MELDOLA MEDAL LECTURES* Molecular Shapes By J. K. Burdettt DEPARTMENT OF INORGANIC CHEMISTRY, THE UNIVERSITY OF NEWCASTLE UPON TYNE, NEWCASTLE UPON TYNE, NE1 7RU 1 Introduction For many years chemists have explored structural aspects of simple main group compounds guided by several simple theoretical tools that have proved invaluable. The Valence Shell Electron Pair Repulsion (VSEPR) model, devised by Sidgwick and Powell and consolidated by Nyholm and Gillespie, is a qualitative predictor of the angular geometry of an AHn or AXn system with a main group (A) central atom.’ The ideas of WaIsh,2 published twenty-five years ago, provided a simple molecular orbital rationale of these structures and, in addition, were able to predict the geometries of excited states which VSEPR could not do.One interest- ing feature of the two methods is that whereas the VSEPR scheme emphasizes electron4xtron interactions, and ignores central-atom-ligand interactions, the opposite is true for Walsh’s molecular orbital approach, which is just concerned with the changing magnitudes of central atom orbital-ligand orbital overlap on distortion. A linking piece in this structural jigsaw was provided by Bartell’s adaptation3 of the second-order (or pseudo) Jahn-Teller effect to structural main group chemistry. Shortly afterwards Pearson published4 his symmetry rules for the prediction of molecular geometry, which included and extended Bartell’s work. We shall find, however, that these well proven methods for looking at main group structures need to be replaccd when rationalizing the shapes of transition-metal complexes.Steric effects are often very important in influencing reactions and structures. Recently Glidewell5 has widely applied the ‘hard sphere’ ideas of BartelP to this area and the interplay of steric and electronic controls on geometry is one that we shall return to. Many sets of quantitative molecular orbital calculations have been performed at varying levels of sophisti- cation in efforts to calculate bond angles. It is, however, the purpose of this *These lectures were delivered in April 1978 at the Annual Chemical Congress, University of Liverpool.?Present address: Chemistry Department, University of Chicago, 5735 South Ellis Avenue, Chicago, Illinois 60637, U.S.A.R. J. Gillespie, ‘Molecular Geometry’, Van Nostrand-Rheinhold, London, 1972. ‘A. D. Walsh, J. Chem. Soc., 1953, 2260, and following papers. L. S. Bartell, J. Chem. Educ., 1968, 45, 754. R. G. Pearson, J. Amer. Chem. Soc., 1969, 91, 1252, 4947. C. Glidewell, Znorg. Chim. Acfa, 1975, 12, 219; 1976, 20, 113.’L. S. Bartell, J. Chem. Phys., 1960, 32, 827. I Molecular Shapes lecture to review methods which lead to an understanding of molecular shapes at a basic level. One may recall that in Hoffmann’s view ‘. . . to understand an observable means being able to predict albeit qualitatively the result that a perfcctly reliable calculation would yield for that observable’.’ 2 Shapes of Main Group Molecules VSEPR and Walsh’s Scheme.-The angular geometries of simple main group molecules are well matched by the predictions of the theoretical tools we have just mentioned.(The exceptions are of interest in themselves.) For the AH3 and AX3 molecules (A = B, C, N; X = halogen etc.) the VSEPR method correctly predicts BH3 to be trigonal planar (three pairs of electrons) and NH3 to be pyramidal (four pairs of electrons). The number of electrons which we use in the VSEPR count is all the valence (ns + np) electrons on the central atom plus (usually) one from each of the ligands. Where double bonding is possible between A and X (e.g. X = 0)only the electron from the (J part of the inter- action is included. In some cases two electrons come from each of the ligands.Thus C(PR3)2 contains two filled shell ligands (PR3) which contribute two electrons to the VSEPR count. With the four carbon valence electrons, a total of four ’electron pairs are included in the scheme, which rationalizes the non-linear structure* of the molecule (isolectronic with OF2). The methyl group with three and a half pairs provides a problem since how does half an electron pair behave? The Walsh diagram for the AH3 system is shown in Figure 1. (For AX3 the situ- ation is similar.) It shows how the valence orbitals of the unit qualitatively change in energy as the molecule distorts. Walsh arrived at the angular depend- ence of the orbital energies simply by considering in qualitative terms how the ligand-central atom overlap integrals changed on distortion. (This may be put on asemi-quantitative basis by applying to the main group case9 the ideas we describe below for transition-metal systems.) Parr has shown10 how valence bond methods arrive at similar results.For BH3 with two electrons in the la’l and four in the le’ orbital bending is obviously unfavourable. For NH3 with two electrons in la”2 an overall stabilization on bending is possible. With only one electron in this orbital (CH3), is the stabilization energy afforded this electron on bending sufficient to overcome the opposirigeffect of the lower energy elec- trons? Jordan and Longuet-Higginsll suggcsted that the radical would be planar, whereas Linnett and Poe12 suggested that it would be pyramidal. In fact the result of gas phase electronic spectral studies13 and e.s.r.resuItsl4 on matrix ‘I R. Hoffmann, Accounts Chem. Res., 1971, 4, I. a A. T. Vincent and P. J. Wheatley, J.C.S. Dalton, 1972, 617. J. K. Burdett, Structure and Bonding, 1976, 31, 67. lo G. W. Schnuelle and R. G. Parr, J. Amer. Chem. SOC.,1972,94, 8974. l1 P. C. H. Jordan and H. C. Longuet-Higgins, Mol. Phys., 1962, 5, 121. la J. W. Linnett and A, J. Poe, Trans. Faraday SOC., 1951, 47, 1033. G. Herzberg, Proc. Roy. SOC.,1961, A262, 291. l4 R. W. Fessenden and R. H. Schuler, J. Chem. Phys., 1963, 39, 2147. 