R. .I. Gillespie McMaster University Hamilton, Ontario, Canada
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1
The Electron-Pair Repulsion Model for Molecular Geometry
Despite the many recent developments in the application of molecular orbital theory to the problem of understanding and predicting molecular shape the concept of localized electron pairs first proposed by G. N. Lewis has continued to be a most useful concept. A localized electron pair may be regarded as an electron pair that occupies a localized molecular orbital that extends over only one or two atomic centers although in some electron-deficient systems it may occupy a three or even a four-center orbital. Localized Molecular Orbitals and Electron Pair Arrangements
According to the localized electron pair (or VSEPR) theory' the arrangement of the bonds around any one atomic center depends on the number of electron pairs surrounding this atom, and since each electron pair occupies one localized molecular orbital it depends on the number of localized molecular orbitals in which the atom participates and on the relative sizes and shapes of these orbitals (1-3). The preferred arrangement of a given number of electron pairs in the valency shell of an atom is that which maximizes their distance apart. This can be regarded as a direct consequence of the Pauli exclusion principle, according to which only electrons of opposite spin are allowed to occupy the same region of space while electrons with the same spin must keep apart (4). Each pair cff electrons thus occupies a reasonably well-defined region of space and other electrons are effectively excluded from this space. Hence localized electron pair orbitals behave as if they repel each other, and they adopt that arrangement which maximizes their distance apart. For electron pairs in the same valency shell the arrangements for 2-12 electron pairs which maximize the least distance between any two electron pairs are given in Table 1 (3, 5). It is now well known that these electron pair arrangements enable one to predict the general shape of any molecule AX, simply from the number of bonding and nonbonding electron pairs in the valency shell of the central atom A (1-3). For example, four electron pairs adopt a tetrahedral arrangement. If there are two bonding and two nonbonding pairs (AXzE,) the molecule is angular, e.g., H1O; if there are three bonding pairs and one lone pair (AX,E) the molecule is pyramidal, e.g., NHa, and if there are four bonding pairs (AX4)the molecule is tetrahedral, e.g., CH4 (Fig. 1). The assumption made in deriving Table 1 that the Presented at the 154th Meeting of the American Chemical Society held in Chicago, Illinois, September, 1967.
' Valence shell electron pair repulsion. 18
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Journal of Chemicol Educofion
Table 1. Predicted Arrangements of Electron Pairs in a Volency Shell Two Three Fmr - . ~.Five Six Seven Eight Nine Ten Eleven
linertr equilateral triangle tetrahedron trigonal bipyramid ocCshedron monocapped octshedron square antiprism trieaooed tri~onalprism bicacped &ire antiprism icosahedron minus one apex irosahedron
least distance between any two pairs of electrons is maximined is ecjiliualent to assuming that the electron pairs occupy hard impenetrable orbitals or that they repel each other with a force law of the type F = l/rn where r is the distance between two orbitals and n = m. It is more reasonable to assume that some inter-
@ @ @ ...--
....--
..---
AX,
AX,E
AXtEz
Figure 1. Shapes of singly-bonded molecules containing up to six electron p a i n in the vdency shell. A, central atom; X, ligand; E, unshared electron-pair.
penetration or overlap of the orbitals can occur and this cornponds to a force law with a value of n less than infinity. There seems to be no reliable way of estimating a reasonable value of n for any particular case, but tt is fortunate that except for seven (and probably also 10 electron pairs) the favored arrange. ment is independent of the value of n. For seven pairs it has been shown that as the value of n is decreased, first the monocapped trigonal prism and then the pentagonal hipyramid become the favored arrangements in place of the monocapped octahedron (6). Thus it is not possible to predict the arrangement of seven (and probably also 10) pairs of electrons with as much certainty as in the other cases. Two other assumptions that have been made in deducing the arrangement of electron pairs given in Table 1, namely that the inner shell electrons occupy complete spherical shells and that all the valency shell electrons are at the same average distance from the nucleus, are discussed later. Effectsof the Size and Shape of Localized Molecular Orbitals
This simple theory can be refined to give a qualitative understanding of many of the finer details of molecular shapes by taking into account the fact that tha electron pairs in a valency shell are not all equivalent; they may bond different ligands, they may be unshared ie., lone pairs or they may participate in multiple bonding. It seems reasonable to make the following assumptions. 1) A nonbonding or lone-pair is larger and takes up more
Figure 2. Effect of decreasing electronegotivity of X on the sirs of a bonding electmn pair. ((a1 Eledtonegativity X A; (b) electronegativity X = A; (cI electrenogativity X A; ond Id) electronegafivity X = 0; this represents o lane pair of electrons.
