Tangent-sphere models of molecules, IV Henry A. Bent University of Minnesota Minneaoolis
I
Estimation of hternuclear Distances; The Electronic Structure of Metals
In Parts I and I1 of this series ( 1 ) it was proposed that the Pauli Principle might play a more important role in chemistry than has generally been supposed. The chemical implications of this proposal were explored with a model called the tangent-sphere model. It was found that this model is capable of representing the familiar stereochemical features of conventional covalent bonds. The model is also capable of representing highly strained bonds and mnlticentered bonds; atoms with expanded octets and atoms with contracted octets; intermolecnlar interactions and intramolecular interactions: the effects of electrone~ative groups, lone pairs, and multipIe bonds on molecular geometry, bond properties, and chemical reactivity; and common reaction mechanisms. In Part I11 (2) the model was extended to compounds thst contain second-row elements. The supposition in these cases of large atomic cores, comb'med with Linnett's suggestion regarding the tendency of electrons in different spin sets to become anticoincident (3),was found to yield simple explanations for such wellknown features of the chemistry of second-row (and later) elements as the occurrence of near-90" valence angles; lengthened bonds and lowered ionization potentials; expanded octets; enhanced r:eactivity toward nucleophilic reagents (more generally, heightened odonor properties and r-acceptor properties) the scarcity of multiple bonds to other second-row elements; the low basicity of attached lone pairs; and (less well known) several anomalous bond-length trends, for which a semiquantitative explanation was offered. It has been found that the model can also be used to estimate internuclear distances and effects of backbonding in covalent compounds; to explain, through
-
This study was supported by a. grant from the National Science Foundation. Tangent-sphere model of methyl silane. The methyl group is shown rtt the top, the silyl group s t the bottom of the figure. The tetrahedral cluster of small spheres represents the silicon atom's coincident L-shell. Stippled spheres represent silicon's valence-shell electrons, the bottom three Si-H bonds, the upper one the Si-C bond. The three large spheres a t the top represent the C-H bands of the methyl group. [Reprinted from Figure 2, Part 111($).I
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Journal of Chemical Education
an extension of the classical radius ratio rules for ionic crystals, why first-row elements generally obey the octet mle and second-row elements often do not (and, in this context, to rationalize the nonexistence of several otherwise seemingly plausible compounds); to account semiquantitatively for lengthened bonds to atoms with "inert pairs''; to estimate van der Waals radii and internuclear distances in some ionic compounds; and to initiate steps toward a chemical description of the electronic structure of metals. These uses of the model will now be described. The R Equation
The tangent-sphere model of methyl silanel suggests that useful estimates of internuclear distances might be made by combining estimates of atomic-core sizes with estimates of the sizes of valence-shell electron-pairs. Using for the former values tabulated by Pauliig (4), denoted hereafter r',? one finds from a plot of internuclear distance against r' that the radii of valence-shell electron-pairs increase as the core sizes increase. The relation between the valence-shell electron-pair radius R and the core radius r' is given, approximately, by the equation R
=
0.60
+ 0.40~'
(1)
By equation (I), for example,
+
Rc*+ = 0.60 0.40r'c.+ = 0.66 A. (r'v+
=
0.15
A)
Thus, for the silicon-carbon internuclear distance in methyl silane, one calculates for a spherical-core modela d(Si-C) = 2R
+ r'si" + ?'cat = 1.88 A
The observed value [Table 1,Part I11 ($)I is 1.8668 A. a These values are not necessarily the best values thst might have been selected for the present purpose. To limit the number of adjustable parameters in this treatment, however, it is desirable to use at this point fixed values for the atomic-core radii. Values (A) used in this paper are: (first-row) Lif 0.60, Be'+ 0.31, Ba+0.20, C'+ 0.15, N6+0.11, On+0.09, F'+ 0.07; (second row) Na+ 0.95, Mgn+ 0.65, Ma+0.50, Si4+0.41, P6+0.34, Se+ 0.29, CP+ 0.26; (others) K + 1.33, Ca3+ 0.99, Ass+ 0.47, See+ 0.42, Br7+0.39, Md+0.62, Sb6+ 0.62, Te6+0.56, I'+0.50. In this expression the distance from the center of the banding pair to the carbon nucleus (TO in the figure reprinted in&otnote rlca+ rather than the ( d 3 / 4 ) ~ 1) has been taken as R dictated by the model far atoms whose vden-hell electrons touch each other before they touch their atomic car?-which broadly speaking ia the case for atoms with r' _< 0.15 A,j.e., far finkrow elements to the right of boron. For r' 13 0.1 A, however, ( ~ / Ar;; ) R R rt.
