+ ν1) = 26Dq +2x ---------(5)
Substituting the value of 2x from (5) in (3) we get,
ν3 = 15 B + 6Dq + (ν2+ ν1)-26 Dq --------(6)
ν3 = 15B -20Dq + (ν2+ ν1) --------(7)
substituting the value of 10Dq from (4)
ν3 = 15B -2(ν2- ν1) + (ν2+ ν1)---------(8)
ν3 = 15B +3ν1 - ν2
B = (ν3 + ν2 - 3ν1) / 15 ---------- (9)

14. How to use the Tanabe –Sugano diagrams?
For a d2 system three transitions are
possible.
ν1 = 3T1g→ 3T2g, - 17800 cm-1
ν2 = 3T1g → 3T1g(P) -39600 cm-1
ν3 = 3T1g → 3A2g – 25700 cm-1
Take the ratio v3/v1
25700 / 17800 = 1.44
Now in the diagram we have to
find the Dq/B ratio at which v3/v1
= 1.44
It gives the energies of transitions
form which B value can be
calculated.

15. d2-system
▪ For a d2 system, [V(H2O)6]3+ is an ideal example.
▪ Two bands are observed.
▪ Third one may be too energetic, obscured by the CT bands.
▪ The first band is assigned easily to 3T1g 3T2g which is at 17800 cm-1
▪ If there is no term interaction i.e. x=0, then
ν1
3T1g 3T2 g = 8Dq+x =17800 - 8Dq = 17800
Dq = 2225 cm-1
▪ Wkt ν2 = 18Dq + x = 18Dq = 39600 cm-1
▪ So the other band is appearing at 25700 cm-1 cannot be ν2 it should be ν3

16. Using Tanabe-Sugano diagram for d2 system – Lever’s Method
▪ From the Tanabe-Sugano diagram, in the UV-visible spectra of
[V(H2O)6]3+, two bands are observed with maxima at around
17800 and 25700 cm-1.
▪ There are three possible transitions expected, which include:
– ν1 = 3T1g→ 3T2g,
– ν2 = 3T1g → 3T1g(P)
– ν3 = 3T1g → 3A2g
– but only two are observed , since ν2 is obscured by CT bands.
▪ Now taking the ratio
ν3/ ν1 = 25700 / 17800 = 1.44

17. `` v2/v1 v3/v1 v3/v2 v3/B
0.1 2.234 19.279 8.629 15.62
0.2 2.22 9.93 4.473 16.279
0.3 2.207 6.828 3.094 16.971
0.4 2.195 5.288 2.408 17.692
0.5 2.185 4.37 2 18.439
0.6 2.175 3.763 1.73 19.209
0.7 2.167 3.333 1.538 20
0.8 2.159 3.014 1.396 20.809
0.9 2.151 2.768 1.286 21.633
1 2.145 2.572 1.199 22.472
1.1 2.138 2.414 1.129 23.324
1.2 2.133 2.283 1.071 24.187
1.3 2.127 2.173 1.022 25.06
1.4 2.123 2.08 0.98 25.942
1.5 2.118 2 0.944 26.833
1.6 2.114 1.93 0.913 27.731
1.7 2.11 1.869 0.866 28.636
1.8 2.106 1.816 0.862 29.547
1.9 2.103 1.768 0.841 30.463
2 2.099 1.725 0.822 31.385
2.1 2.096 1.687 0.805 32.311
2.2 2.093 1.652 0.789 33.243
2.3 2.091 1.621 0.775 34.176
2.4 2.088 1.592 0.762 35.114
2.5 2.086 1.566 0.751 36.056
`` v2/v1 v3/v1 v3/v2 v3/B
2.6 2.083 1.542 0.74 37
2.7 2.081 1.519 0.73 37.947
2.8 2.079 1.499 0.721 38.897
2.9 2.077 1.48 0.713 39.85
3 2.075 1.462 0.705 40.804
3.1 2.073 1.466 0.697 41.761
3.2 2.072 1.431 0.691 42.72
3.3 2.07 1.416 0.684 43.681
3.4 2.068 1.403 0.678 44.643
3.5 2.067 1.39 0.673 45.607
3.6 2.066 1.378 0.667 46.573
3.7 2.064 1.367 0.662 47.539
3.8 2.063 1.357 0.658 48.508
3.9 2.062 1.347 0.653 49.477
4 2.06 1.337 0.649 50.448
4.1 2.059 1.328 0.645 51.42
4.2 2.058 1.32 0.641 52.393
4.3 2.057 1.312 0.638 53.367
4.4 2.056 1.304 0.634 54.342
4.5 2.055 1.297 0.631 55.317
4.6 2.024 1.29 0.628 56.294
4.7 2.053 1.283 0.625 57.271
4.8 2.052 1.277 0.622 58.249
4.9 2.051 1.271 0.619 59.228
5 2.05 1.625 0.617 60.208

