VSEPR Theory of Directed Valency

Directed valency is conventionally explained in terms of the directional character of p and d orbitals (1). The explanation of the angular shape of the water molecule as resulting from bond formation by two oxygen 2p orbitals that make an angle of 90" to each other is to be found in most freshman texts. The difficulties encountered in the direct application of this theory to carbon and the consequent introduction of the concept of hybridization are also very familiar. The limitations of this theory and in particular the idea that hybrid orbitals are a description of rather than an explanation for molecular shape are, however, generally not discussed in elementary introductions to the sub- ject. For example, in the case of five-coordination in which the bonding orbitals are described as a set of five sp3 d hybrid orbitals, it is necessary to make an arbitrary choice of the single d-orbital; the d+,. orbital gives a set of hybrids corresponding to a square pyramid shape, while the d , ~ orbital gives a set of hybrids corresponding to a trigonal bipyramid shape.

In the early development of the simple valence-bond picture little attention was paid to the effects of lone- pairs of electrons on molecular shape. As long ago as 1940, however, Sidgwick and Powell (2) drew attention to the fact that the shape of a molecule could be accounted for in terms of the arrangement of all the electron-pairs (both bonding and non-bonding pairs) in the valence-shell of the central atom, a given number of electron pairs always having the same arrangement. Since that time the importance of lone pairs has been stressed by a number of authors. Lennard-Jones and Pople (5) pointed out that water and ammonia could be described in terms of a set of sp3 hybrid orbitals on the central atom which are occupied by two bonding and two lone pairs of electrons in water and three bonding and one lone pair of electrons in ammonia. These ideas were developed further by Mellish and Linnett (4) who consider that a lone pair, being under the influence of ouly one nucleus rather than two, occupies more of the surface of an atom thau the bonding pairs and that therefore the angle between the bonding pairs is reduced to less than the tetrahedral angle as in the case in water and ammonia. Fowles (5) has summarized some of the effects of lone-pair electrons on bond lengths and hond angles and other molecular properties.

Gillespie and Nyholm (6) have shown how the ideas of Sidgwick and Powell (B) can bc combined with the more recently developed ideas on lone pairs to account in a qualitative manner for the general shapes and bond angles of most inorganic molecules. These authors pointed out that the arrangements of electron pairs in a valence shell could be regarded as arising from the mutual interactions between the electron pairs which

R. J. Gillespie McMaster University

Hamilton, Ontario

are a consequence of both electrostatic forces and the operation of the Pauli exclusion principle. The purpose of the present paper is to review recent developments of these ideas and to show that a very satisfactory under- standing of a large number of the features of the structures of inorganic molecules can he understood and rationalized in terms of the repulsions between the electron pairs in valence shells without making any use of the concept of hybrid orbitals. The paper attempts to give a consistent discussion of molecular shape aud stereochemistry from the point of view of this theory only. Discussions of molecular shape from alternative, although sometimes related, points of view have been given in other recent papers in THIS JOURKAL (7-9).

Basic Ideas

The theory proposes that the stereochemistry of an atom is determined primarily by the repulsive interactions between the electron pairs in a valeuce shell. The electrons in a valence shell are regarded as occupying essentially localized orbitals that are oriented in space around the nucleus and the completed inner electron shells so that their average distance apart is maxi- mized (3, 6,10,11). This may be regarded as a consequence of the operation of the Pauli exclusion principle according to which electrons of the same spin tend to keep as far apart as possible (6, 10-12). Thus two electrons of the same spin have a maximum probability of being found on opposite sides of the nucleus, three electrons a t the corners of an equilateral triangle and four a t the corners of a tetrahedron. In the valence shell of an atom in a molecule there are generally equal numbers of electrons of opposite spin. Thus for the common case of a valence shell of eight electrons, there will be a set of four electrons of parallel spin having a maximum probability of being found a t the corners of a tetrahedron, and a second set of electrons of opposite spin also having a maximum probability of being found a t the corners of a second tetrahedron. This would, for example, be the situation for Se , F-, 0= and Ka-. When one or more protons, or other positive ions com- bine with these ions to form neutral molecules two electrons of opposite spin are attracted towards each other and the two tetrahedra are brought into coinci- dence. There are thcn four essentially localized electron-pairs, i.e., four tetrahedrally directed orbitals each containing an electron-pair. These tetrahedral orbitals are approximately equivalent to the sp3-orbitals of valence-bond theory.