508 Burdett . . ...... ... . .. . E \"A. 0 e-+ 35 Figure 1 Walsh diagram for an AH3 molecule within the C,,distortion co-ordinate isolated CH3 suggest that it is, or is very close to being, planar; the second row analogue SiH3, however, is pyramidal.l5 Matrix studies14916~1~ also show that substituted methyl radicals are pyramidal (e.g. CF3, CHF3, cc13). Thus on the VSEPR model three and a half electron pairs behave differently accord- ing to the substituents attached to the main group atom. But the vibrational data on matrix isolated CH3 are intrinsically very interesting. The radical was made initially in two ways [reactions (1)18 and (2)19]. vac U.V. CH, ___+ CH3 + H M + CH3X -+CH, + MX (MX = alkali halide) (2) When made by reaction (1) the out-of-plane bending mode (YZ) of the radical at 611 cm-l was associated with an unusual ratio of vz(H)/vz(D) (Table 1). This could be rationalized either by a bond angle smaller than that in NH3 in its R.L. Morehouse, J. J. Christiansen, and W. Gordy, J. Chem. Phys., 1966, 45, 1751. R. W. Fessenden and R. H. Schuler, J. Chem. Phys., 1965, 43, 2704. I' L. Andrews, J. Chem. Phys., 1968, 48,972; J, Phys. Chem., 1967,71, 2761. la D. E. Milligan and M. E. Jacox, J, Chem. Phys., 1967, 47, 5146. Is L. Andrews and G. C. Pimentel, J. Chem. Phys., 1967, 47, 3637. Molecular Shapes Table 1 Vibrational data for CH3 andperturbed CH3 radicals CH3 * * LiBr 730 1,288 normal positive anharmonicity CH3 * LiI 730 CH3 NaBr 700 CH3 -* NaI 696 I .305 CH3 * KI 680 negative an harmonici ty CH3 611 1.319 al.291 for harmonic oscillator electronic ground state (which is unlikely) or by a large negative anharmonicity of a planar structure.This type of anharmonicity is well documented for a small number of systems. (Amongst these is the out-of-plane bending mode of the first excited state of NH3 [(le’)*(la”~)~(2a’l)~],which is also planar20 and closely related to the present problem.) Intriguingly the radical formed in reaction (2) (MX = LiI) showed a higher value of v2 and a small (normal) positive anhar- monicity (Table 1). In order to understand this behaviour we need to look at the third approach to molecular geometry. CH3 and the Second-order Jahn-Teller Effect.-The strategy of this method uses the perturbation expansion of the energy of a molecule3~4~21-23 on distortion along a co-ordinate, Qi.If we write the perturbed Hamiltonian as equation (3) then from first- and second-order perturbation theory for the electronic ground state (10)) The first term in this energy series is the first-order Jahn-Teller term.24 It will be non-zero for (i) any electronic state if Xi is associated with a totally sym- metric mode, and (ii) an orbitally degenerate electronic state if ZZis associated with a mode which reduces the molecular symmetry.We usually ignore (i) and focus on the predictions for degenerate electronic states where the distortion 2o A. D. Walsh and P. A. Warsop, Trans. Faraday SOC.,1961, 57, 345. 21 R. F. W. Bader, Mol. Phys., 1960, 3, 137. H. C. Longuet-Higgins, Proc. Roy. SOC.,1956, A235, 537. 43 J. K. Burdett, Appl. Spec. Rev., 1970, 4, 43.2p H. A. Jahn and E. Teller, Proc. Roy. SOC.,1937, A161, 220. Burdett removes this degeneracy. A tabulation of the permitted distortions Qi for given geometries and electronic state is given by Jotham and Kettle.25 The second term (of order QP) in equation (4) is the one which will mainly concern us here and represents the force constant associated with the distortion. It consists of two parts, a ‘classical’ force constant, (g),andarelaxationpart-describing how the electronic charge distribution changes’or relaxes so as to reduce the overall force constant. If the summation is truncated at the first excited state In) then the force constant is given by equation (5). If there is a small Lk [control-led by the size of the highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) separation] then the relaxation term may be large and overwhelm the classical force constant.The resulting negative force constant implies that the molecule will spontaneously deform away from that geometry along the co-ordinate Qf, the actual choice of which is regulated simply by the symmetry properties of the ground and first excited electronic states. An alternative approach is the following. We need to find that distortion co- ordinate which will result in the HOMO and LUMO having identical symmetry properties (i.e. belong to the same symmetry species in the distorted molecule). As this distortion progresses the HOMO and LUMO will heavily mix together and the two energy levels will ‘repel’ one another.The lower energy component (HOMO) which contains one or two electrons will be stabilized by such an inter- action as