and then two lone pairs the tendency of lone pairs to take up more space than bonding pairs squeezes the honding pairs together and the angle between them decreases. Similarly in the BrF6 molecule, which has a square pyramid structure with a lone pair in the sixth octahedral position, the angles between the Br--F bonds are less than the octahedral angle of 90' because of the greater space taken up by the lone pair (Fig. 3). The effect of changing the electronegativity of the ligands is illustrated by comparing NH3 and NFa. The greater electronegativity of fluorine causes the N-F bond orbital to he smaller than the N-H bond orbital and in particular it takes up less room on the surface of the nitrogen atom. Thus under the repulsion exerted by the lone pair the angle between the smaller N-F bond orbitals becomes smaller than the angle between the larger N-H bond orbitals (Fig. 4). Figure 3. (right) Structure of the BrFs molecule.
F
Figure 4. (below1 Bond angler in NFs and NHa.
room on the surface of an atom than a honding pair. 2) The size of a bonding electron pair, i.e., the space that it takes up on the surface of an atom, decreases with increasing electronegativity of the ligand. 3) The two electron pairs of a double bond (or the three electron pairs of a triple bond) take up more room on the surface of an atom than the one electron pair of a. single bond.
As a nonbonding electron pair is under the influence of only one nucleus it is larger and more spread out than a honding pair, and it will tend to take up all the space remaining on the surface of an atom (Fig. 2). In the absence of any other electron pairs it would occupy an s type spherical orbital. A simple illustration of the greater size of a lone pair is given by the decrease in the bond angle in the series CHI, NHa, H,O. As a honding pair is replaced successively by one
A multiple bond contains more than one electron pair and therefore takes up more space than a single electron pair. The size of a double bond orbital in relation to a lone-pair orbital is uncertain but they often seem to be of approximately the same size. The larger size of a multiple bond orbital is seen, for example, in planar X2C0 and X2C=CH2 molecules, i n pyramidal XzSO molecules, and in tetrahedral &PO molecules. The data in Ta6le 2 shows that the XCX angle is al-
Table 2. Bond Angles in Some Molecules Containing a Double Bond
FsCO CHsFCO Cl&O H,CO (NHhCO (NHdzCS
xcx
cxo
lP8.0" 11O0 111.3" 115.8' 118" 116'
126.0" 128, 122' 124.3" 122.1° 121" 12Z0
H2C=CHz HzC=CHF H,C=CF2 HIC=CCI~ FzC=CH, F,=CFCI
XCX
XCC
116.8O 115.4" 109.3' 114.0' 110" 114'
122~ 123.3, 120.9' 125.3O 123' 125' 123'
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ways less than 120' and the angle involving the double bond is therefore always greater than 120' and the XSO angle is always larger than the XSX angle. We note also that in most. cases the angles between the ligands X decrease with increasing electronegativity of X. Trigonal Bipyramidal Molecules
The general ideas concerning the effectsof orbital size and shape on molecular structure are well illustrated by recent structural data (7) on the trigonal bipyramidal molecules PFs, (CHD)PFP,and (CH3)aPli; (Fig. 5).