The tendency of electron pairs in a valence-shell to adopt an arrangement which maximizes their average distance apart may be regarded as arising from repulsive interactions between the electron-pairs. Each electron

The Valence-Shell Electron-Pair Repulsion (VSEPR) Theory of Directed Valency

Volume 40, Number 6, June 1963 / 295

pair occupies as much of the available space around the nucleus and inner shells as it can keeping other electron- pairs out of this space. In the case of four such electron- pairs the available space may be regarded as being divided into four tetrahedrally directed segments each of which is occupied by an electron-pair. The repulsions that we are considering between electron-pairs are essentially the same as those that operate between inert gas atoms, or any pair of nou-bonded atoms with filled valence shells. Although these non-bonded repulsions may sometimes he of importance in determining molecular structure. the valence-shell electron-nair renul- sion theory-henceforth called the VSEPR theory- regards the repulsions between the bonding and the non- bonding electron-pairs in the valence-shell of the central atom as being the most important factor in determining the stereochemistry, the interactions between the non- bonding electron-pairs on different ligands being of relatively minor importance except in special situations. This repulsion may be called a van der Waals repulsion, an exchange repulsion, or a spin-correlation repulsion. The repulsive interaction between inert gas atoms may be very approximately represented by a potential function which involves a l /rn term where n has a fairly large value, e.g., 12, as in the Lennard-Jones potential function. It is assumed that the interaction between the electron-pairs in a valence-shell may be approximately represented by a similar potential function. An interaction of this form means that when the overlap between two orbitals is very small they interact to a negligible extent, but as the overlap increases their repulsive interaction increases very rapidly.

I t is evident from general elementary considerations, and it can also be proved rigorously (6,13, 14), that the most probable arrangements of two, three, four, and six electron-pairs are collinear, equilateral triangular, tetrahedral, and octahedral, respectively. For five electron pairs the trigonal bipyramid and the square pyramid are equally probable for n = m but for n < - the trigonal bipyramid has a slightly greater probability than the square pyramid (14,lb).

Higher coordination nnmbers can be considered in the same manner but they will not be further discussed in this article (11,16).

In the case of transition metals there is in general an incomplete d-shell beneath the valence-shell which will not necessarily have the spherical symmetry of a completed shell. In such cases it is necessary to consider the interaction of the valence-shell electron-pairs with the non-spherical d-shell as well as with each other. In the following discussion we will consider only molecules of the non-transition elements and transition elements with symmetrical do, d6 (spin-free) and dlo shells.

Shapes of Molecules

Each of the preferred arrangements of a given number of electron-pairs can in general give rise to several molecular shapes depending on the relative numbers of bonding-pairs and lone-pairs. The following terminol- ogy is useful: A is a central atom, X is a ligand and E is a lone-pair. Thus in a singly bonded molccule AX,E, there are m + n electron-pairs in the valence-shell of which m are bonding pairs and n are non-bonding or lone-pairs. The shape of the molecule is determined by

the preferred arrangement of the m + n electron-pairs. Or, if we consider the electron-pairs to each occupy a two-electron orbital, the shape is determined by the arrangement of the m + n orbitals. The various possible molecular shapes that can be obtained in this way are given in Table 1 and Figure 1. Predicted shapes for the recently discovered molecule XeFP and the XeF, are included. In the cases of AX4 E, AX3E2, AX&, and AX& molecules there are possible alternative positions for the lone-pairs. In fact they are always found in the equatorial positions in AXaE, AX3E2, and AXzEa molecules and in the trans position in AX& molecules as shown in Figure 1. The reason for this is discussed later.

