Valence-Bond Description of

Research: Science and Education

A Simple Qualitative Molecular-Orbital/Valence-Bond Description of the Bonding in Main Group “Hypervalent” Molecules Owen J. Curnow Department of Chemistry, Rutherford Building, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

The participation of d orbitals in the bonding of maingroup elements has been vigorously debated (1–3). Most theoretical chemists now recognize that the primary function of the d orbitals used in calculations is to polarize the p orbitals: The shape of any orbital is perturbed when it is placed in an electric field and the most convenient way to model that perturbation is to treat it with an angular momentum function such as a d orbital (4). This paper does not attempt to address the debate on the nature of d orbital participation, but rather presupposes that d orbitals do not have a valence role in the bonding of main-group elements. When d orbitals are excluded it is no longer possible to have more than eight valence electrons, as there are only four valence atomic orbitals (although, just as an orbital can have fewer than two electrons, an atom can have fewer than eight valence electrons). For more detailed discussions on d-orbital participation the reader is directed to the papers listed in ref 1. Although many chemists have now accepted the demise of a valence role for d orbitals in the bonding of main-group atoms, spd hybrid orbital descriptions continue to be used extensively in teaching and research when addressing issues such as the π-acceptor ability of phosphines and the bonding of the so-called hypervalent molecules like PF5 and SF6 ( 5). The π-acceptor ability of phosphines and silyl systems has recently been explained by a negative hyperconjugation mechanism that employs back donation from filled p orbitals on the substituents to empty σ * orbitals on the phosphine or silyl group (6 ). In most recent inorganic textbooks this mechanism is at least offered as an alternative to the incorrect model of π donation into low-lying empty 3d orbitals on the P or Si atom (7, 8). It has also been shown that multiply bonded molecules such as R3P=O, O=S=O, ClO4᎑, and SO42᎑, which are traditionally described as hypervalent, are in fact better described by use of the alternative ionic VB structures such as R 3P+–O᎑ and O᎑–S2+ –O᎑ (2). These structures do not require any d-orbital contribution to their bonding, and any multiple bond character can be ascribed to negative hyperconjugative effects (2). One of the reasons that hypervalent molecules continue to have their bonding described in most textbooks using spd hybridization schemes is the lack of a simple bonding description that can rationalize the existence, geometries, and bond distances of “hypervalent” molecules and that is both sensible to theoretical chemists and useful to chemists. Ionic resonance descriptions, first introduced by Pauling (9), do not exceed the octet rule; however, they are not able to simply rationalize trends in bond distances. Although excellent molecular orbital (MO) descriptions are available (10, 11), they can be quite complex. The 3-center–4-electron (3c-4e) MO bonding model of Rundle (12) and Pimentel (13) was initially devised to describe the bonding in XeF2. Although this model is readily extended to a range of hypervalent com910

σ* n σ F

Xe

F

Figure 1. The 3-center–4-electron (3c-4e) bond.

pounds, it does not provide a complete bonding description for all classes of hypervalent compounds. The VB description of Linnett (14) gives a simple picture of 3c-4e bonding that allows for ready electron counting and bond-order determination. This paper brings together the ideas of Rundle, Pimentel, and Linnett and extends them to provide a simple account for the bonding in almost all hypervalent compounds as well as a rationalization of trends in bond distances. Discussion

The 3-Center–4-Electron (3c-4e) Bond For XeF2, valence shell electron pair repulsion (VSEPR) theory (15) predicts a trigonal bipyramidal structure with the F atoms in the axial positions and three lone pairs (in sp2 hybridized orbitals) in the equatorial positions. The bonding model of Rundle and Pimentel for this molecule involves an interaction of the Xe pz orbital with σ-bonding orbitals (largely of p character) on the F atoms. This multicenter bonding interaction (Fig. 1) results in a filled bonding orbital (σ), a filled nonbonding orbital (n) in which the electrons are located entirely on the F atoms, and an empty antibonding orbital (σ *). The resultant Xe–F bond order is 0.5. (In the spd hybridization bonding scheme, the Xe pz orbital is hybridized with the dz orbital to form pd hybrid orbitals, which then give rise to 2c-2e covalent bonds with the F atoms). In the conventional ionic VB framework, a valence-bond resonance structure involving one covalent bond and one ionic bond (F–Xe+ F ᎑ ↔ F ᎑ Xe+–F) has been used to ensure that the octet rule is not exceeded (3). Unfortunately, this picture is not entirely satisfactory when extended to more complex molecules; the linearity of the 3c-4e bond is not obvious, and the model does not always allow the development of formal bond orders. An alternative VB picture of Rundle and Pimentel’s MO bonding description is shown below. 2

