GUEST AUTHOR J. J. Zuckerman Cornell University Ithaco, N e w York
Textbook Errors, 58
Crystal Field Splitting Diagrams
I t is now over a decade since the theoretical treatment of transition metal complexes which goes by the general name Crystal Field or Ligand Field Theory came into general acceptance in inorganic chemistry. (In current usage these terms are interchangeable, but more specifically the crystal field theory does not consider the role played by the ligands further than to credit them with producing an electrostatic field, while the ligand field theory includes molecular orbital considerations. Our discussion relates to electrostatic crystal field theory only.) Its use as part of coulse work in inorganic chemistry is now well established, and simplified explanations based upon the usual Crystal Field Splitting Diagram are about to filter down into the freshman curriculum. This article is written in an attempt to bring this much used diagram into better agreement with the realities of the situation.' Central to the idea of the crystal field approach is that the energies of the d-orbitals of a metal ion in a transition metal complex split in the presence of perturbation by the surrounding ligands. This effect in crystal lat,tices mas discussed by Bethe in 1929.2 In a field of any given geometry, certain of the d-orbitals find themselves oriented more in the direction of the field of the ligands than others, and the energies of these orbitals are raised by the electrostatic repulsions of the ligands while the energies of the other d-orbitals are affected to a smaller extent. I t is usual to illustrate the splitting of the five degenerate d-orbitals of a transition metal ion for the case of three representative symmetries: octahedral, tetrahedral, and square planar. A single energy diagram is frequently used to show all three types of splittings with reference to the free ion or field-free ion as the five-fold degenerate ground state of the d-orbitals is usually labeled. This standard Crystal Field or Ligand Field Splitting Diagram must by now be familiar to both students and teachers. A new diagram, (p. 316), is recommended to replace the one now in use. Consider the case of the octahedral field first. If a complex is formed by bringing six identical ligands along x, y, and z axes so as to form a Suggestions of material suitable for this column, and guest columns suitable for publication directly, should be sent with as many details ss possible, and particularly with references to modern textbooks, to Karol J. Mysels, Department of Chemistry, University of Southern California, Los Angeles 7, California. 'Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the source of errors discussed will not be cited. In order to be presented, an error must occur in at lemt two independent recent standard books. BETAE,H., Ann. Physik. 3,133 (1929).
regular octahedron about the metal ion, the charges on the ligands will repel an electron to a greater extent if it is in a d,, or d+,* orbital than if it is in a dxy, dxz, or dyz orbital, since the former point toward the ligands and the latter point between the axes of approach. (The ligands either possess a net negative charge, i.e., are anions, or align themselves so that a negative region of charge is nearest the central atom. I n electrostatic theory, the ligands can be considered as point charges or point dipoles.) Thus for the field-free ion or atom in vacuo, the d-orbitals are energetically degenerate, but their energies become differentiated in a concerted process as the energies of all the orbitals are raised with the approach of the ligands. In the case of octahedral symmetry the original degeneracy is destroyed, and the energies are split into two groups with the da2and d,2.vz orbitals assuming a higher energy and the dxy, dxz, and dyz orbitals a t a lower energy. The two higher energy orbitals are called d r according to van Vleck, e , according to Mulliken, and -ya in Bethe's original notation. The three lower energy orbitals are called dr by van Vleck, t,, by Mulliken, and r s by Bethc3 The fact that all the d-orbital energy levels are raised by the repulsion of the ligands is emphasized. I n fact the differentiation of the d-orbital energy levels is a relatively small effect superimposed on the much larger repulsive effect of the crystal field on the orbitals. This repulsion of the negative electron cloud about the metal ion by the negative charge or dipole of the ligand is in its turn small by con~parison with the overall attraction of the positive metal ion for the ligands. These repulsions are contained in the lattice energy of a crystal or the ligation energy of a metal conlplex. [For example, in octahedral Ti (HzO)s3+the splitting of the two sets of d-orbitals is of the order of 60 kcal/mole while the hydration energy of Ti3+ is approximately 1000 kcal/mole.] The d-orbital energy level splitting is measured in terms of a parameter A, 10 Dq or El-E2, whose magnitude is assumed to be proportional to the strength of the crystal field: the stronger the field, the larger the value of A or Dq or El-EQ. The center of eravitv of the d-orbital enereies renresents the energyYthe &e &orbitals would h&e in'the It is the presence Of a weighted mean energy of the &orbitals in the presence of the ligands. I n another view, if we may break the concerted process of raising the energy and differentiating into two steps, then this is the energy to which the 8 See footnote 7 in the Resource Paper I, "Ligrtnd Field Theory," by F. A. COTTON, THIS JOURNAL 41, 466 (1964).
Volume 42, Number 6, June 7 965
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Table 1.