These arguments can be readily extended to include molecules in which there are one or more ligands bonded to the central atom by double or triple bonds. In such a molecule the electron-pairs are not all free to take up one of the preferred arrangements of Table 1 because

Table 1. Shapesof Molecules of the Non-Transition Elements and of d: d5 (Spin-free) and dlo Transition Elements

AX* coordination: triangular BF3, BCb, BL, B(CH.).F, Gal., I ~ ( C H S ) ~ , [CU(CN)~-],

AX2E coordination: V-shaped SnCI., SnBh, Sn4. PbCI*, PbBr?, Pb12

4 Electron-oair valenccshells: tetrahedral

AX4 coordination: tetrahedral

.4X& coo dination:V-shaped OX., SX2, Sex*, TeX%

5 Electron-pair valwce-shells:trigonal-hipyramidal

AX, coordination: trironal-biovmmidsl

AXIE eoordinatian:irregular tetrahedral SF4, SeF4, R2SeCI,, R2SeBr., RsTeCI,, R9TeBra, TeCh

AX& coordination:T-shaped CIR, BrF8, C6HJC12

AX& coordination:linenr IC1,-, Is-, XeFz

6 Electron-vair valence-shel1s:actahedral

296 / Journal o f Chemical Education

the two electron-pairs of a double bond and the three pairs of a triple bond are constrained to keep close together. From the point of view of their effect on the stereochemistry of a molecule the two electron-pairs of a double bond are best considered as occupying one four-electron orbital and the three electron-pairs of a triple bond as occupying one six-electron orbital. The general shape of a molecule AX& is then determined by the value of m + n where m is the number of ligands X or the number of bonding orbitals, allowing one four- electron orbital for a double bond and one six-electron orbital for a triple bond. Various possible molecular shapes for molecules containing double-bonds are given in Table 2.

Bond Angles and Bond Lengths

Many of the finer details of molecular shapes can he understood on the basis of the following related postu- lates concerning interactions between the valence-shell electron-pairs.

(1) Non-bonding electron-pairs repel adjacent electron-pairs more strongly than bonding electron-pairs. Because they are under the influence of only one nucleus t.he two electrons of a non-bonding pair occupy a rather fatter and less confined orbital than the electrons of a bonding-pair which are under the influence of two nuclei (Fig. 2). Hence the orbital occupied by a non-honding electron-pair will overlap with neighboring orbitals more extensively and therefore will repel the electrons

Figure 1. General rhoper of molecules of the non-tronrition elements and of #,dl (spin-free) and dlo transition elementr

in these neighboring orbitals more strongly than an electron-pair in a bonding orbital. Thus lone-pair electrons will tend to move apart and to squash bonding electron-pairs together. The repulsions between electron- pairs in a valence-shell decrease in the order lone-pair- lone-pair > lone-pair-bond-pair > bond-pair-bond-pair, In the series CHI, NH3, HzO the successive replacement of bonding electron-pairs by lone-pairs causes the bond angle to decrease from 109.5" in CHa to 107.3" in KHs and to 104.5' in H20. In general NX3 and OXa have bond angles less than the tetrahedral angle when X is singly-bonded carbon (aliphatic), nitrogen, oxygen, or fluorine due to the greater repulsion exerted by the lone- pair(s) .

In some molecules in which the central atom has a valence-shell of five or six electron pairs there are possible alternative positions for lone-pain that would give rise to different molecular shapes. The lone-pairs will tend to occupy those positions in the molecule in which their repulsive interactions with other electron-pairs are minimized and since they repel each other more strongly than they repel bond-pairs they will occupy trans rather than cis positions in an octahedral arrangement of six electron pairs. In a trigonal bipyramid arrangement of five electron-pairs the axial pairs have three neighboring pairs a t 90' while the equatorial pairs have only two neighboring pairs a t 90' and two more a t 120". Since

Figure 2. Diagrammatic representotion of the approximote rhoper of I.) a bending orbital and lbl m non-bending or lone-pair orbital.

the repulsion between neighboring electron-pairs falls off very rapidly with the angle between them, it is reasonable to assume that the interaction between electron- pairs a t 120' is negligibly small. Hence, it is clear that the lone-pairs will tend to occupy the equatorial rather than the axial positions in these molecules, giving the shapes shown in Figure 1 rather than the possible alternative shapes with one or more of the non-bonding pairs in an axial position (15).

