5. F. A. Kettle The University Sheffield, S3 7HF, England
Crystal Field Potentials
Although it is now generally accepted that the ligand-field model of the electronic structure of trensition metal complexes is much more accurate than t,hnt provided by crystal field theory, the latter remains of paramount importance. There are two maiu reasons for this. I h t , if quantities which are well-defined in simple crystal field theory are allowed to becomc paramet,ers, theu the effects of covalency may he allowed for. The second reason for the continued interest in crystal field theory is that it may be used to predict &orbital splitting patterns which are useful in the interpretation of (1-l-d hands in electronic spectra. Examples of this are the spectra of complexes in which the coordination number of a transition metal ion is five or seven, a field in mhich there is much current interest. Despite the assumptions inherent in simple cryst,al field theory, it appears that it usually predicts correctly the relative energies of the &orbitals and, what is more, their relat,ive separations are also predicted fairly accurately. T h e calculation of a d-orhitial splitting pattern, within the crystal field approximation, falls into two parts (1). The first step is to obtain an expression for the electrostatic potential generated a t the metal ion by a suitable array of point charges (or dipoles) which represent the ligands. The second step is to use this potential field to determine the relative energies of the &orbitals. This second step has been discussed in THIS JOURNAL (I), and so in the present article we shall confine our discussion to a qualitative consideration of the form of the potential function. The problem we shall first consider is that of finding an expression for the crystal field potential around an atom mhich is a t the center of an octahedral arrangement of six identical ligands. Because me are interested in what happens around the metal atom it is convenient t o express the potential in terms of some function or functions which are centered on the metal atom. The most convenient functions are the spherical harmonics. Spherical harmonics are the functions that one usually draws when drawing a picture of a n orbital. As the reader may have discovered, there are several ways in which a p orbital, say, may be drawn. Two ways arc shown in I'igurc 1. Suppose that from the nuclcus of the adorn containing the p orbital we imagine that therc is a large numher of lines drawn radially outwards, so that the atom resembles a rolled-up hcdgchog or porcupine. Wc measure the amplitude of thc 71 orhitel along each of the lines and then mark off IL lcngt,h from ihc nuclcus proportional to this :~mplit.udc. l'hcsc marks would he found to define tho surr:~oosof t,wo sphcrcs i n contact, as shown in i A The ir, orbital shown in I'igure 1B is a
contour diagram in which points with the same 1 amplitudel are joined (only if it is the contour of zero amplitude should the two halves of the orbital touch). An important practical difference between Figures 1A and 1B is that the latter can only be drawn if a radial function is specified. The form of diagrams such as Figure 1A does not depend on the radial part of the orbital wavefunction. Of these two representations of p orbitals the former is equally valid as a picture of the spherical harmonic which is labelled
-.
Y S -0 .
To return to the crystal field potential. From the point of view of the metal ion the first, very crude, approximation to the six surrounding negative charges is to regard them as being uniformily smeared out over the surface of a sphere. That is, the first spherical harmonic in the expansion is that corresponding to a n s orbital (Yoo).This term is multiplied by a factor, k,, which, among other things, is proportional to the number of ligands ( e number of point charges). This is easily understood, for the charge density on our spherical shell will increase in direct proportion to the number of point charges. What of the spherical harmonics characteristic of p orbitals? As Figure 1A shows their presence would indicate that the potential along the +z axis differs from spherical symmetry (due to the Y o oterm that we already have) in the opposite way to the potential along the -z axis (and similarly +x and -x; +y and -y). But, by symmetry, for an octahedral complex the potential along the +z axis must equal that along -z (and that along x, -x, y, and -y). We conclude that there will be no p orbital spherical harmonic contribution to the potential. But exactly the same reasoning excludes all spherical harmonics corresponding to orbitals of u symmetry. This
Figure 1.
Two representations of the somep orbital.
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means that we may exclude j-, h-, j-, . . . , orbital spherical harmonics and confine our attention to those corresponding to d-, g-, i-, . . . , orbitals. Of the d orbitals we may at once exclude those spherical harmonics corresponding to the d,,, d,,, and dnE,, orbitals. There must be identical potentials a t corresponding points in the four quadrants of the xy plane, for example. The effect of a contribution from the d,, spherical harmonic, however, would be to increase the potential in two of these quadrants and decrease it in two others, and so this harmonic is excluded. The d,. and d+,. spherical harmonics may also be excluded. That corresponding to d,z would decrease the potential in the xy plane and increase it along the z axis (relative to the Yoo term). This, of itself, is not a valid reason for excluding it, for some other harmonic may be able to compensate for it and make the potentials at +x, -x, +y, and - y equal to Only the d,2-r. harmonic is availthat at +z and -2. able for this and, while able to compensate at +x and -x, only makes the discrepancy worse at +y and -y (Fig. 2). We conclude that none of the d-orbital harmonics will occur in the expansion. At this point is it worthwhile to stop and consider the form of the sort of spherical harmonic for which we are looking. It is one such that all corresponding points in the octahedron will experience the same potential. That is, all of those symmetry operations which turn an octahedron into itself will turn the spherical harmonic into itself. Evidently, no spherical harmonic corresponding to a degenerate set of orbitals will satisfy this condition for it is always the case that for such orbitals some symmetry operations will interconvert or mix them. What we need are orbitals of the same symmetry as a metal s orbital in an octahedral complex, that is, al.. Suitable orbitals are to be found among g, i, k, and m orbitals (each of which provides one) and o orbitals (which provide two).
That is, we expect terms in Yp, Yo, Y8, YIO,Y12 (twice) and so on. Of these, only the first is of importance in the crystal field theory of d electrons and corresponds to the orbital g~.+~.+.. (Fig. 3). For f electrons the
i
i
+
Figure 3. (left) T h e " g d + .I+ .4"rpherical harmonic, Y4' 45/141~4'+ Y,-')I= 3/16& [ I 3 5 cor' 9 - 30 corn E 31 5 sin4 E cos' +]. There is only one negative lobe, consisting of eight interconnected pmtruberances, one in the center of each foce of the octahedron. Each protuberonce has of the lomplitudel of one of the lobes dong the coordinote ore%
+ +
'Ir6
Figure 4. (right) The "g,: E - 30 COOE 311.
+
spherical harmonic, Y
