Introductory courses on coordination chemistry traditionally introduce Crystal Field Theory as a useful model for simple interpretation of spectra and magnetic properties of firstrow transition metal complexes. In addition, Crystal Field Stabilisation Energy (CFSE) calculations are often used to explain the variation of their radii and various thermodynamic properties. Such calculations predict that for octahedral systems d^{3} and d^{8} should be the most stable and for tetrahedral systems, although always less stable than the corresponding octahedral systems, the d^{2} and d^{7} would be the most favourable.
A more detailed interpretation of spectra relies on the development of the concept of multielectron energy states and RussellSaunders coupling. Most textbooks [19] pictorially present the expected electronic transitions by the use of Orgel diagrams or TanabeSugano diagrams [10], or a combination of both. To this end, nearly all inorganic textbooks include TanabeSugano diagrams, often as an Appendix.
At UWI, we have used Orgel diagrams to cover highspin octahedral and tetrahedral configurations, except those with a d^{2} octahedral configuration or d^{5} ions (either stereochemistry). For d^{5}, no spinallowed transitions occur hence no Orgel diagram is possible and the TanabeSugano diagram is introduced to help interpret the spinforbidden bands. For d^{2} octahedral, where characterisation of ν2 and ν3 is made difficult since generally only 2 of the 3 expected transitions are observed and the lines due to ^{3}A_{2g} and ^{ 3}T_{1g}(P) cross, we have once again used a TanabeSugano diagram.
To make use of the TanabeSugano diagrams provided in textbooks for all configurations, it would be expected that they should at least be able to cope with typical spectra for d^{3}, d^{8} octahedral and d^{2}, d^{7} tetrahedral systems. This is not the case. The diagrams presented are impractical, being far too small. To make matters worse, the diagram for chromium(III) d^{3} systems is extremely limited (Δ/B range of 0  30) and for simple NH_{3} or acac complexes would require a small amount of extrapolation, whereas for the [Cr(CN)_{6}]^{3} ion, Δ/B corresponds to greater than 50! The final blow is that the diagrams for d^{5} and d^{6} in the original TanabeSugano paper were incorrect and these errors has been perpetuated in most textbooks.
TanabeSugano diagrams for nonoctahedral stereochemistry are generally not available, although for tetrahedral systems it is possible to use the d^{10n} octahedral diagrams if it is remembered that Δ is likely to be roughly half of the octahedral value and that the g subscripts should be dropped. Alternatively, spectral interpretation of cobalt(II) d^{7} tetrahedral systems can be done by reverting to Orgel diagrams. (Examples of d^{2} tetrahedral complexes are not very common.)
A set of UV/Vis spectra (in JCAMPDX format) as well as spreadsheets and JAVA applets giving the TanabeSugano diagrams are made available and a comparison of interpretation methods presented.
Note: Viewing the UV/Vis spectra requires the SUN Java plugin vs 1.5+: http://www.sun.com (this link opens in a new window). 
Crystal (and with extension, Ligand) Field Theory has proved to be an extremely simple but useful method of introducing the bonding, spectra and magnetism of firstrow transition metal complexes.
To quote from the Preface in the 1969 text by Schlafer and Gliemann[9]:"It is hardly possible today to discuss the chemistry of the transition metals, as offered in general lectures on inorganic chemistry, without employing ligand field theory. It is difficult to find a better example of how useful meaningfully chosen models can be for understanding a large body of exceedingly varied experimental results. Nevertheless, only comparatively few basic concepts are required for a first qualitative understanding of the theory".
In the interpretation of spectra, it is usual to start with an octahedral Ti^{3+} complex with a d^{1} electronic configuration. Crystal Field Theory predicts that because of the different spatial distribution of charge arising from the filling of the five dorbitals, those orbitals pointing towards bond axes will be destabilised and those pointing between axes will be stabilised.
The t_{2g} and e_{g} subsets are then populated from the lower level first which for d^{1} gives a final configuration of t_{2g}^{1} e_{g}^{0}.
The energy separation of the two subsets equals the splitting value Δ and ligands can be arranged in order of increasing Δ which is called the spectrochemical series and is essentially independent of metal ion.
For ALL octahedral complexes except high spin d^{5}, simple CFT would therefore predict that only 1 band should appear in the electronic spectrum corresponding to the absorption of energy equivalent to Δ. If we ignore spinforbidden lines, this is borne out by d^{1}, d^{9} as well as to d^{4}, d^{6}.
The observation of 2 or 3 peaks in the electronic spectra of d^{2}, d^{3}, d^{7} and d^{8} high spin octahedral complexes requires further treatment involving electronelectron interactions. Using the RussellSaunders (LS) coupling scheme, these free ion configurations give rise to F ground states which in octahedral and tetrahedral fields are split into terms designated by the symbols A_{2(g)}, T_{2(g)} and T_{1(g)}.
To derive the energies of these terms and the transition energies between them is beyond the needs of introductory level courses and is not covered in general textbooks[10,11]. A listing of some of them is given here as an Appendix. What is necessary is an understanding of how to use the diagrams, created to display the energy levels, in the interpretation of spectra.
Two types of diagram are available: Orgel and TanabeSugano diagrams.
A simplified Orgel diagram (not to scale) showing the terms arising from the splitting of an F state is given below. The spin multiplicity and the g subscripts are dropped to make the diagram more general for different configurations.
The lines showing the A_{2} and T_{2} terms are linear and depend solely on Δ. The lines for the two T_{1} terms are curved to obey the noncrossing rule and as a result introduce a configuration interaction in the transition energy equations.
The lefthand side is applicable to d^{3} , d^{8} octahedral complexes and d^{7} tetrahedral complexes. The righthand side is applicable to d^{2} , d^{7} octahedral complexes.
Looking at the d^{3} octahedral case first, 3 peaks can be predicted which would correspond to the following transitions:
Here C.I. represents the configuration interaction which is generally either taken to be small enough to be ignored or taken as a constant for each complex.
In the laboratory component of the course we measure the absorption spectra of some typical chromium(III) complexes and calculate the spectrochemical splitting factor, Δ. This corresponds to the energy found from the first transition above and as shown in Table 1 is generally between 15,000 cm^{1} and 27,000 cm^{1} depending on the type of ligand present.
Table 1. Peak positions for some octahedral Cr(III) complexes (in cm^{1}).
Complex

