The Crystal Field Splitting Parameter Dq: Calculation and Significance

29 The Crystal Field Splitting Parameter Dq: Calculation and Significance

Downloaded by NORTH CAROLINA STATE UNIV on January 17, 2013 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0062.ch029

A. B. P. L E V E R Department of Chemistry, University of Manchester Institute of Science and Technology, Manchester, Great Britain

This paper reviews methods which have been used to elucidate the quantity Dq in cubic and non-cubic mole cules. A novel graphical method allowing the rapid de termination of Dq and B from spin-allowed absorption bands of high-spin d , d , d , and d complexes is intro duced. It is possible to calculate both in-plane and out-of-plane ligand field strengths in tetragonal complexes. Calculations with tetragonal nickel(II) complexes indicate that the axial field strengths in complexes involving bulky in-plane ligands can be remarkably weak. There are many difficulties in obtaining a meaningful value of Dq; different methods of analyzing the same spectrum may yield different Dq quantities. Despite such uncertainties, detailed studies of series of analogous complexes provide useful chemical information. 2

3

7

8

V V T h i l e Werner's classic paper (49) of 1891 can be regarded as the foundation of modern inorganic coordination chemistry, i t has been only during the past 10 to 15 years that a rationalization of the many stereochemistries possible for a coordination complex has become possible— through the modern techniques of spectroscopy and magnetism. E v e n today, however, many problems remain to be solved, and it is the purpose of this review to outline some of these difficulties. The major feature of a transition metal coordination complex is the presence of an incomplete d shell in which the energies of the d orbitals, which are no longer fivefold degenerate, are determined by the stereo chemistry of the complex. The electronic absorption spectrum and, there fore, the color of the complex depends upon the arrangement and energies 430 In Werner Centennial; Kauffman, G.; Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

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Dq Calculation


of the d electrons within the d shell. Conversely, a study of the spectrum can reveal the stereochemistry of, and the electronic levels within, a coor dination complex. The basis for understanding electronic absorption spectra was laid down i n 1929 by Bethe (8) i n his development of crystal field theory, but modern inorganic electronic spectroscopy, as we know it, was not really underway until the early 1950's. Since then the develop ment has been rapid (2, 5, 12, 23, 35, 48). Crystal field theory has now been largely superseded by ligand field theory in which the interaction between the metal ion and ligand orbitals is taken into account. Of major importance in either theory is the crystal field (or ligand field) parameter Dq, where lODq is the energy separation between the e and U d orbitals i n a cubic environment (2, 12, 23). The term cubic environment covers 4-coordinate tetrahedral, 6-coordinate octahedral, and 8-coordinate cubic complexes. The g subscripts are not necessary for a tetrahedral environment which lacks a center of symmetry. Henceforth, for simplicity, these subscripts will be dropped. This paper reviews the measurement of this parameter, both i n cubic and i n non-cubic complexes, and discusses its significance. A novel method for the rapid calculation of Dq and B from the spectra of cubic molecules of high-spin d , d , d , and d ions is introduced, and the spectra of some tetragonally distorted nickel (II) complexes are interpreted i n terms of unusually low, crystal field strengths for the axial ligands. It should be mentioned that although Dq is formally an ionic parameter and crystal field theory is not held in very high acclaim today, it is still important as an empirical ligand field parameter. There are, however, many difficulties involved in obtaining a meaningful value of Dq i n a given system. g

g

2

z

8

7

Ionic Model On the basis of crystal field theory, the octahedral crystal field potential Dq is as follows: Dq =

6a

0

(1)

6

(r\ is the average quartic radial displacement of a d electron; a is the metal ligand distance; and a is called a radial integral (3, 6,10,19).) It is well-known (12) that inserting appropriate values of the various parameters into this expression does not reproduce the experimentally observed value, primarily because the ionic model is an unreal approximation to the bond ing i n the complex. When covalency is taken into account, the calculated and observed values can be made to agree well (42)* 0

4

In Werner Centennial; Kauffman, G.; Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