–2/3 F

+4/3

–2/3

Xe

F

Journal of Chemical Education • Vol. 75 No. 7 July 1998 • JChemEd.chem.wisc.edu

Research: Science & Education

For the 3c-4e bond, one dot is placed between each F atom and the Xe atom (these represent the two bonding electrons that are shared between the three atoms) and one dot is placed on each F atom to represent the two nonbonding electrons which, in the MO description, are delocalized over the two F atoms. This valence picture is similar to the nonpaired spatial orbital (NPSO) VB picture proposed by Linnett (14). The only possible shortcoming with this description is that, in the case of XeF2, it gives formal charges of ᎑0.5 for the F atoms and +1.0 for the Xe atom, whereas an analysis of the MO description gives formal charges of ᎑ 2/3 for the two F atoms and + 4/3 for the Xe atom. However, in a qualitative bonding analysis, the magnitudes of the formal charges (which are just that—formal atomic charges) are not important. Perhaps more importantly, this VB description is consistent with the result from the molecular orbital description that the octet rule is not exceeded. It has been argued that 3c-4e bonds break the octet rule because Xe is using five pairs of electrons in its valence shell (16 ). However, because one of those five electron pairs is delocalized entirely on the F atoms in the nonbonding orbital n, only four pairs are in fact delocalized on the Xe atom. Xe has four valence atomic orbitals that can contain no more than eight electrons. This also applies to the F atoms. Furthermore, consider the molecule F2: based on the conventional MO description, the F atoms share seven pairs of valence electrons (four in bonding orbitals and three in antibonding orbitals), but the VB description does not break the octet rule. By placing the Lewis dots for XeF2 as shown above, one can see that there is a trans relationship for the atoms involved in the 3c-4e bonds, that all the atoms obey the octet rule, and that the Xe–F bond order is 0.5. For the molecules ClF3 and SF4, it is necessary for the equatorial F atoms to be bound by regular localized 2c-2e bonds via sp2 hybridized orbitals on the central atom and for the lone pairs to be in sp2 hybridized orbitals. This necessity arises because stereochemically active lone pairs must be in sp, sp2, or sp3 hybridized orbitals (the nonbonding electrons in multicenter bonding orbitals are stereochemically inactive), and sp or sp3 hybridization in the equatorial plane of either of these molecules would not give the most favorable

bonding interaction to the F atoms. The axial F atoms must then be in 3c-4e bonds. The bond order for each axial bond is thus 0.5, whereas the bond order for each equatorial bond is 1.0. This bond order difference provides a rationale for the observed differences in bond distances in which the axial bond distances are significantly greater than the equatorial 2c-2e bond distances (Cl–F ax = 169.8 pm, Cl–Feq = 159.8 pm [17 ]; S–Fax = 164.6 pm, S–Feq = 154.5 pm [18]). It is also instructive to consider the formal charges: for the VB model, each axial atom has a formal charge of ᎑1/2, whereas each equatorial atom has a formal charge of zero. Consequently, in mixed ligand systems, the more electronegative substituents are found in the 3c-4e bonds in the axial positions, where there is a higher formal negative charge. F

F F

Cl

F

F F

F

AX5 Molecules and Multicenter Bonding The similarity of the axial and equatorial P–F bond distances in PF5 (P–Fax = 157.7 pm and P–Feq = 154.3 pm [19]) has been difficult to explain. One description of the bonding is that the equatorial atoms are bound to the P atom by regular 2c-2e bonds using sp2 hybridized orbitals on the P atom and that the axial F atoms are bound via a 3c-4e bond. Unfortunately, this gives formal bond orders of 1.0 and 0.5 for the equatorial and axial bonds, respectively, and provides no rationale for the observed bond distances. However, unlike XeF2, ClF3, and SF4, which have at least one stereochemically active lone pair of electrons, there is no requirement for sp2 hybridization of phosphorus in PF5; it can alternatively be sp-hybridized to form two regular sp covalent bonds to the axial F atoms, which leaves the px and py orbitals to form a 4-center–6-electron (4c-6e) bond. The orbital interactions for 4c-6e bonding are shown in Figure 2, and the corresponding Lewis dot structures are shown below. F

F F

F

P

a

e

Figure 2. The 4-center–6-electron (4c-6e) bond.