positions where there is more room for them. Thus, the most electronegative ligands, which have the smallest bonding electron pairs, always go into the axial positions and the less electronegative ligands-i.e , the methyl groups in the molecules we are considering, occupy the equatorial positions. This is a quite general rule, e.g., in PF3C12,PEL, and P(CF3)Cla the more electronegative F or CFa groups occupy the axial positions (10). The substitution of a fluorine by a methyl group in CH3PFPdecreases the effective electronegativity of the phosphorus and allows all the bonding pairs to move away from the phosphorus thus increasing all the bond lengths. I n addition, however, the axial fluorine bonds are closer to the large electron pair bonding the methyl group than the equatorial fluorine bonds, hence they suffer a greater repulsion and increase more in length than the equatorial bonds. They are also pushed away from the electron pair bonding the methyl group so that the axial FPF bond angle becomes less than 180°, in just the same way as the large, lone pairs cause the same angle in the SF4 and C1F3 molecules to be less than 180". The molecules that we are considering are, indeed, closely related to SFa and CIF3 in the sense that they are all derived from PFs by replacing one or more fluorines by either CH3 groups or by unshared pairs of electrons. The molecular parameters change in the expected manner as the very small electron pair binding a fluorine increases in size when i t binds the less electronegative CH3 groups and increases in size again when it becomes a lone pair. Three-Center Bonds
Figure 5. Structures bf romo AXs, AX& and AXaE2 m~lec~les.
The most noteworthy features of these molecules are as follows.
Molecules containing three or even four-center bonds conform to the general rule that the electron pairs around any central atom keep as far apart as possible. Thus in BzH6there is a tetrahedral arrangement of four orbitals around each boron, two ordinary two-center orbitals for the terminal B-H bonds, and two threecenter orbitals involving the bridging hydrogen and the other boron (Fig. 6). Because the electron density
1) The molecules are trigonal bipyramids or slightly distorted trigonal bipyramids. 2) The axial bonds are longer than the equatorial bonds. 3) Methyl groups ocoupy equatorial positions. 4) All the bond lengths increase snd the ratio of the length of the axial bonds to the length of the equatorial bonds r.,/r, increases as the number of CHa substituents inoreases. 5 ) Methyl substitution causes the P-F bonds to be bent away from the CH, groups.
As the most probable arrangement of five electron pairs is the trigonal bipyramid, any molecule in which a central atom with spherical inner shells has five electron pairs in its valency shell forming single bonds to five equivalent ligands is expected to be trigonal bipyramid. The axial electron pairs in this arrangement are not equivalent to the equatorial pairs, and, in particular, because the former have three nearest neighbors a t 90" while the latter have only two such neighboring pairs, equilibrium can only be attained if the axial pairs are at a greater distance from the nucleus than the equatorial pairs. Thus, in all such trigonal bipyramid molecules the axial bonds are longer than the equatorial bondx (8, 9). Moreover, the smallest electron pairs which have the smallest interactions with other electron pairs tend to go into the axial positions, and the larger electron pairs occupy the equatorial 20
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Figure 6. Strucfure of t h e BzHs molecule.
in a three-center bond is shared between three nuclei it is reasonable to suppose that the electron density at any one nucleus is less than that for a two-electron bond and therefore the three electron orbital takes up less room on the surface of an atom than a two electron orbital. Thus the angle between the terminal hydrogens (121') is greater than the tetrahedral angle and the angle between the two bridging hydrogens is only 97'. Further examples of molecules containing three-center bonds are given in the next section. Cluster Compounds
The bonding in the well-known cluster compounds Mo6Cls4+and Ta6C1122+ (Fig. 7) has usually been de-
w a
6 b
Figure 7. Structures of (a) the ,Mo&ls4+ ion and (bl the ToLlml+ ion.
scribed by molecular orbital theory (11, 12). However, a localized electron pair description has also been given (13) in which there are twelve localized twocenter bonds along each of the twelve edges of the octahedron in MosCls4+ and eight localized threecenter bonds, one in each of the eight triangular faces of the octahedron in TasC1,22+(Fig. 8). In both cases each metal atom forms four localized metal-metal bonds, and an approximately square antiprism arrangement of bonding orbitals around each metal is completed' by four bonds to ligand chlorine atoms (Fig. 8). Kettle has in fact shown that the usual molecular orbitals which are used to describe the bonding in the metal cluster may be transformed into the localized two-center and three-center molecular orbitals that we have described (14). Another simple example is the tetrahedral BaCL molecule in which there is a three-center bond in each face of the tetrahedron so that each boron is surrounded by an approximately tetrahedral arrangement of three 3-center orbitals and one ¢er orbital forming the bond to a chlorine (Fig. 9). The ResCllz3-anion has a triangular arrangement of three rhenium atoms (15). The bonding between the
Figure 9. Struchlre of the B&lr molecule.