C.N. 1
2 3 4
4 5 5 6 7 a
The d-Orbital Enerav Levels in Crvstal Fields of Different Svrnrnetrier
Structure
ih&
trigonalb tetrahedral s q u m lrtnarb triganarbipyrsmid' square pyramid' octahedran ~entaeonalbvovrimidC
d,._.*
-3.1409 -6.28 5.46 -2.67 12.28 -0.82 9.14 6.00 2.82
Bonds lie along z axis. Bonds in the zy plane.
five degenerate d-orbitals would be raised before splitting took place. I t is the zero of energy in the perturbed system, and the algebraic sum of all energy shifts from this level is zero. This is a simple statement of the "preservation of the center of gravity" rule from quantum mechanics. The rule is quite general for all splittings where the forces are purely electrostatic and where only the sets of levels being split is considered. To take the octahedral case where the total splitting is A..,, the upper two orbitals lie 3/s A,, above and the lower three lie 2/s AWt below the energy the orbitals would have in a spherically symmetric field. Note that this zero level is not the free ion or field-free ion energy which is much lower. By analogous and by now familiar argument the splittings for the tetrahedral and square planar cases are worked out.4 In the presence of a tetrahedra1 field the orbitals most affected in the octahedral example are least raised, and vice versa. Here the upper three orbit,als lie a t 2/s A,.* above the center of gravity and the lower two a t 3/s A*, below. Further, it can be shown that, all else being equal (cation, ligands, cation-ligand distance, etc.), the total tetrahedral splitting, A,,, is 4/lg of AOeC, so that the magnitude of the crystal field splitting in an octahedral complex will be over twice 4 Splitting diagrams far the linear, trigonal, trigonal bipyramidd. and cubic fields have been recentlv eiven bv A. L. COM-
I
%
-3.140~ -6.28 5.46 1.78 2.28 -0.82 -0.86 -4.00 2.82
dzc 0.57Dq 1.14 -3.86 1.78 -5.14 -2.72 -4.57 -4.00 -5.28
dw 0.57Dq 1.14 -3.86 1.78 -5.14 -2.72 -4.57 -4.00 -5.2s
Pyramid base in w plane. Only electrostatic ~erturhationsare considered
(
that in a tetrahedral conlplex with the same charge per ligand and the same bond distances as in the octahedral one. &Orbital energy levels in crystal fields of various symmetries can be calculated, and some results of single electron energies are listed in Table 1.5 These values were used in constructing the figure. A further note to Figure 1 concerns the positions of the centers of gravity for each of the three types of fields commonly represented. Since the center of gravity of each of the splittings represents the condition of the ion in the presence of a spherically synnnetric field of the same strength (all other factors being held constant) as the actual field of the ligands, then it can be seen that for identical ligands the field strength will be a simple function of the number of ligands, i.e., of the coordination number of the complex. Thus for similar ligands the center of gravity of an octahedral complex will lie a t higher energy than that of a tetrahedral or square planar con~plex,and these last two will have identical centers of gravity. This is shown in the figure. The actual magnitude of the splitting (we have called A in the figure) is proportional in the first instance to the strength of the crystal field generated by the ligands which depends upon the charge or dipole moment, as well as size, polarizahility and ability to form a-bonds.6 I n addition, the following generalizations can be made: (1) A,,, is approximately 45% larger when 4d orbitals are used and 75% larger for 5d (assuming metal of identical valence in the same periodic group). (2) A,,, is 40430% larger for the trivalent ion than the divalent ion of the samemetal. (3) rouchlv between 2 0 4 0 kcal/mole for , , An., ~ .varies . most divslent hexaco&&ated 3d con~plexes.' se(4) The common ligands may be arranged quence of rigular increase of A.., values for their complexes with any given metal ion. This dependence of A.., values on the identity of the ligands is known as the spectrocheinical series. Bnsom., F.. . AND PEARSON.R.. "Mechanisms
D
of Inorganic Reactions," John wile; 61 Sans, New York, 1958, p. 55. 8 "It seems noteworthy, although not a point to
c m W
1 3 16
dw
d2
5.14Dp 10.28 -3.21 -2.67 -4.28 7.07 0.86 6.00 4.93
be stressed unduly, that by taking into account both types oi crystal field parameters-shifts in the eenter of gravity far &orbitals as well as splittings of their enereim-the diaaram m the figure describes quaiit~tivelythe &sults of even the most sophisticated molecular orbital studies of the energy levels far the d-like orbitals of transition metal complex ions and molecules." (Private communication. Dr. G. L. Goodman, 1964.)
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Journol o f Chernicol Education
A note on the limitations in the use of Crystal Field Splitting Diagrams is in order in conclusion. First, the differentiation of the d-orbital energy levels, as important as it may be for explaining differences between various complexes ma,kes up only a small fraction (5.10%) of the total binding energy of the cornplex. A much more serious limitation comes about because the d-orbital energy levels for various crystal fields are readily obtained only for the case of a single d-electron, and diagrams such as Figure 1 are strictly applicable, therefore, only to the one electron case. Systems with mole than one d-electron are complicated by electron-electron interactions. I t turns out that the diagrams can he used equally well for the dl and d6 cases (for reasons which we will not go into), and also to approximate the d4 and d9 cases. The diagrams for configurations d2 and C and d3 and d8 are much more complex. The changes in the diagrams as one goes
through the configurations d' to d9 are quite drastic, involving not only changes in the relative magnitudes of the splittings, but the appearance of additional splittings and even reversals in the energies of the d-orbitals as well. The situation is further complicated by distortions from perfect octahedral or other geometry. These distortions are quite common and produce additional splittings. Thus to represent correctly the situation in each of the wide range of possible electronic and geometric configurations in transition metal complexes would require a family of Crystal Field Splitting Diagrams. Any one diagram which attempts to serve all situations must be used with care. Acknowledgment
The author thanks Professor R. C. Fay of this university for helpful discussions.
Volume 42, Number 6, June 1965
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