ELECTRON CORRELATION AND BOND PROPERTIES IN SOME SELECTED SULFUR COMPOUNDS

CHAPTER 1

ELECTRON CORRELATION AND BOND PROPERTIES IN SOME SELECTED SULFUR COMPOUNDS HENRY A. BENT School of Chemistry, University of Minnesota, Minneapolis, Minnesota Abstract—From a survey of the effects on molecular properties of multiple bonds and electronegative groups an empirical rule is formulated to describe the character of electron correlation in molecules. This rule states that the s character of a combined atom tends to concentrate in orbitals that the atom uses in bonds toward electropositive substituents. To account for this rule and for certain noted exceptions to it, there is introduced a model called the tangent-sphere model. This model, which has been found useful in a variety of chemical problems, is applied to the following topics: the stereochemistry of the elements with electron-pair coordination numbers ranging from one through six, intermolecular interactions during chemical reactions and in crystalline solids, intramoleculgr interactions, tilted methyl groups, the effect of lone pairs and multiple bonds and electronegative groups on bond angles and bond lengths, and the variable effect of unsgtureticn upon bond lengths. Also discussed in some detail are some qualitative applications of the Hellmann-Feynman theorem, anticpincidence, the structures of sulfur tetrafluoride and hydrogen sulfide and carbon disulfide, and the nature of the sulfur-oxygen bond.

PART I THE correlation of bond angles with orbital hybridization ratios is well known.1, 2 When the s p hybridization ratio in two equivalent orbitals increases from 1/3 (spa orbitals; 25 per cent s character in each orbital) to 1 (sp orbitals; 50 per cent s character in each orbital), the corresponding interorbital angle, and often as well the corresponding bond angle, increases from 109° 28' to 180°. That with such a change in orbital hybridization (and bond angles) there may be important changes also in bond lengths, bond strengths, and such other bond properties as carbon(13)proton coupling constants and inductive constants now seems clear, too. The evidence for this may be summarized as follows. Bond lengths It appears to be well established that bond lengths tend to decrease as the s-content of the corresponding bonding atomic orbitals increase. For carbon-carbon single bonds, this decrease may amount to 0.03-0.04 A when the hybridization of one of the atoms participating in the bond changes from spa to sr2 or from sr2 to sp. Thus whereas the C(sp3)—C(sp 3) interatomic distance is typically 1.53-1.54 A (diamond and the normal paraffins), the C(sp3)—C(sp 2) distance is typically only 1.50 A (acetaldehyde and propene), the C(sp3)—C(sp) distance 1.46 A (methyl acetylene and methyl cyanide), the C(sp2)—C(sp 2) distance 1.46-1.48 A (1,3,5,7-cyclooctatetraene and 1,3-butadiene, respectively), the C(sp2)—C(sp) distance 1.42 A (vinyl cyanide), and the C(sp)—C(sp) distance 1.38 A (cyanogen and diacetylene). Similarly the length of the carbon-carbon double bond decreases as the hybridization of the sigma orbitals of the participating carbon atoms changes from sr2 to sp. Thus the normal carbonB

B

HENRY A. BENT

2

carbon internuclear distance for an sp2 —sp 2 sigma bond with a superimposed jr-bond (to use the conventional notation for a double bond) is close to 1.34 A (ethylene, propylene, 1,3,5,7-cyclooctatetraene, and 1,3-butadiene), but the C(sp2)—C(sp) + p distance is typically only 1.31 A (allene and ketene) and the C(sp)—C(sp) }distance is typically only 1.28 A (butatiene and carbon suboxide). These facts are summarized in Table 1. TABLE 1. EFFECT OF ATOM HYBRIDIZATION ON THE C—C DISTANCE C—C hybridization

1. sp 3—sp

Valence-bond structure

\

3

/

—C—C7

2. sp3—spe

3. sp 3—sp

,

\

\

—C—C

=

\

%

/

4. spe—spe

C—C

\

%

\

5. spe—sp

7. spz—sp

2

p

+

\

1.50

1.46

1.46

C—C~~

1.42

C—C—

1.38

%

6. sp—sp

1.54 A

\

\

—C—C

Characteristic C—C distance

/

C=C

1.34

N C=C= /

1.31

=C=C=

1.28

—CC-

1.205

/

8. sp 2 —sp + 9. sp—sp +

p

IT

10. sp—sp + 2p

Experimental evidence that not all carbon-carbon single bonds are the same length seems first to have been obtained by Lonsdale in 1929 in a study by X-ray diffraction of the crystal and molecular structure of hexamethylbenzene.3 An early discussion of the correlation between orbital hybridization ratios and internuclear distances was given by Coulson.2 Tabulations similar to Table 1 have been given recently by Brown,4 Costain and Stoicheff,5 Dewar and Schmeising,s Somayajulu, 7 Bent,s Bastiansen and 10 Skancke,a Bastiansen and Traetteberg, and Lide.11

Electron Correlation and Bond Properties in some Selected Sulfur Compounds TABLE

3

2. EFFECT OF ATOM HYBRIDIZATION ON BOND-STRETCHING FORCE CONSTANTS Molecule

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

CH3—C H3 CH3—CN CH3—C=CH NC —Cl H2C =0 O=C=C=C=O O=C=O C=0 CH3—Cl, Br, I, H NC—Cl, Br, I, H

Bond type 3— SR 3 sp3 —sp sp3 —sp sp—sp — 2 C(sp Z ) O(sp ) + p 2 C(sR)— O(sp ) + 2 C(SP)— O(sp ) + C(sp)—O(sp) + 2p C(sr3 )—C1, Br, I, H C(sp)—Cl, Br, I, H SR

K (millidynes/A) 4.5 5.3 5.5 6.7 12.3 14.2 15.5 18.9 3.4, 2.9, 2.3, 5.0 5.3, 4.2, 2.9, 5.9

Bond strength The data in Table 2 show that bond-stretching force constants may be classified in a manner very similar to bond lengths.8 The carbon-carbon single-bond stretching force constants of methyl cyanide and methylacetylene (b and c) bear a closer resemblance to each other than they do to either the force constant of ethane (a) or cyanogen (d). Similarly, the carbon-oxygen stretching force constant in carbon dioxide (g) resembles the stretching force constant of the carbon-oxygen double bond in carbon suboxide (f) more closely than it does the stretching force constant of the carbonoxygen double bond in either formaldehyde (e) or carbon monoxide (h). Bond dissociation energies show a similar trend. Values for some carbon-carbon single bonds are (in kcal/mole):12 CHs—CHs, 83; CH3—C2H3, 92.5 and CH3—C6 H6, 93; CH3—CN, 103 and CH,C-CH, 104; C2H3—C2H3, 104; NC —Cl, 112. Carbon-hydrogen bonds have been discussed according to an analogous classification by Bernstein who has shown that the value of the bond dissociation energy of a carbon-hydrogen bond can be estimated from the average value of the corresponding 3 carbon-hydrogen stretching frequencies.» Generally speaking, the more s character in a bonding orbital, the shorter the bond, the larger the bond-stretching force constant, and the larger the bond dissociation energy. Inductive constants Early evidence that the electron withdrawing power of a carbon valency may be greatly modified by the existence of adjacent unsaturation in the molecule has been cited by ingold14 and is illustrated also by the data in Table 3, which shows the effect of a change in orbital hybridization ratio in carbon on the acidity of a proton three atoms removed from the inductive center. The order of electronegativities determined by the data in Table 3 is C(sp, in cyanide) > Cl > C(sp2, in vinyl) > H > C(sp3, in methyl). Taft's survey of the vast literature relating to the evaluation of inductive constants from kinetic data15has yielded a set of s* values that, again, supports the early conten17 tion of Mulliken —and, later, of Walsh 16 and Moffit —that a change in hybridization ratio may have a powerful effect on the electronegativity of an atomic valency.

4

HENRY A. BENT

Some s* values given by Taft are (in parentheses) : N C—C H — (+1.30); C1CH2(+1.05); CH3C(O)CH2— (+0.60); and CHs—CH2— (-1.00). 54

TABLE 3. EFFECT OF UNSATURATION ON ACIDITY Acid

Ka x 105 (in H20)

HCOOH CH3COOH CH3CH2COOH H

17.12 1.75 1.33

CH2 =C

4.62

/

CH2COOH N - C—CH2COOH CL—CH2COOH

360.0 137.8

Dipole moments provide additional evidence for the existence of a correlation between orbital hybridization and orbital electronegativity. Unsaturated halides have been known for many years to have smaller dipole moments than their saturated analogues.18 Vinyl and ethyl bromide, for example, have dipole moments of 1.48 and 2.09 debyes, respectively. More striking still is the case of toluene and methyl acetylene. Were all carbon-carbon bonds nonpolar and all carbon-hydrogen bond moments equal, these two molecules would have no dipole moment. Starting from the fact that they do have dipole moments and using for carbon-hydrogen bonds variable bond 10 moments derived from infrared dispersion data on methane, Petro has derived as shown in Table 4 a set of carbon-carbon bond moments, which, while perhaps not entirely free from criticism,20 nevertheless seems to agree well with the chemical facts cited above. Measurements on the intensities of the vibrational transitions in dimethyl acetylene21 appear to be in essential agreement with Petro's estimate of the polarity of the C(sr3)—C(s r) bond. TABLE 4. EFFECT OF Atsm HYBRIDIZATION ON BOND MOMENTS Carbon-carbon bond Bond moment (and polarity) (in debyes) V(sp3) +—C (sp2)C(sP 2)+—C(s r)— C(sr3)+—C(sP)

0.68 1.15 1.48

The familiar facts about the strengths of common chemicals as acids and bases lend further support to this general picture. Base strength, for example, generally increases as the s-character in the orbital occupied by the unshared electrons decreases. Thus ketones and aldehydes, whose unshared electrons on oxygen are in

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

5

2

sr -type hybrid orbitals, are generally weaker bases (poorer proton acceptors) than ethers and alcohols, whose unshared electrons are in oxygen spa-type hybrid orbitals; similarly, 12 is a weaker base than pyridine, which is a weaker base than ammonia (the unshared electrons in these compounds are, respectively, in nitrogen sp-, sr2-, and spa-type hybrid orbitals); likewise HC-C: is a weaker base than H2C=CH:, which is a weaker base than H3C: -. Carbon(13) proton coupling constants The carbon(13) nucleus has a magnetic moment which can couple with the magnetic moment of a bound proton via magnetic interactions with the intervening bonding electrons. Theory predicts that the magnitude of the magnetic interaction constant, Jch, between a carbon(13) nucleus and adjacent proton should depend on the probability of finding the bonding electrons at the two nuclei in question. Since an electron in a pure p orbital of carbon has zero probability of being found at the nucleus of the carbon atom, whereas this probability is finite for an electron in an s orbital,22 it seems plausible to suppose that the coupling between a carbon(13) nucleus and a directly bound proton should depend on the state of hybridization of the carbon atom. The data23 in Table 5 support this supposition. The greater the s character in the carbon valency toward hydrogen, and, hence, the greater the probability of finding the bonding electrons at the carbon(13) nucleus, the greater the coupling constant Jch• TABLE 5. EFFECT OF ATOM HYBRIDIZATION ON CARBON(13)-PROTON COUPLING CONSTANTS Molecule Methane Benzene Methylacetylene