+
+
In the previous calculation the radius used for the silicon-carbon bonding pair was calculated from the r' value for the smaller atomic core. This procedure reproduces bond lengths better than does the use of average values for ?'. Also, it yields reasonable values for the radii of lone pairs, which represent the limit approached by bonding pairs as the radius of the atomic core on one side of the bonding pair approaches infinity. For nitrogen and oxygen it has been found by calculation that the radii of the lone pairs are about the same size as the radii of bonding pairs on the same atoms (6). This method of calculating R implies that the decrease in bond length in going from methyl phosphine to methyl sulfide to methyl chloride should directly reflect the decrease in core size in going from phosphorus to sulfur to chlorine. The latter changes are 0.05 and 0.04 A, respectively; the corresponding bond-length changes [Table 1, Part I11 (8)l are 0.040 and 0.034A. I t implies, also, that single bonds between different atoms should generally be shorter than the sum of their covalent radii; for when the atomic cores of two atoms ditrer in size, the sum of the core radii plus twice the Rvalue calculated for the smaller atom will be less than the model's covalent-radius sum, the latter being simply the sum of the r' and R values for the two separate atoms. This conclusion can easily be checked. Atoms that differ in core size generally will differ also in electronegativity. As already noted by Shomaker and Stevenson, bonds between atoms that differ in electronegativity are usually shorter than the sums of their covalent radii (6). Nevertheless, covalent radii are useful guides in the interpretation of internuclear distances. Some values obtained from the relation
which follows from equation (I), (with values from Pauling in parentheses) are C = 0.81 (0.77), N = 0.75 (0.70), Si = 1.17 (1.17), P = 1.08 (1.10), C1 = 0.96 (0.99),As = 1.21 (1.21), Se = 1.19 (1.17), andBr = 1.15 (1.14). Table 1 is a brief survey of internuclear distances calculated by equation (1). The calculated values generallydiffer from the experimentalvalues by less than 5%. Deviations greater than 5% do occur, however. These large deviations often correspond to wellrecognized chemical effects. Back-Bonding
Relatively large deviations between calculated and observed bond lengths are found for BF3 (+IS%), SiF, (+12%), Si-0 (+15%), P-0 (+lo%), and S-0 (+9.4%). I n each case the calculated single-bond length is greater than the observed bond length, correspondmg to the generally accepted picture that bonds between these atoms have some partial double-bond character (7). The discrepancies between the calculated and observed bond lengths serve, in fact, as quantitative measures of the double-bond character of these bonds. It is interesting, in this regard, to compare the calculated and observed bond-length change that occurs in passing from BFI to BF4-. The value of R for these
Table 1.
Comparison of Calculated and Observed Internuclear Distances
----Internuclear Calculated Observed
Structure
(oxygen)" (nitrogen) r, (phosphorus) r,, (arsenio) r , (iodine) T,, T.V
r,,
=
1.44 1.42 1.81 2.05 2.10
distance-----
%
Error
1. 50 (Pauling) 1.40 (Pauling) 1.90 (PauFng) 2.00 (Paulmg) 2.15 (Pauline)
van der Wads radius.
