Henry A. Bent University of Minnesota Minneapolis
1
1
Distribution of Atomic
Molecules and Its Chemical Implications
W h a t determines the shape ofa molecule? Methane, we know, is a regular tetrahedron; the interhond (HCII) angle in CH4 is exactly 10Q028',the tetrahedral value. Ammonia, a molecule isoelectronic with methane (both molecules contain 10 electrons, 2 in the inner 1s shell of the heavy atom and 8 in its valence shell), has, however, a somewhat smaller valence angle; the HNH angle in NH3 is 106'46'. And in Hz0 (a molecule that contains 10 electrons, also), the HOH valence angle is 1M027'. We may ask why the valence angles in ammonia and water are less than tetrahedral. In ethylene, on the other hand, the HCH angle is nearly (although not quite) 120" (1); and acetylene is h e a r . About acetylene, it is known, also, that it is a stronger Bronsted acid (better proton donor) than ethylene, although, indeed, neither compound is renowned for its acidity; also, ethylene is a stronger (Bmnsted) acid than methane (2). Insight into such facts as these can be gained by examining in direct, empirical fashion first the distribution of atomic s character in molecules and second the dependence of the properties of an atomic orbital, or a bonding orbital, on s content. Hybrid Orbitals (3)
By "s content" or "s character" we shall mean the following. For atoms that satisfy the Octet Rule, and it is with such atoms (e.g., C, N, 0, F) that this discussion is primarily concerned, the four valenceshell orbitals of an atom may be thought of as compounded from one spherically symmetrical 2s atomic orbital and three mutually perpendicular dumbbellshaped 2p orbitals, designated 2p,, 2p,, 2p,. Since the 2s orbital is lower in energy than the 2p orbitals, the ground state of a carbon atom, for example, is ls2 2s2 2pz 2pv, not ls2 2s 2p, 2p, 2p,. However, in a molecule such as methane, the 2s orbital of carbon may be considered to mix (or "hybridize") with the three 2p orbitals to form four equivalent orbitals that point to the corners of a regular tetrahedron and make angles with each other of 10Q028'; each of these tetrahedrally-directed hybrid orbitals contains '/a s character and p character, abbreviated s1I4 pa/4, or spa, or te (for tetrahedral). In ethylene, on the other
Table 1 .
s Character in
hand, the 2s orbital of carbon may be considered to mix with but two of the three 2p orbitals to form three equivalent orbitals that lie in a plane and make angles with each other of 120'; these three hybrids, which contain 33% s character, are abbreviated s1Ia pZI3, sp2, or tr (for trigonal); perpendicular to the plane of these three orbitals is the axis of cylindrical symmetry of the remaining pure 2p orbital. The geometry of acetylene is often explained by assuming that the carbon 2s orbital mixes with but one of the three 2p orbitals, forming thereby two di (for digonal) hybrids that point in diametrically opposed directions; these are abbreviated s ' ' ~p1i2, or sp. These features of orbital hybridization are summarized in Table 1. The term r-orbital is often used when referring to a hybrid orbital. The nnhyhridized 2p orbitals (in number 0, 1, and 2 for tetrahedral, trigonal, and digonal hybridization, respectively) are termed T-orbitals. Examination of Table 1 reveals a useful fact. As the s character in two equivalent hybrid orbitals increases (te to tr to di), the interorbital angle increases. Conversely, a relatively large (or small) angle between equivalent hybrid orbitals implies the s content of the orbitals is relatively large (or small). If the two orbitals in question overlap orbitals from adjacent atoms, thereby forming chemical bonds, and if these bonds are not highly strained (as they would be in cyclopropane, since the smallest inter-(s-p hybrid)-orbital angle possible is 90°), the orbitals may be assumed to point toward the adjacent, bound atoms. In such a case, molecular geometry provides a direct clue to the interorbital angle and, hence (Table I), to the distribution of atomic s character in the molecule. For example, from the bond angles in ammonia and water cited above, one infers that the nitrogen atom of NHa devotes slightly more s character to its bonding orbitals than does the oxygen atom of H20. The traditional notation of valence-bond theory provides a convenient and concise summary of the main features of the distribution of atomic s character in molecules. For atoms that satisfy the Octet Rule, four major structural types exist. The atom may be bound in a molecule by (1) single bonds only, (2) a double bond, (3) two double bonds, or (4) a triple bond.
Distribution of r Character ~ m o & Equivalent Hybrid Orbitals for Atoms That Satisfy the Octet Rule Per cent
Orbital designation
hybrid orbitals
Tetrahedrttl Trigonal Digonal
616 / Journal o f Chemical Educafian
Abbreviations
in hybrid orbitals
Xumher oi
orbital angle
brbi,his remaming
These four possibilities are shown below.