shown in Figure 2. If the dynamics of the system are controlled by the energetic behaviour of the HOMO, then the molecule as a whole will be stabilized by such a distortion. From group theory we can determine the symmetry of Qi such that the inte- grand l(Ol#iln)12 may be non-zero. We may express the numerator in terms of the transition density+o*#n, where 40is the orbital in the ground state and #,, is the orbital in the excited state which hold the ‘excited’ electron. Figure 3(a) shows schematically how the symmetry species of Qt is determined for AH3 molecules (A = B or N) and also for ClF3. The direct product of the symmetry species of & and 9.must contain either a”2 or e’ for the molecule to distort, as shown in Figure 3(b). Hence the planar structure of BH3, pyramidal structure of NH3, and T-shape of ClFs are neatly rationalized. (It may be noted here that the latter geometry was one not mentioned by Walsh.) The same method can be used to reproduce the geometries of CF4, SF4, XeF4 etc., and usually similar predictions to those from VSEPR are obtained. However, this method does require that we have a knowledge of the MO structure of the symmetric geometry before we can begin. Note that in Figure 3(a) it is the lowest energy ‘transition’ which deter- mines the geometry. A higher energy transition (le’ +2a’l) would give rise to a transition density (and Qi)of species e’ for BH3, which in practice does not give rise to any static distortion.A manifestation of this contribution does however ** R.W.Jotham and S. F.A. Kettle, Inorg. Chim. Acra, 1971, 5, 183. 511 I Molecular Shapes A A€ t E I symmetric distortion co-ordinate- geometry Figure 2 The stabilization of the HOMO brought about by the distortion co-ordinate which causes the HOMO and LUMO to have identical symmetry properties turn up in the vibrational force field of the planar molecule.21 The stretching bond-bond interaction force constant is positive, which shows that the relaxation term of equation (4) is not zero but not large enough to overcome the classical constant.The result of variation of ligand electronegativity can be seen by its effect on the size of the energy gap dc. For the NXs case this will be largely set by the energy difference between la”2 and h’1. The 1a”a orbital remains approximately unchanged in energy but the orbitals involved in u interactions (e.g. &’I) drop in energy and dc becomes smaller. The driving force away from planar for NX3 species should then be larger as X becomes more electronegative. An alter- native way of looking at the increasing tendency for AX3 molecules to pyrami- dalize as the electronegativity of the X ligands increase is provided26 by the theory of isovalent hybridi~ation.~~ Briefly, the more electronegative the ligand X the more polarized the AX bond.In MO language this means that the AX bond will contain more A atomp-character. Increasing central atomp-character in a hybrid leads to smaller X-A-X angles (recall sp, 180”;sp2, 120”;sps, 109”28’;ps, 90”) and thus the molecule should be driven further away from planar as the total ligand electronegativity increases. CH3 has similar HOMO-LUMO properties to NH3 and a distortion from the trigonal planar geometry is predicted but with a value for the relaxation term of half2892Q that for NH3 (half the number of a”2 electrons). The radical is perhaps loJ. H. Current and J. K. Burdett, J. Phys. Chem., 1969,73, 3505. R. S. Mulliken, J. Phys. Chem., 1937, 41, 318; 1952,56,295. J. K. Burdett, J. Chem. Phys., 1970, 52, 2983. J. K. Burdett, J. Mol.Spec., 1970, 36, 365. Burdett CIF,BH3 e' e' ....6; a; -a: a:'+-.'## a: e'$0 X 9"-e" distortion none pyramid T-shape Figure 3 (a) Second-order Jahn-Teller approach to AH8 geometries; (b) the routes bywhich the and e' bending vibrations ofa trigonal planar AHSmolecule lead to pyramidaland T-shape structures rather tenuously planar with a vibrational frequency (VZ = 61 1 cm-l) much less than in BH3 (1 125 cm-l). As the ligand eiectronegativity increases (e.g. to CF3) the molecules become pyramidal in accord with the ideas described above for NF3. By means of the fourth-order perturbation term in the expansions of equations (3) and (4) we may tackle the problem of the anharmonicity of the planar radical (and that of planar NH3 in its first excited state).The anhar- monicity may be shown*s to consist of a 'classical' term and a relaxation term. I Molecular Shapes The latter is simply given by equation (6), which suggests that the larger the I<Ol*rln>l* .<nl&rrln> (6)A4 second-order ‘softening’ of the vibrational force constant, the larger the positive contribution to the quartic term in Qz which gives rise to the negative anhar- monicity. By a mechanism29 which we will not discuss here the effect of perturba- tion by alkali halides is to reduce the relaxation term and thus increase the vibrational force constant and frequency v2 with a commensurate replacement of the negative anharmonicity by a normal (small) positive one (Table 1).A molecule approachable along similar lines which also has a large second- order softening is XeFs, a fluxional molecule in the gas phase with a large ampli- tude and highly anharmonic flu bending mode.30 The VSEPR scheme would give the molecule (seven pairs) a much larger distortion than that actually observed. Steric Effects.