F

F F

F

P

F F

e*

S

F F

P

F F

F F

P

F F

F F

The VB description of the 4c-6e bond has three resonance structures because there are four bonding and two nonbonding electrons to share between three bonds. Together with the 3c-4e bonding description, there are therefore a total of four resonance structures that satisfy the octet rule. For the sp/4c-6e bonding description, the bond orders of the P–F eq bonds are 2/3 and the bond orders of the axial bonds are 1.0. From a consideration of the molecular orbital occupancies shown in Figure 2, the formal charge on the P atom is 1.83, while each of the equatorial F atoms has a formal charge of ᎑0.61 and the axial F atoms have zero formal charge. In the VB picture the average formal charges are +1, ᎑1/3, and zero, respectively. Just as we cannot simply determine the relative contribution of resonance structures in a molecule like N2O, it is not obvious what the relative contributions of the resonance structures are in PF5. However, mixing of the sp2/3c-4e and sp/4c-6e bonding descriptions in a 1:1 ratio would give av-

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erage bond orders of 0.75 and 0.83 for the axial and equatorial bonds, respectively, and would provide a rationale for the similarity of the P–F bond distances. The bond distances in the series of compounds PF5, MePF4, and Me2PF3 can also be rationalized by considering the relative contributions of the sp2/3c-4e and sp/4c-6e bonding descriptions: Me groups, being much less electronegative than F, prefer to be in regular spx 2c-2e bonds, as multicenter bonding induces a higher negative charge on the substituents. Therefore, the contribution from sp2/3c-4e bonding is expected to increase and give a larger increase in the length of the axial bonds than in the equatorial bonds. As Table 1 (20) shows, this is the observed trend. The only 10-electron AX5 compounds observed to date that do not have a trigonal bipyramidal structure are [InCl 5 ]2᎑ and [TlCl5 ]2᎑, which have square pyramidal structures. These molecules can most simply be described as octahedral-like species with an empty sp orbital. Thus the apical atom is in a regular sp 2c-2e bond with a bond order of 1.0, while the basal atoms are in 3c-4e bonds with bond orders of 0.5. Indeed, the apical bond distance is found to be shorter than the basal bond distances: In–Clap = 241.5 pm and In–Clbas = 245.6 pm (21). It might be expected that the basal atoms would bend towards the apical atom so as to maximize the 3c-4e bonding interaction, but in fact the average Clap-In-Clbas angle of 103.9° is close to the optimum angle for minimizing steric interactions (103.6°) (21). As can be seen in the Lewis dot picture below, the In atom has six valence electrons in this bonding description and is formally electron deficient. [InCl5]2᎑ may be stabilized by back-bonding from the lone pairs on the halide atoms into empty orbitals on the In atom, just as the electron-deficient molecule BF3 is stabilized by backbonding from the F lone pairs. In this case, the major backbonding interaction would be into the empty sp orbital (there may also be some from the apical atom lone pairs into the 3c-4e antibonding orbitals). It may be that competing electronic interactions result in the sterically optimized Clap -In-Clbas angle.

orbital with the basal atoms results in Xap-A-Xbas angles near 80°. Analogously to [InCl5]2᎑, the apical bond distances are shorter than the basal bond distances by 3 to 30 pm (22). Also consistent with this bonding description is that in mixed substituent species the more electropositive substituent is in the apical position (10). F F