In this notation, X represents an atom that satisfies the Octet Rule (e.g., C, N, 0, F) and dots represent electrons unshared or in single bonds. The abbreviation te after a pair of electrons, structure (I), indicates that the orbital occupied by the pair is, t o a first approximation, a tetrahedral hybrid (25y0 s character); tr, structure (2), indicates an approximately trigonal hybrid (33% s character); di, structures (3) and (4), indicates an approximately digonal hybrid (500/0 s character). The te hybrid orbitals in (1) point to the corners of a regular tetrahedron. The tr hybrid orbitals in (2) lie in a plane. If we call this plane, the plane of the paper, the yz plane, with, say, the z axis horizontal and the y axis vertical, and the axis perpendicular t,o this plane the x axis, the lone, unhybridized 2p orbital in structure (2), indicated as 2p,, points
along the x axis. In structures (3) and (4) the colinear di hybrids have been taken as pointing along the z axis; the two unhyhridized 2p orbitals then point along the x and y axes. We may illustrate the utility of structures (I), (2), ( 3 , and (4) by examining the shapes of several families of isoelectronic molecules. Isoelecfronic Structures
Methane-Neon Family, Structure (1). Imagine, if you will, that one of the extranuclear protons in methane, the proton of one of the hydrogen atoms, was somehow removed from its position on the fringe of the molerule and squeezed into the nucleus of the carhon atom. By this alchemy, one pair of the four pairs of valence electrons about the heavy atom would be left unshared, three pairs would remain shared with hydrogen atoms, and the atomic number of the heavy atom i~ouldincrease from 6 (carbon) to 7 (nitrogen). This is ammonia. During the change, the hydrogen(heavy atom)-hydrogen bond angle would change, albeit only slightly, from the value that makes the energy of methane a minimum (109'28') to the value that minimizes the energy of ammonia (106°46'). Continuing this process: we may form water from ammonia, hydrogen fluoride from water, and, finally, neon from hydrogen fluoride. This is the simplest isoelectronic sequence that satisfies the Octet Rule. In addition to the species just mentioned (CK, NHs, H20, HF, Ne), it includes such important ions as NHa+, H30+, CHJ-, OH-, 0-2, and F-. These, together with other isoelectronic species, are illustrated in Figure 1. In this figure, the atomic number of the heavy atom
is plotted along the ordinate against the number of extranuclear protons along the abscissa. The eight valence-shell electrons are shown as being tetrahedrally disposed in four pairs about the heavy atom, as if the positions of the electron pairs were the same whether they were shared or not (4). Actually, the analysis of the bond angles in ammonia and water suggests that the s character of the heavy atom tends to concentrate somewhat in orbitals occupied by unshared electrons; this effect is greater, so far as the bonding orbitals are concerned, the greater the number of unshared pairs about the heavy atom. In the species NH-2, OH-, HF, and NeH+, which have three lone pairs, this shift in s character to the orbitals occupied by the unshared electrons may he nearly complete.
",,&
CHI NH4+
Figure 1.
CHaNHs HsO+
NHsHPO+ HzF
NH-2 OH HF NeH +
-
C N 0
-' -"
FNe
The Methone-Neon fomily.
Ethylene-(Oxygen) Family, Structure ( 2 ) . Starting with ethylene, H,C=CH2, we may form the isoelectronic species H,C=NH, H,C=O, HN=O ( 5 ) , HN= NH (6) and 0=0. With the exception of the last molecule (02), whose electron cloud has cylindrical symmetry about the internuclear axis, we may picture the six pairs of valence electrons in these molecules as distributed in approximately the following manner (electrons indicated by dots may be unshared or in single bonds).
..-.. .. ...
Allene-Carbon Dioxide Family, Structure (3). Starting with allene, H2C=C=CH2, we may form such species as HpC=C=O (ketene), HN=C=O (hydrogen isocyanate), O=C=O (carbon dioxide), N=N=O (nitrous oxide), N=N=NH (hydrazoic acid), N=N=i'(azide ion), and O=N=O+ (nitronium ion). It is interesting to note that the colors of these compounds in the infrared, a stringent test of the distribution of charge and mass in a molecule, are remarkably alike. AcetyleneNitrogen Family, Structure ( 4 ) . Starting with acetylene, H-C=GH, we may form H-C=N:, H-N=C:, :N=N:, : C e O : , and such ions as : C k C : -2 (as in calcium carbide, a sufficiently strong base to remove protons from water) and :N=O: + (in NO. C10,). Interesting isoelectronic families may be manufactured virtually without end. From CHICHa one obtains CH3-NH2, CH-OH, CH3-F, NH2-NH,, NHs-OH, XH2-F, HO-OH, (HO-F), F-F, B H r NH8, and ions formed by the addition or removal of one or more protons to or from these species; from CHrCH=CH2 one obtains CH-CH=O, CHrN= 0, HsN-CH=O, HO-CH=O, HO--N=O, F-N= 0, and ozone; and from isobutylene may be derived some sixty-odd molecules, including urea, acetamide, guanidine, boric acid, nitromethane, nitric acid, acetic acid, and nitryl fluoride. Volume 37, Number 12, December 1960
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This survey of molecular structure supports the surmise that whenever the Octet Rule is satisfied, a condition generally met by first-row elements, the distribution of electrons about each atom in a molecule and the local molecular geometry about each atom can be classified into four broad categories. If, as is the usual practice in drawing valence-bond structures, all pairs of shared electrons are indicated by lines and all unshared pairs in the valence shell by dots, these four categories may be elaborated to the 11 structures listed below.