-The importance of steric effects is illustrated by a single example. For many years the pyramidal structure of N(CH3)3 but planar geometry of N(SiH3)3 has been ascribed to .rr-bonding between the p,-orbital on the N atom and a Si d-orbital. However, recently Glidewell, using the ideas developed earlier by Bartell of intramolecular van der Waals forces6 and applied especially to hydrocarbon structures, has convincingly argued5 that the non-bonded repul- sions of the large SiH3 groups are large at the pyramidal geometry but are relieved at the planar. In support of this view is the fact that P(SiH3)3 is pyra- midal.The longer P-Si bonds compared with those of N-Si result in a less tightly packed environment. N(SCF3)3 also has a planar NS3 skeleton31 but with shortenedN-S bonds, which are suggested to arise throughn-bonding. The inter- play between steric and electronic controls on molecular geometry is clearly an interesting one. Summary.-The use of several different approaches leads to quite a good appreciation of the factors determining main group geometries and it is possible in some cases (e.g. CH3) to understand very simply rather subtle features of the potential energy surface associated with angular deformations. We have con- centrated on the geometries of covalently bound species. Structures which do not fit into the VSEPR scheme are also found for the more ionically bound members of the MX2series (M= Ca-Ba).In order to understand this behaviour and also some of the basic dynamics behind the operation of the Walsh diagrams, we refer the reader to a recent study by 3 Shapes of Transition-metal Complexes The Failure of Existing Models.-In contrast to main group chemistry, the struc- L. S. Bartell and R. M. Gavin, J. Chem. Phys., 1968, 48, 2466. 31 C. J. Marsden and L. S. Bartell, J.C.S. Dalton, 1977, 1582. 3a M. B. Hall, J. Amer. Chem. SOC.,1978. 100, 6333; fnorg. Chem., 1978, 17, 2261.514 Burdett tures of relatively few transition metal complexes are known in the gas phase. Most of the available ones are either octahedral [Cr(CO)s, MFs] or tetrahedral (vc14, TiC14). The structures of complexes in crystals should be interpreted with care since the influence of the medium is not well understood, and an observed geometry in the solid may not represent the lowest energy configuration of the free molecule. It is also probably true that transition metal structures are more susceptible to distortion in the solid state so that a wider variety of structures are found. For example, CuC142- may adopt the D2d or square planar geometry depending upon the counterion. Pressure also reversibly converts one into the other.A significant advance in the range of available transition-metal systems with a variety of co-ordination numbers and d-electron configurations was the production in low temperature matrices of binary M(CO), and M(N& com-plexes.33 Most of the quantitative structural determinations on these molecules have been performed in laboratories in Newcastle upon Tyne and Toronto. Some of this work is described by Poliakoff in the next article (see p. 527). The matrix does not seem to exert a strong force influencing the geometry of the molecule, and in many ways we may regard them as being pseudo gas-phase structures. Fe(CO)3 (1) and Cr(C0)3 (2) are both pyramidal molecules with bond angles determined using the band intensity method. They were both made by the careful matrix photolysis of the parent molecules Fe(c0)~~~ and Cr(CO)s.35 Ni(C0)3 (3) is a trigonal planar molecule.3~ Fe(C0)3 is probably a triplet species .,....... . . .. . . ... .. . . . . . . . ..... .. . ... . ...\A,,yfi,, -\ Fe Cr Ni [magnetic circular dichroism studies on Fe(C0)4 (see p. 531) show it to have S = 11 and we shall refer to its electronic configuration as hs d8(hs = high spin); Cr (co)3 is certainly 1s d6 (Is = low spin). With three ligand (T pairs the molecules Cr(C0)3, Fe(C0)3, and Ni(C0)3 have a total of six, seven, and eight pairs sur- rounding the central atom, respectively. It is possible to find VSEPR polyhedra to rationalize their shapes (octahedron for six pairs, CsVcapped octahedron for seven pairs, and distorted square antiprism for eight pairs) but these polyhedra may not be used to rationalize other transition-metal structures; CI(CO)S also with eight pairs is a square pyramid3' which does not fit into the square anti- 33 J.K. Burdett, Cuurd. Chem. Rev., 1978,27, 1. M. Poliakoff, J.C.S. Dalton., 1974, 210. s6 R. N. Perutz and J. J. Turner, J. Amer. Chem. SOC.,1975,97,4800. s6 R. L. DeKock, Inorg. Chem., 1971, 10, 1205. ST R. N. Perutz and J. J. Turner, Inorg. Chem., 1975, 14, 262. I Molecular Shapes prism concept. Gillespiel suggested that with these carbonyls the d-electrons should be neglected and just the number of ligand CJ pairs included. Although this correctly predicts Cr(C0)a to be octahedral and Ni(C0)4 to be tetrahedral, all three tricarbonyls should be trigonal planar according to this model, which is not the case.The Jahn-Teller theorem may often be used to rationalize the observed geometries of transition metal complexes, although as a predictor of molecular shape it is usually not very specific. One area where this approach fails is for those