INTERPRETATION OF THE SPECTRA OF FIRST-ROW TRANSITION METAL COMPLEXES (textbook problems)

Introduction

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 first-row 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 Ti³⁺ complex with a d¹ electronic configuration. Crystal Field Theory predicts that because of the different spatial distribution of charge arising from the filling of the five d-orbitals, 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 d¹ gives a final configuration of t_2g¹ e_g⁰.

The energy separation of the two subsets equals the splitting value D and ligands can be arranged in order of increasing D which is called the spectrochemical series and is essentially independent of metal ion.

For ALL octahedral complexes except high spin d⁵, simple CFT would therefore predict that only 1 band should appear in the electronic spectrum corresponding to the absorption of energy equivalent to D. If we ignore spin-forbidden lines, this applies to d¹, d⁹ as well as to d⁴, d⁶.

The observation of 2 or 3 peaks in the electronic spectra of d², d³, d⁷ and d⁸ high spin octahedral complexes requires further treatment involving electron-electron interactions. Using the Russell-Saunders (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 Mulliken 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 Tanabe-Sugano diagrams.

Use of Orgel 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 A₂ and T₂ terms are linear and depend solely on D. The lines for the two T₁ terms are curved to obey the non-crossing rule and as a result introduce a configuration interaction in the transition energy equations.

The left-hand side is applicable to d³ , d⁸ octahedral complexes and d⁷ tetrahedral complexes. The right-hand side is applicable to d² , d⁷ octahedral complexes.

Looking at the d³ octahedral case first, 3 peaks can be predicted which would correspond to the following transitions:

1. ⁴T_2g ¬ ⁴A_2g            transition energy = D
2. ⁴T_1g(F) ¬ ⁴A_2g       transition energy = 9/5 *D - C.I.
3. ⁴T_1g(P) ¬ ⁴A_2g       transition energy = 6/5 *D + 15B' + C.I.

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, D. This corresponds to the energy found from the first transition above and as shown in Table 1 is generally between 15,000 cm^-1 (for weak field complexes) and 27,000 cm^-1 (for strong field complexes).

Table 1. Peak positions for some octahedral Cr(III) complexes (in cm^-1).

For octahedral Ni(II) complexes the transitions would be:

1. ³T_2g ¬ ³A_2g            transition energy = D
2. ³T_1g(F) ¬ ³A_2g       transition energy = 9/5 *D - C.I.
3. ³T_1g(P) ¬ ³A_2g       transition energy = 6/5 *D + 15B' + C.I.

where C.I. again is the configuration interaction and as before the first transition corresponds exactly to D.

For M(II) the size of D is much less than for M(III) 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).

For d² octahedral complexes, few examples have been published. One such is V³⁺ doped in Al₂O₃ where the vanadium ion is generally regarded as octahedral, Table 3.

Table 3. Peak positions for an octahedral V(III) complex (in cm^-1).

Interpretation of the spectrum highlights the difficulty of using the right-hand side of the Orgel diagram above for many d² cases where none of the transitions correspond exactly to D and often only 2 of the 3 transitions are clearly observed.

The first transition can be unambiguously assigned as:

³T_2g ¬ ³T_1g           transition energy = 4/5 *D + C.I.

But, depending on the size of the ligand field (D) the second transition may be due to:

³A_2g ¬ ³T_1g           transition energy = 9/5 *D + C.I.

for a weak field or

³T_1g(P) ¬ ³T_1g       transition energy = 3/5 *D + 15B' + 2 * C.I.

for a strong field.

The transition energies of these terms are clearly different and it is often necessary to calculate (or estimate) values of B, D and C.I. for both arrangements and then evaluate the answers to see which fits better.

The difference between the ³A_2g and the ³T_2g (F) lines should give D. In this case D is equal to either:
25200 - 17400 = 7800 cm^-1
or 34500 - 17400 = 17100 cm^-1.
Given that we expect D to be greater than 15000 cm^-1 then we must interpret the second transition as to the ³T_2g(P) and the third to ³A_2g. Further evaluation of the expressions then gives C.I. as 3720 cm^-1and B' as 567 cm^-1.

Solving the equations like this for the three unknowns can ONLY be done if the 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 D. This approach still requires some evaluation of the numbers to ensure a valid fit. For this reason, Tanabe-Sugano diagrams become a better method for interpreting spectra of d² octahedral complexes.

Using Tanabe-Sugano diagrams

The first obvious difference to the Orgel diagrams shown in general textbooks is that Tanabe-Sugano diagrams are calculated such that the ground term lies on the X-axis, which is given in units of D/B. The second is that spin-forbidden terms are shown and third that low-spin complexes can be interpreted as well, since for the d⁴ - d⁷ diagrams a vertical line is drawn separating the high and low spin terms.

The procedure used to interpret the spectra of complexes using Tanabe-Sugano diagrams is to find the ratio of the energies of the second to first absorption peak and from this locate the position along the X-axis from which D/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 spin-allowed and spin-forbidden 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(NH₃)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 D/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 ³T_2g ¬ ³T_1g and the second and third to ³T_1g (P) ¬ ³T_1g (blue line) and ³A_2g ¬ ³T_1g (green line) respectively.

The average value of B' calculated from the three Y-intercepts is 598 cm^-1 hence D equals 18420 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 1000 cm^-1 so as long as it is possible to assign peaks unambiguously the techniques are valuable.

Use of spreadsheets and JAVA applets

To overcome the problem of small diagrams, it was decided to generate our own Tanabe-Sugano diagrams using spreadsheets. This has been done, using the transition energies given in the Appendix, for the spin-allowed transition expected for:

• for d2 (oct)
• for d3 (oct)
• for d8 (oct)
• for d7 (tet)

Even so, the method of finding the correct X-intercept is somewhat tedious and time-consuming and a different approach was devised using JAVA applets.