(2) The repulsions exerted by bonding electron- pairs decrease with increasing electronegativity of the ligand. This can he regarded as an extension of (I). An unshared pair has no ligand and therefore occupies a larger, "fatter" orbital than a bonding pair; it may be regarded as a bonding pair attaching a ligand of zero electronegativity. Any ligand attracts the bonding electron-pair to some extent contracting the orbital and drawing it in toward itself. The greater the electronegativity of the ligand the more contracted is the bonding electron orbital and the nearer to the ligand is the average position of the bond-pair (6, 12). Hence, as the electronegativity of the ligand increases, the amount of overlap between the bonding orbital and neighboring orbitals decreases. In agreement with this idea it is found that the bond angle in OFz (103.2") is less than that in H,O (104.5") and the bond angle in NF3 (102') is less than in NH3 (107.3") (Table 3). In the series PC13 (100°), PBr3 (101.53, PIa (102") the bond angle increases as the electronegativity of the halogen decreases. A similar variation in bond angle is


found in the series AsC13 (98.4"), AsBrs (100.5°) and Asia (101.5') (Table 4)

(3) Multiple-bond orbitals repel other orbitals more strongly than single bond orbitals. The repulsions exerted by multiple bond orbitals on other orbitals decrease in the order triple bond > double bond > single bond. This generalization may be regarded as a further extension of the ideas already discussed in (1) and (2). Triple bond orbitals which contain three electron-pairs are necessarily larger and therefore overlap more with nearby orbitals than double bond orbitals which contain only two electron-pairs, and double bond orbitals are in turn generally larger than single bond orbitals. Thus angles involving multiple bonds tend to be larger than angles involving single bonds only. Some examples are given below. For AX8 molecules the "ideal" bond angle is 120' but when one of the ligands is a double-bonded atom the angles involving this ligand are always greater than 120" and the other

angles correspondingly smaller (Table 5). For AX4 molecules the "ideal" bond angle is 109.5' but when one of the ligands is a double-bonded oxygen or -. L

~ o b l e 3. Bond Angles (in degrees) in the Hydrides of Groms V and VI

Table 4. Bond Angles (in degrees) for Group V Halides

NFa PFa A s h 102.1 104 102

Table 2. Shapes of Molecules Containing Double-Bonds

2 Orbital valence-shells : linear

AX2 coordination : linear

3 Orbital valence-shells : triangular

AX. coordination : triang lar

/X 0 -0 0

\+ \+ O=C X,C=CX, 0=S

//

\x -0 /"-" 0 /"=" .4X2? coordination : V-shaped

F

/ x -0 (/ \OF 0 "" / F /

4 Orbital valence-shells : tetrahedral

AX8E coordination : trigonal pyramidal

A L E 2 coordination : V-shaped

5 Orbital valencoshells : trigonal hipyramidal 6 Orbital valence-shells : octahedral

AX: coordination : trigonal hipyramidal

F AX. coordination : octahedral

F AX5E cmrdination : square pyramidal

AXrE coordination : irregular tetrahedral

sulfur atom the augles involving this ligand are always larger than the other bond angles (Table 6). In AX4 molecules containing two double-bonds the largest angle is always found to be the angle between the double bonds (Table 6). Again, in AX3E molecules containing one double bond, the angle involving this bond is always larger than the other angles (Table 6).

(4) Repulsions between electron-pairs in filled shells are larger than those between electron-pairs in incompletely filled shells. The orbitals in a filled shell effectively occupy all the available space. Any influence tending to reduce the angle between the orbitals will cause appreciable orbital overlap and hence will be strongly resisted. The orbitals in an incompletely filled shell however do not occupy all the available space and they are therefore more susceptible to factors which can reduce the angle between the orbitals as some distortion of the angles between the electron pairs can occur mith- out causing any appreciable overlap (17).

The valence-shells of the elements of the first row of the periodic table, in particular B, C, N, 0, and F are completely filled by four electron-pairs. Thus quite generally for AX4 molecules where A is a first row ele- ment and for AX3 E and ALEl molecules where X is also a first row atom the bond angles are always within a few degrees of the ideal angle of 109.5".