dipole moment of 0.75 debyes (9). This fact may be explained by assuming that not all C-H bonds have the same polarity (one would expect the carbon in the =C-H bond to be more negative than the carbon in the C-H bonds of the methyl group) and by assuming that not all carbon-carbon bonds are nonpolar (one would expect the HC-C= bond, which is formed by the overlap of a CSa orbital of the methyl group with the C,, orbital of the acetylenic group, to be polarized in the direction H3C+--C=). With these assumptions, the contributions of individual bonds to the dipole moment of a molecule, called bond moments, have been calculated for several types of carbon-carbon single bonds (9). Interestingly, it is found that chlorine Type of C-C bond (md polarity) C(sp8)+-C(spa)C(sps)+-C(sp)C(sp"+-C(sp)-
Bond moment (9) (in debyes) 0.68 1.15
1.48
constitutes the positive end of the C C l bond moment in HC=C-C1. I n this tabulation, the structures have been arranged so that the number of unshared pairs in each column is the same. Typically, carbon atoms carry no unshared pairs, nitrogen atoms one, oxygen atoms two, fluorine atoms three, and neon atoms four. In this way, formal charges are kept a t zero. However, it is vossible for other structures to exist. Nitrogen atoms, for example, may be found in any one of the eleven environments, e.g., NH4+, NHs NHz-, NH-2, N-a: HON02, FNO, CN2r2; RNC, Nz; NO%+. The a~nroximatestates of hvbridization of these structures a& indicated a t the left each row.
of
Atom Hybridization and Electronegativity
I t was mentioned that a 2s orbital lies lower in energy than a 2 p orbital. This implies that the electronegativity of an atomic valency should increase as the s character in that valency increases (7). A carbon sp valency, for example, should be more electronegative than a carbon sp2 valency, which, in turn, should be more electronegative than a carbon spa valency. Inductive constants, constants that measure the electron-withdrawing tendency of an atom or group of atoms, confirm this. Values for the cyanide group (carbon sp valency), the acetyl group (C,? valency), and the methyl group (Cep3valency) have been determined by Taft (8). Gmup NC-
Carbon hybrid
di
hduetive constant (8) +1.300
s Character and Base Strength
I t was mentioned that acetylene (and, it could have been added, hydrogen cyanide) is a stronger acid (better proton donor) than ethylene, which is a stronger acid than methane. Or, what is the same thing, only in terms of base strength (ability to accept protons), the acetylide ion, HC=C: - (and, it could he added, :N=C: -), is a weaker base than HzC=CH-, which is a weaker base than H3C:-. The orbitals in which reside the electrons responsible for the basic properties of these ions are, respectively, di-, tr-, and te-type hybrid orbitals. This example suggests that the more s character an atom devotes to an orbital occupied by an unshared pair, and the better, thereby, the provision by the atom of low potential energy space for the lone pair, the poorer the electron-donor properties of the system. Aldehydes and ketones, for example, are less basic (poorer electron donors, or proton acceptors) than ethers and alcohols. I n these two pairs of compounds, the lone pair electrons reside in O(sp2)- and O(sp3)-type hybrid orbitals, respectively. Similarly, nitrogen is less basic than pyridine, which in turn is less basic than ammonia. In this set of compounds, the lone pair electrons reside in N(sp)-, N(sp2)-, and N(sp3)-type hybrid orbitals, respectively. In each case, as the s character in the orbital occupied by the unshared pairs increases, the base strength decreases. Effect of Electronegative Substituents on Atom Hybridization
A positive value indicates that the group is electron withdrawing, compared to hydrogen; a minus sign indicates that it is electron releasing. On this scale, atomic chlorine has an inductive constant of +1.10. Bond moments provide additional evidence of the effect of hybridization on the electronegativity of an atomic orbital. If, for example, all carbon-hydrogen bonds had the same polarity (either C+-Hor C--H+), and if all carbon-carbon bonds were nonpolar, the molecule methyl acetylene, HsC-C=C-H, would have no dipole moment; yet this molecule has a 618
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Probably only a relatively small fraction of all molecules contain atoms that are hybridized exactly sp, sp2or spS. In strain-free molecules, as mentioned, the actual state of hybridization of an atom can often be determined by examining the interbond angles about the atom (10). When this is done, it is found that electronegative substituents have a noticeable, and systematic, effect on atom hybridization (11). Table 2 contains data (1%) to illustrate the effect of electronegative groups on bond angles; additional examples may be found in an article by Mellish and Linnett (13). These data show that an atom tends to
concentrate its s character in the orbital or orbitals occupied by unshared electrons as the electronegativity of the substituents increases (labcd, 2ab, 3abc, 3de, 3fg) or as the electronegativity of the atom itself decreased (lb and 2a; lc and 2b; 3a, d, and f; 3b, e, and g). I t is an interesting fact that changes in interbond angles are sometimes in the reverse direction from what one would expect if repulsions between atoms not directly bonded to each other (nonbonded atoms) were the most important effect operating (lc and id, 3b and 3c, 4b and ethylene). Additional information regarding the effect of electronegative substituents on atom hybridization can be obtained from a study of bond lengths. However, before such data can be interpreted, it is necessary to know in what manner bond lengths depend on atom hybridization. Table 2 .
0 a. CHJ
b. CHs
c. H d. F . 2.
