Page 269 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

P1: LDK/GJK P2: GQT/Final Pages
Encyclopedia of Physical Science and Technology EN012G-576 July 28, 2001 12:44

Physical Organic Chemistry 215

D. Molecular Orbitals
Formoleculestheorbitalsbecomeevenmorecomplicated.
In general, they are not simply s or p orbitals on a single
atom, but orbitals that spread over the entire molecule. FIGURE 3 Two p orbitals forming (a) sigma and (b) pi molecular
In some cases they can be approximated in terms of two orbitals.
atomic orbitals on two adjacent atoms. For example, in
the simplest case of the hydrogen molecule, the molecu- tracted to two positive charges. However, because s and p
lar orbital ψ MO can be expressed as a linear combination atomic orbitals are symmetric, there is a limit to the ex-
(“sum”) of the s orbitals on each of the two atoms, labeled tent to which that probability can increase. For any s or
A and B, as follows (where c A and c B are numerical co- p orbital on atom A half of its probability is on the side
efficients that can be evaluated by quantum mechanical opposite to atom B. To overcome this limitation, the s
calculation): and p orbitals on an atom can be constructed into hybrid
ψ MO = c A ψ A + c B ψ B . (4) orbitals, defined as follows, where ψ s , ψ x , ψ y , and ψ z are
respectively s, p x , p y , and p z orbitals (Fig. 1), and the c’s
According to the Pauli exclusion principle, only two elec-
are coefficients:
trons are permitted in this or any molecular orbital.
This molecular orbital could be illustrated by drawing ψ hybrid = c s ψ s + c x ψ x + c y ψ y + c z ψ z . (5)
its surface of constant ψ. It is more common just to show
Figure 4 shows that adding s and p y orbitals leads to an
the stylized surfaces of the two atomic orbitals and imag-
increased probability of finding the electron to the right
ine that the surface of the molecular orbital is the boundary
of the atom because in that region of space the s and p y
envelope of those atomic orbitals, as suggested in Fig. 2. If
orbitals reinforce, whereas to the left the positive value of
the probability, Eq. (3), associated with ψ MO is evaluated,
the s orbital tends to cancel the negative value of the p y
it is found that the two electrons in this molecular orbital orbital and reduce the value of ψ hybrid and also its square,
are more likely to be found in the overlap region, the re- which is the probability. The surface of constant ψ hybrid is
gion of space between the two nuclei where the atomic readily drawn, but usually a stylized version is adequate,
orbitals overlap. Since this is a region where the elec- as in Fig. 4.
trons are attracted to both positively charged nuclei, the These hybrids are then used to form the molecular or-
probability increase represents a stabilization. This is the bitals, as in Eq. (4) but with a ψ hybrid formed from atomic
molecular orbital counterpart of the stabilization associ- orbitals on atom A or B in place of ψ A or ψ B . If those
ated with each hydrogen’s achievement of its “octet.” hybrids point toward each other, as in Fig. 5, they can
Molecular orbitals [Eq. (4)] can be formed between two form a sigma bond of increased strength and stability be-
atomic orbitals on any adjacent atoms A and B. With p or- cause of an increased probability for the electrons to be
bitals there are two possible orientations of the atomic
found in the overlap region. The surface of constant ψ MO
orbitals. They can overlap end-on, as in Fig. 3a, or they
is quite elaborate, and only the individual hybrid orbitals
can overlap sideways, as in Fig. 3b. In the former case
are shown, to suggest their boundary envelope.
the molecular orbital is designated as sigma ( σ, “cylindri-
Many hybrids are possible, depending on how the coef-
cal”), and in the latter it is designated as pi (π) because the
ficientsarechosen.However,thenumberofdistincthybrid
nodal plane of the atomic orbitals is preserved. Pi molec-
orbitals must equal the number of atomic orbitals used to
ular orbitals must be used for the second bond of a dou- λ
construct them. In a hybrid designated as sp , the super-
ble bond and for the second and third bonds of a triple
script λ represents the ratio of p character to s character in
bond.
terms of the coefficients that enter Eq. (5):

2 2 2 1/2
λ = c + c + c c s . (6)
E. Hybrid Orbitals x y z
1
The most common hybrids are sp (customarily written
Placing two electrons into any of these molecular orbitals
2
3
simply sp), sp , and sp . No hybridization is possible to
leads to stabilization since the electron probability is in-
improve sideways overlap, so that one p orbital must be
creased in the overlap region, where the electrons are at-
FIGURE 2 Molecular orbital of the H 2 molecule, composed from
the two individual atomic orbitals. FIGURE 4 Hybrid atomic orbital formed from s and p y orbitals.