432

WERNER


Determination

of Dq-Cubic

CENTENNIAL

Molecules

There are two principle methods for determining Dq spectroscopically —the method used depending upon the metal ion in question. W i t h some ions the lowest energy, spin-allowed transition corresponds exactly with lODq, at least to a first order approximation. In these circumstances, lODq is simply the energy of this absorption band and is obtained without recourse to calculation. A l l other spin-allowed transitions in cubic mole cules will be functions of lODq and also of the Racah interelectronic re pulsion parameters B and/or C (2) (or alternatively the Slater-Condon F parameters (40)). Where lODq cannot be directly estimated from an absorption band energy, two or more observed transition energies must be used to solve the appropriate secular equations. Tanabe and Sugano first calculated the required energy (44) matrices which have since been repeated by other authors (17, 34). Simplified equations have been presented by J0rgensen (24). These latter equations are not always exact solutions to the Tanabe-Sugano secular determinants because the assumption is made that C = 41? to aid simplification. Tanabe-Sugano diagrams are graphical representations (17, 34, 44) of the energy matrices in which the ratio Dq/B is plotted against E/B. Plots of Dq vs. E, sometimes called Orgel diagrams, in which a value of B is assumed, have also been published (37). The Tanabe-Sugano equa tions were calculated on a strong field basis, while Orgel used a weak field basis. The terms "strong field" and "weak field" refer to the situations in which the crystal field is either large or small, compared with the inter electronic repulsion, respectively. In the weak field calculation, interelec tronic repulsion is considered first, then the free ion spectroscopic terms of the central ion are derived. Subsequently, the crystal field potential is considered as a perturbation upon these terms, the degeneracies of which may now be partially or wholly lifted to give a number of levels described by group theoretical labels. In the strong field treatment, the effect of the crystal field upon the d orbitals is first considered. The possible electron configurations in the particular molecular environment are obtained, and then the interelectronic repulsion within each configuration is evaluated. Provided that configurational interaction is included, both approaches lead to the same conclusions (2, 12). The various ions may be conveniently divided into a number of categories. k

High-spin Complexes in which the Central Ion-free Ion-ground Spectroscopic Term is D; d (?D), d*(*D), d ( D), d ( D)—e.g., Ti(III), V(IV), Mn(III), Fe(II), Cu(II), etc. In all cases, in cubic symmetry, the de generacy of the D term is lifted to give two levels, T and E, the energy separation of which is lODq to a first approximation (Figure 1 ) . In this n

l

6

5

9

2

2


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Dq Calculation

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E

(b)

(a)

(c)

Figure 1. Energy level diagram for high-spin ions with a D-free ion ground term, a) Free ion in a spherical potential; b) splitting diagram for d and d* ions in an octahedral field, and d and d* ions in a tetrahedral field; c) splitting diagram for d* and d ions in an octahedral field, and for d and d ions in a tetrahedral field. 1

4

9

1

6

situation then, the band energy of the only spin-allowed transition gives a direct measure of lODq. Those ions with T ground states (e.g., octahedral T i ( I I I ) , V (IV), Co(III) high-spin, etc.) generally give rise to an asymmetric spectrum consisting of a peak with a shoulder on the low energy side. This phenom enon is believed to occur as a consequence of the lifting of the degeneracy of the excited E term by the Jahn-Teller effect (12). Although the sense of the splitting is not known, the energy of the peak is usually taken to represent lODq. Complexes with E ground states (e.g., octahedral Cu(II) and Cr(II)) are usually so tetragonally distorted that they cannot be considered to have cubic symmetry; their spectra are generally extremely broad, and a precise value for lODq cannot be assigned without a detailed spectral analysis. (See tetragonal complexes below.) 2