The elements of the second short period of the periodic table (Na-Cl), however, can in principle have up to nine electron-pairs in their valence-shell. In fact, for reasons which are not completely understood, the valence-shells of these elements are effectively completed by fewer electron pairs. If all the electron pairs are unshared as in argon, the maximum number appears to be four, but when all the electron-pairs are shared the maximum number appears to be six as in PF,- and SF6. As might be expected we find the highest coordination numbers with the most electronegative elements, since these cause the bonding pairs to occupy the most contracted orhitals and therefore more of them

Table 5. Bond Angles (in degrees) for Some AX8 Molecules Containing Double Bonds

-----X2C=O------ -XE=CX2 XCX XCO XCX XCC

F?CO 112.5 123.2 (CHa)nC=C!CH3)s 109 125 CICO 111.3 124.3 (CHx)zC=CH1 lo!) 125 H9C0 118 121 (NH>)&O 118 121

Table 6. Bond Angles (in degrees) in AX4 and AXaE Molecules Containing Double Bonds

XPX XPX

can be packed around the central atom. Thus, for these elements it is reasonable to assume that bondiug electron-pairs do not interact with each other very strongly until the angle between them is the same as in the octahedron, i.e., 90'. Thus in passing from SiHI to pHa the greater repulsion exerted by the lone-pair in PH, causes a much greater distortion of the bond angle (Table 3) than the lone-pair does in ammonia, because in phosphine interaction between the bonding-pairs does not begin to be appreciable until the bond-angle approaches 90". I11 XH3, however, there is already appreciable interaction betweeo the bonding-pairs at the tetrahedral angle. The repulsions exerted by the two lone-pairs in HIS would be expected to make the bond angle approach even more closely to 90' as is indeed observed. The shape of the H2S molecule is closely related to that of SO2F2; the double-bond orhitals of S02F2 replace the lone-pairs of H2S (Fig. 3).

Figure 3. Approximately letrahedral arrangement of two lone-poir orbitals and two bond orbitols in HnS and two double-bond orbit& ond two single bond orbitals in SOzFe

When the valence-shells of P and S are filled with six electron-pairs the hond angles do not differ significantly from 90' as observed, for example, in S2Flo and SF60F (18, 19).

With decreasing electronegativity of the central atom the bonding electron-pairs move closer to the ligands thus further decreasing their interaction and therefore the hond angle decreases slightly from H2S to H2Se and from pH3 to ASH$.

(5 ) When an atom with a filled valence-shell and one or more unshared electron-pairs is bonded to an atom with an incomplete valence-shell there is a tendency for the unshared electron-pairs to be transferred from the filled shell to the incomplete shell (17). This may be regarded as a consequence of the repulsions between electron-pairs in the filled shell being considerably larger than the repulsioils between the electron- pairs in the incomplete shell. This effect causes the bonds in fluorides such as BF3 and SiF4 to be shorter than expected for single bonds. These bonds have considerable double-bond character as a consequence of the delocalization of the fluorine unshared electron- pairs into the incomplete valence-shell of the central atom, i.e., there are important contributions from resonance structures such as (I) and (11). This effect

+F \-

R-F

also has a considerable inflnence on the shapes of molecules. In the series of molecules PF3, PCL, PBr, and PI3 and AsF8, AsCl8, AsBr3 and As13 it might be expected, because of the decreasing electronegativity of the halogen atom, that the bond angle would increase steadily from the fluoride to the iodide. The expected


increase in hond angle is observed from the chloride to the iodide hut in both cases the fluoride is anomalous having the largest rather than the smallest angle (Table 4). This may be attributed to the effect described above. The unshared electron-pairs on the fluorine have a strong tendency to delocalize into the incomplete shell of the central atom, thereby giving the honds some double-bond character. Because of the consequent increased size of the bonding orbitals, they repel each other more strongly; and the hond angle increases accordingly. For the other halides the halogen atoms have incomplete shells and so there is little tendency for their electron-pairs to delocalize into the incomplete shell of the central atom and hence there is little or no double-bond character in the honds in these cases. In other words it is suggested that the large electron- pair repulsions in the valence-shell of the fluorine atom compared with the other halogens causes the contribu- tien of structures such as (111)

-.p +/ I \ F F F

111

to he considerably more important in the fluorides than in the other halides.