> CHJ H H F
Effects of Electronegative Groups on Bond Angles ( 1 2 )
LXOY 111" 109" 105' 103'
,s.
/ \ X Y a. CH, H h. H H
LXSY 10O0 Q2'=
3. XYI a. N(CH& b.NH8 e. NFs d. P(CHa)a e. PH. f. As(CH& g. AsH3 4. X,Yin CF.CXY a. H.H b. F,F
L YXY 10QO 106"46' 10Z030' 100" 93'18' 96" 91°31' LFCF 110" 114'
the single-bond radius of carbon that are not less than 0.035 A. Recent experimental data confirm this suggestion, but imply that the variation of the atomic radius of carbon may in many cases be 2 4 times greater than the lower limit set by Coulson (10). Table 3 presents a summary of the effect of atom hybridization on the carbon-carbon interatomic distances. In this table, bond distances cited as representative of each class reproduce the reported interatomic distances for essentially all representatives of that class to within 0.02 A, and for most members the values lie within 0.01 A of the listed value. This summary shows that valence-bond structures are useful in classifying bonds and predicting bond lengths. I n this classification, the separation between classes is several times greater than the variation in bond lengths among members within each given class. A useful rule of thumb is that the carbon-carbon single-bond distance decreases by 0.04 A each time one of the participating carbon atoms changes hybridization from sp3 to sp2or from sp2 to sp. When a r-bond (formed by the overlap of unhybridized 2p, or 2p, orbitals on adjacent atoms) exists superimposed on the u-bond, a better figure to use is 0.03 A. The data in Table 3 suggest that the superposition of a r-bond on a a-bond compresses the latter (20) and that this compression is less, the greater the s character of the u-bond. This is seen by the comparison of 4 with 7, 5 with 8, 6 with 9, and these with each other. The changes are: sp2-sp2 to spZ-sp2 ?r, 0.12 A; spZ--sp2 ty sp2-sp T , 0.11 A; and sp--sp to sp-sp r, 0.10 A. Superposition of a second r-bond causes a further, but smaller, decrease in bond distance (6,9, and 10). Superposition of less than a full =-bond appears t o compress the a-bond proportionately less. Two particularly good examples are provided by benzene and graphite. Following the prescriptions of valence-bond theory, one must take the carbon skeletons of these
+
Effect of Atom Hybridization on Bond Length
Experimental evidence that all carbon-carbon single bonds are not the same length seems first t o have been obtained by Lonsdale in 1929 in a study by X-ray diffraction of the crystal and molecular structure of bexamethyl benzene (14). This result was questioned in 1937 by Pauling and Brockway, who examined by electron diffraction the structures of 13 hydrocarbons and concluded that the carbon-carbon single-bond distance has the constant value of 1.54 + 0.02 A, being unaffected by the presence of an adjacent double bond (provided that it does not form part of a conjugated system). I t was concluded, also, that the carbon-carbon double bonds in allene and ethylene are the same length (15). This conclusion leads to the view, which has found wide application to this day, that the covalent radius of carbon in single bonds to other elements is 0.77 A. In 1939, however, Pauling, Springall, and Palmer confirmed the fact, which had been obtained spectroscopically (16), that the carboncarbon single-bond distance in methyl acetylene is only 1.46 A. At the same time, they proposed for this shortening two explanations (17). One explanation developed subsequently into the theory of hyperconjugation (18). The other explanation, which is adopted here, directed attention to the possibility of bond-length changes due to changes in hybridization ratios in the u-bonds. The effect of hybridization on bond lengths was reconsidered nearly a decade later by Coulson (10). From a consideration of C H distances, Coulson concluded that differencesin hybridization ratio between different types of single a-type bonds cause changes in
Table 3.
+
+
Effect of Atom Hybridization on the C-C Distance
C - 4 hybridkation
6. sp-sp
9. sp-ap 10. sp-sp
+r
+ 2*
Valence-bond structure
Characteristic C C distttnce
d-C=
1.38
=C=C= -C=C-
1.28 1.20~
Volume 37, Number 12, December 1960
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compounds to be sp2-hybridized with rr-bond orders of exactly and I/$, respectively. With these facts in mind, linear interpolation between classes 4 and 7 yields the following predictions for the C-C interatomic distances in benzene and graphite. spLsp' spa-spa
+ ('/&
+ ('/a)*
Benzene 1 . 4 0 i Graphite 1.42 A
Linear interpolation between classes 1 and 7 would yield rather different values. The observed values are benzene 1.397 A and graphite 1.421 A. On these grounds, one would predict that carboncarbon bond lengths in aromatic compounds always should be found to lie between 1.46 A (class 4) and 1.34 A (class 7). In naphthalene and anthracene, for example, the reported bond lengths fall within the ranges 1.425-1.365 A and 1.436-1.370 A, respectively (21). Effect of Electronegative Substituents on Bond Lengths
If one accepts the interpretation of the previous section regarding the effect of atom hybridization on bond lengths, namely that bond lengths decrease as the percentage s character increases, and the conclusion of the section before regarding the effect of electronegative substituents on atom hybridization, namely, that s character tends to concentrate in orbitals occupied by unshared electrons (which may be regarded as electrons in a bond to an atom of zero electronegativity), one is led to infer that in the absence of obvious steric effects, the introduction of an electronegative group into a molecule should lead to a noticeable shortening of adjacent bonds. Existing data appear to confkm this conjecture (12). A particularly striking example is the shortening of the G--F bond that occurs when hydrogen in the molecule CHaF is replaced by fluorine. The C-F distance in CH3F is 1.391 A, whereas this distance is only 1.323 A in CF, (22). Again, it is interesting to note that the effect is the opposite of what one would expect were repulsions between nonbonded atoms the important effect operating. I t may he noted, also, that the effect here is that of an atom on a bond one atom removed. As such, it is closely related to the relay of electrical effects through a molecule known to organic chemists as the inductive effect. Coupled with the effect of atom hybridization on bond angles and bond lengths, the effect of electronegative substituents on molecular geometry reveals that atomic s character tends to concentrate i n orbitals that are directed toward electropositive* groups. Before considering some of the obvious chemical implications of this statement, a brief discussion will be given of a simple quantum-mechanical model that serves to explain why s character concentrates in orbitals to electropositive groups. Coupled Perturbations in Polar Bonds
The model described in this section contains three ingredients. First, use is made of the generally accepted facts regarding electronegativity. Second, use is made of the fact that s electrons are bound more * I t is the Mtme thing to say that atomic p character tends to concentrate in orbitals that are directed toward eleetronegstive groups.
