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216 Physical Organic Chemistry

F. Antibonding Orbitals
When two atoms A and B interact, there are two solutions
FIGURE 5 Two hybrid orbitals forming a sigma molecular orbital.
to Eq. (4), or two sets of coefficients c A and c B . One of
those solutions is the bonding molecular orbital that allows
dedicated to each pi bond. The remaining orbitals are hy- electrondensitytoaccumulatebetweenAandB.Theother
bridized to improve directionality. Then the hybridization solution has a nodal surface between these atoms, and
of any (octet) atom is given by the following equation, in electrons are excluded from this region. This molecular
terms of the number of pi bonds or multiple bonds that the orbital is said to be antibonding.
atom is forming: Usually such orbitals are empty, but this conclusion de-
pends on the number of electrons available. If two helium
Hybridization = sp (3−N pi ) = sp (3−N double −2N triple ) . (7)
atoms interact, there are a total of four electrons, only two
of which are permitted in the bonding molecular orbital
According to this formulation, the double bond of
according to the Pauli exclusion principle. The other two
ethene (4) is composed of both sigma and pi molecular
must occupy the antibonding orbital. According to calcu-
orbitals. The pi orbital is formed from sideways overlap of
lations, this orbital is more antibonding than the bonding
two p atomic orbitals on the carbons, as shown in Fig. 6a.
According to Eq. (7), the remaining orbitals are of sp 2 orbital is bonding. The result is a net repulsion between
the two helium atoms whenever they approach so closely
hybridization. One of those hybrids on each carbon forms
that their orbitals overlap. Such repulsion between stable
the C C sigma bond, as shown in Fig. 6b. The other two
2
sp hybrids on each carbon, along with the s orbitals on atoms or molecules is quite general. It is often called van
der Waals repulsion or steric repulsion.
the hydrogens, are used to form two C H sigma bonds,
shown in Fig. 6c.
Hybridization affects bond lengths and electronegativ- G. Molecular Orbital Theory
ity. Because a p orbital has a node at the nucleus, an elec-
tron in that orbital has zero probability of being found at A unique advantage of molecular orbitals is that they pro-
the nucleus, but there is no such restriction on an electron vide an alternative to resonance theory for describing de-
localized electrons. Equation (3) is generalized as follows,
in an s orbital. Consequently, an s electron is closer to the
where ψ 1 ,ψ 2 ,..., and ψ N are atomic orbitals on atoms
nucleus and is held more tightly. Moreover, the distance
1 through N and the coefficients c 1 –c N are evaluated by
of an electron from the nucleus and its ease of removal in-
quantum mechanical calculation:
crease with the degree of p character. One manifestation of
˚
this feature is that C H bond lengths decrease from 1.11 A ψ MO = c 1 ψ 1 + c 2 ψ 2 +· · ·+ c N ψ N . (8)
˚
3
in ethane (CH 3 CH 3 , sp ) to 1.10 A in ethene (CH 2 CH 2 ,
˚
2
sp ) to 1.08 A in ethyne (HC CH, sp). Another is that the Usually only p orbitals are treated this way, and the sigma
3
electronegativity of the carbon increases from sp to sp 2 bonds are treated by one of the previous approaches. There
to sp. are actually N different molecular orbitals, each with its
The advantage of the orbital approach is that it provides set of coefficients. Some of these carry two electrons each
2
a geometric picture. Since the sp hybrid orbitals in ethene and some are empty. For any of these the surface of con-
(Fig. 6) are constructed from p orbitals perpendicular to stant ψ MO is quite elaborate. Figure 7 suggests the bound-
the one used for the pi bond, the molecule must be planar. ary envelopes of two of the three filled pi molecular or-
However, for most purposes this approach is equivalent to bitals of benzene (21). Because these molecular orbitals
the Lewis approach, where a molecular orbital [Eq. (4)] extend over the entire molecule, they automatically allow
is symbolized simply by a line joining atoms A and B, for the electrons to be delocalized.
regardless of whether it is sigma or pi. Calculations lead to the conclusion that cyclic sys-
tems containing 4n + 2(n = 0, 1, 2,...) pi electrons are

FIGURE 6 Molecular orbital description of ethene, CH 2 CH 2 ,
viewed lying in a horizontal plane. (a) Atomic orbitals forming the
2
pi molecular orbital. (b) Hybrid sp orbitals forming the C C sigma FIGURE 7 Two filled pi molecular orbitals of benzene, viewed
2
molecular orbital. (c) Hybrid sp orbitals on carbon and s orbitals lying in a horizontal plane and showing the p atomic orbitals. Hy-
on hydrogens forming the C H sigma molecular orbitals. drogens are omitted and carbons are not labeled.