ν1

ν2

ν3

ν2/ν1

ν1/ν2

Δ/B

Ref

Cr^{3+} in emerald 
16260

23700

37740

1.46

0.686

20.4

13 
K_{2}NaCrF_{6} 
16050

23260

35460

1.45

0.690

21.4

13 
[Cr(H_{2}O)_{6}]^{3+} 
17000

24000

37500

1.41

0.708

24.5

This work 
Chrome alum 
17400

24500

37800

1.36

0.710

29.2

4 
[Cr(C_{2}O_{4})_{3}]^{3} 
17544

23866

? 
1.37

0.735

28.0

This work 
[Cr(NCS)_{6}]^{3} 
17800

23800

? 
1.34

0.748

31.1

4 
[Cr(acac)_{3}] 
17860

23800

? 
1.33

0.752

31.5

This work 
[Cr(NH_{3})_{6}]^{3+} 
21550

28500

? 
1.32

0.756

32.6

4 
[Cr(en)_{3}]^{3+} 
21600

28500

? 
1.32

0.758

33.0

4 
[Cr(CN)_{6}]^{3} 
26700

32200

? 
1.21

0.829

52.4

4 
For octahedral Ni(II) complexes the transitions would be:
where C.I. again is the configuration interaction and as before the first transition corresponds exactly to Δ.
For M(II) ions the size of Δ is much less than for M(III) ions (around 2/3) and typical values for Ni(II) are 6500 to 13000 cm^{1} as shown in Table 2.
Table 2. Peak positions for some octahedral Ni(II) complexes (in cm^{1}).
Complex

ν1

ν2

ν3

ν2/ν1

ν1/ν2

Δ/B

Ref

NiBr_{2} 
6800

11800

20600

1.74

0.576

5

13 
[Ni(H_{2}O)_{6}]^{2+} 
8500

13800

25300

1.62

0.616

11.6

13 
[Ni(gly)3]^{} 
10100

16600

27600

1.64

0.608

10.6

13 
[Ni(NH_{3})_{6}]^{2+} 
10750

17500

28200

1.63

0.614

11.2

13 
[Ni(en)_{3}]^{2+} 
11200

18350

29000

1.64

0.610

10.6

3 
[Ni(bipy)_{3}]^{2+} 
12650

19200

? 
1.52

0.659

17

3 
For d^{2} octahedral complexes, few examples have been published. One such is V^{3+} doped in Al_{2}O_{3} where the vanadium ion is generally regarded as octahedral, Table 3.
Table 3. Peak positions for an octahedral V(III) complex (in cm^{1}).
Complex