cases where a distorted geometry is found but the highest symmetry structure is not orbitally degenerate and therefore is Jahn-Teller stable. Examples of this type are found in both Cr(C0)3 and Fe(C0)3. For application of Jahn-Teller arguments we need a MO diagram for the D3h structure. Two extended Huckel calculations38J9 have derived slightly different results as to the order of the energy levels in the trigonal planar structure [Figures 4(a) and 4(b)].We note that both a Is d6and hs d8 system on the scheme of Figure 4(a) would be Jahn-Teller unstable, but distortion to a T-shape (not to a pyramid) is predicted on group theoretical arguments.25 On the scheme of Figure 4(b) both molecules would be Jahn-Teller stable. On both schemes a pyramidal (CsV)geometry is unlikely for the low spin d8 Fe(C0)3 molecule since here it would be Jahn-Teller unstable. On either scheme the Cr(C0)3 and Fe(C0)3 molecules are predicted to be unstable on second-order Jahn-Teller grounds. The distortion co-ordinate is of species e’ which predicts a distortion to a T-shape, which is patently not the case. The higher energy transition d‘ -e’ does give rise to a transition density of species a”2 for the Cr and Fe examples but the rule discussed in Section 2, p.508 required that the lowest energy transition was most important. Here the structural predictions of the second-order Jab-Teller effect are not reliable. This does not mean, of course, that the perturbation approach to the analysis of MO energy changes on distortion is in general invalid. It does mean, however, that the energy changes associated with a group of valence orbitals on distortion must be considered rather than that associated with one orbital in particular. For Ni(C0)3 the e” e’ transition is not allowed since the 4 d manifold is full (dlo). Figure 4 also shows some quantitative calculations of the orbital energy changes on distortion.We can readily see that double occupation of the 4’1 orbital in both Is d6 and hs d8 molecules strongly encourages pyramidalization. The smaller distortion for Fe(C0)3 away from the trigonal planar geometry com- pared with that for Cr(C0)3 is simply understood since here there is double occupation of the e’ orbital which is destabilized on distortion. In Ni(C0)3, four electrons in this orbital ensure planarity. Figure 4(c) shows that distortion to a T-shape from the trigonal plane is also favoured for the electronic configurations Is d6, hs d8.Do we need then to rely on quantitative calculations in order to pre- dict molecular geometry? The most comprehensive sets of calculations on these angular geometries have used the extended Huckel method.In general these reproduce the observed geometries with remarkable fidelity. Both sets of 38 J. K. Burdett, J.C.S. Furaday fI, 1974, 70, 1599. 3s M. Elian and R. Hoffmann, Inorg. Chern., 1975, 14, 1058. 516 ...... . . . . . .. . . . eA88 e' e" 0 120 H-'Figure 4 MO energy level diagrams for bending within; (a) C,,co-ordinate, from ref. 39; (b) C,, co-ordinate,from ref. 38 ;(c) C,,co-ordinate, i&fromref. 38 Cr 4 .x I Molecular Shapes calculations predict a D2d geometry for the d9 M(C0)4 species but with a CaV geometry close in energy. Both forms have been made in low temperature matrices. Similarly although (hs d8)Fe(C0)4 is predicted and observed to have a CZ,structure, a CsV or Csgeometry lies close in energy above it.As described in the following article, laser i.r. photolysis experiments confirm a thermal re- arrangement pathway via a transition state of this symmetry. For the tricarbonyl series 8is predicted to be: 30", 33", [obs. 25" for Mo(CO)3]; 17", -, [obs. 18" in Fe(C0)3]; 0,8", [obs. 0"in Ni(C0)3] from the two sets of calculations described in refs. 38 and 39, respectively. There is clearly a need for a simple model with which to view these structures. The Angular Overlap Approach.-Our simple molecular orbital approach is based on the angular overlap model (AOM),40-42 which has been used mostly in the past in the interpretation of the electronic spectra and magnetic properties of transi- tion-metal complexes. Basically it provides the energies of the (mainly) transition metal d orbitals in an MLn complex of given geometry in terms of two para- meters, one describing cr-and the other v-type interactions.[Similar to the d or Dq of the crystal field theory (CFT).] Once these energies have been obtained then the weighted sum of the d-orbital energies (weighted by the number of electrons in these orbitals) as a function of the molecular geometry provides the oppor- tunity to explore the configurational potential surface and find the most stable geometry demanded by metal d-ligand interactions for a particular electronic configuration. The AOM is based on an approximation involving the inter- action energy between two orbitals (42, +,) on different atoms. Here +t rep-resents a metal d-orbital and +* a single ligand orbital or symmetry-adapted combination.The stabilization energy E of the bonding component may be written as a perturbation sum [equation (7)] where k is a constant, Sij is the overlap integral between~$iand +j, and deij their unperturbed energy separation. Inthe following discussion we shall concentrate on the leading term in the expansion with an occasional