F I

F

F

12-Electron AX4 Molecules: Two 3c-4e Bonds or One 5c-8e Bond? XeF4 has a square planar structure in which there is a lone pair in each axial position. These lone pairs are located in sp hybridized orbitals, leaving two p orbitals on the Xe atom for bonding with the F atoms. This can be achieved by either two 3c-4e bonding interactions or one 5c-8e bonding interaction. These bonding descriptions are in fact identical and will both give two filled bonding MOs, two filled nonbonding MOs, and two empty antibonding MOs. The Xe–F bond order is thus 0.5 and the formal charges are ᎑1/2 for the F atoms and +2 for the Xe atom. The average Xe–F bond distance of 195.2 pm (23) is between the average Xe–F bond

2

Cl

3c-4e anti-bonding orbitals

t1u*

In Cl Cl

Cl

Why is the square pyramidal structure favored for these compounds? The group 13 dianions have the highest overall negative charge of all the 10-electron AX5 compounds; and of the group 13 compounds, the indium and thallium chlorides have the largest differences in electronegativity between A and X. Thus the relatively high-energy valence orbitals on the central atom favor an sp orbital being empty. Also, the formal charge on the central atom in a dianionic trigonal bipyramidal structure would be ᎑1, whereas in a square pyramidal structure the formal charge on the central atom is zero. The increase in multicenter bonding may help to distribute the negative charge away from the electropositive indium or thallium atom and towards the electronegative chlorine atoms. Twelve-electron AX5 compounds such as IF5, XeF5+, and [SbCl5]2᎑ also have square pyramidal structures, but now the sp orbital is occupied and an unfavorable interaction of this 912

a1g*

Cl

3c-4e non-bonding orbitals

eg

3c-4e bonding orbitals

t1u

a1g

Figure 3. MO diagram for octahedaral AX6 molecules with 12 valence electrons.



distances of XeF2 (197.7 pm) (24) and XeF6 (189.0 pm) (25) in which the Xe–F bond orders are also 0.5. The changes in oxidation state and formal charge readily account for these differences.

are shown in Fig. 4) and regular sp 2c-2e bonds for the axial atoms. As there are four bonding electrons and six non-bonding electrons, the VB structure, shown below for IF 7, has five resonance structures (only one is shown). F F

F

F

F Xe

I

F F

F

F F

12-Electron AX6 Molecules The bonding in SF 6 can be described as three 3c-4e bonds with one more bonding MO being formed from the 3s orbital on S and the fully symmetric nonbonding combination of the 3c-4e nonbonding orbitals (a1g in Fig. 3). The corresponding antibonding combination, a1g*, is empty. A simple alternative bonding description involving a resonance of three VB structures, shown below, consists of two 3c-4e bonds and two 2c-2e bonds formed using sp hybridized orbitals. In the MO description the only electrons associated with the S atom are in the a1g and t1u orbitals; this gives a total of eight valence electrons on the S atom and does not exceed the octet rule. The VB structures also satisfy the octet rule. F

F

F

S F

F F

e1* F

F

S F

F

F

F

S

This description results in axial bond orders of 1.0 and equatorial bond orders of 0.4. As is then expected, the axial bonds are shorter than the equatorial bonds: for IF7, I–Fax = 178.6 pm and I–Feq = 185.8 pm (28); and for [IF6O]᎑, I–Fax = 182.0 pm and I–Feq = 187.4 pm (29). The formal charge on each of the equatorial atoms of IF7, is –0.6, while the I atom has a formal charge of +3. The axial atoms have formal charges of zero. It should be noted that for AXnO oxide species such as

F

F

F

F

F

F F

From the VB description, the average S–F bond order is 2/3, which is in agreement with the MO bonding scheme. The S–F bond distance of 156.4 pm (26 ) falls between the S–F bond distances of SF4 (154.5 and 164.4 pm) (18), in which the bond orders are 0.5 and 1.0. One difference between the MO and VB descriptions is that the calculated formal charge on the S atom is +3.71 and +2, respectively.

14-Electron Molecules with 6c-10e Bonding VSEPR theory predicts pentagonal bipyramidal structures for 14-electron compounds. However, only a few compounds exhibit this structure (27–29). The bonding in IF7, [TeF7] ᎑, [IF6O] ᎑, and [TeF6O]2᎑ can be described by invoking a 6c-10e bond for the five equatorial atoms (the orbital interactions

e2 a

e1 Figure 4. The 6-center–10-electron (6c-10e) bond.