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tightly than p electrons. Third, use is made of the result from perturbation theory that to a first approximation the change in energy W' of a system whose potential energy is perturbed by an amount H' and whose wave function is initially $o is simply the average of H' over the unperturbed stat,e (25). Thus the larger the overlap of the perturbation with the wave function of the system, the larger the effect of the perturbation on the energy of the system. Consider, for example, a particle, such as an electron, bound by the potential depicted in Figure 2 by the curve labelled V. The wave function for the lowest bound state will look like $,. This is drawn to show that the particle tends to spend most of its time in the region of space where V is small (most negative). Thus a perturbation such as Hfzthat occurs where V is most highly negative has a greater effect on the energy of the particle than a differently placed but otherwise equal perturbation H',.
Figure 2. Ground-state wave funcfion ~erturbotionH'.
T. for
potensol V, with sample
Consider, now, an electron in a bond between two atoms A and B that differ in electronegativity. We shall suppose that B is more electronegative than A . I n such a bond, the potential energy and ground-state wave function vary from point to point along the internuclear axis in the manner illustrated in Figure 3.
Figure 3.
y A
One-dimensional model of o polar bond.
The general tilt of these curves is the feature of particular Interest here. As long as R is more electronegative than A , the potential energy curve slants dowu toward B, and the camel-humped curve $has its maximum value at that point. With the exception of the extrennm that ocrurs near the center of each curve, the general features of these curves may be summarized schematically by figures of the type labelled V and $o in Figure 2. This convention provides a convenient, visual representation of a polar bond in potential energy space. Bonds between groups that differ in electronegativity are visualized as tilting downward tox~ard the more electronegative group.
As an illustration, consider again two atoms or groups A and B that differ in electronegativity, B more electronegative than A , and suppose that these are bonded to an atom X of intermediate electronegativity, X being more electronegative than A but less electronegative than B. The system A-X-B might, for example, represent the array CHa-N-C1 in N(CH& Cl, or the sequence unshared pair-oxygen atomfluorine atom in OF,. Figuratively, we represent such a system in potential-energy space by orbitals directed downward from A to X and from X to B. Considering the electrons in each bond separately, the usual procedure of valence-bond theory, we obtain the repre sentation of V and J. for the system A-X-B shown in Figure 4.
attached groups differ among themselves in electronegativity, the central atom does not distribute its s character uniformly among its hybrid o-orbitals. The line beneath V and $ labelled H' in Figure 4 illustrates schematically an explanation for this dependence of atom hybridization on the electronegativities of the adjacent groups. The figure depicts the transfer by X of some s character from the orbital directed toward B to the orbital directed toward A. This lowers thelow end of the A-X hond and heightens the high end of the X-B bond (24). Because the electrons are more in the lower end of the A-X bond than they are in the upper end of the X-B bond, this perturbation decreases the energy of the A-X pair more than it increases the energy of the B-X pair. The net effect is a decrease in the energy of the molecule. Were the electronegativity of X assumed to be less than that of A , or greater than that of B, the conclusion would still be reached that the s character of X tends to concentrate in the hond to A , which is the more electropositive group. The Inductive Effect
Figure 4. One-dimensional model of the structure A (the electronegativity of Bl X, XA.