Jcrr(sec -1) 125 159 248

The changes described above in bond properties have been correlated here with gross changes in hybridization, i.e. with changes in what may be called first-order hybridization of the type sr3, sr2, or sp. It seems likely, however, that only a relatively small fraction of all molecules contain atoms that may be described as hybridized exactly sr3, sr2, or sp. The carbon atom in methane is generally described as hybridized sr3, but other members of the isoelectronic sequence, ammonia, water, and still more hydrogen fluoride, are hybridized somewhat differently; nor are the carbon atoms in ethylene hybridized exactly sr2. Numerous other examples could be cited. Departures of this type from exact sr3-, sr2- or sp-type hybridization have been 26 termed second-order hybridization,25 or isovalent hybridization. Familiar rules exist for determining the first-order hybridization of an atom in a molecule, particularly for atoms from the first row of the periodic table. These rules are the rules of structural chemistry: e.g. the octet rule, and rules governing formal charges. One may ask whether supplementary rules exist for predicting the secondorder hybridization of a combined atom. It will be suggested in the following section that second-order hybridization does

6

HENRY A. BENT

in fact follow a predictable pattern and that this pattern can be summarized in a simple rule.8'27 The evidence for this rule will be presented, following which the rule will be illustrated by its application to several sulfur compounds.27 In Part II the underlying reasons for this rule will be explored in an effort to establish a valid physical basis for the rule. In so doing, a model will be presented that summarizes previously cited empirical generalizations regarding bond properties of compounds of 1rst-row elements and that permits these generalizations to be extended, as will be demonstrated, to compounds of second-row elements. BOND PROPERTIES AND ELECTRONEGATIVE SUBSTITUENTS

Examination of the effect of electronegative substituents on bond angles, bond lengths, inductive constants, and carbon(13)–proton coupling constants suggests this rule :8, 25 The s character of an atom tends to concentrate in orbitals that the atom uses in bonds toward electropositive substituents.± In applications of this rule, lonepair electrons are regarded as electrons in bonds to very electropositive atoms (atoms of zero electronegativity). Experimental evidence for this rule, which describes the direction of second-order hybridization, may be listed as follows. Evidence from bond angles In compounds of the type AC2 and AC3, the valence angle C— A— C often appears to be correlated with the electronegativity of X. In the absence of obvious steric effects and/or d-orbital participation, the valency angle generally decreases as the electronegativity of the substituent C increases. Illustrative examples are given in 29 Table 6. Additional examples have been cited by Mellish and Linnett28 and by Walsh. These data suggest that as the electronegativity of the substituent C increases, the central atom (A) diverts increasing amounts of s character to the orbital, or orbitals, occupied by its lone-pair electrons. TABLE 6. EFFECTS OF ELECTRONEGATIVE GROUPS ON BOND ANGLES O 1. / \ C U a. CH3 CH3 b. CH3 H c. H H d. F F S 2. / \ X U a. CH3 H b. H H

Angle CO U 111° 109° 105° 103° Angle XS U 0

100 92°

3. CU3 a. N(CH3)3 b. NH3 c. 1F3 d. P(CH3)3 e. PHS f. As(CH3)3 g. AsH3

Angle YX U 109° 106° 46' 102° 30' 100° 93° 18' 96° 91°31

4. C, U in CF2CCU a. H, H b. F, F

Angle FCF 110° 114°

It is an interesting fact that these changes in interbond angles are sometimes in the reverse direction from what one would expect were repulsions between nonbonded t It is the same thing to say that The p character of an atom tends to concentrate in orbitals that the atom uses in bonds towards electronegative substituents.

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

7

atoms the most important effect operating.^o One explanation for this directs attention to the electrostatic interactions that exist between substituents and the lone-pair electrons.31 The latter would attract protons, but repel fluorine atoms. However, this explanation appears to leave unanswered (see below) the eflfect of electronegative substituents on b o n d lengths. Another explanation of bond angles that has been widely discussed^^, 33 focuses attention on the Coulomb and Pauli repulsions that must exist between electron pairs in an atom's valence shell. With the reasonable assumption that the order of (electron-pair) — (electron-pair) repulsions is (lone-pair) — (lone-pair) > (lone-pair) — (bonding-pair) > (bonding-pair) — (bonding-pair) and the also reasonable assumption that the more electronegative the substituent X, the further removed from the atomic core of atom A are the A — X bonding electrons, and with some less expUcit assumptions regarding the orientation of electron pairs about an atomic core (see Part II), it has been shown that a large number of facts about the detailed configuration of covalent molecules can be quaUtatively accounted for. N o t explained, however, is why the equilibrium conformation of hydrogen peroxide is such that the lone-pair electrons on adjacent oxygen atoms are gauche to each other, or perhaps even eclipsed nor does this theory of geminal electron-pair interactions make it immediately apparent why the symmetry axis of the methyl group in such compounds as methanol, methyl amine, and dimethyl sulfide is tilted toward the lone-pair electrons on the adjacent atom.^^ O n the other hand, it should be noted, t o o , that the rule given above regarding the distribution of atomic s character in molecules fails to explain why in dimethyl ether the C—Ο—C bond angle is greater than tetrahedral;^^ nor does the rule as it stands explain why the X — A — X angle is often larger t h a n normal when a t o m A is a firstrow element and substituent X is a second-row element.^s Particularly striking in this respect is the compound (SiH3)20 with a Si—Ο—Si bond angle of about 155°37 and the compound N(SiH3) with a Si—N—Si bond angle of 120°.38 xhese facts together with the problem of the methyl group tilt and a quantitative estimate of the energetics of the geminal electron-pair model will be discussed in more detail in Part II. Several compounds containing second- and third-row elements have been included in Table 6. The data on these compounds suggest that unshared electrons capture an increasing share of the s character of the central atom as the electronegativity of this atom decreases ( l b and 2a; Ic and 2 b ; 3a, d, and f; 3b, e, and g). This observation will be discussed in more detail at several points later in this chapter. Evidence from bond lengths It is at first sight surprising to find that in the series C H 3 F , C H 2 F 2 , C H F 3 , C F 4 , the C—F bond decreases monotonically in length. Similar, albeit smaller, effects are observed with chlorine and bromine (Table 7). Several other examples^ of the shortening in the structure X — C — o f the X — C b o n d when Y is replaced by an atom or group of atoms of greater electronegativity are cited in Table 7. Similar trends have been observed and commented upon for further halogen substitution in chloroform^^ and the chlorinated and brominated silanes,^^ the methyl tin chlorides, bromides and iodides, and in the methyl arsenic chlorides.^o, 4 i evidence seems to point to a definite phenomenon.

8

HENRY A. BENT

TABLE 7. EFFECT OF ELECTRONEGATIVE GROUPS ON THE LENGTH OF AN ADJACENT BOND Compound

Bond C—F

CH3F CH2F2 CHF3 CF4 CH3C1 CH2Cl2 CHCl3 CCl4 CH3Br CHBra CH3C1 CCI2F2 CF3C1 G2H ~~ C2F6

C—Cl

C—Br C—Cl C-C

Bond length (A) 1.391-1.385 1.358 1.332-1.326 1.323 1.784-1.781 1.772 1.767-1.761 1.766-1.760 1.939 1.930 1.784-1.781 1.775 1.751 1.536 1.51

Again, it is an interesting fact that these changes in geometry are sometimes in the reverse direction from what one would expect were repulsions between nonbonded atoms the most important effect operating. This problem has been reviewed by Pritchard and Skinner.42 Explanations that invoke participation of double-bonded structures have been criticized by Skinner and Sutton,41 Wells,43 and Burawoy.44 Explanations in terms of ionic-covalent resonance45 and inner-orbital repulsions46 have been criticized by Duchesne,47 who focuses attention on changes in hybridization of the substituent halogen atoms, but then encounters difficulty in accounting for the bond angles in the hydrides of Group V. It seems possible that the shortening of adjacent bonds by electronegative substituents may be described, at least in part, by the statement that an atom tends to concentrate s character in orbitals toward electropositive groups. As an illustrative example, consider the carbon-fluorine bond in CH3F and CH2F2. The length of this bond in these two structures is 1.39 A and 1.36 A, respectively (Table 7). It is the bond labelled b in Fig. 1.

k H—C—F

6

—H

H (A) d = 1.39 ~~

k

~ 6

-i F—C—F

k (B) b= 1.36 A

FIG. 1. The carbon-fluorine bond in CH3F and CH2F2.

Suppose one asks, "What are the perturbations on the carbon and fluorine atoms of the C—F bond in going from structure (A) to structure (B) ?" The perturbation on carbon is clearly an important one; it is the perturbation caused by the replacement of

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

9

one of the three hydrogen atoms about carbon by fluorine ( F ' in Fig. 1) T o a first approximation, there is no perturbation of the fluorine a t o m of the original C — F bond, since the a t o m attached to fluorine has not been changed; it is still a carbon atom. To a second approximation, the perturbation on fluorine is the perturbation caused by changing from the group C H 3 — to the group C H 2 F — ; this is a second-order perturbation compared to the perturbation on the carbon atom. If one judges the effect of the aforementioned perturbation on carbon by the change that it produces in the length of the original C — F bond, it would appear that the s character at the carbon end of the C—F b o n d has increased in going from C H 3 F to C H 2 F 2 . Interpolation between C H 3 F , where presumably the carbon orbital toward fluorine contains less than 25 per cent s character, and C F 4 , where by symmetry the carbon Orbitals toward fluorine may be presumed to contain the full 25 per cent s character of a tetrahedral orbital, leads to the same conclusion. The effects described above are next-nearest neighbor effects; i.e. they involve the effect of an atom on a bond one a t o m removed. In this respect, the effect of electronegative substituents on b o n d lengths is similar to the inductive effect. It may be noted, also, that the variations in b o n d length in Table 7 are not those predicted by the Schomaker-Stevenson Rule;^^ the variations are, in fact, opposite in direction from what one would expect from a logical extension of the SchomakerStevenson Rule.49 Finally, it should be mentioned in this discussion of substituent effects on bond lengths that in the structure X — A — Y b o n d X — A may sometimes be abnormally long when atom Y has on it lone pairs, particularly if the anion X ~ is a weak base. The length of the carbon-chlorine bond in acetyl chloride (X = chlorine, A = carbon, Y = carbonyl oxygen), for example, is longer t h a n the length of the same b o n d in vinyl chloride (Y = CH2-group) (the lengths of the two bonds are 1.789 and 1.736 Ä, respectively). Additional examples have been summarized elsewhere.^o Often (although perhaps not always) this bond-lengthening effect appears to be ascribable to an interaction of a lone pair on Y with the interior lobe of the antibonding orbital that is associated with the X — A bond. Broadly speaking, this interaction corresponds to the well-known structural diagrams that are used to represent departures from perfect pairing. X—A-Y-