reference to the others. The stabilization of the bonding combination becomes E = /3,Sij2, where PA is k21Aej (A = CT, T,etc.). In our simple model it is also assumed that the destabilization energy of the antibonding orbital is equal to the stabilization energy of its bonding partner. The tremendous power of the model lies in the fact that the SZ~are, in general, dependant on simple geometric expressions as the angular metal-ligand geometry is adjusted while maintaining the same bond length. This means that the following calculations may be simply performed using 'back-of-envelope' computations.Table 2 gives functions for overlap of a ligando orbital (located at the polar position 8,#)with the d-orbitals, *O J. K. Burdett, Adv. Znorg. Chem. Radiochem., 1978, 21, 113. Q1 C. E. Schaffer and C. K. Jerrgensen, Mol. Phys., 1965, 9, 401. rsC.E. Schaffer, Struct. Bonding, 1973, 14, 69. 518 Burdett Table 2 Angular dependance of ligand a-metal d-orbital overlap integral as a junction of the polar co-ordinates of the ligand d-orbital S 1-(3H2 -l)Su2 3+ x2 -y2 -(F2 -G2)Su2 xz 3*FHS, YZ 3*GHS, XY 3+FGS, F = sinOcos4, G = sinOsin4, H = cod and shows some of the overlap integrals which are particularly useful.Thus the interaction energy of a ligand a orbital with the z2orbital is given by the function 1 E = /3,Su2 -(3cos28 -1)2, where 18, is introduced as the proportionality 4 constant describing a-type interactions. The notation used is that of Kettle;43 a simpler way of expressing these values is to put puSa2 = eu, see for example ref. 41. V-Type interactions are described by analogous equations but we will concentrate on a-type interactions in our discussion since these are generally considered to be of large magnitude. We are thus in a position to be able to write the interaction energy of a pair of orbitals as E = hBuSg2,where h is a calculable number and the product &Sa2 the AOM parameter.One way of evaluating the interaction energy is to write down a symmetry-adapted ligand CT combination and calculate its overlap integral Stj with the relevant d-orbital. A quicker method of calculation for an MLn complex is to use the ligand additivity [equation (8)] over all n ligands co-ordinated to the central metal atom. The total a stabilization energyC(a) is then very simply given by equation (9),wherehi is the number of electron holes in the ith d-orbital. Equation (9) arises simply because 43 S. F. A. Kettle, J. Chem. SOC.(A), 1966, 420. I Molecular Shapes when filling d-orbitals with electrons we are filling metal-ligand antibonding orbitals. It is only the empty d-orbitals that have filled metal-ligand bonding counterparts which contribute to the stabilization energ~.4~,~5 An interesting sum rule [equation (lo)] applies to the orbital energies derived from equation (8), CEi = n&,s,Smz (10)i where n is the number of (5 orbitals (ligands) surrounding the central atom.This can be used to check the arithmetic involved in evaluating orbital interaction energies. An exactly analogous prescription applies to the evaluation of .rr-bonding interactions. Care must be taken here to distinguish betweenv-donors (the mainly ligand located components are ML bonding but the mainly metal d-orbitals are ML antibonding) and n acceptors (the mainly metal d-located orbitals are ML bonding and the mainly ligand-located orbitals, which are usually empty, ML antibonding). The d-orbitals are destabilized in the former but stabilized in the latter case.The AOM is then much easier to apply than the crystal field method, especially in lower symmetry environments. The CFT also differs from the drawback that (T,Tbonding, which are vital concepts in modern inorganic chemistry, cannot be included in the model. Another problem with the CFT is that in lower than cubic environments two parameters, Dq and Cp are needed to describe the energy levels. The two are related via a parameter p. In most places where the CFT is used in low symmetry situations a value of p = 1 is arbitrarily used.46 From spectroscopic measurements larger values are probably more accurate but the factors governing the exact choice of p in different environments is far from clear.Figure 5 shows the d-orbital energy levels of geometries of interest obtained from simple calculations involving the overlap integrals. (A slight complication occurs in the T-shape geometry where two d-orbitals have the same symmetry, and allowance for mixing of these orbitals needs to be made.47) Geometries of Complexes.-We have shown elsewhere that the variation in the heats of hydration of the M2+ ions across the first row transition-metal series, one of the classic successes of the CFT, is similarly described by our molecular orbital mode1.46 The forces contributing to are the metal nd-ligand interaction augmented by contributions from ligand interactions with metal (n + 1) s,p-orbitals.These observations suggest that the observed angular geometry will be a balance between that demanded by s,p interactions and that by the d-orbital interactions with the ligands. In general an MLn complex contains no pairs of electrons which are involved in s,p (and d) interactions. 44 J. K. Burdett, Inorg. Chern., 1975, 14, 375. 4bJ. K. Burdett, Znorg. Chem., 1976, 15, 212. p6 J. K. Burdett, J.C.S. Dalton, 1976, 1725. 