In O X n b n d NX3' molecules, where Xi is a ligand having an incomplete valence-shell, delocalization of the lone-pair electrons on O or N occurs giving partial double-bond character to the OX'and NXL honds which, together with the partial removal of the lone-pair electrons from the central O or N, causes a considerable increase in the bond angles in these molecules. Thus the SiOSi hond angle generally falls in the range 130- 140" (Table 7) while the N(SiH& molecule is planar with 120" hond angles (20). In SX2' molecules, however, because both S and the ligand X have incomplete valence-shells, there is little tendency for the lone-pairs on S to be transferred to the ligand X> and hence the hond angles are less rather than larger than tetrahedral as in the corresponding oxygen compounds. When the ligand Xi in an OX2' molecule is a very strong acceptor, e.g., if it is a transition metal, the transfer of the lone- pain from the oxygen into the OX" honds may be essentially complete in which case a linear molecule re- sults (IV). Collinear X'OX' bonds have been found in the molecules

[ClsRuORuC1S]4- and [TiCL (CsHs)I2O (21, 22). Similar delocalization of the oxygen lone-pair elec-

trons occurs in the ion O(HgC1)3+ which has a plane triangular shape (28). We conclude that there must be

Table 7. Si-0-Si and Si-S-Si Bond Angles (in dearees)

complete transfer of the oxygen lone-pair electrons into the OHg bonds, i.e., the most important resonance structures are three structures such as (V).

Similar effects are observed when the ligand X, although having a complete shell, is unsaturated, i.e., is forming one or more multiple bonds to other atoms, e.g.,

I n such a case X behaves as if i t had an incomplete shell since one of the electron-pairs of the X=Y double-bond may be partially transferred to Y leaving X electmn- deficient and thus facilitating the transfer of lone-pairs on O into the 0-X hond. Hence, for aliphatic compounds the COC bond angle is always very close to and generally a few degrees smaller than the tetrahedral angle, hut for aromatic compounds the COC angle is generally in the range 120-125' (Table 8). This may

Table 8. C--0-C Bond Anales (in degrees)

be attributed to the partial double-bond character of the 0-C bonds, i.e., to the importance of contributing structures such as (VI).

The planar 120° arrangement of the honds around nitrogen in urea and formamide (24,25) can also be similarly explained.

The large SOS bond angle found in many molecules may also he accounted for in the same way (Table 9).

Table 9. SOS, SSS, POP, and PSP Angles (in degrees) - S d - L n - S _ S -

300 / Journal of Chemical Education

The S atoms in these molecules have filled valence- shells of six electron-pairs but they are unsaturated in the sense that they are double-bonded to one or more oxygen atoms. Hence, resonance structures such as (VII) make an important contribution to the actual structure. The bond angle in the imino-disulfonate ion NH(SO&- has a similarly large value of 125.5 (85) due to the important contribution of structures such as (VIII.)

\// \ / / -&S S-O-

\ //+\/ -0s S=0

I II // R 6 o 6 h

VII VIII

I t is interesting to note that in a molecule in which sulfur has a complete valence-shell of six electron-pairs and does not have any double bonds, e.g., in F5SOOSF6 the S-0-0 bond angle is only l O j 0 (87). The numerous phosphorous compounds in which the POP angle is in the range of 120-130' have P atoms with only four or five electron-pairs and therefore incompletely filled shells. They are also unsaturated because of double- bonds to oxygen atoms. However, S-S-S and P- S-P bond angles are, as expected, generally smaller than 109 (Table 9).

(6) In a valency shell containing a number of electron pairs, such as five or seven, in which all the electron pairs cannot have the same number of nearest neighbors, then those electron pairs with the greatest number of nearest neighbors will be at a greater average distance from the nucleus than the other electron-pairs. In AX,, AX,, AX,, aud AX6 molecules symmetry re- quires all the bond lengths to be identical but in AXs molecules the axial and equatorial bonds are not required by symmetry to be the same. The axial electron-pairs have three neighboring pairs a t an angle of 9O0, while the equatorial pairs have only two neighhor- ing electron-pairs a t 90' and two more a t 120". Since the interaction between two electron-pairs a t 120' is much smaller than that between two pairs a t 90' it is clear that one would not have an equilibrium situation since the total repulsive force on the axial pairs would be greater than on the equatorial pairs. An equilibrium situation is only obtained if the axial pairs are a t a greater distance from the nucleus than the equatorial pairs, thus reducing the axial-equatorial interactions. Hence, the axial bonds would be expected to be longer than the equatorial bonds. This conclusion would also he expected to he valid for AXIE and AXaE2 molecules. Indeed, the greater repulsion exerted by the lone-pairs should increase the difference between the equatorial and axial bond lengths. As the data in Table 10 show

Table 10. Bond Lengths in AX5, AXIE, and AXaE2 Molecules (A).