•X:

A=Y

or

The vicinal interaction between electron pairs described by these diagrams affects not only molecular geometry but the rates of organic^i and inorganic^^ reactions as well and probably accounts, in part, for the difficulty described by Bastiansen and Traettebergio in locating systematic trends in the effects of substituents on the lengths of carbon-carbon bonds. It also lends support to the conclusions expressed by Mulliken^^ and more recently by Lide^i to the effect that variations in the lengths of single bonds probably cannot be described solely in terms of variations in orbital hybridization ratios. Β·

10

HENRY Α . BENT

Evidence from inductive constants Taft's54 inductive constants for the groups C ( C H 3 ) 3 , C H 3 , C H 2 C I , C H C I 2 , and C C I 3 are, respectively, — 0 . 3 0 , 0 . 0 0 , + 1 . 0 5 , + 1 . 9 4 , and + 2 . 6 5 . These d a t a illustrate the well-

known fact that replacement in the structure X — A — Y of Y by a n atom or group of atoms more electronegative than Y increases the effective electronegativity of atom A toward the substituent X . RecaUing that the electronegativity of a n atomic orbital increases with increasing s content, one infers that the change described above causes atom A to rehybridize in a manner such as to increase the s content in its orbital toward X. This inference leads t o the following description of the inductive effect.^^ In the structure X—A—Y, as the electronegativity of group Y increases, atom A rehybridizes so as to shift s character from the bond toward Y t o the bond toward X where the relatively low potential energy space that is characteristic of a n s orbital will be used to greater advantage. I n effect, the electronegativity of A toward X becomes greater. I n turn, by a n identical, b u t smaller, operation of the same mechanism, the electronegativity of X toward attached groups (other than A) becomes greater. I n this manner, the original perturbation is relayed in a n attenuated manner throughout the bonded system. It may be noted that at each atomic center that engages in a significant redistribution of atomic s character there should exist coupled t o the inductive effect definite, albiet perhaps small, b u t n o t necessarily chemically insignificant, changes in bond angles and bond lengths. This statement has certain chemical implications. Since a change in hybridization ratio corresponds to a change in both the effective electronegativity of a n atomic orbital and the effective bonding radius of the orbital, it should be possible, for example, to predict trends in bond lengths from inductive constants, or conversely, to infer from measured bond lengths trends in group inductive constants. Evidence from carbon(l3)-proton coupling constants Replacement of Y in Η—C—^Y by a n atom, or group, more electronegative than Y generally increases the couphng constant JCK of the C — Η bond. The couphng TABLE 8 . EFFECT OF ELECTRONEGATIVE GROUPS ON CARBON(13)-PROTON COUPLING CONSTANTS

X inCHsX Jen Η CHO CO2H C^CH CCI3 OH NO2 F

(sec-i) 125 127 130 132 134 141 147 149

constant in the series CH4, CH3CI, CH2CI2, CHCI3, for example, increases in the order 125, 150, 178, 209 sec-i;^^ concomitantly the C—Η distance decreases in the order 1.11, 1.0959, 1.082, 1.07 Ä. Additional examples of the effect of electronegative groups on carbon(13)-proton coupling constants are cited in Table 8. These data

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

11

are taken from the work of Muller and Pritchard, who have discussed in detail the correlation of Jen with r(C—H).^^ Recalling that Jen increases with increasing s content at the carbon end of the C—Η bond, one infers from these data that replacement in the structure H — C — Y of Y by an atom, or group, more electronegative than Y leads to a rehybridization of the carbon a t o m in a manner which, in agreement with the rule stated at the beginning of this section, is such as to increase the s content of the C—Η bond. In summary, it appears that many of the effects of electronegative substituents on bond properties can be accounted for and the direction of these effects often predicted if it is assumed (1) that the radial part of an s orbital is more contracted than that of a ρ orbital and (2) that atomic s character tends to concentrate in orbitals where it will do the most good; i.e. in orbitals that are directed toward electropositive substituents. These assumptions describe a form of electron correlation that is illustrated diagrammatically in Fig. 2.

FIG. 2. Summary of the displacement of the electron cloud about an atom relative to the atom's nucleus (the small sphere) when the electronegativity of the substituents bound by the two valencies that in the Figure point downward increases. The central atom's ρ character in these two valencies increases and correspondingly its s character in the two upward-directed valencies increases. In drawing this figure the assumption has been made that the radial function of an s orbital is more contracted than that of a /? orbital.

Figure 2 might represent, for example, the change in the electron cloud about an oxygen atom that occurs in going from O H 2 (bond angle 105°) to O F 2 (bond angle 103.8°). In this case, the small sphere in the cube center would represent the oxygen nucleus, the two lower spheres would represent the bonding pairs of electrons (in either Ο—Η or Ο—F bonds), and the two upper spheres would represent the unshared pairs on the oxygen atom. In O F 2 vs. O H 2 the bonding pairs are further removed from the oxygen nucleus, and the angle subtended by them at that nucleus diminishes; correspondingly the unshared pairs are brought closer to the oxygen nucleus and the angle subtended by them at the oxygen nucleus increases. The dotted lines indicate

HENRY A. BENT

12

these new inter-orbital angles. If the wave function of a molecule can be satisfactorily expressed in terms of spherical harmonics centered at the nuclei, one can say that the change from OH2 to OF2 corresponds to shifting oxygen s-character from its two bonding orbitals to the orbitals occupied by its unshared electrons. In the following section several sulfur-containing molecules are examined with a view to showing in a qualitative manner the extent to which effects arising from this kind of second-order hybridization among s-bonds may be important in determining details of molecular geometry about atoms of the second row of the periodic table. BOND PROPERTIES IN SOME SELECTED SULFUR COMPOUNDS

Thionyl and sulfuryl fluoride 58 Accurate structural parameters are known for thionyl and sulfuryl fluoride.59 These are given in Fig. 3 beneath the Lewis octet structures for these two molecules. Not shown in these valance-bond structures are d„ r„ bonds between sulfur and the surrounding atoms. It seems likely that such bonds are formed between sulfur and oxygen, but not between sulfur and fluorine. Accordingly, changes in the S—F bond in passing from one compound to the other may be viewed as arising, at least in part, from changes in hybridization of the s-orbitals of sulfur. As shown in Fig. 3, the shapes of SOF2 and S02F2 are determined primarily by the o--orbitals of sulfur. The two molecules differ chiefly in the fact that the lone pair on sulfur in SOF2 is shared with an oxygen atom in S02F2. In view of the strong s-seeking s-bonds character of unshared electrons, it is to be anticipated that the S—O and S—F in SO F2 will receive less s character from sulfur than do the corresponding bonds in the more fully oxygenated compound. One observes that in fact the S—O and S—F interatomic distances are greater, and the FSF angle is smaller, in SOF2 than in S02F2, in agreement with the effects anticipated for second-order rehybridization of the sulfur atom. Trends similar to those observed for thionyl and sulfuryl fluoride occur in the 27 trihalides of phosphorus and their phosphoryl and thiophosphoryl derivatives.

F—S—I 1 F

Thionyl fluoride58 S—O distance 1.412 A S—F distance 1.585 A FSF angle 92° 49'

O 1 F—S—I 1 F Sulfuryl fluoride 59 1.405A 1.530 A 96° 5'

Flo. 3. Lewis octet structures for thionyl fluoride and sulfuryl fluoride.

X2YSO molecules In compounds of the type X2YSO, there is a correlation between the length of the sulfur-oxygen bond and the electronegativities of the substituents X and Y. This correlation is in the direction that one would expect from the previous discussion: The greater the electronegativities of X and Y, the shorter the S-0 bond. This is illustrated by the series Mee (unshared pair) SO, C12 (unshared pair) SO, F2 (unshared pair) SO, F20S0. Lewis structures and S—O bond lengths for these molecules are

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

13

given in Fig. 4. Also, in going from sulfuryl chloride to sulfuryl fluoride there is as expected a decrease in the S-0 bond length and an increase in the ISO bond angle. Me I Me—S---I X U S—O bond length

Cl I Cl—S—O

F I F—S-0

F I F—S—O I O Fluorine Oxygen 1.405

Methyl Chlorine Fluorine Unshared pair Unshared pair Unshared pair 1.47 A 1.45 A 1.412 A

FIG. 4. S—O bond lengths in X2YS0 compounds for the series S11e2, SOCl2, SOF2, S02F2.

Another series of this type, 02SS02- ; 020S02- ; and F20S0 is illustrated in Fig. 5. 0

X Y S—O bond length

[O--O] I S Oxygen Sulfur 1.48 A

2-

F I F—S—O I O Fluorine Oxygen 1.405 A

2-

0

[0.--O] I O Oxygen Oxygen 1.44 A

FIG. 5. S-0 bond lengths in 52032-, S042-, and S02F2.

The examples that have been discussed in this section have so far been limited to two second-row elements: sulfur and phosphorus. For these elements, it has been seen in several instances that in structures of the X—A—Y, A=S or P, the A—X bond becomes shorter as the electronegativity of Y increases. The data in Fig. 6 suggest that this rule may hold also for A=C1 (and X=0). (All structures in Fig. 6 carry a net charge of —1.)

:ci— o I

0

CI—O bond length

1.64A

O—cl—

I

O I o—c i— o I

o

o

o

1.57 A

FIG. 6. Bond lengths in the ions 0„ (unshared pair)a_n: C10

1.50 A -1

, n = 1, 2, 3.

We may summarize this rule in the statement that for Lewis structures of the type X—A—Y, A=a first- or second-row element, d[d(A —X)] 0. < dcy Here d(A—X) is the length of the A—X bond and Cy is the electronegativity of Y. Several factors might be listed as underlying causes for this rule. Of the three (1) s-orbital hybridization (2) formal charges (3) d„rp bonding it may be noted that both the first factor (s-character tends to concentrate in orbitals toward electropositive groups; and the more s-character in a bond, the shorter that

14

HENRY

A.

