William A. Bernetll University of Illinois Urbono, 61801
A Unified Theory of Bonding for Cyclopropanes
Since their inception in the early 19307s, the concepts of sp, sp2, and sp3 hybridization and of r and s bonds have formed a foundation for the structural theory of organic chemistry ((1-4). The use of these three states of hybridization in constructing models of organic compounds has offered simple pictures of bonding which enhance one's insight into the basis of chemical behavior and physical properties. The method of deriving localized hybrid orbitals on carbon has the advantage that it is mathematically simple and straightforward, involving only linear combinations of the carbon 2s, 2p,, 2p,, and 2piatomic orbitals. The Trigonally Hybridized Cyclopropane Model
I n keeping with this concept of hybridization, and in an effort to provide an explanation for the unusual properties and reactivity of cyclopropanes relative to other saturated con~pounds,Walsh constructed a model of cyclopropane which employed a state of sp2 hybridization at each carbon center (5). The Walsh model of cyclopropane is shown in Figure 1 (the G H bonds are omitted from the figure for simplicity). The G C bonds of t,he ring are formed from the intra-annular overlap of one of the sp2-hybridized orhitals of each carbon atom and three p orhitals. The structure shown in Figure 1 is actually one of three resonance structures necessitated by the fact that one of the three overlaps between p orbitals is antibonding. The bonds formed by the overlap of the p orbitals are occupied by four electrons while two electrons are associated with the overlap of the three spz-hybridizedorbitals in the center of the ring. The orbitals used in forming the C-H bonds are pictured as being sp2-hybridized. Although this model of cyclopropane may lack certain aesthetic qualities, since the hole is now gone from the center of the ring, it has been useful. This model provides an explanation for the chemical reactivity of the C-C bonds, which often undergo addition reactions rather than substitution, suggesting that there is more p character in these bonds than in those of saturated compounds (6'). This increased p character is also supported by dipole moment and nuclear quadrupole resonance studies of cyclopropyl chloride (7, 3). This model also provides an explanation for the reported ability of the cyclopropyl group to enter into "pseudo" conjugation with s-electron systems (9-11), and to stabilize adjacent carbonium ions when properly oriented (12, 15) ; the postulated ring current (14); and the shortening of the G C bond length relative to that) in saturated rompounds (15). A state of hybridization I
Present address: 3M Co., St,. Paul, Minn
in the G H bonds close to sp2 is supported by the chemical reactivity of these bonds (67, the G H bond length and HCH bond angle (15), the C-H stretching frequency (force constant) (16), and the ClrH spin-spin coupling constant (17: 13). The Bent-Bond Cyclopropane Model
As an alternative to this picture of cyclopropane, Coulsou and Moffitt derived a bent-bond model by applying the valence bond perfect-pairing approximation and minimizing the energy in the bonds formed-the energy of which is a function of the relative amounts of s and p character in the bonds (see Fig. 2; the G H bonds are omitted from the figure for simplicity) (19). This bent-bond picture was derived in a mathematically Figure I. The more complicated manner Walrh model .t ,,I,~,. than that required by the PO"=. Walsh model. I n the Coulson and Moffitt model, the orbitals associated with the ~i~~~~ 2. A G C bonds are calculated to bent bond be sp4.'z-hybridized, while model of cyclothose associated with the propane. G H bonds are ~ p ~ . ~ ~ - h y bridized (20). The angle between the axes of the spL12-hybridizedorbitals is 104' rather than t,he idealized 60" for an equilateral triangle so that the bonds are "bent" by 22'. Coulson and Goodwin later re-examined cyclopropane applying the principle of maximum orbital overlap (21). This was accomplished by varying the state of hybridization of the orbitals until the sum of the overlap integrals was maximized, and led to an optimum configuration with bonds bent by 21%' (H). This calculation has been repeated by Randii: and Maksii:, who introduced a scale factor into the wave functions to account for the fact that C-C and G H bonds do not have the same energy (23). The optimum configuration was obtained when the orbitals forming the C-C bonds were sps-hybridized and those associated with the C-H bonds were sp2-hybridized. The angle between the axes of the two sps-hyhridizcd orbitals on each carbon is 101°32', leading to bonds which are bent by 20'46'.
@
Transformation and Equivalence of Cyclopropane Models
The purposes of this article are: (1) to show that the Walsh model of cyclopropane can be easily and simply transformed into the bent-bond model of cyclopropane, and to comment on t,he transformation of the bent-bond model into the molecular orbital model, and (2) to Volume 44, Number I , January 1967
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explore the utility of the bent-bond model for explaining the behavior of cyclopropane and the cyclopropyl group. The method of arriving a t a benbbond orbital model for cyclopropane employs the use of familiar states of carbon hybridization, and requires no complicated mathematics or elaborate computer programs. I t should be pointed out, however, that we are talking about a simple model rather than a picture of bonding derived from an elaborate quantum mechanical calculation. Hybrid orbital wave functions (Vs) can be formed from linear combinations of the carbon $8, Spz, Sp,, and Sp, atomic orbital wave functions (6's) and written in the general form:
Figure 3. tion.