Axial Equatorial

the axial bonds are indeed longer than the equatorial bonds in all AXs, AXIE, and AX3E2 molecules for which accurate data are available. Even when the ligands in the axial and equatorial positions are different, subtraction of the covalent radii of the ligand atoms from the observed bond lengths shows that, as expected, the covalent radius of the central atom is greater in the axial than in the equatorial directions (Table 11).

Table 11 . Equatorial and Axial Radii of the Central Atom in AXS, AX*, and AXaE2 Molecules with Different Axial and

Equatorial Lieands (A).

Literature Cited

( 1 ) PAULING, L., "The Nature of the Chemical Bond," Cornell University Press, Ithaee, New York, 3rd ed., 1962.

( 2 ) ~IDGWKK, N . V., AND POWELL, H. M., PVX. Roy. SOC., A176,153 (1940).

(3) LENNARDJONES, J. E., AND POPLE, J. A,, PTOC. Roy. Sac. ( I ~ n d o n ) , A202.166 (1950).

( 4 ) MELLISH, C. E., AND LINNETT, J. W., T~an-8. Foraday Soe., 50, 657 (1954).

(5) FOWLES, G. W. A., J. CHEM. EDUC., 34, 187 (1957). ( 6 ) GILLESPIE, R. J . , AND NYHOLM, R. S., Quart. Reu. (London),

11. RRQ (1Q.57) --. --- ~ - - - . , - ( 7 ) STEWART, G. H. , AND EYRING, H., J. CHEM. EDUC., 35,550

(1958). ( 8 ) BENT, H. A,, J. CHEM. EDUC., 37,616 (1960). ( 9 ) SANDERSON, R. T. , J. CHEM. EDUC., 38, 382 (1961).

(10) GILLESPIE, R. J., AND NYHOLM, R. S., "Progress in Stereo- Chemistry," Val. 2, KLYNE, W., AND DE LA MARE, P.B.D., eds., Butterworths, London, 1958, p. 261.

( 1 1 ) GILLESPIE, R. J., Can. J . Chem., 38, 818 (1960). (12) DICHENS, P. G. , AND LINNETT, J. W., Quart. Rev. (London),

11, 291 (1957). (13) ZIMMERMAN, H. K., AND VAN RYSSELBERGH, P., J. Chem.

Phys., 17,598 (1949). (14) LINNETT, J. W . , AND MELLISH, C. E., Trans. Faladay Sac.,

50, 665 (1954). (15) GILLESPIE, R. J., Can. J. Chem., 39, 318 (1961). (16) GILLESFIE. R. J.. Can. J . Chem.. 39. 2336 (1961) ;17j GILLESPIE; R. J.; J . Am. Chem. SOC.; 83, 5Sj8 1960). (18) HARVEY, R. B., AND BAUER, S. H., J . Am. Chem. Soc., 75 ,

ZR4n ( I W R ) ~ \----,-

(19) CRAWFORD, R. A., et al., J . Am. Chem. Sac., 81,5287(1959). (20) HEDBERG, K., J . Am. Chem. Soe., 77,6491 (1955). (21) MATHIESON, A. M., M E L I ~ R , D . P., AND STEPHENSON,

N . C., Acta Crysl., 5 , 185 (1952). (22) CORRADINI, P., AND ALLEGRA, G. , J. Am. Chem. Soc., 82,

1883 (1960). (23) SCAVINCAR, S., AND GREDNIC, D., Acta Crysl., 8 , 275

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859 (1954).

+ Volume 40, Number 6, June 1963 / 301

VSEPR Theory of Directed Valency

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