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[IF6O]᎑ and [TeF6O]2᎑, the overall contribution of the O atom to the electron count of the central atom is zero whether it is considered as a Lewis base/acid adduct, X n A→O, or as an ionic VB structure, X nA+–O ᎑. The 14-electron molecules [Sb(C2O4)3]3᎑ and [XeOF5]᎑ both have pentagonal pyramidal structures with a lone pair in an axial sp hybridized orbital. Each oxalato ligand is bidentate and contributes two electrons to the central Sb atom. As would be expected for [Sb(C2O4) 3] 3᎑, the apical bond distance of 208 pm is shorter than the basal bond distances, which average 229 pm (30). Also consistent with a bond order of 0.4, the average Xe–F bond distance of 199.5 pm in [XeOF5]᎑ is longer than all Xe–F bond distances in which the formal bond orders are greater than 0.4 (see Table 2) (31). Fourteen-electron molecules with pentagonal planar structures, in which there is a lone pair in each axial position, are very rare: [Te(S2COEt)3]᎑ and [XeF5]᎑ are the only examples known to date (32). [Te(S2COEt)3]᎑ contains one monodentate ligand that contributes one electron to the central Te atom and two bidentate ligands that each contribute three electrons to the Te atom. The long average Xe–F bond distance in [XeF5]᎑ of 201.2 pm is even longer than that of [XeOF5]᎑. Although both have Xe–F bond orders of 0.4, [XeF5] ᎑ has a lower oxidation state (+4 vs + 6) and formal charge (+2 vs +3).

Figure 5. Molecules with arrangements of lone pairs that are not possible.

14-Electron AX6 Molecules The 14-electron AX6 molecules have proved to be an interesting problem. By analogy with the 14-electron compounds described above, one would expect a pentagonal bipyramidal structure with a lone pair in an axial sp hybridized orbital. However, most AX6 molecules have an octahedral or distorted octahedral structure. Addition of one pair of electrons to the MO diagram for the 12-electron octahedral AX6 molecules (Fig. 3) indicates that the extra pair of electrons is located in an antibonding orbital made up of the s orbital on A and the fully symmetric combination of nonbonding 3c-4e orbitals on the X atoms (a1g* in Fig. 3). The A–X bond order is, therefore, 0.5; and for neutral AX6 molecules, the formal charges are ᎑2/3 for the X atoms and + 4.0 for the A atom. (From the VB picture, the formal charges are –1/2 and +3, respectively.) The average Xe–F bond distance of 189.0 pm in XeF6 is shorter than is observed in XeF2 (197.7 pm) and XeF4 (195.2 pm), which have the same Xe—F bond order but lower oxidation states and formal charges for the Xe atom. The electrons from the s orbital of A are essentially nonbonding electrons because they are involved in both the bonding a1g and antibonding a1g* orbitals. They can be pictured as a spherically symmetric, stereochemically inactive lone pair:

Hypervalent, Hypercoordinate, or Hypobound? The question of whether molecules such as PF5 and SF6 are “hypervalent” depends on the definition of hypervalency. The conventional definition is that such molecules exceed the octet rule. However, the octet rule is not exceeded for these molecules. An alternative definition is that “hypervalent” molecules have unusually high oxidation (or valence) states. The question then becomes: What is an unusually high oxidation state? The +3 oxidation state of the In atom in [InCl5] 2᎑, for example, is not unusual. Some authors have recently coined the term “hypercoordinate” (34), but this term does not fit comfortably with molecules like ClF 3 and XeF2. An alternative term, proposed here, is “hypobound” (derived from Greek for “less than”), as these molecules all have at least one bond with a bond order of less than one.

F

F

F Xe

F

F F

It is difficult to rationalize the bonding in [Pb(S2P(OiPr) 2)2]n. The geometry about the Pb atom is best

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described as a pentagonal bipyramid with a lone pair of electrons in the equatorial plane (33). It may be that relativistic effects, f orbitals, or the nature of the polymeric structure could account for this unusual coordination geometry.