>
>
- X - B when XB
As shown by the curve for J., the energy minimization procedure of quantum mechanics tends to concentrate electrons about regions of low potential energy. Consequently, owing to the tilt to the localized molecular orbital between A and X, the electrons in that orbital are polarized in the direction of the ionic structure A+X-. Similarly, because of the assumed difference in electronegativity between X and B, the electrons in the X-B bond tend to concentrate down a t the B end of that bond. If either of these polar bonds were to dissociate into two free radicals, it is apparent that while one electron would remain in the region of relatively low potential energy associated with the atom on the right, one of the bonding electrons would have to receive some extra energy to promote it to full occupancy in the atomic orbital of the less electronegative atom on the left. This extra energy should contribute to the stability of the hond. The greater the tilt to the molecular orbital, presumably the greater this contribution to bond stabilization. This manner of thinking may be turned about. Pauling has found that the excess in A-X bond energy over the arithmetic, or geometric, mean of the A-A and X-X bond energies provides a useful, quantitative measure of the difference in electronegativity of A and X. If we so choose, we may consider that the model is in this way able to provide its own electronegativity scale. (Fig. 4), In the potential energy curve for A-X-B the two bonds are shown as meeting a t X, in accordance with the convention of beginning the analysis of the distribution of s character in molecules with the approximation that the s character of each atom is distributed equally among sp, sp2, or spa hybrids, as the case may be. However, it is known that when the
We have seen that electronegative substituents have an effect on atom hybridization (the s character in a valency decreases as the electronegativity of the attached group increases). If this effect is combined with the effect of atom hybridization on electronegativity (the electronegativity of a valency increases with increasing s character of the valency), one obtains directly a simple mechanism for the inductive effect. Suppose, for example, that the electronegativity of B in the system A-X-B increases; i.e., that the orbital between X and B becomes tilted more steeply downward toward B. This change will cause X to rehybridize slightly, so as to shift s character from the bond to B to the bond to A, where the low potential energy space characteristic of an s orbital will be used to greater advantage. This increase in s character a t the X end of the X-A bond represents an increase in electronegativity of X with respect to A. I n turn, by an identical, though smaller, operation of the same mechanism, the electronegativity of A toward adjoining groups increases. In this manner, the original perturbation is relayed in an attenuated manner throughout the bonded system. It may be added that a t each atomic center which engages in a signscant rehybridization there should exist coupled to the inductive effect certain definite, albeit perhaps small, changes in bond angles and bond lengths. Carbonium Ions
The principle that s character is used where it does the most good may be illustrated by the spontaneous isomerization of certain classical carbonium ions. Hydride-ion shifts such as the one below have been observed in solution (25) and in the gas phase ($6'). Hammett has summarized these arrangements in the
statement: "The new carbonium ion generally carries a greater number of alkyl groups than the old one." Efficient utilization of s character implies, in the first Volume 37, Number 12, December 1960
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place, that a11 s character has been removed from the empty orbital of the carbonium ion, i.e., that the configuration about the carbonium ion is as nearly planar as possible. Should this be prevented by steric effects, as, for example, a t a bridgehead, the carbonium ion ceases to be au important reaction intermediate (87). In the second place, efficient utilization of s character requires that, where possible, the rich-in-s-character and electronegative sp2-hybrids of the carbonium carbon be used to best a d v a n t a g e t h a t is to say, in bonds to the most electropositive groups in the molecule. Thus, in structure (5) above, the carbonium carbon is bonded t o one electro~ositivealkvl mouu and two hydrogen atoms,whereas iuthe morestable s&~cture(6), it is bonded to one hydrogen atom and two alkyl groups. Additions to Multiple Bonds
When a reagent A-B, such as hydrogen, a halogen, or hydrogen bromide, adds to a double bond X=Y, a significant change occurs in the hybridization of the two atoms X and Y. In particular, the four orbitals labelled below 1,2,3,4 change from sp2 to spa hybrids, i.e., the s content of these orbitals decreases. Thus, in the destruction of the double bond, electrons in bonds AB
\I
X
/3
13 41
1,2,3,4 are promoted in energy. How important this promotion is to the net heat of reaction will depend on how important the potential-energy space in the immediate vicinity of X and Y is to the electrons in bonds 1,2,3,4. The more electropositive the substituents attached through these four bonds, and, therefore, the greater the use made by the bonding electrons of the s character of atoms X and Y, the more will the loss in s character be felt, and the less the observed evolution of energy. For example, it is well known that heats of hydrogenation of olefins decrease with increasing methyl substitution on the double bond. When this fact is restated as: The double bond of the more stable isomer of two isomeric olefins carries a greater number of alkyl groups than does the double bond of the less stable isomer, the situation with regard to additions across a double bond is seen to be closely analogous to that described by Hammett with regard to the stability of carbonium ions. In general, the more electrrmegatiue the substituents attached to a multiple bond, the greater the heat of additim. Substitution of chlorine for hydrogen, or fluorine for chlorine, for example, increases the heat of chlorination, the heat of hromination, and the heat of hydrobromination of the carbon-carbon bond (88). Triple bonds may be treated in an analogous manner. The Carbonyl Group
Many of the effects described above have been observed in carbonyl compounds. For example, suppose the electronegativities of X and Y in the compound XCOY increase.
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1. Closure of the X C Y angle, as carbon withdraws s character from the C X and C Y bonds; 2. diminution of the C O distance, as carbon feeds freed 8 character into the orbitals toward oxygen; 3. enhancement of the C=O stretching force constant (Badger's Rule); and 4. rehybridization of oxygen, as this atom takes advantage of the enhanced electronegativity of carbon toward it and transfers some of its s character to the orbitals occupied by the unshared electrons, thua diminishing the electrondonor strength of the system and leading to a. a decrease in base strength and h. an increase in ionization potential.