47 D. S. McClure, in ‘Advances in the Chemistry of the Coordination Compounds’, ed. S. Kirschner, Macmillan, New York, 1961, p. 498. 520 Burdett 22, x2 -y2 x2-Y2 3 x2-y'Z22.75 22 x2 -y21.125 1 22 5, &h5,c4v '22' 2.5 Z2 2.37 Z?X2 -y* xy, yz, xz 1.5 XY 1.5 1.33 'x2 -y2'0.63 4,c2v 3,c2v 3,C3" Figure 5 A40 diagrams in the d-orbital region for some geometries of interest (energy units #luSoaor eu).The co-ordination number and molecular point group are given under each diagram. In the geometries with an angular degree offreedom (e.g. 4, C2v)the n ligands are placed at the vertices of an octahedron such that LML angles are either 90"or 180" Using the VSEPR method (which often dealt successfully with the shapes of molecules with s,p-orbitals alone on the central atom) the geometry demanded by interactions with these higher orbitals will be trigonal planar (&I,) for ML3 and tetrahedral (Td)for ML4 etc. We note that in these molecules the VSEPR geometry is the one with minimum ligand pair repulsions (Pauli avoidance) and also the one containing minimum non-bonded repulsions between the ligands themselves.We call these combined forces, ligand-ligand terms. From Figure 5 we can readily calculate the d-orbital stabilization energy for the various three-co-ordinate geometries which we include here as a function of d-electron configuration. For Is de, Is d8, hs d8, and d10 the results are given in Table 3. There is of course no d-orbital stabilization for Ni(CO)3 at any geometry since all bonding and antibonding orbitals are filled. For the two other mole- cules the T-shape and 'octahedral fac trivacant' (& pyramid) geometries have the same d-orbital energy. We need to turn to the fourth-order term in equation (7) to resolve them.The general result is that the structure with the largest number of cis ligands is more stable. (If the ligands are tl acceptors, tl stabiliza-tion is also maximized at this geometry.) The driving force away from the D3h geometry is larger for Cr(CO)3 (1.5pAc2) Cr (C0)3 than for Fe(C0)3 (0.75/?uSu2). has the structure which is most distorted (0 = 25") from trigonal planar. Ni(C0)3 521 4 Molecular Shapes Table 3 d-Orbital stabilization energies for some three-co-ordinate structures (unitsP# or e,) d-electron configurationa C3Ub D3A CZV" example Is d6 (22200) 6.0 4.5 6.0 Cr(CO)3 hs d* (22211) Is d8 (22220) 3.O 3.O 2.25 2.25 3.O 4.73 Fe(CO)3 Rh(P Ph3)3+ d9 (22221) 1.5 1.125 2.37 d1O (22222) 0 0 0 Ni( C0)3 ad-Orbital occupation numbers in parentheses, lowest energy orbital first; bLML angles 90" (octahedral fac 'trivacant'); CLML angles 90°,180"(T-shape) has no driving force away from the D3h geometry and thus remains planar, held there by ligand-ligand forces.Recently the T-shaped structure predicted for the three-co-ordinate Is d8 ML3 system has been observed48 in a crystallographic environment for Rh(PPh&+. All the observed geometries are in encouraging agreement with those predicted. Of course all we have done really is to calculate, using a simple model, the relative energies of the e and a1 orbitals shown in Figures 4(a) and (b) compared with those of the orbitals of Figure 4(c).Similar arguments may be used to understand the geometries of four-co- ordinate molecules,44 using Table 4. Cr(C0)g with a larger driving force away Table 4 d-Orbital stabilization energies for some four-co-ordinate structures (units &SUBor e,) d-electron configuration Dela Ta c2va example 1s d6 (22200) 8.O 5.3 8.O Cr(C014 Is d8 (22220) 6.0 2.67 5 .O Ni(CN)d2-hs d8 (22211) 4.0 2.67 5 .O Fe(C0)4 d9 (22221) 3.O 1.33 2.5 cuc142-d10 (22222) 0 0 0 Ni(C0)g aLML angles 90",1SO" (octahedral cis 'divacant') from tetrahedral than Fe(C0)4 gives the more distorted geometry [the cis-divacant is more stable than the square planar for this configuration by consider-ing the fourth-order terms of equation (7)]. Ni(CN)42- is found as the square planar molecule but the d9 system CuC1g2- has a smaller driving force from tetra- 4aY.W.Yared, s. L. Miles, R. Bau, and C. A. Reed, J. Amer. Chem. SOC.,1977, 99,7076. Burdett hedral and is sometimes found in square planar and sometimes in D2d environ-ments49 (Figure 6). The reluctance of the 1s d8 square planar geometry to add Is d6 hsd6 lSd7+ D4h Isd8 hsd8 d9 4 D4h Figure 6 Observed geometries of four co-ordinate molecules as a frtnctiorl of'd-electron configuration. Examples: ds, Cr(C0)4 (Is), FeC142-(hs); d7 Rh {S2C2(CN)2}22-(Is),COCI,~-(hs); d8 Ni(CN)42-(Is), Fe(CO), (As); ds CUC~,~-(a variety of geometries are found for this electronic configuration); d10 Ni(CO),, two more ligands to complete an octahedron is another structural feature of these molecules that we can view using our method.50 In addition the kinetic behaviour of ligand substitution in this system (trans effect) is another field where this simple parametrized model is succe~sful.~~ For five-co-ordinate molecules, ground state Cr(C0)5 (Is d6)has the largest stabilization energy for the square pyramid compared with trigonal bipyramid geometry.Fe(C0)5 with only a small distortion energy is found as the trigonal bipyramid although it is fluxional, probably via a square pyramid transition state. The trigonal bipyramid is predicted for the first excited state of Cr(C0)5 4s J. R. Ferraro and J. Long, Accounts Chem. Res., 1975, 8, 171. J. K. Burdett, Inorg. Chem., 1975, 14, 931. 