rhenium atoms may be described in terms of molecular orbital theory (16) or alternatively, as Kettle has pointed out, the molecular orbitals may be transformed to an equivalent set of localized orbitals (17). I n this description there is a double bond between each pair of rheniums, i.e., two localized but bent orbitals. Each rhenium then forms four localized bonds to other rhenium atoms and five bonds to chlorine atoms, the nine bonds having approximately the expected tricapped trigonal prism arrangement (Fig. 10). I n the RezClsz- anion there is a quadruple bond (i.e., four shared electron pairs) between the two rhenium (18) and each rhenium also forms four bonds to four chlorines resulting in eight localized pairs around each rhenium. These adopt the expected square antiprism arrangement leading to the chlorines on one rhenium adopting the eclipsed conformation with respect to the chlorines on the other rheninm(Fig. 11).
d
8
0
b
Figure 10. lo) Structureof tho ReaCLpa-ion. (bl Representotionof localized bonds in ReaCllF.
D
Figure 8. Localized electron pair models for 10) TosClnaf and lbl MosCIe4+ (c) View down 4-fold axir of octahedron showing approximate rqvore oneprirm arrangement of eight bonds formed b y the metd atom; heavy lines represent M-CI bonds; dotted lines, M-M bonds.
b
Figure 11. (a) Structure of tho ResCla2- ion. I b l View down Re-Re axir in the RenCls2- onion showing the approximote square antiprim a r r a n g e men1 of fovr Re-CI bonds ond the fovr elsctron pairs, represented b y circler, of the quodrvple bond.
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Transition Elements
The electron pair arrangements considered above assume that the inner shells of electrons are spherical and have no influence on the arrangement of the valency shell electron pairs. Thus the arrangements of Table 1 and all the deductions made from them are valid for the main group elements and also for transition elements that have spherical do, d5, or dl0 shells. I n other cases the effect of the incomplete and therefore unsymmetrical d shell on the arrangement of the ligand electron pairs must be considered. This is best discussed in terms of the ligand-field theory and the Jahn-Teller effect. Any molecule with a degenerate electronic sta!e will distort in such a manner as to remove this degeneracy and to reduce the symmetry of the molecule. This can be visualized as resulting from the repulsion between the d-shell and the valenoy-shell electron pairs. For example, an AXs molecule in the absence of any interaction with a d shell, or if the d shell is spherically symmetric, will be a regular octahedral molecule. According to ligand-field theory an octahedral arrangement of ligands splits the d shell into two levels, containing the d,,, d,,, and d,, or & orbitals and the & - @ and dn or e, orbitals, respectively. Since the k , orbitals concentrate electron density between the bonding electron pairs while the e , orbitals concentrate electron density in the direction of the bonding electron pairs, the former are of lower energy than the latter. If either of these two sub-levels is unsymmetrically occupied then some of the bond-pairs suffer a greater repulsion than others and the octahedron is distorted, although the effects produced by the tl, orbitals are expected to he very small and have not been observed. In general a tetragonal distortion occurs so that the symmetry of the molecule is reduced from Oh to D M but it cannot be predicted whether this distortion will give rise to an elongated or a compressed octahedron. Very frequently an elongated octahedron is observed for 6,da, and d9systems. I n these cases it is a reasonable approximation to regard the d shell as having an ellipsoidal (either prolate or oblate) shape. The arrangement of six electron pairs around a prolate ellipsoid gives an elongated octahedron and around an oblate ellipsoid gives a compressed octahedron (Fig. 12). Strong interaction with the d shell may cause an octa-
Figure 12. Distortion of the octahedral orrangemen1 of rir bonding pain of electrons b y o nonspherisal d shell; 1.1 elongated octahedron; lb) compressed octahedron.
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hedral complex to lose its axial ligands and their electron pairs thus giving rise to a planar AXa complex. Alternatively, such a structure can be regarded as arising from the interaction of an ellipsoidal d shell with a tetrahedral AXl molecule. I n some cases where the interaction with the d shell does not completely dominate the interaction between the four bonding electron-pairs a distorted tetrahedral structure might be expected, as has indeed heen observed for C U C ~ ~ ~ and CuBm2- (Fig. 13). The predicted distortions of an AX trigonal bipyramid molecule have been discussed in detail elsewhere (19).
