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Although it is now generally acceptedth at the ligand-field model of th e electronic structu reof tren sitio n metal complexes is much more acc urat ethan t ,hnt provided by crystal f ield theory, the latterremains of paramount importance. There are twomaiu reasons for th is. I h t , if quant it ies which arewell-defined in simple cry stal field theory are allowed tobecomc paramet,ers, theu the effects of covalency mayhe allowed for. Th e second reason for the continuedinterest in c rystal field theory is th at i t may be used topredict &orbital splitting patterns which are useful inthe interpretation of (1-l-d hands in electronic spectra.Exam ples of this are th e spec tra of complexes in whichthe coordination number of a transition metal ion isfive or seven, a field in mhich there is much currentinterest. Despite the assumptions inheren t in simplecryst,al field theory, it appears that it usually predictscorrectly the relative energies of the &orbitals and,wh at is more, their relat,ive separations a re also pre-dicted fairly accurately.The calculation of a d-orhit ial spli t t ing pattern,within the crystal field approximation, falls into twopa rts (1). Th e first step is to obtain an expressionfor the electrostatic potential generated a t th e metalion by a suitable array of point charges (or dipoles)which represent the ligands. T he second step is touse this potential field to de termine th e relative energiesof the &orbitals. Th is second step has been discussedin THIS J O U R N A LI), and so in the present art icle weshall confine our discussion to a qualitativ e considera-tion of the form of the potential function.T h e problem we shall first consider is th at of findingan expression for the crystal field potential around anato m mhich is a t the center of a n octahedral arrange-m en t of six identic al ligands.Because me are interested in what happens aroundthe metal a tom it is convenient t o express the potentialin terms of some function or functions which arecentered on the metal atom . Th e most convenientfunctions ar e the spherical harmonics.Spherical harmonics are the functions that oneusually draws when drawing a picture of an orbital.As the reader may have discovered, there are severalways in which a p orbital , say, may be drawn. Tw oways arc shown in I'igurc 1. Suppose that f rom thenuclcus of t he adorn contain ing th e p orbital we imaginethat therc is a large numher of lines drawn radiallyoutwards, so that the atom resembles a rolled-uphcdgchog or porcupine. W c measure the amp litudeof thc 71 orhitel along each of t he lines and t hen markof f IL lcngt,h from ihc nuclcus proportional to this:~mplit.udc. l 'hcsc mark s would he found t o definetho surr:~oosof t,wo sphcrcs i n contact, as shown ini A T h e ir , orbital shown in I'igure 1B is a

5. F. A . KettleTh e Universi ty

Sheffield, S3 7HF, England

contour diagram in which points with the same 1 am-plitudel are joined (on ly if it is th e con tour of zeroamp litude should th e two halves of the orb ital touch).An im por tant p ractical difference between Figures1A and 1B is that the latter can only be drawn if aradial function is specified. Th e form of diag ramssuch as Figure 1A does not depend on the radial partof th e orbital wavefunction. Of these two representa-tions of p orbitals the former is equally valid as apicture of the spherical harmonic which is labelledYS0 .

Crystal Field Potentials

- .To return to the crystal field potential. From thepoi nt of view of th e metal ion th e first, very crude,approximation to the six surrounding negative chargesis to regard them as being uniformily smeared o utover the surface of a sphere. T ha t is, the f irst sphericalharmonic in the expansion is th at corresponding t o a ns orbital (Yoo). his term is multiplied by a factor,k,, which, among other things, is proportional to then um b er o f lig an ds ( e u m be r of po in t c ha rg es ).This is easily understood, for the charge density onour spherical shell will increase in direct proportion toth e number of poin t charges.What of the spherical harmonics characteristic ofp orbitals? As Figure 1A shows their presence wouldindicate that the potential along the +z axis differsfrom spherical symmetry (due to the Y oo erm tha twe already have) in the opposite way to th e potentialalong the -z axis (and similarly +x a nd -x; +yand -y). But, by symmetry, for an octahedralcomplex the potential along the +z axis must equalthat a long -z (and that a long x, -x, y, and -y).We conclude that there will be no p orbital sphericalharmonic contribution to the potential . Bu t exactlythe same reasoning excludes all spherical harmonicscorresponding to orbitals of u symmetry. This

Figure 1. Tw o representations of t h e somep orbital.

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means that we may exclude j-, h- , j-, . . . ,orbitalspherical harmonics and confine our attention to thosecorresponding to d-, g-, i-, . . ,orbitals.Of the d orbitals we may a t once exclude thosespherical harmonics corresponding to the d,,, d,,,and dnE,,rbitals. There must be identical potentialsa t corresponding points in the four quadrants of the xyplane, for example. The effect of a contributionfrom the d,, spherical harmonic, however, would be toincrease the potential in two of these quadrants anddecrease it in two others, and so this harmonic isexcluded. Th e d,. and d+,. spherical harmonics mayalso be excluded. That corresponding to d,z woulddecrease the potential in the xy plane and increase italong the z axis (relative to the Yoo term). This, ofitself, is not a valid reason for excluding it, for someother harmonic may be able to compensate for it andmake the potentials at +x, -x, +y, and - y equal tothat at +z and -2. Only the d,2-r. harmonic is avail-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-orbitalharmonics will occur in the expansion.At this point is it worthwhile to stop and considerthe form of the sort of spherical harmonic for which weare looking. It is one such that all correspondingpoints in the octahedron will experience the samepotential. That is, all of those symmetry operationswhich turn an octahedron into itself will turn thespherical harmonic into itself. Evidently, no sphericalharmonic corresponding to a degenerate set of orbitalswill satisfy this condition for it is always the casethat for such orbitals some symmetry operations willinterconvert or mix them. What we need are orbitalsof the same symmetry as a metal s orbital in an octa-hedral complex, that is, al.. Suitable orbitals are to befound among g, i , k, and m orbitals (each of whichprovides one) and o orbitals (which provide two).