two nlolecules is determined by the 1.' value of the fluorine substituent; for both BFa and BFp-, therefore, R = 0.63 A. Now, for an atomic core in a trigonal (ti) environment of electron pairs, whose radii are 0.63 A, the nonrattling condition is T' (tr environment) 0.154 (0.63 A) = 0.097 A. Simiirly, for nonrattling in a tetrahedralo(te) environment, T' (te environmeut) 2 0.225(0.63 A) = 0.142 A;\. The Ba+ core with a radius of 0.20 (4) will not rattle in either environment. In the absence of back-bonding, one would therefore predict on the basis of the present model that the B-F bond lengths in BFa and BF4- would be the same [cf. SbF, and SbFS2-, Part I11 ($)I. On the other hand, if the valence-shell electrons about the B8+ core are taken as touching each other in both BF, and BF,- (the small-core model), the predicted bond-length change, again in the absence of back-bonding, is (0.63 &(1.225 - 1.154) = 0.04 A. The observed bond-length chttnge (B-F in BF4- less B-F in BF,) is about 1.43 A 1.295 A = 0.13 A. Evidently there is more backbonding in BF1 than in BFI-. The effect of this additional back-bonding in BF, is to shorten its B-F bond by about 0.13 A - 0.04 = 0.09 A. The bond-shortening effect discussed above appears to diminish as one moves to the right in a given period of the periodic table.' The reason for this may lie in the corresponding decrease in core size, which in the present model would lead to a decrease in separation of the a-bond electrons and, hence, to a decrease in the space available in the valence-shell for additional electrons. As the nuclear charge of a rr-acceptor atom increases, auxiliary p, - d, bonding electrons are, so to speak, squeezed out of the atom's valence-shell by the atom's primary bonding pairs as the latter press inward upon the contracting atomic core.
>
' P - 4 , S--0, and C1--0 bond-stretching farce conatants (9.05, 9.07, and 8.24 millidynea/A) lend some support to this K ,W. J., view. For a contrary discussion, see C R U I C K S E ~D. J . C h . Soc., 5486 (1961). Volume 42, Number 7, July 1965
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349
Contraction of the back-bonding space with increasing nuclear charge may explain, thus, why bonds are longer in CIFathan in SF4,why they are longer in C10,than in S O P (note, too, the instability of CIOa-), and why the C1-0 distance in ClzO is longer than the "single-bond" distances between oxygen and silicon, oxygen and phosphorus, and oxygen and sulfur.
The nonrattling condition for octahedral electronpair coordination is
Steric Effects and Electron-Pair-Coordination-Numbers
This calculation assumes that the radii of the valenceshell electron-pairs are determined by the central atom. If fluorine is the ligand, the nonrattling condition is
In most cases bond lengths calculated by equation (1) are larger than the observed values; in several cases, however, the calculated values are noticeably smaller. Carbon tetrabromide is one such case, nitrogen trichloride another. I t is tempting to suppose that these cases reflect the effect upon bond lengths of steric repulsions between electrons on nonbonded atoms. Scale models suggest that such effects would be rather larger than normal, also, in the nonexistent dimers of boron trimethyl and boron trichloride (S),but not appreciably so in AL(CH&, A12Cl6, and (BeClJ,, owing to the large size of the aluminum and beryllium cores.
r' 2 0.414R.
(3)
Substituting for R from equation (I), one finds that to coordinate six electron-pairs without rattling r'
2 0.30 A.
r' 2 0.26 A.