The Nature of Lone Pairs An interesting aspect of multicenter bonding is that the nonbonding electrons have no stereochemical influence, whereas a pair of nonbonding electrons located in an s orbital hybridized with one or more p atomic orbitals (an sp, sp2, or sp3 orbital) results in a stereochemically active lone pair. Consequently, main-group atoms cannot display any of the coordination geometries that are shown in Figure 5, and to date, none have been observed.

Relevance to VSEPR Theory and Lewis Structures If the octet rule is not exceeded for hypobound molecules, does this mean that there is something wrong with VSEPR theory? Not really. It is a misconception that VSEPR says anything about the number of valence-shell electron pairs on the central atom; it only says what the geometric distribution of the substituents and lone pairs is likely to be, based on the Lewis structure. Lewis structures work because four electrons are accounted for both in a 3c-4e bond and in two regular 2c-2e spx hybrid bonds, so the number of electrons left to form stereochemical lone pairs will be the same and the VSEPR structure will be the same. As far as the octet rule is concerned, it should be remembered that there cannot be more than eight electrons in four valence orbitals—although, of course, there can be fewer than eight.



Conclusions One of the major arguments used in favor of d-orbital hybridization was the apparent lack of “hypervalent” compounds formed by the first-row elements, but in fact there are rare examples of molecules containing five and six coordinate atoms from the first row (35). As has already been mentioned by a number of workers (3, 36 ), two major factors disfavor the formation of hypobound compounds by firstrow elements. One is the significantly smaller size of the firstrow elements, which results in a high electron density on the atom. The second is their much higher electronegativity, compared with the second- and later-row elements, which also leads to a high electron density on the atom. Although multicenter bonding alleviates high electron densities, it is rarely enough to offset these two factors for the first-row elements. The use of 3c-4e, 4c-6e, 5c-8e, and 6c-10 multicenter bonding interactions to describe the so-called “hypervalent” molecules offers an approach that is more accurate than, and as simple as, current models that describe bonding via spd hybridization. Multicenter bonding also allows the determination of bond orders and a rationalization of bond-distance trends. It should also be pointed out that it is not necessary to be intimately familiar with the multicenter MO diagrams to obtain useful information from such a bonding description. Simply fill in the Lewis dot structure so as not to exceed the octet rule and to satisfy the geometric requirements of spx hybridization, and the bond orders and formal charges will then follow. The term “hypobound” is proposed to replace “hypervalent”, which has been shown to be inappropriate. Acknowledgments Bryce E. Williamson and Robert G. A. R. Maclagan from the Department of Chemistry at the University of Canterbury are to be thanked for useful discussions. Literature Cited 1. Gilheany, D. G. Chem. Rev. 1994, 94, 1339. Reed, A. E.; Weinhold, F. J. Am. Chem. Soc. 1986, 108, 3586. Magnusson, E. J. Am. Chem. Soc. 1990, 112, 7940. 2. Reed, A. E.; Schleyer, P. v. R. J. Am. Chem. Soc. 1990, 112, 1434. Suidan, L.; Badenhoop, J. K.; Glendening, E. D.; Weinhold, F. J. Chem. Educ. 1995, 72, 583. 3. Kutzelnigg, W. Angew. Chem., Int. Ed. Engl. 1984, 23, 272; Angew. Chem. 1984, 96, 262. 4. Coulson, C. A. Nature 1969, 221, 1106. 5. Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper and Row: New York, 1983. Cotton, F. A.; Wilkinson, G.; Gauss, P. L. Basic Inorganic Chemistry, 3rd ed.; Wiley: New York, 1995. King, R. B. Inorganic Chemistry of Main Group Elements; VCH: New York, 1995. Winter, M. J. Chemical Bonding; Oxford University Press: Oxford, 1994. 6. Schleyer, P. v. R.; Kos, A. J. Tetrahedron 1983, 39, 1141. Farnham, W. B.; Smart, B. E.; Middleton, W. J.; Calabrese, J. C.; Dixon, D. A. J. Am. Chem. Soc. 1985, 107, 4565. Grein, F.; Lawlor, L. J. Theor. Chim. Acta 1983, 63, 161. 7. Shriver, D. F.; Atkins, P. W.; Langford, C. H. Inorganic Chemistry, 2nd ed.; Oxford University Press: Oxford, 1994.

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Valence-Bond Description of

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