18. v2/v1
v3/v1
v3/v2
Chart Title
ν3/ν1 = 25700 / 17800 = 1.443
From the diagram,
ν3/ν1 = 1.443 corresponds to Dq/B = 3.2
ν3/B against 3.2 is 42.72
ν3 /B’ = 42.72
B’ = ν3 /42.72
B’ = 25700 /42.72
B’ = 601.59 cm-1
Dq/B’ = 3.2
Dq = 3.2 * 601.59
Dq = 1925cm-1
10Dq = 19250cm-1
β = 601.59 / 860 = 0.699

19. • From the Tanabe-Sugano
diagram we can read the ν1/B’
or ν2/B’ or ν3/B’ from which
the B’ value can be calculated.
•Knowing the value of B’, Dq
and x values can be computed.

20. d7- system
▪ Example: [Co(H2O)6]2+
▪ Ground term 4T1g
▪ Expected transitions three
– 4T1g  4T2g v1 = 8000 cm-1
– 4T1g  4A2g v2 = 19600 cm-1
– 4T1g  4T1g (P) v3 = 21600 cm-1

21. `` v2/v1 v3/v1 v3/v2 v3/B
0.1 2.234 19.279 8.629 15.62
0.2 2.22 9.93 4.473 16.279
0.3 2.207 6.828 3.094 16.971
0.4 2.195 5.288 2.408 17.692
0.5 2.185 4.37 2 18.439
0.6 2.175 3.763 1.73 19.209
0.7 2.167 3.333 1.538 20
0.8 2.159 3.014 1.396 20.809
0.9 2.151 2.768 1.286 21.633
1 2.145 2.572 1.199 22.472
1.1 2.138 2.414 1.129 23.324
1.2 2.133 2.283 1.071 24.187
1.3 2.127 2.173 1.022 25.06
1.4 2.123 2.08 0.98 25.942
1.5 2.118 2 0.944 26.833
1.6 2.114 1.93 0.913 27.731
1.7 2.11 1.869 0.866 28.636
1.8 2.106 1.816 0.862 29.547
1.9 2.103 1.768 0.841 30.463
2 2.099 1.725 0.822 31.385
2.1 2.096 1.687 0.805 32.311
2.2 2.093 1.652 0.789 33.243
2.3 2.091 1.621 0.775 34.176
2.4 2.088 1.592 0.762 35.114
2.5 2.086 1.566 0.751 36.056
`` v2/v1 v3/v1 v3/v2 v3/B
2.6 2.083 1.542 0.74 37
2.7 2.081 1.519 0.73 37.947
2.8 2.079 1.499 0.721 38.897
2.9 2.077 1.48 0.713 39.85
3 2.075 1.462 0.705 40.804
3.1 2.073 1.466 0.697 41.761
3.2 2.072 1.431 0.691 42.72
3.3 2.07 1.416 0.684 43.681
3.4 2.068 1.403 0.678 44.643
3.5 2.067 1.39 0.673 45.607
3.6 2.066 1.378 0.667 46.573
3.7 2.064 1.367 0.662 47.539
3.8 2.063 1.357 0.658 48.508
3.9 2.062 1.347 0.653 49.477
4 2.06 1.337 0.649 50.448
4.1 2.059 1.328 0.645 51.42
4.2 2.058 1.32 0.641 52.393
4.3 2.057 1.312 0.638 53.367
4.4 2.056 1.304 0.634 54.342
4.5 2.055 1.297 0.631 55.317
4.6 2.024 1.29 0.628 56.294
4.7 2.053 1.283 0.625 57.271
4.8 2.052 1.277 0.622 58.249
4.9 2.051 1.271 0.619 59.228
5 2.05 1.625 0.617 60.208