BENT

bond tends to be) and the second factor (the A—O bond tends to become shorter as the formal charge on A increases) are consistent with the trends exhibited in Figs. 4, 5, and 6. The third factor, important though it may be in some cases, cannot explain the rule when A is a first-row element. Also, of the first two factors, only one of them, the first one, is consistent with the trend in the S—S bond length illustrated in Fig. 7. (All structures in Fig. 7 carry a net charge of —2.) 00 1 1 0—S--S—O

00 I I 0—S— I s-0

00 I I 0—S—S—o 1 1 00

0

S—S

2.389 A

bond length

FIG. 7. S—S bond length in the ions 5204

2.209 A

2.15-2.16

A

2-

, S2052-, and 52062-.

The hydrogen sulfide problem The angle between two bonds that meet at an atom from Group V or Group VI generally decreases as the atomic number of the atom from Group V or Group VI increases. In hydrogen sulfide, for example, the bond angle is less than that in the water molecule; and in Group V, the bond angles decrease in the order IC3 > RX3 > AsX3 for X=H or CH3 (Table 6). Explanation of these trends has been a troublesome problem in valence theory.60 We consider here the role that s and p electrons might be expected to play in these compounds.8 To place this problem in the context of the present discussion, attention is focused upon molecules of the type (I) BCA3 and (II) B2CA2, where C represents a Group V or Group VI element and A and B represent substituents that may differ in electronegativity. Compounds of Group V are of type I, B representing a lone pair; compounds of Group VI are of type II. If, as before, unshared electrons are regarded as electrons in bonds to a substituent of zero electronegativity, the following account B B B X

X

AAA

A A

(I)

(II)

may be given of the effect on bond angles of the electronegativities of atoms A, B, and C. 1. If A and B are identical the four hybrid orbitals of X are, of course, equivalent. 2. If A is more electronegative than B, C should concentrate its s character in those orbitals that it directs toward B, thus diminishing the AXA angle below and increasing the BXB angle above the tetrahedral value 109° 28'. 3. If B represents an unshared pair, and the electronegativity of A remains fixed, a decrease in the electronegativity of C has the sane effect on the AXA angle as an increase in the electronegativity of A (Table 6).t This seems reasonable. As the ability t Walsh has stated this observation in these terms: "Substitution of an atom of higher atomic weight but in the same sub-group of the periodic table for A in AH2 is in one respect like substitution of halogen atoms for hydrogen atoms when A is fixed."29

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

15

of C to effectively contribute low potential energy space to the electrons in the A— C bond diminishes, owing to either an increase in the electronegativity of A or to a decrease in the electronegativity of C, C should concentrate more and more of its s character in the orbitals occupied by its unshared electrons; and, correspondingly, the AXA angle should become smaller, as observed. SUMMARY

The suggestion that orbital hybridization may have a marked effect on molecular properties receives support from the present review of data on bond angles, inductive constants, carbon(13)-proton coupling constants, and parameters generally associated with the strengths of chemical bonds: viz, bond dissociation energies, bond stretching force constants, and bond lengths. As a general rule, bond strength appears to increase with increasing s-content. For atoms from the first row of the periodic table, the s-content of a hybrid s-orbital may be assessed in a first approximation by means of a widely used rule that consists of separating the manifold variations of hybridization that occur in practice into three classes: sp, sp2 and spa. The advantages of this classification are that it is simple, that it corresponds rather well to the known facts, that definite and familiar rules exist whereby it can be quickly determined to which class an atom belongs, and that such a classification lends itself to a simple refinement. This refinement states that the s character of an atom tends to concentrate in orbitals that the atom uses in bonds toward electropositive substituents. Evidence for this is found in the effects of electronegative substituents on bond angles, bond lengths, inductive constants, and carbon(13)-proton coupling constants. While much of the data presented here concerning the effects of orbital hybridization on molecular properties, and the direction of second-order hybridization, are for atoms from the first row of the periodic table free from complications of participating d-orbitals, it seems likely that data of similar accuracy for heavier atoms would reveal similar trends. Provisional verification of this conjecture has been presented 2here for S—F, S-0, Cl-0, and S—S bonds. S—S bonds in S414, 5204 , S2O62-, and S8 have been discussed from a similar point of view by Lindquist, the belief being expressed that bond lengths in these compounds depend principally on the s character of the hybrid orbitals that form the s-bonds.~1 Other discussions of bonds to sulfur atoms have been given by Moffitt (sulfur-oxygen bonds),62 Abrahams (the sulfoxide 63 66 group), Meschi and Myers84 and Giguere~5 (S2O), Craig and Zauli (SF6), and Burg (Vol. I). PART II Tke preceding discussion of bond properties has been couched in conventional terms and has been largely empirical in nature. It would probably, therefore, be fair to say that whereas this discussion may describe a form of electron correlation in molecules as seen from atomic nuclei, it does not explain this correlation. Also, as noted previously, certain features of molecular architecture such as tilted methyl groups and larger-than-normal valence angles remain unexplained. There are other questions as well. Why does the valence angle in going down the Group VI hydrides drop suddenly from its approximately tetrahedral value in H2O to nearly 90° in H25 and remain at essentially 90° for the heavier members of the group. Why does a

16

HENRY Α . BENT

change in the formal hybridization ratio of a carbon orbital appear to affect the length of a carbon-hydrogen bond only about one-half as much as it does the length of a carbon-carbon bond, whereas the effect on a carbon-chlorine bond is almost twice as great as that for a carbon-carbon bond ?^ A n d why are the lengths of certain sulfuroxygen bonds (that in classical theory have different bond orders) nearly identical? Possible answers to these questions will be considered in several of the following sections of this chapter. These answers arise from considerations sketched below of the fundamental reasons for the nature of the electron correlation in molecules. The discussion of the nature of electron correlation in molecules that follows is based upon a direct use of the total electron density distribution in molecules and upon two fundamental principles that presumably govern (or describe) the nature of this distribution: the Exclusion Principle and the Hellmann-Feynman theorem-^"^ The Hellmann-Feynman theorem is a consequence of the Schrodinger wave equation. The Exclusion Principle is not. The procedure usually adopted in discussions of molecular structure is to begin with the Schrodinger wave equation and to introduce at some later stage the Exclusion Principle. It is sometimes convenient as will be illustrated here to introduce these two fundamental statements in the opposite order. THE

EXCLUSION

PRINCIPLE

A N D THE FERMI

HOLE

Mathematical statement We seek a statement of the Exclusion Principle whose validity is not contingent upon the validity of the idea of "one-electron orbitals", for in reality in a many-electron system there is n o such thing as a "one-electron orbital"; indeed, the idea of "oneelectron orbitals" is contrary to the intrinsic character of the Exclusion Principle. The Exclusion Principle is essentially a statement concerning the collective-particle behavior of a system of electrons, whereas the idea of "one-electron orbitals", to be truly valid, must presume that the system exhibits in an essential manner the characteristics of a collection of independent-particles. Both Coulomb and Pauli forces, however, prevent electrons in atoms and molecules from behaving as independentparticles. One general way of stating the Exclusion Principle is to say that the wave function φ must be antisymmetric with respect t o a n interchange in the coordinates of any two electrons. F o r example, for the interchange of coordinates (spatial and spin) of electrons 1 and 2, the Exclusion Principle states that 0 ( X l , X 2 , X 3 , . . . ) = = —Φ{Χ2,

.

XU ^ 3 , . . . )

.

.

(1)

In this expression, xi = n, ξι, where η represents the spatial coordinates of electron i (say its Cartesian coordinates xu yu zt) and ξί represents its spin coordinate. A consequence of Equation (1) is that the wave function must vanish whenever electrons 1 and 2 have the same coordinates (spatial and spin). F o r example, if x i and X2 are equal to some common value, x, it follows from (1) that Φ(Χ,

X, X3, . . . ) =

—Φ(Χ,

X, -^3» . . . )

and, hence, that φ ( χ , χ , Χ 3 , . . , ) = 0

.

.

.

(2)

It should be noted that the Exclusion Principle does not say that the wave function must vanish when η = if the two electrons have opposite spins; for then ξι ^

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

17

and xi ^ X2. N o r , in fact, does the Schrodinger Equation {H — Ε)φ = 0 require that φ vanish under these conditions, even though the inter-electronic repuLsion term e^lri2 of the Hamiltonian does become infinite. This matter has been discussed for the hydrogen molecule by James and Coohdge.es The statement summarized in Equation (2) is in a sense a stronger statement regarding the nature of electron correlation in molecules than is the usual statement of the Exclusion Principle in which it is said that electrons with parallel spins cannot occupy the same orbital. For many orthogonal orbitals—for example, hydrogen-like 2s and 2p orbitals, or the usual σ and π components of a multiple bond—overlap extensively in space. Of course, any properly antisymmetrized product function will satisfy Equation (2); however, it is important to reaUze that the mathematical procedure of rendering antisymmetric a product of one-electron orbitals destroys the simple orbital picture one might have started with whenever the starting one-electron orbitals overlap in space.^^ If the physical basis of the orbital picture is to be retained at all, Equation (2) suggests that the orbitals used ought to be localized, non-overlapping orbitals.'^^ Physical statement The physical significance of Equation (2) is illustrated^^ in Fig. 8. A b o u t each electron of given spin exists an excluded volume or "Fermi h o l e " consisting of a deficiency of

FIG. 8. The Fermi hole. A figure such as this was given first in 1933 by Wigner and Seitz in a discussion of the electronic structure of sodium metal. The area cross-hatched represents the electron cloud of a many-electron system. The small circle represents the location of a specific electron within whose near-neighborhood the probability of finding another electron of the same spin is very small.

charge of the same spin as the electron in question.'^2 effect, each electron behaves toward other electrons of the same spin-type as one hard sphere does to another. A question of considerable chemical interest is, " H o w large are these hard spheres?" THE T A N G E N T — S P H E R E

MODEL

Consider a molecule such as methane. I n the carbon atom's valence shell are four electrons whose spins are parallel to each other. Taken in pairs or otherwise these electrons can never be in the same place at the same time (Eq. (2)). In other words, taken together the four electrons constitute a collection of hard objects. Would it not therefore be physically reasonable t o represent each electron of the collection not as a point particle as one does in setting u p the Schrodinger equation but, rather, as a sphere whose non-vanishing radius represents the effective radius of the electron's Fermi hole? If this suggestion is adopted, one is led to ask the further question: H o w much of the valence shell is occupied by these spheres ?

