makes use of atomic orbitals of the valence-bonding sort. We will assume dur-
ing this discussion that the bonding that occurs between atoms in a molecule
exists between atomic orbitals in the valence shell only.
As an example, consider the methane molecule, CH 4. It consists of four hy-
drogens, each of which contributes a 1svalence orbital to the molecule, and a
carbon atom, which in its valence shell has the 2sorbital and three 2porbitals,
which we represent as 2px,2py, and 2pzorbitals. The fact that carbon has only
two 2pelectrons in that subshell does not mean that there are only two 2por-
bitals. The mathematics of quantum mechanics requires that porbitals come
in triplicate. Recall that these porbitals can point in specific directions arbi-
trarily defined as the x-axis, the y-axis, and the z-axis. The three porbitals are
perpendicular to each other, 90° apart.
Methane is a covalent molecule known to have the shape of a tetrahedron,
where the bonds make angles of 109.45° with the other bonds.* Figure 13.21
illustrates a potential problem: although the methane molecule has the shape
of a tetrahedron, the valence orbitals of the carbon atom don’t. How can
methane have its shape? This question isn’t confined to organic molecules. The
water molecule is bent at 104.5° and NH 3 has 107° bond angles, not the 90° of
the porbitals.
In the 1930s, the development of valence bond theory (most notably by
Linus Pauling) was extended to include linear combinations of the valence or-
bitals themselves. Such linear combinations are called hybrid orbitals.
Specifically, the combination of a certain number of atomic orbitals provides
linear combination hybrid orbitals that collectively have the proper symmetry.
This single fact is what makes valence bond theory and hybrid orbitals such a
useful interpretational tool in chemistry (whether or not such orbitals actually
exist).
A crucial concept about hybrid orbitals is that the number of hybrid or-
bitals must equal the number of atomic orbitals that are combined. Thus,
two atomic orbitals will combine to make two hybrid orbitals, four atomic
orbitals will combine to make four hybrid orbitals, and so forth. Orbitals that
do not combine to make hybrid orbitals continue to act as regular atomic
orbitals.
As an example, consider the carbon atom in a methane molecule. The car-
bon atom is making four bonds to the surrounding hydrogen atoms. One sand
three porbitals make up the valence shell of the carbon atom. Instead of con-
sidering them individually, suppose we assume that they combine in linear
combinations. With four atomic orbitals, we can define four independent lin-
ear combinations. The first, simplest such combination is
1 c 1 sc 2 pxc 3 pyc 4 pz
where is the lowercase Greek letter eta. As with any linear combinations, each
wavefunction is multiplied by an expansion coefficient ci(i1 to 4, in this
case). The values of these constants need to be determined before the function
is complete, so that 1 is normalized. However, it can be shown that all of the
constants are equal to^12. Therefore,
1 ^12 (spxpypz)13.11 Hybrid Orbitals 451*In fact, the demonstration of that fact was a major milestone in the advancement of
organic chemistry. The argument was first made by the French chemist Joseph Le Bel and
the Dutch chemist Jacobus van’t Hoff in 1874.2 px2 py2 pz2 sHCHHHFigure 13.21 The atomic orbitals of carbon
do not point in a tetrahedral direction. However,
the methane molecule is known to be tetrahedral.
The proper symmetry-adapted linear combina-
tions of carbon’s atomic orbitals do have tetrahe-
dral symmetry.