High-spin Complexes in which the Central Ion-free Ion-ground Spectroscopic Term is »F; d ( F ) , d ( F ) , d ( F ) , d ( F)—e.g., V(II), V(III), Mn(IV), Mo(III), Cr(IIl), Co(II), Ni(II), etc. The degeneracy of the F term is lifted in a cubic environment to yield the group theoretical terms A 2, T , and T (Figure 2). In all cases, there is an P term lying 152? above the F term. The P term will transform as Ti i n a cubic environment. Two basic situations arise: a) the A term lies lowest in energy; b) the Ti(F) term lies lowest i n energy (Figure 2). 2

3

3

4

7

4

8

3

t

n

x

2

n

2

A 2 T E R M A S G R O U N D S T A T E , O C T A H E D R A L CP A N D d , s

TETRAHEDRAL

d A N D d? ( A L L H I G H - S P I N ) . The lowest energy, spin-allowed transition, T and vz, respectively (Figure 2). The method employed is discussed below. 2

T E R M AS G R O U N D STATE, OCTAHEDRAL d

Ti(F) HEDRAL d

2

z

A N D d* ( A L L H I G H - S P I N ) .

In

the

AND d,

limit of

7

TETRA

a very

weak

crystalline field, the energy of the first spin-allowed transition is 8Dq (2, 28). However, in the complexes commonly investigated the crystal field strength is such that this approximation is not valid. In the so-called intermediate or medium field region the T\ terms derived from the F and P free ion terms will mix together so that the energy of the ground state, with reference to the free ion in a spherical field, will be not only a function of Dq, but also of the Racah parameter B. Values of lODq, and also of B, may be found by solving the appropriate secular equations, utilizing two of the spin-allowed transition energies. Direct solution of the rather complicated equations involved (2, 23) is rather tedious, especially so when the ground state is T\ and a graphical solution is to be preferred. The method commonly used involves the computation of the ratio of two of the spin-allowed transition energies,


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Dq Calculation

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435

say v and v and the location on the appropriate Tanabe-Sugano diagram of a position where the vz/v ratio agrees with the experimental ratio. Dq/B, E{vz)/B, and/or E(v )/B can then be directly obtained from the diagram and values of Dq and B evaluated (12). 2

Zj

2

2

This method has the disadvantage that virtually all published TanabeSugano diagrams are physically rather small so that the location of a position, where the vz/v ratios tally and the reading of the Dq/B and E/B ratios are inaccurate. Moreover, the actual process of locating the correct position on the diagram is slow. Downloaded by NORTH CAROLINA STATE UNIV on January 17, 2013 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0062.ch029

2

These disadvantages are eliminated by diagrams in which the transition energy ratios themselves are directly plotted against Dq/B and against E/B. These are illustrated for all high-spin ions with F ground terms in Figures 3 - 5 . Larger scale diagrams may be found elsewhere (25). Consider the three following examples: 1) T h e octahedral complex ion Ni(dimethylsulfoxide) has absorption bands at 7728 (*>i), 13000 (v ), and 24040 (v ) c m . - The observation of v\ means that Dq is 773 c m . A value for B may be found from the v /v\ ratio, which is 1.682. Reading from Figure 3, Dq/B = 0.82, hence B = 942 c m . Direct solution of the secular equation yields B = 948 cm." T h e energies of v and vz may also be used to compute Dq and B. The ratio vz/v is 1.854, hence from Figure 3, Dq/B = 0.84, and from Figure 4, E(v )/B = 26.05, hence Dq = 775 and B = 923 c m . - Direct solution of the secular equation gives Dq = 773 and B = 921 c m . " +2

2

1

z

- 1

2

- 1

1

2

2

1

z

1

Figure 3. Transition energy ratios, vz/v*, v /vi (left-hand scale), and vz/vi (right-hand scale) vs. Dq/B for A (F) ground stale ions 2

2


436

WERNER CENTENNIAL

2) The tetrahedral complex ion C o C l has bands centered at 6200 (j>) and 16000 (v ) c m r The ratio v /v is 2.581. Using Figure 3, Dq/B = 0.465 and from Figure 4, E(v )/B = 20.8, hence Dq = 357 and B = 769. Direct solution of the secular equation yields Dq = 356 and B = 768 c m . " 4