18

HENRY A. BENT

Let us designate as the free volume the volume of the valence shell not occupied by the Fermi holes. Then broadly speaking there are these three possibilities to consider. 1. Fermi volumes << Free volume 2. Fermi volumes ti Free volume 3. Fermi volumes > Free volume The first possibility corresponds to picturing the mutual behavior of the valence electrons, at least so far as exclusion effects are concerned, as corresponding to that of a collection of almost-independent-particles analogous to a dilute van der Waals' gas. In these terms, the second possibility corresponds to a compressed van der Waals' gas and the third possibility to a condensed van der Waals' gas. The last possibility may appear at first glance to be the least credible of the three. In fact, it turns out to be a very useful model. This we take to be sufficient justification for considering it further. Another perhaps less empirical reason for seriously considering the third possibility may be expressed as follows. Consider once again the four electrons of parallel spin in the valence shell of a methane molecule and suppose that in the space occupied by these four electrons we wish to draw four spheres such that the probability of finding at least one but no more than one electron in each of the four spheres is a maximum. Two questions immediately arise. First, Where should the sphere centers be placed? Theoretical calculations 73 and chemical evidence 74 suggest the same answer: At the corners of a regular tetrahedron, Fig. 9.

FIG. 9. The configuration of maximum probability for the four electrons of parallel spin in the valence shell of the carbon atom in methane. The carbon nucleus is located at the center of the tetrahedron.

Secondly, how large should the spheres be? Make them very small and the probability of finding even one electron in the entire set will be very small. On the other hand, the probability of finding only one electron in a very large sphere is also very small. Clearly there must exist some optimum size for the spheres. 75 Clearly, too, this size must be such that the spheres do not overlap. They could be tangent to each other, however. We assume they are, Fig. 10. In this way we arrive at the tangent-sphere model. In this model the electron pairs in the valence shell of a combined atom are represented (in general) as tangent spheres. The one person who seems seriously to have considered this model previously is George Kimball, who has carried through calculations with several students on the hydrides of the first-row elements. 76 A brief account of their work has been given by Platt.77

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

19

FIG. 10. The tangent-sphere model of methane. Each sphere represents two spin-paired electrons. Imbedded in each sphere is a proton. The tetrahedral hole formed by the four tangent spheres is occupied by a fifth sphere much smaller in size than the other four. This sphere represents the K-shell electrons of the carbon atom. At its center is the carbon nucleus.

SURVEY OF THE IMPLICATIONS OF THE TANGENT-SPHERE MODEL

Stereochemistry Perhaps the outstanding feature of the tangent-sphere model is its ability to predict properly the gross geometrical features of many different kinds of molecules. The tetrahedral atom is of course an immediate consequence of the coordination by an atomic core of four pairs of electrons. Also correctly given are the stereochemical configurations about atoms with electron-pair coordination numbers of 1 (H, He), 2 (BeH2), 3 (BH , CH3+), 5 (PCI5, SF4, C1F3, 13-), and 6 (SF6, NSF3, IC14 -, XeF4), Fig. 11.

FIG. 11. A summary of the stereochemistry of combined atoms for all electron-pair coordination numbers commonly encountered in practice. Coordination about an atomic core of one, two, three, four, five, or six electron pairs in the manner illustrated corresponds to the use, respectively, of the hybrid orbitals s, sp, sp 2, spa, dsp 3, or d 2sp 3.

Each model enjoys the further advantage that it may be used to represent any one of several isoelectronic molecules. The methane model, for example, represents also the electronic structure of ammonia and water. In the latter cases one or two of the valenceshell spheres would be unprotonated. Calculations suggest that this does not greatly alter the effective size of a sphere. 76 The proton of a protonated sphere lies about half-way between the center of its sphere and this sphere's outer edge along a line that passes through (or nearly through) the sphere center and the heavy-atom nucleus to which the proton is bonded.

20

HENRY A. BENT

FIG. 12. Tangent-sphere model of ethane or of any molecule such as methylamine, methyl alcohol, methyl fluoride, hydrazine, or hydrogen peroxide that is isoelectronic with ethane. The light spheres represent either lone pairs or bonds to hydrogen.

Figure 12 shows the tangent-sphere model of ethane or of any other molecule that like ethane contains two heavy atoms that satisfy the octet rule and that has altogether in the valence shells of its atoms seven pairs of electrons, one of which (the dark one in Fig. 12) is shared in common by the two tetrahedral holes that contain the heavy atom cores. This pair represents the (heavy-atom)—(heavy-atom) bond. The molecule is shown in its staggered configuration. This corresponds to stacking the electron pairs as a fragment of a cubic-close-packed lattice. For simplicity all valence-shell electron pairs are shown the same size.

FIG. 13. Tangent-sphere model of ethylene or any molecule or ion isoelectronic with it. The two

components of the double bond are represented as equivalent bent bonds.

Figures 13 and 14 illustrate the well-known geometrical fact that two tetrahedral

FIG. 14. Tangent-sphere model of the family of isoelectronic molecules that includes acetylene, nitrogen, carbon monoxide, and hydrogen cyanide. Electron pairs that comprise the triple bond may be partially anticoincident (see text).

holes may share two spheres in common or three spheres in common. These configurations give rise to the tangent-sphere representation of double and triple bonds.

FIG. 15. Tangent-sphere model of the equilibrium conformation of propylene or acetaldehyde. As in ethane the electrons of the C—H bonds of the methyl group have been staggered with respect to the three electron pairs at the opposite end of the carbon-carbon single bond. This places one of the methyl protons near the carbon atom of the methylene group or the oxygen atom of the aldehyde group. It seems possible that this is one of the reasons why the barrier to rotation of the methyl group in these compounds is smaller than in ethane.

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

21

The equiUbrium orientation of a methyl group adjacent to a double b o n d - -as, for example, in propylene'^8 or acetaldehyde'^^—is shown in Fig. 15. Again, the electron pairs across the c a r b o n - c a r b o n b o n d are staggered with respect to each other and the stacking of electron pairs is therefore that of a fragment of a cubic-close-packed lattice. The same statement appUes to the electron pairs in ^-trans-l,3-butadiene

(b)

FIG. 16. Tangent sphere models of three conformations of 1,3-butadiene: {a) s-trans, φ) ^-gauche, and (c) 5-Cis.

(Fig. 16a). However, in the s-cis conformer (Fig. 16c), which is often mentioned in connection with the more stable trans isomer, the electron pairs across the carboncarbon single bond are eclipsed. One of the gauche conformers (Fig. 16b) would appear to be a more likely alternative.

FIG. 17. Tangent-sphere model of trans-Decalin. The six-membered rings are in the chair configuration.

22

HENRY Α . BENT

Figure 17 shows a tangent-sphere model of the most stable isomer of decalin.

FIG. 1 8 . Tangent-sphere model of cyclopropene. In this model all carbon-carbon bonds are bent bonds.

FIG. 19. Tangent-sphere model of benzyne. Legend: White sphere—Carbon-hydrogen bond. Closehatched sphere—Carbon-carbon single bond. Cross-hatched sphere—Component of a carbon-carbon double bond. Open-hatched sphere—Component of a carbon-<;arbon triple bond.

Highly strained systems, such as cyclopropene (Fig. 18) and benzyne (Fig. 19), can be represented also and departures from normal bond angles and bond lengths estimated. Finally, it is geometrically possible for a sphere to be shared by more than two

FIG. 2 0 . Tangent-sphere model of a non-classical carbonium ion, C3H7+. The hatched sphere that is uppermost is part of three tetrahedral holes. It represents, therefore, a three-centered bond.

heavy-atom cores. In this way multicentered orbitals can be represented (Fig. 20). Lithium hydride, for example, may be considered as a cubic-close-packed array of electron pairs in which each electron pair has a proton at its center (the H " ion) and six hthium ions surrounding it in the octahedral holes of the close-packed array. F r o m the point of view of this model, the electron-pair orbitals in lithium hydride are seven-centered orbitals. Inter-molecular interactions Inspection of the models discussed above suggests that the interaction of an unshared pair on one molecule with the force field of a nucleus in another molecule should occur most readily at one of the external "pocket" sites in the second molecule. If

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

23

the interaction between the unshared pair and the nucleus it is attacking is strong enough to cause the nucleus to move from its normal position in its tetrahedral hole in the direction of the attacking reagent, the relative shift that occurs in the nuclear position with respect t o its electronic environment may be illustrated as follows.

/

\

The models thus provide a concrete and perhaps not altogether imprecise picture in three dimensions of that branch of two-dimensional "pencil-and-paper"-chemistry that for want of a better term is often referred to as "arrow bending" or "curly arrow chemistry". The same form of pair-pocket interaction can be used to account for such facts as the structure of crystalline iodine and the structures of the iodine complexes of amines and ethers and ketones,^^ the formation of hnear chains in the halogen cyanides,^^ the formation V-shaped Ιδ"^^ and Z-shaped Is^"^^ ions, the occurrence in certain molecular crystals of abnormally short inter-molecular contacts,^^ and, in general, the occurrence in condensed systems of variable van der Waals' radii^^ and directional van der Waals' forces. Intra-molecular interactions Interaction of an unshared pair with an adjacent internal pocket corresponds to the electronic interaction described earlier as the interaction of an unshared pair with the anti-bonding orbital of a vicinally located bond. A strong interaction of this type may lead to the formation of a new chemical linkage and heterolytic scission of the vicinal b o n d ; i.e. t o /ra«5'-elimination. Weaker interactions may manifest themselves as lengthened vicinal bonds or, if the vicinal bond is the C—Η bond of a methyl group, to a tilt of the methyl group's symmetry axis toward the lone pair^^ with a concomitant increase in the corresponding valence angle(s) (c/. dimethyl ether).^^ Other features By avoiding the use of overlapping one-electron orbitals, the tangent-sphere model avoids on the computational side the very great and perhaps entirely artificial mathematical problem of exchange integrals.e^ On the other hand, the model does partake o f t h a t part of molecular orbital theory that is generally thought to be s o u n d : namely, the requirement that the electron cloud be considered not as parts in isolation but as a whole. This happens automatically since the spheres by supposition are tangent to one another. Remove one of them and the shape of the set that remains, or the orientation of the atomic nuclei within the set, will generally change. The implications of this fact appear to be consistent with what is known about the shapes of radicals and of molecules in excited states.^e By making some assumption regarding the nature of the charge distribution within each sphere—e.g. that it is uniform'^^—it is possible to make quantitative estimates of the energy differences between various coincident and anti-coincident electronic configurations in the sense that Linnett has used these terms.'^ Tliis fact will be discussed in more detail later. We may note here that full anti-coincidence is not to be expected; nor is full coincidence.