The JAVA applets display the spin-allowed transitions and when the user clicks on any region of the graph then the values of n₂/n₁ and n₃/n₁ are displayed. In addition, the values of D/B and the Y-intercepts are given as well. This simplifies the process of determining the best fit for D/B.

The expected ranges for the ratio of n₂/n₁ are:

• for d2 (oct) 2.2 - 1.3 for D/B 10 - 35
• for d3 (oct) 1.8 - 1.2 for D/B  0 - 50
• for d8 (oct) 1.8 - 1.5 for D/B  0 - 18
• for d7 (tet)  1.8 - 1.5 for D/B  0 - 15

These ratios show the need for a certain degree of precision in attempting to analyse the spectra. It has been suggested that instead of using n₂/n₁ 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 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 lines are observed then this is not an option.

Changes in JAVA development kits and compilers have meant two different links to the applets are needed although the CLASS files are the same in each case.

• The first and alternate first can be applied to d3, d8 (oct) d2, d7 (tet) systems.
• the second and alternate second is for d2, d7 (high spin only) (oct), d3, d8 (tet).

Further information for use in laboratory classes is available.

References

1. Basic Inorganic Chemistry, F.A.Cotton, G.Wilkinson and P.L.Gaus, 3rd edition, John Wiley and Sons, Inc. New York, 1995.
2. Physical Inorganic Chemistry, S.F.A.Kettle, Oxford University Press, New York, 1998.
3. Complexes and First-Row Transition Elements, D.Nicholls, Macmillan Press Ltd, London 1971.
4. The Chemistry of the Elements, N.N.Greenwood and A.Earnshaw, Pergamon Press, Oxford, 1984.
5. Concepts and Models of Inorganic Chemistry, B.E.Douglas, D.H.McDaniel and J.J.Alexander 2nd edition, John Wiley & Sons, New York, 1983.
6. Inorganic Chemistry, J.A.Huheey, 3rd edition, Harper & Row, New York, 1983.
7. Inorganic Chemistry, G.L.Meissler and D.A.Tarr, 2nd edition, Prentice Hall, New Jersey, 1998.
8. Inorganic Chemistry, D.F.Shriver and P.W.Atkins, 3rd edition, W.H.Freeman, New York, 1999.
9. Basic Principles of Ligand Field Theory, H.L.Schlafer and G.Gliemann, Wiley-Interscience, New York, 1969.
10. Y.Tanabe and S.Sugano, J. Phys. Soc. Japan, 9, 1954, 753 and 766.
11(a). Inorganic Electronic Spectroscopy, A.B.P.Lever, 2nd Edition, Elsevier Publishing Co., Amsterdam, 1984.
11(b). A.B.P.Lever in Werner Centennial, Adv. in Chem Series, 62, 1967, Chapter 29, 430.
12. Introduction to Ligand Fields, B.N.Figgis, Wiley, New York, 1966.
13. E.Konig, Structure and Bonding, 9, 1971, 175.
14. Y. Dou, J. Chem. Educ, 67, 1990, 134.

Appendix

Transitions calculated for spin-allowed terms in the Tanabe-Sugano diagrams.

Octahedral d³ (e.g. Chromium(III) ).

⁴T_2g ¬ ⁴A_2g ,         n₁/B= D/B

⁴T_1g(F) ¬ ⁴A_2g,    n₂/B= ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) }

⁴T_1g(P) ¬ ⁴A_2g,    n₃/B= ½{15 + 3(D/B) + Ö(225 - 18(D/B) + (D/B)² ) }

from this, the ratio n₂/n₁ would become:

                    ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) } /  D/B

and the range of D/B required is from ~15 to ~55

Octahedral d⁸ (e.g. Nickel(II) ).

³T_2g ¬ ³A_2g,                n1/B= D/B

³T_1g(F) ¬ ³A_2g,       n2/B= ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) }

³T_1g(P) ¬ ³A_2g,       n3/B= ½{15 + 3(D/B) + Ö(225 - 18(D/B) + (D/B)² ) }

from this the ratio n₂/n₁ would become:

                    ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) } / D/B

and the range of D/B required is from ~5 to ~17

Octahedral d² (e.g. Vanadium(III) ).

³T_2g ¬ ³T_1g ,        n1/B= ½{(D/B) - 15 + Ö(225 + 18(D/B) + (D/B)² ) }

³T_1g(P) ¬ ³T_1g,  n2/B= Ö(225 + 18(D/B) + (D/B)² )

³A_2g ¬ ³T_1g,      n3/B= ½{ 3 (D/B) -15 + Ö(225 + 18(D/B) + (D/B)² ) }

from this the ratio n₂/n₁ would become:

                  Ö(225 + 18(D/B) + (D/B)² ) /  ½{(D/B) - 15 + Ö(225 + 18(D/B) + (D/B)² ) }

and the range of D/B required is from ~15 to ~35

Tetrahedral d⁷ (e.g. Cobalt(II) ).

⁴T₂ ¬ ⁴A₂,          n1/B= D/B

⁴T₁ (F) ¬ ⁴A₂,   n2/B= ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) }

⁴T₁ (P) ¬ ⁴A₂,    n3/B= ½{15 + 3(D/B) + Ö(225 - 18(D/B) + (D/B)² ) }

from this the ratio n₂/n₁ would become:

                ½{15 + 3(D/B) - Ö(225 - 18(D/B) + (D/B)² ) } /  D/B

and the range of D/B required is from ~3 to ~8.

Acknowledgments

To the 135 students in the 1998/1999 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, for developing the JAVA applets.