ν1

ν2

ν3

ν2/ν1

ν1/ν2

Δ/B

Ref

V^{3+} in Al_{2}O_{3} 
17400

25200

34500

1.45

0.690

30.8

13 
Interpretation of the spectrum highlights the difficulty of using the righthand side of the Orgel diagram above since none of the transitions correspond exactly to Δ and often only 2 of the 3 transitions are clearly observed.
The first transition can be unambiguously assigned as:
^{3}T_{2g} ← ^{ 3}T_{1g} transition energy = 4/5 * Δ + C.I.But, depending on the size of the ligand field (Δ) the second transition may be due to:
^{3}A_{2g} ← ^{ 3}T_{1g} transition energy = 9/5 * Δ + C.I.for a weak field (Δ/B' < 13.5) or
^{3}T_{1g}(P) ← ^{ 3}T_{1g} transition energy = 3/5 * Δ + 15B' + 2 * C.I.for a strong field (Δ/B' > 13.5).
The transition energies of these terms are clearly different and it is often necessary to calculate (or estimate) values of B, Δ and C.I. for both arrangements and then evaluate the answers to see which fits better.
The difference between the ^{3}A_{2g} and the ^{3}T_{2g} (F) lines is equivalent to Δ so in this case Δ is equal to either:Solving the equations like this for the three unknowns can ONLY be done if ALL three transitions are observed. When only two transitions are observed, a series of equations[14] have been determined that can be used to calculate both B and Δ. Note though that the paper ignores the fact that the identification of the lines changes after the crossing over point so that Δ is then ν3ν1 and not ν2ν1. This approach therefore still requires some evaluation of the numbers to ensure a valid fit. For this reason, TanabeSugano diagrams become a better method for interpreting spectra of d^{2} octahedral complexes.
The first obvious difference to the Orgel diagrams shown in general textbooks is that TanabeSugano diagrams are calculated such that the ground term lies on the Xaxis, which is given in units of Δ/B. The second is that spinforbidden terms are shown and third that lowspin complexes can be interpreted as well, since for the d^{4}  d^{7} diagrams a vertical line is drawn separating the high and low spin terms.
The procedure used to interpret the spectra of complexes using TanabeSugano diagrams is to find the ratio of the energies of the second to first absorption peak and from this locate the position along the Xaxis from which Δ/B can be determined. Having found this value, then tracing a vertical line up the diagram will give the values (in E/B units) of all spinallowed and spinforbidden transitions.
N.B. Another approach has been to use the inverse of this ratio, ie of the first to second transition and so both values are recorded in the Tables.
As an example, using the observed peaks found for [Cr(NH_{3})_{6}]^{3+} in Table 1 above then, from the JAVA applet described below, D/B' is found at 32.6. The E/B' for the first transition is given as 32.6 from which B' can be calculated as 661 cm^{1}. The third peak can then be predicted to occur at 69.64 * 661 = 46030 cm^{1} or 217 nm (well in the UV region and probably hidden by charge transfer or solvent bands).
For the V(III) example treated previously using an Orgel diagram, the value of Δ/B' determined from the appropriate JAVA applet is around 30.8.
Following the vertical line upwards leads to the assignment of the first transition to ^{3}T_{2g} ← ^{3}T_{1g} and the second and third to ^{3}T_{1g} (P) ← ^{3}T_{1g} (blue line) and ^{ 3}A_{2g} ← ^{ 3}T_{1g} (green line) respectively.
The average value of B' calculated from the three Yintercepts is ~600 cm^{1} giving Δ a value of roughly 18600 cm^{1}, significantly larger than the 17100 cm^{1} calculated above and shows the sort of variation expected from these methods.