24

HENRY A.BENT

The models also make it possible to utilize the Hellmann -Feynman theorem in one's qualitative thinking about the chemical and physical behavior of molecular systems (see below). The most useful feature of the model, however, probably resides in the fact that there exists a simple isomorphism between the three-dimensional tangent-sphere model of a molecule and the molecule's valence-bond structure. Every line or pair of dots in the valence-bond representation of a molecule becomes in the model a sphere and vice-versa. In effect the model is a three-dimensional reconstruction of the twodimensional valence-bond diagram. Conversely, a valence-bond diagram may be viewed as a two-dimensional projection of a three dimensional model in which lines are used for protonated spheres and also for spheres that are part of two tetrahedral holes, dots are used for unprotonated spheres that are part of only one tetrahedral hole, and letters are used to indicate the locations of protons and other, more complex atomic cores. QUALITATIVE APPLICATIONS OF THE HELLMANN-FEYNMAN THEOREM

The Hellmann—Feynman theorem states that all forces on atomic nuclei in an unperturbed molecule can be considered as purely classical electrostatic attractions and repulsions involving Coulomb's Law. The force on a nucleus is just the classical electrostatic force that would be exerted on this nucleus by the other nuclei in the molecule and by the molecule's electron cloud considered as a classical charge cloud that is prevented from collapsing by obeying Schrödinger's equation. 67 To use this theorem it is not necessary to have a detailed knowledge of the wave function. A knowledge of the ordinary three-space electron density is suffficient. Some qualitative applications of the Hellmann—Feynman theorem to several current problems in valence theory are given in the following paragraphs of this section. Effect of lone pairs on bond angles The bond angle between two bonds that meet at an atom that has on it lone pairs is generally less than tetrahedral. This fact has been summarized by saying that lone pairs behave as s seeking centers.8 This behavior is consistent with the HellmannFeynman theorem as applied to the tangent-sphere model. In methane, for example, the carbon nucleus experiences no net force upon it, due to repulsions of the protons and to attractions of the bonding pairs, when it is in the center of its tetrahedral hole. (Calculations suggest that nuclei of first-row elements can be assumed to be at the centers of their Ise shells," which in the present context may therefore be considered as part of the nucleus.) In ammonia, however, the center of the tetrahedral holet is not a position of electrostatic equilibrium for the nitrogen nucleus. To reach such a position the nitrogen nucleus must move away from the protons in the direction of the lone pair. This diminishes the HIH angle and requires that less p character and more s character be included in our description of the lone pair orbital if this orbital is to be described in terms of spherical harmonics that are centered at the nitrogen nucleus. t For simplicity (see also ref. 76) we have assumed here that all valence spheres are the same size.

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

25

Ejfect of electronegative groups on bond lengths and bond angles The preceding discussion suggests a simple interpretation of the empirical findings that have been summarized in Fig. 2 and in the rule that the ρ character of an atom tends to concentrate in orbitals that the a t o m uses in bonds to electronegative atoms. F r o m the standpoint of the H e l l m a n n - F e y n m a n theorem, an electronegative atom—i.e. an atom with a relatively large effective nuclear charge (e.g. a proton compared to no proton at all; or a fluorine atom compared to a hydrogen atom)—should tend to drive adjacent nuclei (the small circle in Fig. 2) in the direction of the electron pairs opposite the electronegative atom, thus tending to diminish (i) the lengths of the bonds that these electron pairs represent and (ii) b o n d angles between bonds to electronegative atoms. Effect of multiple bonds on bond lengths and bond angles The ratio of the length of a single bond to that of a double bond and that of a double bond to that of a triple b o n d is easy to calculate in the tangent-sphere model. In the zeroth approximation the two ratios are simply the ratios of the distances between the centers of two tetrahedra that are sharing a corner compared to an edge and an edge compared to a face. The value of the ratio in both cases is 1.732. This calculated value is much larger than the experimental value for these two ratios. F o r carbon —C—C—

\/

/

\//\/

C=C

\/

= 1 . 1 5 and

\/

C=C

/ ( — C = C — ) = 1.11.

\//

In both cases the zeroth order calculation greatly overestimates the decrease that occurs in b o n d length when the b o n d order of the bond between two carbon atoms increases by unity. The calculated value of the ratio of the two ratios (1.000) is nearly correct, however. This suggests that b o t h calculations err for the same reason, or reasons. There are probably two reasons. First, it is umealistic to suppose that the spheres corresponding to a multiple bond are the same size as the corresponding single-bond sphere. They are probably larger. Second, it is unrealistic to suppose that the carbon nuclei in ethylene and acetylene are in the same positions as those in ethane with respect to the adjacent c a r b o n hydrogen electrons. In ethylene with respect to ethane and in acetylene compared to ethylene, the carbon nuclei come closer together and will tend to drive each other in the direction of the C—Η electrons. This displacement will diminish the C—Η distance and increase the H C H angle (in ethylene). Figs. 21 and 22. It also should render the carbon nuclei in these unsaturated molecules susceptible to nucleophihc attack, both from within the molecule and from without. This description of the characteristic physical and chemical properties of a double bond is useful in explaining the curious fact^^' ^'^ that the carbon-carbon single bond in planar l,3-butadiene88 is longer t h a n the corresponding b o n d in non-planar l,3,5,7-cyclooctatetraene.8^ Also, it illustrates the justification in molecular problems of considering that the radial part of an .y-orbital is more contracted t h a n that of a /7-oribtal. A n d it provides an explanation for the fact that the change from a single bond to a double bond or from a double bond to a triple bond changes the length of an adjacent carbon-substituent bond relatively Httle when the substituent is hydrogen and relatively a lot when the substituent is chlorine.^ F o r as the carbon nucleus moves toward the substituent (—H, —CH3, or —CI), the substituent will ride out along the

HENRY A. BENT

26

CH4

C2H4

HCH

I09.5

18°

C—H

I.095

I.086

'ICH

I25

I55

s

0.00

0.36

Fic. 21. A tangent-sphere model of ethylene that shows the approximate location of the protons (the small solid circles) and the carbon nuclei (the larger solid circles). The carbon nuclei are pushed by each other in the direction of the electrons of the two adjacent C—H bonds. The effect of this displacement on several properties of these bonds is indicated beneath the Figure. The straight lines in the drawing illustrate the close correspondence that exists between a molecule's tangent-sphere model and its valence-bond structure.

C—H

I.055

JcH

248

s.

I.35

FiG. 22. Tangent-sphere model of acetylene showing the effect of the outward displacement of the carbon nuclei on several properties of the carbon–hydrogen bond.

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

27

bond—corresponding to the picture given earHer of the inductive effect—the more so the larger the force constant of the carbon-substituent bond. One expects to find, therefore, an inverse correlation between the stretching force constant of a bond and the susceptibility of the bond to a change in length with a change in adjacent un¬ saturation. This is what is observed. Typical stretching force constants for C—H, C—CH3, and C—CI bonds are, respectively, 5.0, 4.5, and 3.4 milUdynes/Ä. Structure of sulfur tetrafluoride The sulfur-fluorine bonds in SF4, SFe, and NSF3 are slightly shorter than the sum of the covalent radii for sulfur and fluorine. It therefore seems reasonable to suppose that the sulfur-fluorine bonds in these molecules are conventional covalent single bonds. This impHes that the sulfur a t o m in sulfur tetrafluoride has an electron-pair coordination number of five (four pairs in bonds to fluorine and one pair unshared). The structure of this molecule has been determined recently by Tolles and Gwinn.^o

FIG. 2 3 . Structure of SF4.

Their results are given in Fig. 23. The molecule can be considered to be a trigonal bipyramid in which one of the equatorial positions is occupied by a lone pair. The structure of this molecule appears to be consistent with the expectations of the tangent-sphere model. Location of the lone pair at an equatorial position is reasonable since this places the s seeking pair as close as possible to the sulfur nucleus; at the same time it keeps

28

HENRY Α . BENT

the average value of the bonded distance between the sulfur nucleus and the fluorine nuclei as large as is possible with bonding pairs of a given size. The bonds to the axial fluorines (Fa) are slightly longer than those to the equatorial fluorines (F^). This, too, is to be expected if the plausible assumption is made that the sulfur nucleus lies in the plane defined by the centers of the spheres that represent the equatorial electron pairs. That the actual ratio of bond lengths, (Faxiai—sulfur)/(Fequatoriai—sulfur)-=1.07, is less than the zeroth order calculated value, approximately 1.20, is to be expected, also. For unhke the axial fluorine nuclei, which exert practically no net force on the sulfur nucleus, the equatorial fluorine nuclei exert a large resultant force on the sulfur nucleus in the direction of the lone pair. A displacement of the sulfur nucleus toward the lone pair would increase the (Faxiai—sulfur) distance less rapidly than the (Fequatorial—sulfur) distance and would therefore diminish the value of the ratio of the first distance to the second. Such a displacement would also make the angle FeSFe less than 120°, as observed, and would cause the angle F a S F « to depart from 180° in the observed direction. Similar considerations apply to the molecule CIFs,^^ Fig. 24.

FIG. 24. Structure of CIF3.

Displacement of the sulfur nucleus 0.29 Ä in the direction of the lone pairs would bring down the angle F^SFe from 120° to the observed value 101° 33' and would bring down the calculated value for the bond length ratio from 1.20 to 1.09. This is still slightly above the observed value. Also, in the absence of additional adjustments in the molecule, this displacement would raise the angle F a S F a from 180° to 200° 19'.

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

29

Both of these calculations suggest that the axial fluorine atoms have their lone pairs tilted toward the pocket space about sulfur that is especially well developed by virtue of a Hellmann-Feynman-type displacement of the sulfur nucleus by the equatorial fluorine nuclei. The magnitude of the tilt required to fit the experimental data, about six degree for each axial fluorine atom, is comparable to the tilts found experimentally for compounds that contain lone pairs adjacent to methyl groups. The lengths of the sulfur-fluorine bonds in SF4 appear to bracket the length of the corresponding bond in SF6 as calculations based upon the tangent-sphere model suggest they should. In NSF3, whose structure has recently been determined by Kirchhoff and Wilson,92 the close proximity of the nitrogen nucleus to the sulfur nucleus appears to be reflected in the fact that the S—F distance in this molecule is probably slightly less, and the FSF angle slightly greater, than in SF6. Concluding remarks Forces between adjacent atomic cores in a molecule have played a prominent part in the present exploratory discussion of the applications of the Hellmann-Feynman theorem to problems of molecular structure. Electrostatic calculations suggest that for typical first-row elements these forces are comparable in magnitude to the force exerted upon an atonic core by a pair of valence-shell electrons. Such forces are five to ten times greater in magnitude than the electrostatic forces between geminal electron pairs. ANTICOINCIDENCE