Trigonol
x* =
4%*, - 4%**
Substituting in the values of where the (c,,) are the coefficients of the atomic orbital wave functions. Since these are nonhomogeneous linear equations it is often convenient to write them in matrix form:
*
where is a 4 by 1 column vector of hybrid orbital wave functions (Vs), C = (clJ is a 4 by 4 matrix of the coefficients of the atomic orbital wave functions, and is a 4 by 1 column vector of atomic orbital wave functions (4's). A matrix Cis unitary if:
*
Figure 4. Pqin of equivalent sp2 ond rp6 orbitolr.
$pa hybridira-
+I
and $2 we have:
If the coefficients are squared it is seen that these two equivalent orbitals are sp6-hybridized. Thc remaining two equivalent sprhybridized orbitals are left unaltered and thus, two pairs of equivalent orbitals are formed, one pair being sp5-hybridized and the other pair spe-hybridized (Fig. 4). I n effect the matrix multiplication shown below has been carried out:
CCT = I
where CT is the transpose of C, and I is an identity matrix. The transpose of a matrix C is obtained from C by changing the rows into columns and the column into rows. An identity matrix has the number 1 for its diagonal elements and 0 for all other elements. A unitary matrix has the property that it is orthonormal. The unitary matrix transformation shown below is that which transforms the carbon Ss, Sp,, Zp,, and Sp8 atomic orbitals into three sp2-hybridizedorhitals, leaving the 2p, orbital unhybridized.
The resulting unitary matrix can be used to transform the atomic orbital basis set into a pair of equivalent sp5hybrids and a pair of equivalent sp2hybrids. Thus:
4% 4% 4%
4% O -4%
%*
4%"- 4 z -4%
The resulting state of sp2 hybridization is that used.hy Walsh in his model for cyclopropane and is shown in Figure 3. If a linear comb.ination of the 2pa orbitd (+*) and the sp2 orbital which lies along the x axis (h)is now taken, then by properly choosing the coefficients, two equivalent orbitals can be formed. By inspection, it is seen that each of these two resulting orbitals will be sp5-hybridized.% In mathematical terms the two equivalent orbitals formed are: XI =
477
$1
+ 472 $*
18 / Journal of Chemical Education
Since the matrix is unitary, the resulting hybrid orbital wave functions (x's) meet the requirements for normalization: and orthogonality: 'Since we started with one orbital which wss p and ruratber which was and slap we have, in effect, one part s character and five parts p character, i.e., sp5 hybrids.
f xixjdr
= 0 or aiaj
+ bibj + cicj + didj = 0 ..
I,] =
1,2,3,4
i #j
where the (a)s, (b)s, (c)s, and (d)s are the coefficients of the 29, 2p,, Zp,, and 2p, orbital wave functions (4's) respectively. Since it is more difficult, however, to arrive a priori at the latter unitary transformation, the wave functions for the pairs of equivalent orbitals are more easily derived by starting with a simple state of sp2hybridization. If such a transformatiou is carried out at each carbon center in the Walsh model of cyclopropane this picture is transformed into a bent-bond model, and perhaps fortuitously, identical to that derived by RandiC and MaksiE in a more complicated manner. The angle between the axes of the two sp6-hybridized orbitals of each carbon can he derived, without using the law of cosines, from the values of the coefficients for the 2p. and 2p, atomic orbital wave functions in the hybrid orbital wave function xl, as shown in Figure 5.
i 012 = 500 46'
0 =
m
X
lolo32'
with the G H bonds.3 Although the localized orbitals associated with the G H bonds in the bent-bond model were not transformed into symmetry orbitals, Hall and Lennard-Jones have outlined the general procedure necessary to do this (24). It is apparent that the bent-bond model, and therefore the Walsh model, can be transformed into a molecular orbital model of cyclopropane. The relationships between these three models of cyclopropane has now been shown. Two questions may be raised at this point: (1) which picture, the Walsh or the bent bond, of the bonding in cyclopropane is more physically correct, and (2) which is a more useful model for explaining the molecule's behavior? The Reality of Bent Bonds
Although the reality of bent bonds has been debated ($67, Fritchie reportedly has found evidence from X-ray diffraction studies for bent bonds in the cyclopropyl ring of 2,5-dimethyl-7,7-dicyanonorcaradiene, I (27).