Figure 13. Distortion of the tetrahedral arrangement of four bonding pmir. of electrons b y a nonspherical d shcll.
Exceptions lo the Valence-Shell Electron Pair Repulsion Rules and Possible Effects of Ligond-Ligand Repulsions
The theory of valence-shell electron-pair repulsions assumes that the stereochemistry of a molecule is determined solely by the interaction between the electron pairs in the valency shell and that ligand-ligand repulsions are generally of lesser importance. For molecules in which all the valency shell electron pairs are used in bonds to ligands consideration of either the repulsion between bonding electron pairs or the repulsion between ligands leads to the prediction of the same molecular shape. However when the central atom of a molecule has one or more lone pairs in its valency shell then consideration of valence-shell electron-pair repulsions predicts a different shape from a consideration of ligandligand repulsions. I n almost every case the valenceshell electron-pair repulsion theory predicts the correct molecular shape, i.e., in almost every case lone pairs appear to be stereochemically active. However, for high coordination numbers and for sufficiently large ligands or sufficiently small central atoms it is clear that repulsions between ligands could be of importance in determining stereochemistry. For example the ions TeCls2-, TeBrs2-, SbBreS-, and some closely related molecules have regular octahedral structures ($0, $1) although they all have valency shells containing seven electron pairs including one lone pair and should therefore have structures based on a preferred arrangement of seven electron pairs. Although it is not possible to pedict the most likely arrangement of seven electron pairs with complete certainty the six ligands would not be expected to have an octahedral arrangement. It seems reasonable to assume therefore that the ligandligand repulsions dominate the stereochemistry in these cases. I n the T$Bra2- ion, for example, the Br-Br distance is 3.81 A which is slightly smaller than the van der Waal's distance of 3.9 A. Although this probably does not imply that there is any appreciable repulsion between the bromine ligands it is clear that they
are essentially touching each other and that there is therefore no room for the lone-pair. If the lone pair did occupy the valency shell in TeBrs2-, for example, some of the bromine-bromine separations would be very small, e.g., 2.9 A in a pentagonal bipyramid arrangement of the seven electron pairs, and such a small Br-Br distance would imply very considerable repulsions between the bromines. I t would appear therefore that the lone-pair is forced inside the valency shell into a spherical s type orbital. The observed bond length of 2.75 in TeBrs2- compared with 2.51 predicted from the sum of the covalent radii is consistent with the idea that the lone pair is inside the valency shell and thus contributes considerably to the shielding of the bonding electron pairs from the tellurium nucleus, hence effectively decreasing the electronegativity of tellurium and increasing the bond lengths. Bartell (88) bas pointed out that the chlorine-chlorine distance in TeCla in which the lone pair is stere~chemically~active is only 3.3 A compared with the value of 3.5 A in TeCls2- and he argues therefore that ligand-ligand repulsion cannot be the only factor responsible for the octahedral geometry of TeCls2-. However, as has already been pointed out for the case of TeBrs2-, $he C1-C1 distance would be much smaller than 3.3 A if the lone-pair were included in the valency shell. Although TeFs2- is not known the related species IF6- and XeF6 do exist, and they appear to have nonoctahedral structures based on an arrangement of seven electron pairs in the valency shell including the lone pair (83, 84). The exact structures of these species are still not certain but the most likely structure would appear to be the monocapped octahedron in which the lone pair occupies the unique position opposite the middle of one face of an octahedron which is somewhat distorted by the presence of the seventh pair (Fig. 14).