It is important to remember that the width of many of these peaks is often 12,000 cm^{1} so as long as it is possible to assign peaks unambiguously, the techniques are valuable.
To overcome the problem of small diagrams, it was decided to generate our own TanabeSugano diagrams using spreadsheets. This was done, using the transition energies given in the Appendix, for the spinallowed transitions and using CAMMAG to generate the energies of the spinforbidden transitions.
Even so, the method of finding the correct Xintercept is somewhat tedious and timeconsuming and a different approach was devised using JAVA applets.
The JAVA applets display the spinallowed and lowlying spinforbidden transitions and when the user clicks on any region of the graph then the values of ν_{2}/ν_{1} and ν_{3}/ν_{1} are displayed. In addition, the values of Δ/B and the Yintercepts are given as well. This simplifies the process of determining the best fit for Δ/B.
The expected ranges for the ratio of ν_{2}/ν_{1} are:
These ratios show the need for a certain degree of precision in attempting to analyse the spectra, especially for d^{7} and d^{8}. It has been suggested that instead of using ν_{2}/ν_{1} that any two ratios can be used and graphs of these plots were produced by Lever in the 1960's[11]. Once again though the published diagrams are rather small and so the spreadsheets above contain these charts which can be printed in larger scale. The slopes of the various ratio lines vary greatly and it is useful to examine the region of interest first before deciding on which set of lines should be used for analysis. If only 2 peaks are observed then this is not an option.
Further information for use in laboratory classes is available and some example calculations are available as well.
Transitions calculated for spinallowed terms in the TanabeSugano diagrams.
Octahedral d^{3} (e.g. Chromium(III) ).
^{4}T_{2g} ← ^{ 4}A_{2g} , ν_{1}/B= Δ/B
^{4}T_{1g}(F) ← ^{4}A_{2g}, ν_{2}/B= ½{15 + 3( Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) }
^{4}T_{1g}(P) ← ^{4}A_{2g}, ν_{3}/B= ½{15 + 3( Δ/B) + √(225  18(Δ/B) + ( Δ/B)^{2} ) }
from this, the ratio ν_{2}/ν_{1} would become:
½{15 + 3( Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) } / Δ/B
and the range of Δ/B required is from ~15 to ~55
Octahedral d^{8} (e.g. Nickel(II) ).
^{3}T_{2g} ← ^{ 3}A_{2g},_{ } ν1/B= Δ/B
^{3}T_{1g}(F) ← ^{3}A_{2g}, ν2/B= ½{15 + 3(Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) }
^{3}T_{1g}(P) ← ^{3}A_{2g}, ν3/B= ½{15 + 3(Δ/B) + √(225  18(Δ/B) + ( Δ/B)^{2} ) }
from this the ratio ν_{2}/ν_{1} would become:
½{15 + 3( Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) } / Δ/B
and the range of Δ/B required is from ~5 to ~17
Octahedral d^{2} (e.g. Vanadium(III) ).
^{3}T_{2g} ← ^{ 3}T_{1g} , ν1/B= ½{(Δ/B)  15 + √(225 + 18(Δ/B) + ( Δ/B)^{2} ) }
^{3}T_{1g}(P) ← ^{3}T_{1g}, ν2/B= √(225 + 18(Δ/B) + (Δ/B)^{2} )
^{3}A_{2g} ← ^{ 3}T_{1g}, ν3/B= ½{ 3 (Δ/B) 15 + √(225 + 18(Δ/B) + ( Δ/B)^{2} ) }
from this the ratio ν_{2}/ν_{1} would become:
√(225 + 18(Δ/B) + ( Δ/B)^{2} ) / ½{( Δ/B)  15 + √(225 + 18(Δ/B) + ( Δ/B)^{2} ) }
and the range of Δ/B required is from ~15 to ~35
Tetrahedral d^{7} (e.g. Cobalt(II) ).
^{4}T_{2} ← ^{ 4}A_{2}, ν1/B= Δ/B
^{4}T_{1} (F) ← ^{4}A_{2}, ν2/B= ½{15 + 3(Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) }
^{4}T_{1} (P) ← ^{4}A_{2}, ν3/B= ½{15 + 3(Δ/B) + √(225  18(Δ/B) + ( Δ/B)^{2} ) }
from this the ratio ν_{2}/ν_{1} would become:
½{15 + 3(Δ/B)  √(225  18(Δ/B) + ( Δ/B)^{2} ) } / Δ/B
and the range of Δ/B required is from ~3 to ~8.
To the students in my C21J class who have unwittingly helped formulate my ideas on how to teach this material, I am deeply grateful.
Thanks are due as well to Christopher Muir and Debbie Facey for help in developing the JAVA applets.
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