X (0) YFigwe 2. A, The "d,? spherkol harmonic. YZ0I= l/1457;;(3 cosPB- I,mgles being defined or g h o w n e nsert. B , The " d A , ~ "phericalharmonic Ys2+ Y2cs = '/%15 / 2 r s i n 9 E cor 281.

That is, we expect terms in Yp, Yo, Y8, YIO,Y12 (twice)and so on. Of these, only the first is of importance inthe crystal field theory of d electrons and correspondsto the orbital g ~ . + ~ . + . . (Fig. 3). For f electrons the

i i

Figure 3. (left) Th e" gd + .I+ .4"rpherical harmonic, Y4'+ 4 5 / 1 4 1 ~ 4 ' +Y,-')I= 3/ 16 & [I3 5 cor' 9 - 30 cornE + 31 + 5 sin4 E cos' +].There is only one negative lobe, consisting of eight interconnected pm-truberances, one in the center of each foce of the octahedron. Each pro-tuberonce ha s 'Ir6 f the lomplitudel of one of the lobes don g the co-ordinote ore%

Figure 4. (right) The "g: spherical harmonic, Y

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of the ligands as being represented by point charges.Other models ma y be used (point dipoles, for example).The only difference would be to alter the numericalvalues of the 16s in the abov e expression. Since theIc's are, in an y case, best regarded as param eters, weshall not discuss t ,hem further beyond noting th at theyar e also function s of th e metal-ligand s epar ation .We now consider complex es of low er-than-octah edra1symmetry. Th ere are two ways by which we mayproceed. Eith er we may appeal to a physical pictureor use formal group theory. Th e former has conceptualadvantages but the latter has a smaller probabilityof error. As in our discussion of octah edral complexes,we shall use the app roaches successively.Consider a tetragonally elongated octahedral complex(Fig. 6). The potential is equal at +x, -x, +y, and-y. Along the z axis the potential will be smallerthan along x and y b u t t h a t a t +z is still equal tot h a t a t -z. There are two ways in which we mayadjust the expression given above for the octahedralcase in order to adapt it to the present situation.The simplest is to recognize that if we add a sphericalharmonic corresponding to -4% Fig. 2A ) we decreasethe potential along +z an d -z and increase it along+z, -x, +y, and -y. To compensate for this latterincrease we must decrease the co ntrib ution of th eYo o term (th at is, decrease k ~ ) . Bu t we would expectto do this anyhow , for the tetragon al elongation would,in th e limit, give us a square planar complex for whichthere would be four point charges instead of six, andwe hav e already recognized th at Ic, is proportional toth e number of point charges.The second way of allowing for the elongation arisesfrom our dissection of the [Yd D+ -\/5/14(yp4 +Y4-"l term. If the mixing coefficient n this expressionwere greater than d2/i/14he potential would begreater at +x, -x, +y, an d -y t h an a t +z and -2.We conclude th at th e elongation will "unmix" th eYdo and (Y44 + Y~F ') terms. This separation isrelated to a feature which has been discussed in THISJOURNALn the context o f Jorbitals ( 8 ,S ) bu t which is ageneral featu re of orbitals of high nodality. Namely,th at a "cubic" set of orbital may have to be replaced

by another set when moving to a lower-symmetryenvironment.We conclude that the crystal field potential functionfo r a tetragonally elongated octahedral complex hasthe general formV' = k,'Y." - k,'Y," + k,'Y,' + k,'(Y,' + Y,c4)+ . . .

where, for simplicity, we have only included terms th atwill cause splittings between d orbitals. Th is is alsothe general form of the potential for square planarcomplexes.I n general, terms will appear in the potential corres-ponding to all orbitals transforming as the totallysymmetric representation of the corresponding pointgroup. T ha t is, orbitals which have symm etry a,

al, ,', al. (as approp riate). These are usually listed insets of ch aracte r tables (for example in the conv enientcompilation by Cotton (4). Data for f orbitals(and hence the corresponding spherical harmonics)has recently been published (6). We note at thispoint that terms corresponding to p and f orbitals m ayonly occur for complexes lacking a center of sym-metry. No complete set of d at a is available fo r gorbitals, but a check to see whether any terms occur(other than those discussed above) is provided by theuse of correlation tables (6) together with the infor-mation that the set of nine g orbitals transform asal , + e, + 11, + tl, under the operations of the O hpoint group. In addition, some limited da ta is avail-able on g orbitals in th e literature (7).Literature Cited(1) COMPINION, . L., A N D K O MA B Y N SK Y ,. A., J. CHEM.E o uc , 41 ,257 ( 1964) .(2 ) FR I E D MA N ,. G., CHOPP IN,. It., AND FEUERBICHER,.G .J. CHEM. DUC., 1 ,354 (1964) .(3) BECKICR,., J. CHE M. DUC., 1 ,35 8 (1964) .(4 ) COTTON,F. A,, "Chemical Applications of Group Theory,"Interscience, (division of John Wiley & Sons, Ino.) NewYork, 1963.ib ) KETTLE.. F. A, . A N D SMI T H , . J.. J . Chem. Soc. (A), 688,~ . (1967j.(6) WILSON, . B. , DECIZTG,. C ., A N D CROSS,P. C., "MolecularVibrations," McGraw Hill, New York fi955.( 7 ) A L T MA N N ,. L., ~ O C .ambs. Phil. SOC., 3, 343 (1957), andreferences therein.

Figure 6. A tetragonolly extended octahedron.Volume 46,Number 6, une 1969 / 341