These changes lead to the following correlations: The bond length of the carbonyl group should decrease as the ionization potential of the nonhonding oxygen electrons increases (item 2 with 4b) (89); in the absence of conjugative effects, the carbonyl stretching frequency should increase as the electronegativities of the substituents on the carbonyl group increase (item 3) (SO); and the basic properties of the carbonyl group should decrease, while the carbonyl stretching frequency should increase, as the ionization potent.ia1 increases (item 4a with 4b, and 3 with 4b, respectively) (31).
11 21
A-X-Y-B
4\
This should cause the following to occur:
Carbon Dioxide
As a h a 1 example of the role of atomic s character in determining molecular properties, we consider the structure of carbon dioxide. Reasoning from bond lengths in carbon dioxide and ketones, Pauling has introduced the generally accepted suggestion that the structures +O=C-0and -0k O + contribute importantly to the ground state of carbon dioxide (32). In selecting a valence-bond structure or set of structures for carbon dioxide, it is instructive to compare the C-0 distance in this molecule with the C-0 distances in the isoelectronic molecules ketene (H2C=C=O) and hydrogen isocyanate (HN=C=O). In each of these molecules, the u-component in the bond to oxygen is a carbon digonal-type hybrid, instead of the usual trigonal-type hybrid of a carbonyl group. A similar situation occurs in carbon suboxide (C302). Interestingly, the C-0 distances in these four molecules are remarkably alike. Molecule HG=O O=Cd H ~ ~ = HN=C=O O=C=C==C==O C 4
O
C - 4 d i ~ t m c e (A) , 1.225 1.162 1.16 1.171 1.16 1.128
Qualitatively, these effects are exactly analogous to those described in greater detail for C-C bonds in an earlier section. It may be noted, furthermore, that the pattern followed by the C-0 distances above is repeated by the stretching force constants for these same bonds (33); and, also, that the characteristic ultraviolet absorption bands of aldehydes and ketones is missing in ketene, the isocyanates, and carbon dioxide (34). On these grounds, it would appear that the ground state of carbon dioxide is well represented by the classical structure O=C=O.
Summary
A classification of molecular shapes has been given and illustrated by several isoelectronic sequences. I t is observed that the electron cloud in a molecule is not seriously altered when a proton is moved from the nucleus of a heavy atom to an unshared pair. Structural inorganic and organic chemistry may therefore be described by a common set of rules. I t is shown that once it is known how an atom distributes its s character among its four orbitals, the local geometry about the atom, with regard to both bond angles and bond lengths, can be predicted, and the relative electronegativity and basicity of its valencies assessed. A widely used approximation with regard to the manner of atom hyhridization in molecules consists in separating the manifold variations of hybridization that occur in practice into three classes: sp, sp2, and sp3. This corresponds to asserting that the s character of an atom is concentrated equally in two, three, or four hybrid orbitals. The advantages of this classification are that it is simple, that it does in fact correspond remarkably well to reality, and that definite and familiar rules exist whereby it can be quickly determined to which class an atom belongs. Furthermore, the classification lends itself to a simple refinement. Combining the fact that introduction into a molecule of an electronegative group causes bonds adjacent to the group to become shorter and interbond angles opposite it to become larger with the fact that bond lengths decrease with increasing s content, one is led to the inference that when groups of diierent electronegativity are attached to an atom, deviations from simple sp-,sp2-, or sps-type hybridization occur which are in the direction that concentrates s character in orbitals directed toward the groups of lowest electronegativity. A quantum-mechanical model based on simple first-order perturbation theory was introduced to explain this effect and used to discuss an explicit mechanism for the inductive effect, carbonium-ion rearrangements, energetics of additions to multiple bonds, interactions within the carbonyl group, and the structure of carbon dioxide. Addendum
The previous sections of this paper have dwelt on the rich fund of information that may be gleaned from the valence-bond structure of a molecule. A brief discussion follows on the construction of these. In the construction of any valence-bond structure-we shall restrict our attention here to atoms that satisfy the Octet R u l e i t is useful foreknowledge to know how many pairs of electrons are involved in bonds. Let us call this number B, and the number of unshared pairs U . B
=
number of pairs of electrons in bonds.
U = number of unshared pairs. I n any molecule that contains V valence electrons (V is assumed to be 8n even number), N heavy atoms, each of which satisfies the Octet Rule, and n hydrogen atoms, each of which requires a pair of clectrons,
there are then
+
('/I)V = B U psirs of valence electrons, and n = 2B U atomic orbitals.
4N
+
+
Eliminating U from these two relations, we find that B = (4N
+ n) - (%)V.
This simple and useful relation states that the difference between the number of pairs of electrons required by the Octet Rule for an assemblage of N n atoms, (4N n), and the number of pairs of electrons actually present, (1/2)V,is the number of pairs of electrons that must be placed between atoms to be counted twice toward the Octet Rule. For example, for HNCO. B = (4X3+1)-'/2(1+5+4+6) =13-8=5. This may be interpreted to mean that any valencebond representation of the molecule HNCO that provides for a complete octet about each heavy atom must contain five pairs of bonding electrons.