51 J.K. Burdett, Inorg. Chem., 1977, 16, 3013. I Molecular Shapes Table 5 d-Orbital stabilization energies for some five-co-ordinate structures (units &Sa2 or e,) d-electron configuration C4v D3n diference example 1s d6 (22200) Is d6(221 10) 10.0 8.0 7.75 7.75 2.75 0.25 Cr(C0)5 Cr(C0)5* 1s d7 (22210) 8.0 6.625 1.375 Mn(C0)5 Is d*(22220) 6.0 5.5 0.5 Fe(CO)5 since the driving force away from this geometry is smaller than for Fe(C0)5. Matrix experiments with this molecule using polarized spectroscopy and photo- lysis, Scheme 1,52 and visible photolysis studies on Cr(C0)4CS, Scheme 253 show that this is very probable. Scheme 1 pseudorotated cs * ;ic =Ahv' axial CS Scheme 2 basal CS Thus the idea of a balance between VSEPR steric and d-orbital demands in controlling the shape of the molecule is a very satisfying one.A general observa- tion is that the larger the d-orbital driving force away from the ligand-ligand determined geometry the closer the observed geometry is to the d-orbital only prediction. Thus Ni(CN)42- (driving force = 3.3 PuS,2) is a regular square plane, Cr(C0)4 (2.25 puSu2)and Cr(C0)5 (2.33 &,2) have bond angles close to 90" 6a J. K. Burdett, J. M. Grzybowski, R. N. Perutz, M. Poliakoff, J. J. Turner, and R. F. Turner, Inorg. Chem., 1978, 17, 147. 6a M. Poliakoff, fnorg. Chem., 1976, 15, 2022, 2892. 5 24 Burdett and 180" but Cr(C0)3 (1.5 pgSg2)is less than two thirds the way to the fac octahedral trivacant structure.Steric Effects.-Table 6 shows the result of calculations designed to reveal the relative stability of a series of eight-co-ordinate geometries.54 Extended Huckel Table 6 geometry electronic stabilizationa steric energy EHMO energyfrom AOMof do results on L& MLs /kcal mol-l dodecahedron 77.2 3.5 square antiprism 76.4 Ob square prism (cube) 85.2 27 hexagonal bipyramid 86.8 97 CsVbicapped trigonal prism 71.4 166 C2, bicapped trigonal prism 75.2 24 %nits are k4Su4(Aqj)-) = yuSu4 from equation (7); *i.e. on steric grounds the square anti- prism is the most stable geometry calculations on the L@-system itself (MLs but without the metal atom) gave the relative steric (a contributor to the ligand-ligand terms above) merits of the various structures.Evaluation of the fourth-order terms of equation (7) gave the electronic advantages for each geometry [since these are included with a minus sign in equation (7) the smaller this contribution the more favourable the structure]. From the sum rule of equation (10) the second-order terms are equal for all geometries if we assume do configurations. The superposition of the two series gives a good description of the popularity of the various structures. There are a large number of dodecahedra1 and square antiprismatic structures-good on both steric and electronic grounds. An increasing number of CzVBTPgeometries are being identified as a result of the use of various crystallographic shape parameters. These geometries and intermediate versions make up the vast majority of eight-co-ordinate examples.The cube and hexagonal bipyramid are not very good on either basis; only three examples of the former are known and a handful of hexagonal bipyramid structures if the special case of U02 containing systems is excluded. The D3h bicapped trigonal prism is a combination of excellent electronic but very poor steric stability. There are no characterized examples with transition metal ions. Steric effects are therefore clearly important in the geometry field, especially with the higher co-ordination numbers, and may often work against electronic factors. The molecular mechanics results of Kepert55 and the existence of small co-ordination number molecules with bulky O4 J.K. Burdett, R. C. Fay, and R. Hoffmann, Znorg. Chem., 1978, 17, 2553. 66 D. L. Kepert, Progr. Znorg. Chem., 1977, 23, 1. Molecular Shapes ligands as demonstrated by Bradley56 also strikingly reveal the importance of these non-bonded effects. Finally we must also mention here Johnson’s intriguing method57 for deter- mining the stereochemistry of Mn(CO)mspecies, the elucidation of the number and position of terminal, doubly, and triply bridging carbonyl groups, and the geometry of the metal skeleton. The spatial arrangement of the CO groups is found to be one of the close packed arrangements of mspheres. The metal atoms are located at positions set by the best arrangement of an Mnpolyhedron within this structure. The resulting geometrical relationships between each M and a given CO determine whether the latter is in a terminal, doubly, or triply bridging position.This essentially steric argument is the first theoretical method which is able to predict polynuclear carbonyl stereochemistry with any success. It remains to be seen if a MO alternative can be developed. I would like to thank my friends and colleagues who, over the years, have provided a stimulating environment in which to work, especially J. J. Turner and M. Poliakoff for a period of particularly exciting interaction in which many new ideas were conceived. 66 D. C. Bradley, Chem. in Britain, 1975, 11, 393; P. G. Eller, D. C. Bradley, M. B. Hurst-house, and D. W. Meek, Coord. Chem. Rev., 1977, 24, 1. 67 B. F. G. Johnson,Chem. Comm., 1976, 211.