Introduction The energy of the helium atom can be calculated by assuming that the electron pair is a uniformly charged sphere whose kinetic energy is proportional to the reciprocal of the square of the sphere's radius. Considering the simplicity of the calculation, one obtains a fairly good value for the energy. With a little extra effort this value can be significantly improved by relaxing the condition that the two electrons have identical radii. It is then found that one electron cloud contracts and the other expands so that the two electrons become partially (in fact, about fifty per cent) anticoincident. In more complex chemical systems electrons of opposite spin can become partially anticoincident by having the centers of their Fermi holes become anticoincident. 74 This form of anticoincidence has—in effect—been recently discussed by Linnett. It can occur in any system in which the spin sets about an atomic core are not tied down by two or more valence bonds; for example: in the halide atoms of organic halides; in the oxygen atoms of alkoxide radicals and ions; in the fluoride ion; and in the neon atom. It can occur also in aromatic systems and in other chemical systems that can be represented by more than one classical valence-bond structure and it can occur in such molecules and ions as 02, CO2, and 13- in which there is "threedimensional" resonance and it is probably a factor to be considered in the assessment of the stability of triple bonds. Anticoincidence implies that at any moment the spin density at many points in a molecule will tend not to vanish. Moreover, it implies that in magnitude and sign the spin density in one part of a molecule will tend to be correlated with the instantaneous spin densities in other parts of the molecule. It seems likely that this fact may eventually provide a basis for understanding spin-spin coupling constants. A discussion of that,

30

HENRY A. BENT

however, is beyond the scope of this chapter. We shall be content here with the more modest program of examining qualitatively the implications of the existence of anticoincidence in two specific problems: the problem of explaining the valence angle in hydrogen sulfide and the problem of explaining the moderating effect toward free radicals of carbon disulfide. Hydrogen sulfide The oxygen atom in a water molecule has a relatively simple, helium-like core. The sulfur atom in hydrogen sulfide has a more complex, neon-like core. In neon itself the most probable disposition of the two spin sets with respect to each other is shown in Fig. 25. Two spin-paired, localized electrons that are attempting to approach the

¤ _i Fto. 25. The most probable configuration of the two spin-sets in the L-shell of neon. A crude electrostatic calculation suggests that this configuration is approximately 500 kcal/mole more stable than a fully coincident one.

center of such a core configuration as closely as the exclusion principle permits must avoid so far as possible electrons in both spin sets. This can be achieved most readily by approaching the core configuration broadside to one of its faces. Location of a second localized pair on an adjacent face of the core would then yield the almostninety-degree valence angle found in hydrides of all elements in Groups V and VI that have immediately inside their valence shells a double-quartet of electrons. The idea of interactions between the inner-shell and outer-shell electrons of an atom is not new. Interactions such as this have been considered responsible for the magnetic properties of atoms93 and they play an important role in modern descriptions of the chemical and physical properties of coordination compounds of transition metal ions. It has been suggested that the small valence angles in hydrogen sulfide and phosphine arise from repulsions of the bonding pairs by the lone pairs. Why the valence angles should be almost exactly ninety degrees is not stated, however. With the assumption—which has not been proved—that the spheres that represent the valenceshell electrons are tangent both to one another and to the inner core (see below), it can be shown that for phosphine the minimum angle subtended at the core center by the centers of a pair of bonding electrons is 97° 10'. Carbon disulfide Russell has shown that as a reagent for complexing chlorine atoms carbon disulfide

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

31

94

seems to be particularly effective. The discussion given above suggests that this property of carbon disulfide may stem from the coordinate action of three features of its electron cloud. A chlorine atom, it may be noted, has in its valence shell a four-membered spin-set and an incomplete, three-membered spin-set. Like the two spin-sets in BH3, the threemembered spin-set shoulde make the chlorine atom strongly electrophilic, particularly toward a reagent like carbon disulfide that can donate a single electron to the incomplete spin-set in the chlorine atom without the need of any major electronic rearrangement within the electron donor itself since the two spin-sets in the donor molecule are already anticoincident. Moreover, the divalent sulfur atom in carbon disulfide has about it good potential energy space with which to accommodate the non-bonding electrons on the chlorine atom. The importance of this factor in chemical kinetics has been discussed recently by Edwards and Pearson 52 This 3-d orbital space or pocket space about the sulfur atom should be particularly well-developed in carbon disulfide by virtue of the fact that the sulfur nucleus in this molecule is pushed strongly in the direction of its exterior, non-bonding electrons by the close-lying carbon core to which it is bound by a double bond. Carbon disulfide should thus be a particularly effective reagent for complexing chlorine atoms because (1) its two spin-sets are anticoincident, because (2) sulfur is a second-row element and its valence shell can therefore be expanded with relative ease, particularly in this case because (3) the sulfur nucleus in CS2 is pushed outward by the close-lying carbon nucleus. THE NATURE OF THE SULFUR-OXYGEN BOND IN SULFUR OXIDES

In sulfur dioxide the length of the sulfur—oxygen bond is 1.432 L. The length of the same bond in sulfur trioxide is 1.43 + 0.02 A. This near identity in length is anomalous in classical theory, which postulates for sulfur dioxide two resonance structures and a bond order of 1.5 and for sulfur trioxide three resonance structures and a bond order of only 1.33. From this point of view it is also curious that both bonds are significantly shorter than the sulfur—oxygen distance in sulfur monoxide, 1.493 A. Explanations of such facts as these in terms of d„43r„ bonding have been thoroughly reviewed by Wells and rejected as unsatisfactory. One is led to ask, "What new light, if any, is shed upon this problem by the tangentsphere model?" From the standpoint of the tangent-sphere model the electronic structures of firstrow elements and second-row elements differ in one important respect. This difference concerns the relation between the valence-shell electrons and their atomic cores. For first-row elements the valence-shell electrons have virtually the size they would have were the atomic cores infinitesimal in size. For second-row elements, however, the situation is radically different from this. To be tangent to one another the valenceshell electrons about the cores of second-row elements must be significantly larger than they would be were the atomic cores of infinitesimal in size. So-to-speak the valence-shell electrons touch the cores before they touch each other; in effect they are pushed out by the cores. This accords with the fact that bonds to second-row elements are longer than those to first-row elements. It also means that it should be relatively easy to squeeze more than four pairs of electrons into the valence-shells of second-row elements.

32

HENRY A. BENT

These considerations suggest the following picture of the sulfur-oxygen bond. The bond between a sulfur atom and a lone oxygen atomt is a double bond, but a double bond of a special kind. It is a double bond that is formed by two spinsets that are anticoincident. Thus, in effect, the electron cloud about the oxygen atom is cylindrically symmetric about the sulfur-oxygen internuclear line. From the standpoint of the tangent-sphere model the electrons of this sulfuroxygen bond may be considered to be equivalent to a single sphere of a rather larger than normal radius. This picture of the sulfur-oxygen bond suggests that the bond should be relatively short (e.g. shorter than the sulfur-fluorine bond) and relatively constant in length (a typical value is 1.43 A) with probably the normal +0.03 A variation in length when there are substituents on the sulfur atom that are highly electronegative or highly electropositive. The picture also suggests that the ISO angle in compounds should be larger than normal (a typical value is 123°). And it suggests that the oxygen atom of the S=0 bond should not be readily protonated both because this oxygen atom has a low (zero) formal charge and because the two spin-sets about the oxygen atom would have to be brought into coincidence to form a good bond to a proton; moreover, in forming this bond there would be no net increase in the number of electron pairs involved in bonds. The hypothesis of a cylindrically symmetric double bond also provides a simple alternative to the ad hoc hypothesis that the stereochemistry of second-row elements is determined largely by their sigma bonds rather than by their pi bonds (which are assumed to affect bond lengths but not bond angles). Finally, we may note that from the standpoint of the present discussion both SO2 and SO3 are similar to BH3 with respect to the electronic environment about the central atom; and like the latter they should be strong Lewis acids. Further chemical and physical implications of anticoincidence and of the large-core 95 model for second row elements have been discussed by the author in a current article. ACKNOWLEDGMENTS

The research reported in this chapter has been supported through a grant from the Graduate School of the University of Minnesota of a Faculty Summer Research Appointment and through a du Pont Grant-in-aid to the Department of Chemistry of the University of Minnesota. This support is gratefully acknowledged. t By this we mean any oxygen atom that is bound only to the sulfur atom in question and not to any other atoms. REFERENCES 1. L. PAULING, The Nature of the Chemical Bond, Third Edition, Cornell Univ. Press. Ithaca, N.Y., Chapter 4 (1960). 2. C. A. CoutsoN, Victor Henri Commemoratif Volume, Maison Desoer, Liege, p. 15 (1947-1948). 3. K. LONSDALE, Proc. Roy. Soc. (London) A123, 494 (1929). 4. M. G. BROWN, Trans. Faraday Soc. 55, 694 (1959). 5. C. C. Cosrnin and B. P. SrotckeFF, J. Chem. Phys. 30, 777 (1959). 6. M. J. S. DEWAR and H. N. SCHMEISING, Tetrahedron 5, 166 (1959); ibid. 11, 96 (1960). 7. G. R. SOMAYAIULU, J. Chem. Phys. 31, 919 (1959). 8. H. A. BENT, Chem. Rev. 61, 275 (1961). 9. O. BASTIANSEN and P. N. SKAICKE, Ada. in Chem. Phys., Vol. III, Edited by I. FRIGOGINE, Interscience, New York, p. 323 (1961).