1

z

2

z

- 2

2

z


1

Figure 4- Transition energy ratios, vzlvi (for A ground state ions) and vzlvi and vz/vi (for Ti(F) ground state ions) vs. E(v ) IB. Left-hand scale for E(v ) IB less than 24.6; right-hand scale for E(vz)/B greater than 24-6 2

z

z

3) The octahedral complex ion C o ( H 0 ) 6 has absorption bands at 8100 (vi), 16000 0 ), and 19400 (v ) cmr Because this ion has a T ground term, the first band does not correspond with lODq. Dq may be obtained from the energy ratios v /v\ or v /v . The energy ratio v /v\ is almost independent of Dq/B and should not be used. The ratio v /v is 1.213, hence from Figure 5, Dq/B = 0.99 and from Figure 4, E(v )/B = 22.3, hence Dq = 861 and B = 870 c m . The energy ratio v /v\ is 2.395, from which Dq/B = 1.11 and E(v )/B = 23.4; therefore, Dq = 920 and B = 829 c m . " + 2

2

1

z

2

z

z

x

2

2

z

2

z

- 1

z

z

1

In this manner values of Dq and B may be rapidly and accurately obtained. Where the use of two different ratios gives rise to two different sets of Dq and B values, it is customary to accept the set which gives the best fit to the spectrum for all three bands. Using the cobalt hexahydrate complex as an example, if Dq/B = 0.99 as determined from the v /v ratio, the position of v\ can be rapidly calculated by reading off the v /v\ ratio from Figure 3 at Dq/B = 0.99. In this way, v is predicted to lie at 7550 c m . , i n fair agreement with the experiment. z

2

z

x

- 1



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437

Dq Calculation

Figure 5.

Transition energy ratios v /v and v^/v* vs. Dq/B for ground state ions z

x

Ti(F)

From the second set of data, with Dq/B = 1.11, vz/v is seen to be 1.13, hence v lies at 17,170 c m . , in rather poor agreement with experi ment. Therefore, the first set of data fits better to the spectrum as a whole than the second set. The reasons different sets of Dq and B values might be obtained by using two different analyses are discussed later. 2

2

- 1

It should be emphasized that inaccurate values of Dq and B are likely to be derived if the transition energy ratio used is relatively insensitive to Dq/B. F o r example, considering Figure 3, use of the ratio v$/v would be inadvisable in systems where this ratio lies between 1.55 and 1.7. Similarly, the vz/v\ ratio will provide very inaccurate values if it is less than about 2.2. It is equally true that inaccurate data may be obtained if the energy ratio is highly sensitive to Dq/B because, in this case, a small experimental error will lead to a large change in Dq/B. 2

Of course if exact results are required, nongraphical methods should be employed. The computer provides the easiest way of processing a large amount of data. However, in view of the experimental uncertainty in the positions of the absorption bands observed, the graphical method described above will provide correct solutions to the secular equations to within experimental error. Spin-orbit coupling has been neglected in this treatment because the spin-orbit coupling coefficient, \, is relatively small in first row, transition metal ions. Large scale energy level diagrams for d , d , d , and d ions, 2

3

7


8

438

WERNER

CENTENNIAL

relating Dq and E, and taking spin-orbit coupling into account, have been published (81, 82). A s far as determining Dq is concerned, however, they suffer the same disadvantage as the graphs published by Orgel (87), in that they are constructed with a constant value of B.


Spin-orbit coupling may be of greater importance in tetrahedral com plexes which frequently exhibit multi-component bands (17, 41, 47). B e cause such structure is not understood, at present it is best to use the cen ter of such bands for Dq determinations (7). Ions Not Falling into the Above Categories; d Ions, Low-spin Complexes, etc. I n low-spin complexes and in complexes of the d ions, both low-spin and high-spin, the transition energies are generally functions not only of Dq and B, but also of C. C is generally taken to be equal to 4 B and therefore eliminated from the equations. This is a rather dangerous, if not a common, procedure. There is evidence (50) that the C/B ratio can be considerably higher than 4. Moreover, B is not constant but varies from one spin state to another (23). h