22. v3/v1 = 2.7 from the above table Dq/B” = 0.9
=> v3/B’ = 21.633
B’ = v3/21.633
= 21600 / 21.633
B’ = 998 cm-1
It is known that Dq/B’ = 0.9
So Dq = 0.9 x 998 = 898.2 cm-1
10Dq = 898.2 x 10 = 8982 cm-1
β = 898.2 / 1120 = 0.802

23. Dq/B
v2/v1
v3/v1
v3/v2
v3/B Dq/B
When the ground term is A

24. Dq/B v2/v1 v3/v1 v3/v2 v3/B
0.1 1.799 16.211 9.062 16.21
0.2 1.777 8.273 4.909 17.446
0.3 1.764 6.236 3.535 18.708
0.4 1.75 5 2.857 20
0.5 1.735 4.265 2.458 21.325
0.6 1.719 3.781 2.199 22.685
0.7 1.702 3.44 2.021 24.083
0.8 1.685 3.19 1.893 25.521
0.9 1.667 3 1.8 27
1 1.648 2.852 1.731 28.521
1.1 1.629 2.735 1.679 30.083
1.2 1.61 2.64 1.64 31.685
1.3 1.59 2.563 1.612 33.325
1.4 1.57 2.5 1.591 35
1.5 1.553 2.447 1.576 36.708
1.6 1.535 2.403 1.566 38.446
1.7 1.517 2.365 1.559 40.211
1.8 1.5 2.333 1.556 42
1.9 1.484 2.306 1.554 43.81
2 1.468 2.282 1.552 45.639
2.1 1.453 2.261 1.556 47.485
2.2 1.439 2.243 1.559 49.346
2.3 1.425 2.227 1.563 51.22
2.4 1.412 2.213 1.567 53.105
2.5 1.4 2.2 1.571 55
Dq/B v2/v1 v3/v1 v3/v2 v3/B
2.6 1.388 2.189 1.576 56.904
2.7 1.377 2.178 1.582 58.817
2.8 1.367 2.169 1.587 60.736
2.9 1.356 2.161 1.593 62.662
3 1.347 2.153 1.599 65.493
3.1 1.338 2.146 1.604 66.53
3.2 1.329 2.14 1.61 68.471
3.3 1.321 2.134 1.616 70.416
3.4 1.313 2.128 1.621 72.365
3.5 1.305 2.123 1.627 74.318
3.6 1.298 2.119 1.632 76.273
3.7 1.291 2.114 1.638 78.232
3.8 1.284 2.11 1.643 80.192
3.9 1.278 2.107 1.648 82.155
4 1.272 2.103 1.653 84.121
4.1 1.266 2.1 1.658 86.088
4.2 1.261 2.097 1.663 88.057
4.3 1.255 2.094 1.668 90.028
4.4 1.25 2.091 1.673 92
4.5 1.245 2.088 1.677 93.974
4.6 1.24 2.086 1.682 95.949
4.7 1.236 2.084 1.686 97.925
4.8 1.231 2.081 1.69 99.902
4.9 1.227 2.079 1.695 101.881
5 1.223 2.077 1.699 103.86

25. Systems with A2 ground state- d3 system
▪ Example: [Cr(H2O)6]3+
▪ Ground term is 4A2g
▪ Three transitions expected
– 4A2g  4T2g (v1) = 10Dq
– 4A2g  4T1g (F) (v2) = 18Dq-x
– 4A2g  4T1g (P) (v3) = 12Dq+15B’+x
▪ v1 occurs at 17,400 cm-1
▪ v2 occurs at 24,600 cm-1
▪ v3 occurs at 37,800 cm-1