Electron Correlation and Bond Properties in some Selected Sulfur Compounds

33

10. O. BASTIANSEN and M. TRAETTEBERG, Tetrahedron 17, 147 (1962). 11. D. R. LIDE, JR., Tetrahedron 17, 125 (1962). 12. T. L. COTTRELL, The Strengths of Chemical Bonds, Academic Press, New York (1954); A. G. HAiu ison and F. P. LOSSING, J. Am. Chem. Soc. 82, 519 (1960). 13. H. J. BERNSTEIN, Spectrochimica Acta 18, 161 (1962). 14. C. K. INGGLD, Structure and Mechanism in Organic Chemistry, Cornell Univ. Press, Ithaca, New York, p. 70 (1953). 15. R. S. MULLIKEN, J. Phys. Chem. 41, 318 (1937). 16. A. D. WALSH, Disc. Faraday Soc. 2, 18 (1947). 17. W. MOFFITT, Proc. Roy. Soc. (London) A202, 534 (1952). 18. J. F. J. DIPPY, Chem. Rev. 25, 151 (1939). 19. A. J. PETRI, J. Am. Chem. Soc. 80, 4230 (1958). 20. D. R. LIDE, JR., J. Chem. Phys. 33, 1879 (1960). 21. I. M. MILLS and H. W. Thomrson, Proc. Roy. Soc. (London) A228, 287 (1955). 22. A. A. J. HOROWITZ and M. H. L. PRYCE, Proc. Roy. Soc. (London) A230, 169 (1955). 23. N. MULLER and D. E. PRITCHARD, J. Chem. Phys. 31, 768 (1959). 24. L. S. BARTELL and R. A. BoiiAM, J. Chem. Phys. 27, 1414 (1957); H. C. ALLEN, JR., and E. K. PLYLER, J. Am. Chem. Soc. 80, 2673 (1958). 25. W. MOFFITT, Proc. Roy. Soc. (London) A202, 534 (1950). 26. R. S. MuLLIKEN, J. Phys. Chem. 56, 295 (1952). 27. H. A. BENT, J. Inorg. lucl. Chem. 19, 43 (1961). 28. C. E. MELLIsu and J. W. LINNETT, Trans. Faraday Soc. 50, 657 (1954). 29. A. D. WALSH, Prog. in Stereochem. Vol. I, Edited by W. KLYNE, Academic Press, New York, p. 1 (1954). 30. L. S. BARTELL, J. Chem. Phys. 32, 827 (1960). 31. V. SCHoMAKER and C. Lu, J. Am. Chem. Soc. 72, 1182 (1950). 32. R. J. GILLESPIE and R. S. NYHOLI, Quart. Rev. 11, 339 (1957) and Frog. in Stereochem. Vol. II, Edited by W. KLYNE, Academic Press, New York, p. 261 (1958). 33. R. J. GILLESPIE, J. Am. Chem. Soc. 82, 5978 (1960). 34. R. L. REDINGToN, W. B. OLSON and P. C. CROSS, J. Chem. Phys. 36, 1311 (1962). 35. L. PIERCE and M. HAYASHI, J. Chem. Phys. 35, 479 (1961). 36. K. KIMURA and M. KUBO, J. Chem. Phys. 30, 151 (1959); P. H. KASAI and R. J. MYERs, J. Chem. Phys. 30, 1096 (1959). 37. R. F. Cttpt., JR. and K. S. PIrzER, J. Am. Chem. Soc. 80, 2371 (1958). 38. K. HEDBERG, J. Am. Chem. Soc. 77, 6491 (1955). 39. H. J. BERNSTEIN, J. Phys. Chem. 56, 351 (1952). 40. T. L. COTTRELL and L. E. SUTTON, Quart. Rev. 2, 260 (1948). 41. H. A. SKINNER and L. E. SUTTON, Trans. Faraday Soc. 40, 164 (1944). 42. H. O. PRITCHARD and H. A. SKINNER, Chem. Rev. 55, 745 (1955). 43. A. F. WELLS, J. Chem. Soc. 55 (1949). 44. A. BuRRawou, Trans. Faraday Soc. 40, 537 (1944); Victor Henri Commemorat~f Volume, Maison Desoer, Liege, p. 73 (1947-1948). 45. E. WARHURsT, Trans. Faraday Soc. 45, 461 (1949). 46. K. S. PITZER, J. Am. Chem. Soc. 70, 2140 (1948). 47. J. DUCHESNE, Trans. Faraday Soc. 46, 187 (1950). 48. V. SCHOMAKER and D. P. STEVENSON, J. Am. Chem. Soc. 63, 37 (1941). 49. H. A. BENT, J. Chem. Phys. 33, 304 (1960). 50. H. A. BENT, J. Chem. Phys. 36, 1090 (1962). 51. J. HIKE, A. D. KETLEY and K. TANABE, J. Am. Chem. Soc. 82, 1398 (1960); J. HuE and R. J. ROSSCUP, ibid., 82, 6115 (1960). 52. J. O. EDWARDS and R. G. PEARSON, J. Am. Chem. Soc. 84, 16 (1962). 53. R. S. MuLLIKEN, Tetrahedron 5, 253 (1959). 54. R. W. TAFT, JR., Steric Effects in Organic Chemistry, Edited by M. S. NEWMAN, Ch. 13, John Wiley, New York (1956). 55. H. A. BENT, Can. J. Chem. 38, 1235 (1960). 56. H. A. BENT, J. Chem. Phys. 32, 1582 (1960). 57. N. MULLER and D. E. PRITCHARD, J. Chem. Phys. 31, 1471 (1959). 58. R. C. FERGUSON, J. Am. Chem. Soc. 76, 850 (1954). 59. D. R. LIDE, JR., D. E. MANN and R. M. FRISTROM, J. Chem. Phys. 26, 734 (1957). 60. S. MIZUSHIMA and T. SHIMANOUCHI, Ann. Rev. Phys. Chem. 7, 449 (1956). 61. I. LINDQVJST, J. Inorg. Nud. Chem. 6, 159 (1958); I. LINDQVIST and M. MURTSELL, Acta Cryst. 10, 406 (1957). C

34

HENRY A. BENT

62. W. MOFFITT, Proc. Roy. Soc. (London) A200, 409 (1950). 63. S. C. ABRAHAMS, Quart. Rev. 10, 407 (1956). 64. D. J. MEscHI and R. J. MYERS, J. Mot. Spec. 3, 405 (1959). 65. P. A. GIGUERE, J. Phys. Chem. 64, 190 (1960). 66. D. P. CRAIG and C. ZAULI, J. Chem. Phys. 37, 601 (1962). 67. R. P. FEYNMAN, Phys. Rev. 56, 340 (1939); T. J. BERLIN, J. Chem. Phys. 19, 208 (1951); H. C. LONGUET-HIGGINS and D. A. BROWN, J. Jnorg. Nucl. Chem. 1, 60 (1955); R. F. W. BADER and G. A. JONES, Can. J. Chem. 39, 1253 (1961). 68. H. M. JAMES and A. S. COOLIDGE, J. Chem. Phys. 1, 829 (1933). 69. G. E. KIMBALL and E. M. LOEBL, J. Chem. Ed. 36, 233 (1959). 70. J. E. LENNARD-JONES, Proc. Roy. Soc. (Lund.) A198, 1, 14 (1949); G. G. HALL and J. E. LENNARD-JONES, ibid., A202, 155 (1950); J. E. LENNARD-JONES and J. A. POPLE, ibid., 166; J. E. LENNARD-JONES, J. Chem. Phys. 20, 1024 (1952); J. A. POPLE, Quart. Rev. 11, 273 (1957); J. W. LINNETT, Can. J. Chem. 36, 24 (1958). 71. E. WIGNER and F. SEIrz, Phys. Rev. 43, 804 (1933); ibid. 46, 509 (1934). 72. J. C. SLATER, Phys. Rev. 81, 385 (1951). 73. H. K. ZIMMERMAN, JR. and P. VAN RYSSELBERGHE, J. Chem. Phys. 17, 598 (1949); J. W. LINNETT and A. J. POE, Trans. Faraday Soc. 47, 1033 (1951); C. E. MELLISI-I and J. W. LtnnerT, ibid., 50, 657, 665 (1954). 74. J. W. LINNETT, J. Am. Chen. Soc. 83, 2643 (1961). 75. R. DAUDEL, H. BRION and S. IDIOT, J. Chem. Phys. 23, 2080 (1955). 76. L. M. KLEIss, Thesis, Columbia Univ., 1952. H. R. WESTERMAN, Thesis, Columbia Univ., 1952. J. D. HERNITER, Thesis, Columbia Univ., 1956. 77. J. R. PLATT, Handbuch der Physik, Springer-Verlag, Berlin, 1961, Vol. CCCNII/2, pp. 258, 259. 78. D. R. LIDS, JR. and D. E. MANN, J. Chem. Phys. 27, 868 (1957). 79. R. W. KILn, C. C. LII and E. B. WILSON, JR., J. Chem. Phys. 26, 1695 (1957). 80. 0. HASSEL and C. RIMMING, Quart. Rev. 16, 1 (1962). 81. S. GELLER and A. L. SCHAWLOW, J. Chem. Phys. 23, 779 (1955); R. B. HEART and G. B. CARPENTER, Acta Cryst. 9, 889 (1956). 82. J. BROEKEMA, E. E. HAVINGA and E. H. WIEBENGA, Acta Cryst. 10, 596 (1957). 83. E. E. H~nEnoA, K. H. BoswiK and E. H. WIEBENGA, Acta Cryst. 7, 487 (1954); see also E. E. HAVINGA and E. H. WIEBENGA, Rec. Tray. Chin). 78, 724 (1959). 84. L. HELMHOLTZ and M. T. ROGERS, J. Am. Chem. Soc. 62, 1537 (1940) (potassium fluoroiodate); E. M. ARCHER, Acta Cryst. 1, 64 (1948) (pars-chloroiodoxybenzene); E. M. ARCHER and T. G. D. VAN SCHALKWYK, Acta Cryst. 6, 88 (1953) (benzene iododichloride); M. W. DOUGILL and G. A. JEFFREY, Acta Cryst. 6, 831 (1953) (dimethyl oxalate); D. R. DAMES and J. J. BLUM, Acta Cryst. 8, 129 (1955) (parabanic acid); J. A. WtmDekLtCH and W. N. Lirscome, Tetrahedron 11, 219 (1960) (2,4-dibromomenthone); R. N. BROWN, Acta Cryst. 14, 711 (1961) (N-chlorosuccinimide). 85. D. A. BEKOE and K. N. TRUEBLOOD, Z. Kristal. 113, 1 (1960). 86. J. H. CALLOMON, D. M. Siipsoi, and N. SHEPPARD, Ann. Rept. Chem. Soc. (London) 52, 76 (1955); K. J. LAIDLER and D. A. RAMSAY, Can. J. Chem. 36, 1 (1958); D. A. RAMSAY, Adv. in Spec. 1, 1 (1959). 87. D. W. J. CRUICKSHANK, Tetrahedron 17, 155 (1962); C. A. CouLsoI, ibid., 256. 88. A. ALMENNINGEN, O. BASTIANSEN and M. TRAETTEBERG, Acta Chem. Scand. 12, 1221 (1958); D. J. MARAIS, N. SHEPPARD and B. P. STOICHEFF, Tetrahedron 17, 163 (1962). 89. 0. BASTIANSEN, L. HEDBERG and K. HEDBERG, J. Chem. Phys. 27, 1311 (1957). 90. W. M. TOLLEs and W. D. Gwiii, J. Chem. Phys. 36, 1119 (1962). 91. R. D. BURBANK and F. N. BENSEY, J. Chem. Phys. 21, 602 (1953). 92. W. H. KIRCHHOFF and E. B. WILSON, JR. J. Am. Chem. Soc. 84, 334 (1962). 93. A. ABRAGAM, J. HoROWIz and M. H. L. PRVCE, Proc. Roy. Soc. (London) A230, 169 (1955); M. H. L. PRVCE, Disc. Faraday Soc. 26, 21 (1958); J. H. WOOD and G. W. PRAIA, JR., Phys. Rev. 107, 995 (1957); V. HEINE, Phys. Rev. 107, 1002 (1957). 94. G. A. RUSSELL, J. Am. Chem. Soc. 80, 4987 (1958). 95. H. A. BENT, "Tangent Sphere Models of Molecules. III, The Chemistry of Inner-Shell Electrons," J. Chem. Ed. 42, 302, (1965).