(4)
(5)
From equations (4) and (5) one predicts that first-row elements from fluorine (r' = 0.07) through boron (r' = 0.20) should not expand their octets. Beryllium (r' = 0.31) lies near the borderline. For this atom, formal charges work against the formation of six ordinary twocenter u-bonds; such charges would be satisfactory, however, if the atom formed six six-center u-bonds (vide infra). Interestingly, beryllium does form a volatile sandwich compound with cyclopentadiene, (CsHs)l Be. In this compound the beryllium core is located unsymmetrically between the two cyclopentadienyl rings, 1.485 k from one ring and 1.980 k from the other (9). Sulfur (7' = 0.29) and chlorine (r' = 0.26) lie near the borderline, also. It is intercsting to note, therefore, that SC1, does not exist, although PC1,- does exist, and so does SF, (partly this could be a steric effect between noubonded atoms; uide supra). Also, the compound C1F6 has not been reported, although BrF6 exists. Attempts to make ArFl have been unsuccessful ( I O ) . ~ Bond Lengths in SeCls2- and TeCls2- and the "lneri Pair" Effect
Figure 1. One face of the tongent-sphere model of ToaC1t22C. The four m o l l black dots represent C17+ corer; e m h core is tetrohedrdly surrounded b y four localized electron pairs. The larger centrol dot represents a To6+ core; i t is surrounded by a rquare ontiprirmotic orrangement of eight localized electron pain. Stippled spheres represent lone pairs on the 1 2 bridging chlorine otoms, open spheres ordinary two-center chlorinelantalum bondr, inner black sphere, three-center tantal~m-tontalum-tantalum bonds. The rtrvctvre is built up of on inner cube of eight Ta-Ta-To bondr on whore six facer rest squareplanor retr of Ta-CI bondr. On the 1 2 edges of the rervlting structure ore diagonal sets representing chlorine-atom lone pairs. The orrow indicoter one of the 2 4 rteric intermtionr between lone pairs on adjacent nonbonded chlorine otomr. The fractional oxidation number of tontolvm in this rtructure orirer from the presence of the three-center bonds. For an alternote description d the electronic structureof ToaClazf, see COTTON, F. A,, AND HASS,T. E.,Inorg. Chem., 3, 10 (19641.
Similar considerations may account for the nonexistence of numerous compounds in which there would be a large number of electron pairs, 8 for example, about highly charged cores seemingly large enough to coordinate this number of electron pairs without rattling, but which, nonetheless, appear not to do so. Figure 1shows, for example, that if an atomic core is only just large enough to fit without rattling into the M6C1,,2+ structure, steric interactions occur between the lonepairs on adjacent chlorine atoms. A larger core, such as Ta5+is required, not only to occupy adequately the Msites, but, beyond this, to keep apart the lone-pairs on adjacent ligand atoms. 350
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Journal of Chemical Education
With r'%#+ = 0.42 and r ' =~0.56, ~ equation (1) yields for the bond lengths in SeCls2- and TeC&- 2.08 and 2.22 A. The experimental values are much larger: 2.41 and 2.51 A. The discrepancy in this case arises from the presence about the Se6+and Te6+ cores of stereochemically inert ("inert" as to effect on bond angles, not bond lengths) 4s and 5spairs. A further test of the several hypotheses upon which the previous discussion has been based, particularly the hypothesis that the Pauli Principle holds in a strong form, may be made by supposing, as the latter hypothesis requires, that these outer s electrons of selenium and tellurium form spherical shells about their respective atomic cores. If, as a zeroth approximation, the volumes of these s electrons are assumed to be the same as the volumes given by equation (1) for electron pairs about the same 6+ cores, the calculated radiiof the Se4+andTe4+cores are 0.81 and 0.89 I%. The correspondmg calculated bond lengths are 2.48 and 2.56 k, remarkably close to the experimental values. These considerations have a bearing on the shape and
chemical valence theory advanced by the late R. E. Rundle, in "Survey of Progress in Chemistry,'' edited by SCOTT,A. F., Academic Press, New York, 1963.
size to be expected for xenon hexafluoride. The structure of this molecule has not yet been determined. If the bonds in XeFBturn out to be of normal length, comparable to those in XeFp, the electron-pair-coordination-number of the xenon atom should probably be reckoned as 7; correspondingly, the n~oleculeshould be expected to have the shape of an irregular octahedron. If, however, the molecule has the shape of a regular octahedron, the seventh pair about the xenon atom is probably to be viewed as an inner ("inert") s-pair; correspondingly, the molecule should be expected to have bonds that are significantly longer than normal, by analogy with TeC162-. van der Woals Radii and Intermolecular Distances
Table 1 lists some estimates of van der Waals radii. The physical model upon which these estimatesare based is the model illustrated in Figure 2, Part 111 (8)'. For first-row elements, whose valence-shell electrons generally touch each other before they touch their atomic cores, the values listed in Table 1 have been calculated from the fommla TVW
+ 1) R
(small core) = (