5

If three or more bands in the spectrum can be assigned, then an approxi mate value of Dq may be derived by fitting the spectrum to the appro priate Tanabe-Sugano diagram, or by solving the secular equations directly. Derivation of Dq from Charge Transfer Spectra. In addition to crystal field bands, many ions exhibit absorption due to transfer of an electron from the metal to the ligand or from the ligand to the metal—the so-called "charge transfer" or "electron transfer" bands (23). I n many cases, it is possible to assign bands due to the transfer of an electron from the ligand to the e orbital on the metal and from the same ligand orbital to the U orbital on the metal. Clearly the energy separation between bands of this nature is related to lODq (23). Unfortunately, the energy separation is also a function of the difference i n interelectronic repulsion of the various states concerned and, unlike the simple crystal field transi tions, w i l l involve coulomb and exchange integrals between electrons on different atoms. These are difficult to evaluate. Such a method of derivation offers no advantages as far as d ions are concerned but has a very clear advantage for d° ions where crystal field transitions cannot occur. G r a y (16, 46), for example, has interpreted the charge transfer spectra of a series of tetroxo anions and concludes that the crystal field splitting energies in the tetrahedral d° manganate(VII) and chromate(VI) ions are 26000 and 24000 c m . " , respectively. (See Reference 36 for an alternative interpretation.) n

1

Thermodynamic Methods. Since many thermodynamic properties depend upon the crystal field stabilization energy (e.g., lattice energy, solvation energy, electrode potentials, etc.), Dq may in principle be esti mated by thermodynamic rather than spectroscopic means. The agree-


29.

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439

Dq Calculation

ment between the thermodynamic and spectroscopic values of Dq is moderately good for ionic complexes of the lower oxidation state ions (e.g., M(II)hexahydrates) but becomes very poor when the higher oxidation state, more covalent compounds are considered (12, I S ) .


Determination

of Dq in Non-Cubic

Molecules

Dq is proportional to the average quartic radial displacement of the d electrons. In cubic molecules, as we have seen, the energy separations are functions, at least as far as the crystal field potential is concerned, only of this fourth power radius. I n non-cubic molecules, however, the energy separations may also be functions of the average quadratic radial displace ment of the d electrons (8, 6, 10, 19, 85), and under such circumstances Dq is not so readily evaluated. Nevertheless, the spectra of a number of tetragonal complexes have been interpreted to reveal both in-plane and out-of-plane ligand field strengths. It is important to realize first that many formally non-cubic com plexes, such as 6-coordinate derivatives of the type M X Y _ o , give spectra which are almost indistinguishable in type from those of regular octahedral complexes. This is principally due to the fact that spin-allowed transi tions are fairly broad with half-widths of some 500-1000 or more wavenumbers (2, 12). Distortion from octahedral symmetry must be quite large before it manifests itself experimentally. I n other words, although the degeneracy of the orbital doublet and triplet levels described above will be partially or wholly lifted by the lower symmetry component to the ligand field, the splitting must be quite large before it will be observed experimentally. This is particularly true of the and v* transitions of F term ions. Similar arguments apply to the 4-coordinate quasi tetrahedral complexes of the classes M X Y and M X Y , which often have spectra a l most indistinguishable i n type from their regular tetrahedral, M X analogues. 0

3

2

6

2

4

Under such circumstances it is legitimate to obtain solutions of the secular equations as though the complex were cubic. The value of Dq, so obtained, is referred to as the mean crystal field strength Dq' of the various ligands involved. This mean crystal field strength may be related to the crystal field strengths of the ligands concerned by means of the "average environment rule" (4, 4$), which states that i n mixed ligand complexes the central ion will experience a crystal field which is the average of the crystal fields of each of the ligands. I n the case of 6-coordinate complexes M X Y _ o 0

6

Dq' = \\a Dq + (6-a) 2)

The Crystal Field Splitting Parameter Dq: Calculation and Significance

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