+
+
Literature Cited (1) BARTELL,L. S., AND BONHAM, R. A,, J. Chem. Phvs., 27, 1414 (1957); ALLEN,H. C., AND PLYLER,E. K., J. Am. Chem.Soc., 80,2673 (1958). (2) MILLER,S. I., AND LEE, W. G., J. Am. Chem. Sac., 81, 6313 (1959). ~ , A., "Vdence," Oxford Univ. Press, London, (3) C o n ~ s o C. 1953, pp. 25, 89, and chap. 8 ; PAULING, L., "The Nature of the Chemical Band," 3rd ed., Cornell Univ. Press, Ithacs, N. Y., 1960, chap. 4 . N. V., AND POWELL,H. M., Proc. Roy. SOC. (4) SIDDWICK, (London) A176,153 (1940). (5) DALHY,F. W., Can. J. Phys., 36, 1336 (1958); ROBINSON, G. W . , A N D MCCARTY,M., J. Chem. Phys., 28, 350 (1958); BROWN,H. W., AND PIMENTEL, G. C., J. Chem. Phys., 29,883 (1958). S. N., AND HUDSON,R. L., J. Chem. Phys., 28, (6) FONER, 719 (1958). (7) MULLIKEN, R. S., J . Phya. Chem., 41, 318 (1937); WALSH, A. D., Dismssim Fa~adaySoc., 2 , 18 (1947); MOFFIT, W., Pme. Roy. Soe. (London) A202, 534, 548 (1950). (8) TAW,R. W., JR.,in "Steric Effects in Organic Chemistry," Newman, M. S., ed., John Wiley & Sons, Inc., N. Y., 1956, chap. 13. (9) PETRO,A. J., J. Am. Chem. Sac., 80, 4230 (1958). (10) COULSON,C. A,, Victor Henri Cornmemoratif Volume, Maison Desoer, Liege, 194748, p. 15. (11) BENT,H. A,, J. Chem. Phys., in Press. (12) See "Tables of Interatomic Distances and Configuration in Molecules and Ions," SUTTON, L. E., ed., Special Puhlication No. 11, The Chemical Society, Burlington House, W . l , London, 1958. . Sac., (13) MELLISH,C. E., AND LINNETT,J. W., T ~ a n 8Faradag SO, 657 (1954). K., PTOC.Roy. Soc., A123, 494 (1929). (14) LONSDALE, L., AND B R O C K ~ AL.Y ,O., J . Am. Chem. Soc., (15) PAWLING, 59,1223 (1937). G., PATAT,F., AND VERLEGER, H., 3. Phya. (16) HERZBERG, Chem., 41,123 (1937). L.. SPRINGALL. H. D.. AND PALMER. K. J.. J. Am. (17) , , PAULING. Chem. ~ o e : ,61,927 (1939). R. S., RIEKE, C. A,, AND BROWN, W. G., (18) See MULLIKEN, J . Am. Chem. Soe., 6 3 , 41 (1941); also, BAKER,J. W . , "Hyperconjugation," Oxford Clarendon Press, 1952. (19) For a summary of the recent data bearing on this issue, see BROWN, M. G., T r a m . Faraday Soc., 55, 694 (1959);also B. P., J . Chem. Pkys., COSTAIN.C. C.. AND STOICHBFF. 30,777 (i959). C. A,. "Valence." Oxford Univ. Press. London, (20) COUL~ON. -
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(21) ROBERTSON, J. M., "Organic Crystals and Molecules," CornellUniv. Press, Ithaca, N. Y., 1953. L. O., Acta Cryst., 7,682 (1954). (22) BROCK~AY, (23) PAULING,L., AND WILSON,E. B., JH., "Introduction to Quantum Mechttnies," McGraw-Hill Book Co., Inc., New York, 1935, p. 159. (24) Matthew XXV: 29. (25) HAMMETT,L. P., "Physical Organic Chemistry," McGrawHill Book Co., Inc., New York, chap. 10. (26) FIELD,F. H., AND LAMPE,F. W., J. Am. Chem. Soe., 80, F. H., AND FRANKLIN, J. L., 5587 (1958); also FIELD, "Electron Impact Phenomena," Academic Press, Inc., New York, 1957, chap. 6 .
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(27) APPLEQUIST, D. E., AND ROBERTS,J. D., Chem. Rev., 54, 1065 (1954). H. A., "Modern Aspects (28) For supporting data, see SKINNER, of Thermochemistry," Monograph No. 3, The Royal Institute of Chemistry, London, 1958; also L A C ~ E RJ .,
R., EMERY,E., BORMFOLK, E., AND PARK,J. D., J . Phys. Chem., 60,492 (1956). 29) WALSH,A. D., Trans. Faraday Sac., 42, 56 (1946).
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(30) LORD,R. C., AND MILLER, F. A,, Appl. Spec., 10, 115 (1956). (31) COOK,D., J . Am. Chem. Soc., 80, 49 (1958). (32) PAULING, L., "The Nature of the Chemical Bond," 3rd ed., Cornell Univ. Press, Ithaca, N. Y., 1960, p. 267. (33) LINNETT,J. W., Quart. Rev., 1, 73 (1947). G J . Chem. Phys., 3, (34) Sno-Cnow Woo A N D T A - K ~ N LK, 544 (1935).