F i g u r e 5. The angle between two rp6 orbitals. j
The transformations which were carried out show that the Walsh picture and the bent-bond picture of cyclopropane are really equivalent, i.e., they are just two different interpretations of the same total wave function. The Molecular Orbital Cyclopropane Model
Because of the symmetry of cyclopropane the three localized orbitals associated with the three bent C-C bonds, each formed from a linear combination of two sp5-hybridized orbitals, can be transformed into delocalized symmetry orbitals. The matrix transformation employed is that of Hall and Lennard-Jones (24). This transformation gives rise to three symmetry orbitals (SOs) which are associated with the plane of the ring (x,y). Two of the SOs, a degenerate pair, are composed almost entirely of p (Zp, and 2p,) orbitals (92% p character) while the third contains more s character (33%). These results can be compared with those of Hoffmann who has carried out an extended Hiickel MO calculation on cyclopropane (26). The two highest energy occupied MOs (a degenerate pair) are composed almost entirely of carbon 2p orbitals (p, and p,) in the plane of the ring (93% p character). There is no pz character in these MOs. The carbon 2p, orbitals are associated with the G H bonds which lie above and below the plane of the ring (x,y). The hydrogen 1s orbitals make very little contribution to these two MO's. The lowest energy occupied MO in the Hoffmann ca1culation;is composed almost entirely of carbon 2s orbitals (91% s character) and receives very little contribution from the carbon 2p, and 2p, orbitals and hydrogen 1s orbitals, and no contribution from the carbon 2p, orbitals. The other six occupied MO's are associated almost exclusively
The experimental electron density was best matched with an angle of bending of about 20". The cyclopropyl ring itself is inclined at about 73' to the approximately planar Ca ring. The Role of Models
The G H bonds in both the Walsh and bent-bond models have sp2 hybridization. I n actuality the G H bonds in cyclopropane probably have slightly less s character as evidenced by the HCH angle which is less than 120' (114-118") and the C13-H spin-spin coupling constant, which shows approximately 32% s character rather than the 33% which an sp2-hybridized G H bond should show. It is, however, well to point out again that only simple orbital models are being discussed rather than a detailed, specific structure for cyclopropane. To be more precise, it would be necessary to take into consideration such effects as contributions from the 1s state and higher energy states of carbon as well as higher energy states of hydrogen, leading to an expanded basis set and configuration interaction. It is perhaps useful to recall that the C-H bonds in ethylene have approximately 31% s character, based .upon the ClhH spin-spin coupling constant (28), and that the HCH angle is 116' instead of 120" (29), although the G H bonds are pictured as being sp2hybridized in the usual orbital model of ethylene as well as in other olefins. It is obvious that the r a double bond in the ethylene model could also he replaced by two sp5-hybridizedbent bonds. Such a picture would be different from the spa-hybridized "banana" bond model suggested by Pauling (1). I n the case of double bonds, a T h e author is indebted to Professor Hoffmann for kindly furnishing a. copy of the output data from his calculations. These cillculations differed from those in ref. (86)in that s hydrogen 1s orbital exponent of 1.3 instead of 1.0 was used.
Volume 44, Number 1 , Jonuory 1967
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however, this is not a very useful picture as far as explaining their behavior in that, e.g., there is no easy way to explain aromatic behavior in the absence of a bonds. Bent-Bond "Pseudo" Conjugation
The bent-bond model does not offer a simple explanation for a ring current, and it may be necessary to postulate a certain amount of nonorthogonality in the spS orbitals to accommodate a ring current in this model. The idea of unbent, nonorthogonal G C bonds in cyclopropane has been discussed by Handler and Anderson but no calculations were done (SO). On the other hand, many spectral properties of molecules are more easily understood from a molecular orbital picture than from a localized orbital model. The molecular orbital picture does seemingly accommodate a ring current. Like the Walsh model of cyclopropane, the bent-bond picture places a large amount (actually the same amount) of p character in the ring t,o explain the compound's physical properties and chemical reactivity, e.g., its ability to form a charge-transfer complex with iodine (51). The more diffuse nature of spa orbitals as compared to sp3 orbitals is such that they should be able t,o overlap with adjacent p orbitals if properly
adjacent carbon, however, the situation is different. The geometry for maximum overlap, based upon the support mentioned above, is shown in Figure 7A. Although the overlap between one pair of lobes decreases as the cyclopropyl ring is rotated clockwise (Fig. 7B, C) the overlap between the other pair of lobes increases and passes through a maximum (39' later) before decreasing. This suggests that, on the basis
Figure 7. The overlop of adjacent rp6- and p-orbitals or o function of dihedral angle.
