Physical Chemistry , 1st ed.

The wavefunction  1 is a hybrid orbital, composed of one part sorbital and

three parts porbital. The other three hybrid orbitals are

 2 ^12 (spxpypz)

 3 ^12 (spxpypz) (13.15)

 4 ^12 (spxpypz)

(The numbering of the hybrid orbitals is arbitrary.) Because these hybrid or-

bitals are the combinations of one sorbital and 3 porbitals, they are called sp^3

hybrid orbitals.These hybrid orbitals can be represented graphically just as

hydrogen-like orbitals are. Figure 13.22 shows a cross-section of one orbital,

which resembles a porbital but has one enlarged lobe and one shrunken lobe.

The plot of the four sp^3 orbitals in Figure 13.23 shows the spatial relationship

of the four hybrid orbitals. The larger lobes of the four hybrid orbitals make

the shape of a tetrahedron. When these four orbitals overlap with orbitals of

another atom to make four bonds, the bonds made point in the direction of a

tetrahedron. This accounts for the known tetrahedral shape of methane: the

valence orbitals of the carbon atom can be thought of as not “pure”sand por-

bitals, but sp^3 hybrid combinations that together have a tetrahedral shape.

In considering the hybrid orbitals in the valence shell of an atom, one must

provide a hybrid orbital for each (sigma) bond (defined in the previous

chapter as a bond having cylindrical electron density between the nuclei mak-

ing the bond) the atom makes as well as nonbonding valence electron pairs. In

the case of methane, in which the carbon atom makes four bonds, four hy-

brid orbitals are required: the four sp^3 hybrids made from the four atomic or-

bitals from the carbon valence shell.

Different atoms make different numbers of bonds, and also have nontetrahe-

dral shapes. The covalent molecule BeH 2 is linear, with the two bonds made by

the Be atom pointing in opposite directions. This sounds like one of the porbitals,

but the ground-state electron configuration of the Be valance shell is 2s^2 .No por-

bitals are occupied. However, if the sorbital and one of the porbitals combine,

 1 

1

2

(spz)

(13.16)

 1 

1

2

(spz)

then the resulting two hybrid orbitals have relative directions as shown in Figure

13.24: they are oriented 180° from each other. These sp hybrid orbitalscan make

bonds with the hydrogen atoms, yielding a molecule that has a linear shape.

The other two orbitals in Be, the pxand the py, are unaffected by the hybridiza-

tion and are unoccupied. (Using the pzorbital in equations 13.16 was arbitrary.)

In cases where three hybrid orbitals are needed, a combination of the one s

orbital and two of the porbitals is assumed. The choice depends on how one

defines the system. Assuming that one wants a hybrid orbital pointing along

the z-axis, using the pzorbital and (arbitrarily) the pxorbital provides three sp^2

hybrid orbitalsin the xzplane. They have the forms

 1 

1

3

s

2

3

pz

 2 

1

3

s

1

2

px

1

6

pz (13.17)

 3 

1

3

s

1

2

px

1

6

pz

452 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

+=

1 s^ orbital 3 p^ orbitals

4 sp^3 orbitals

Figure 13.23 The four sp^3 orbitals, formed by
the combination of the single sand the three p
atomic orbitals, have tetrahedral symmetry. (For
clarity, the small lobes of each orbital are not
shown.) Their spatial geometry makes sp^3 orbitals
very useful in explaining the structures of organic
molecules.

Figure 13.22 A single sp^3 orbital is reminis-
cent of a porbital, but with unequal lobes. Like a
porbital, it does have one angular node.

+=

1 s^ orbital 1 p^ orbital

2 sp^ orbitals

Figure 13.24 The two sphybrid orbitals,
formed by the combination of one sand one p
atomic orbital, point in opposite directions. They
are used to explain the linear geometries of mol-
ecules like BeH 2 and C 2 H 2.