inner orbital as in TeBrs2-, etc. Moreover, for a large atom such as xenon the differencein energy for an electron pair in a localized orbital in the valency shell and in a large spherical inner orbital would be expected to be relatively small. Another group of molecul~swhich provide exceptions to the simple rules of the electron-pair repulsion theory are the alkaline earth dihalides as these molecules are linear when the cation is small and the anion is large, e.g., BeCl,, but bent when the cation is large and the anion small, e.g., BaF2 (85). For large, highly polarizable atoms such as barium the valence shell is not well separated from the inner shells and a highly polarizing ligand such as fluorine may distort the completed inner shell as well as the two s electrons that occupy the valency shell. I n this case one should probably consider the two outer s electrons plus some or all of the electrons of the outermost inner shell as constituting the effective valency shell. In this case the valency shell in the molecule consists of two bonding pairs and a large cloud of nonbonding electron density containing one or more pairs of electrons. This will give a bent molecule just as in the case of SnC12. Literature Cited
(1) GILLESPIE, R. J., AND NYHOLM, R. S., Quart. Rev., 11, 339 11957). ~-...,~ (2) GILLESPIE, R. J., J. CHEM.EDUC.,40,295 (1963). R. J., Angew. Chemie (inlemt. ed.), 6 , 819 (3) GILLESPIE, (1967). (4) DICKENS,P. G., AND LINNETT,J. W., Quart. Rev., 9, 33 (1955). R. J., Can. J . Chem., 38, 818 (1960). (5) GILLESPIE, D., Can. J. Chem., 41, 1632 (1963); CLAXON, (6) BRITTON, T.A,, AND BENSON,G. C., Can. J . Chem., 44, 157 (1966). L. S., AND HANSEN, K. W., Ino~g.C h . , 4, 1777 '(7) BARTELL, (1965). (8) GILLESPIE, R. J., Can. J . Chem., 39, 318 (1961). R. J., J . Chem. Phys., 37, 2498 (1962). (9) GILLESPIE, E. L., AND SHURN,R. L., Quart. Rev., 20, (10) MUETTERTIES, 245 (1966). L. D., OMEN,D . P.,AND DUFFEY,G. H.. J . (11) CROSSMAN, Chsm. Phus.. 38. 73 11963). , (12) COTTON, F. A,, AND HAAS, E.,Inorg. Chem., 3.10 (1964). R. J., Can. J . Chem., 39, 2336 (1961). (13) GILLESPIE, (14) KETTLE,S. F. A., Theoret. Chim. Acta, 3, 211 (1965). J. A,, COTTON, F. A,, AND DOLLASE, W. A,, (15) BERTRAND, Inorg. Chem., 2 , 1166 (1963). (16) . . COTTON.F. A., AND HAAS,T. E., 1120787. Chem., 3, 1094 (1964j. S. F. A., Thmret. Chim. Acta, 3,282 (1965). (17) KETTLE, F. A,, AND HARRIS,C. H., I ~ o TChem., ~. 4, 330 (18) COTTON, 119R5I - * -- ,. (19) GILLESPIE, R. J., J . Chem. Soc., 4679 (1963). I. D., Can. J . Chem., 42, 2758 (1964). (20) BROWN, S. L., AND JACOBSTON, R. A,, Inorg. Chem., 5 , 743 (21) LAWTON, (1966j. (22) B ~ E L LL., S., J. CHEM.EDUC.,45, (1968). L. S.. AND GAYIN.R. M.. J . Chem. Phus.. (23), BARTELL. - . 48.. 2460 (i96u). ' (24)CHRISTE, K. O., AND SAWODNY, W., Inorg. Clwm., 6, 1783
".
.
f.
\
Figure 14. Predicted rtrvsture of XeFe
I t may also be predicted that the three bonds adjacent to ihe lone pair will be longer than the other three bonds c s is observed for example in BrF5 and related molecules. There has been some discussion as to the extent that the structure of XeFs deviates from that of an octahedron (BS), and it could be that the'distortion is rather small because the lone pair may be at a somewhat smaller distance from the nucleus than the bonding pairs because of a tendency to occupy a spherical
.
(1Qfi7) \-"-. ,.
(25) WHARMN,L., BERG,R. A., AND KLEMPERER, W., J . Chem. J. L., Phys., 39, 2023 (1963); BiicnLER, A,, STAUFFER, KLEMPERER, W., AND WHARMN, L., J . Chem. Phys., 39, A,, STAUFPER, J. L., KLEMPERER. 2299 (1963); B~~CHLER, A., W., J. C h m . Phgs., 40, 3471 (1964); BWCHLER, STAUFFER, J. L., AND KLEMPERER, W., J . Am. Chem. Soc., 86, 4544 (1964).
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