of the bent bond picture, the magnitude of the overlap between a cyclopropyl ring and adjacent p orbitals should, in addition to being smaller, undergo proportionately less change over a rather wide range of dihedral angle than does the magnitude of the overlap between two adjacent p orbitals. This may offer at least a partial explanation for the fact that the UV absorption maximum for compounds I1 and I11 are fairly close together and those of IV and V are nearly identical (55, 56). In the latter case this result was in-
Figure 6. Pseudo conjvgotive overlap of a bent bond cyclopropyl ring with a double bond
oriented-the overlap will not be as great as that between two adjacent p orbit.als with parallel axes, however.4 This would provide an explanation for the ability of the cyclopropyl ring to pseudo-conjugate with a-electron systems (Fig. 6). The optimum geometry for the interaction of a cyclopropyl ring and a p orbital on an adjacent carbon has been shown experimentally to be that where the plane of the ring and the axis of the p orbital are parallel (11-15, 52, 55), and is supported by extended Hiickel MO calculations (25,34). An interesting feature turns up when the overlap of a cyclopropyl ring with an adjacent p orbital is considered as a function of the dihedral angle between t,hem. When two p orbitals on adjacent carbon atoms have parallel axes their overlap is a t a maximum. As the dihedral angle increases the overlap between both pair of lobes decreases. I n the case of the overlap of two sps-hybridized orbitals with a p orbital on an By convention the larger of t,he two lobes in a hybrid orbital, e.g., an @-hybrid orbital, is given a. plus (+) sign and the smaller lobe a minus sign (-). At first sight it might seem that although there could he a bonding overhp between one of the +hybrid orbitals with theplrls (+)loheof an adjacent p orbital that the overlap between the other spehyhrid orhitd (which would normally have a plus (+)sign in the larger lobe) and the minus (-) lobe of the p orbital would be ant,ihonding. This is not the case, however, and both overlaps will he hortding. Although the wave function for such a, hybrid changes sign as it, prisms through the node, it is arbitrary as to which lobe hawhich sign m long as they have opposite signs. The signs of two of the hybrid orbitals in Figure 6 have heen changed to ~eflectthe fact that all of the overlaps are of a honding nattrre.
terpreted as possibly meaning that there was no pseudo conjugation with the a-electron systems, but only an inductive effect (55). There may be an alternative interpretation for these results. If the five-membered ring in IV is puckered then the cyclopropyl ring will not be oriented with a symmetry which will result in the maximum overlap with the a system. In compound V the cyclopropyl ring will be inclined to the plane of the aromatic ring by quite a large angle, so that one of the sp5 orbitals on the carbon cu to the Ce ring could show appreciable overlap with the a-electron system. The total overlap in each case may be similar. Benzene rings have been observed to be less sensitive than ketones to substituent effects in their UV absorption (37). A comparison of the absorption maximum of compounds VI (242 mp) and VII (274 m ) demonstrates the effect which a cyclopropyl group can have on UV absorption (9).
Ring Strain in Cyclopropane
The Kilpatrick and Spitzer criteria can be applied to the bent-bond model of cyclopropane to make a crude estimate of the strain energy in the molecule (58). Kilpatrick and Spitzer assumed that the bond strength of a C-C bond is proportional to the product of the angular part. of the bond orbitals in the inter-
nuclear direction, regardless of whether a maxima for the orbitals lies in that direction. The bond strength is in turn taken to be proportional to the bond energy, with an sp3-hybridized orbital having the relative maximum bond strength of 2.00. This in effect places all of the strain in the C-C bonds of cyclopropane. It
Figure 8. Relation$hip between corterian and polar coordinates with respect to cyclopropane orientation.
7"s
0.20 Jclra
I n the Walsh picture of spiropcntane, IX, the central carbon atom is sp-hybridized while the other four car-
W IX
bons are sp2-hybridized as in cyclopropane. This is illustrated in Figure 9. By taking a linear combination of the p orbital and the sp orbital associated with each ring, four equivalent orbitals can be formed which are
Table 1.
must be assumed that the sp2-hybridized C-H bonds have, in effect,, the same energy as the sp3-hybridized G H bonds in a normal unstrained hydrocarbon. Although sp2-hybridized G H bonds are stronger than sp3-hybridized G H bonds, this effect could be mathematically cancelled out by the eclipsing of the G H bonds in cyclopropane which reduces bond energies. By using x1 and the angular parts of the 2s, 2p,, and 2p, atomic orbital wave functions:
=
Jc,,-H
Compound
Spin-Spin Coupling Constants and Character Jvo-n
(CIS)
yos
%
CHICHI CH2=CHz CII=CH
170 (e)
iar fa)
34
30
we have: As seen in Figure 8, the internuclear directions are such that 0 = GO", @ = 0" (polar coordinates). Evaluating xl,now taken as the bond strength along the internuclear direction, we have: XI =
1.887
Using 78.82 ICcal/molc as the bond energy of the spahybridized C-C bond in ethane5 the G C bond energy in cyclopropane can be estimated:
Each bond is now "strained" by 8.68 I