26. ▪ Considering v2/v1 = 24600 / 17400 = 1.413
▪ From the table ,
for v2/v1 = 1.413 , Dq/B ‘= 2.4
▪ From the d3 Tanabe-Sugano diagram,
– For Dq/B’ = 2.4 v1/B’ = 25
B’ = 696 cm-1
▪ If v2/v1 is used B’ value should be calculated by
using v1/B’ or v2/B’ for consistency.
▪ If v3/v2 is used v3/B’ should be used to calculate B’
▪ Dq/B’ = 2.4
▪ Dq = 2.4 x 696 = 1670 cm-1
▪ 10Dq = 16700 cm-1
▪ From v2 = 18Dq –x, x is calculated as 6270 cm-1
Dq/B v2/v1 v3/v1 v3/v2 v3/B
0.1 1.799 16.211 9.062 16.21
0.2 1.777 8.273 4.909 17.446
0.3 1.764 6.236 3.535 18.708
0.4 1.75 5 2.857 20
0.5 1.735 4.265 2.458 21.325
0.6 1.719 3.781 2.199 22.685
0.7 1.702 3.44 2.021 24.083
0.8 1.685 3.19 1.893 25.521
0.9 1.667 3 1.8 27
1 1.648 2.852 1.731 28.521
1.1 1.629 2.735 1.679 30.083
1.2 1.61 2.64 1.64 31.685
1.3 1.59 2.563 1.612 33.325
1.4 1.57 2.5 1.591 35
1.5 1.553 2.447 1.576 36.708
1.6 1.535 2.403 1.566 38.446
1.7 1.517 2.365 1.559 40.211
1.8 1.5 2.333 1.556 42
1.9 1.484 2.306 1.554 43.81
2 1.468 2.282 1.552 45.639
2.1 1.453 2.261 1.556 47.485
2.2 1.439 2.243 1.559 49.346
2.3 1.425 2.227 1.563 51.22
2.4 1.412 2.213 1.567 53.105
2.5 1.4 2.2 1.571 55

27. d8 system – [Ni(H2O)6]2+
•Example: [Ni(H2O)6]2+
•Ground term is 3A2g
• Three transitions are observed
 3A2g  3T2g (v1) = 10Dq
 3A2g  3T1g (F) (v2) = 18Dq-x
 3A2g  3T1g (P) (v3) = 12Dq+15B’+x
 v1 occurs at 8700cm-1
 v2 occurs at 14500 cm-1
 v3 occurs at 25300 cm-1

28. v3 /v1 = 25300/8700 = 2.9
From the table Dq/B = 0.95
v3/B = 27.755
B= 915 cm-1
Dq/B = 0.95
So Dq = 866 cm-1
10 Dq = 8660 cm-1
x = 1088 cm-1
β = 915 / 1080
= 0.8472
Dq/B v2/v1 v3/v1 v3/v2 v3/B
0.1 1.799 16.211 9.062 16.21
0.2 1.777 8.273 4.909 17.446
0.3 1.764 6.236 3.535 18.708
0.4 1.75 5 2.857 20
0.5 1.735 4.265 2.458 21.325
0.6 1.719 3.781 2.199 22.685
0.7 1.702 3.44 2.021 24.083
0.8 1.685 3.19 1.893 25.521
0.9 1.667 3 1.8 27
1 1.648 2.852 1.731 28.521
1.1 1.629 2.735 1.679 30.083
1.2 1.61 2.64 1.64 31.685
1.3 1.59 2.563 1.612 33.325
1.4 1.57 2.5 1.591 35
1.5 1.553 2.447 1.576 36.708
1.6 1.535 2.403 1.566 38.446
1.7 1.517 2.365 1.559 40.211
1.8 1.5 2.333 1.556 42
1.9 1.484 2.306 1.554 43.81
2 1.468 2.282 1.552 45.639
2.1 1.453 2.261 1.556 47.485
2.2 1.439 2.243 1.559 49.346
2.3 1.425 2.227 1.563 51.22
2.4 1.412 2.213 1.567 53.105
2.5 1.4 2.2 1.571 55