Physical Chemistry , 1st ed.

These three orbitals have the relative spatial orientation shown in Figure 13.25.

They are all in the same plane and make 120° angles with each other. The mol-

ecule BF 3 , where the boron atom makes three bonds to fluorine atoms, can

be described as having sp^2 hybrid orbitals. The remaining porbital retains its

original, hydrogen-like form.

Example 13.15

Show that two of the sp^2 hybrid orbitals on the same atom are orthogonal.

Solution

Using  2 and  3 from equations 13.17 above, consider the following integral:



1

3

s

1

2

px

1

6

pz*

1

3

s

1

2

px

1

6

pzd

Since all of the atomic orbitals are real, the complex conjugate makes no

change. The above expression can be expanded term by term into a sum of

nine simpler integrals:



1

3

s

1

3

sd

1

3

s

1

2

pxd

1

3

s

1

6

pzd



1

2

px

1

3

sd

1

2

px

1

2

pxd

1

2

px

1

6

pzd



1

6

pz

1

3

sd

1

6

pz

1

2

pxd

1

6

pz

1

6

pzd

and the constants can be removed to outside the integral. For each resulting

integral, those that have different atomic orbitals are identically zero (because

atomic orbitals themselves are orthogonal). The only remaining nonzero in-

tegrals are

(^13) s sd (^12) pxpxd (^16) pzpzd
which, because the atomic orbitals are normalized, yields
(^13)  (^12)  (^16)  0
Since it can be shown that any combination of different hybrid orbitals yields
exactly zero, the hybrid orbitals are in fact orthogonal.
For third-row elements and larger atoms, especially those in the pblock, the
existence ofdorbitals introduces other possible hybridization schemes where
the dorbitals themselves participate. The inclusion of one dorbital with the s
and the three porbitals yields five sp^3 d hybrid orbitals,which collectively have
an overall trigonal bipyramidal shape (like in PCl 5 ). The inclusion of two dor-
bitals with the sand porbitals yields six sp^3 d^2 hybrid orbitals, which collec-
tively have an octahedral shape (like in SF 6 ).
These are the most common hybridization schemes, as well as the most
ideal. Many real systems are described as having only partial hybrid character.
For example, the H 2 O molecule has a bond angle of 104.5°, not the 109.45° re-
quired by pure sp^3 hybrid orbitals. This is somewhat closer to the 90° angles of
unhybridized porbitals, suggesting more of a porbital contribution to the hy-
brid orbital description of the O–H bonds. In fact, based solely on the bond
angle, some equations (not given here) predict a hybrid orbital that is 81% p
orbital and only 19% sorbital (as opposed to 75% porbital and 25% sorbital
13.11 Hybrid Orbitals 453
+=
1 s^ orbital 2 p^ orbitals
3 sp^2 orbitals
Figure 13.25 The three sp^2 hybrid orbitals are
formed by the combination of one sand two p
atomic orbitals. They are arranged in a plane and
have the shape of an equilateral triangle. These
hybrid orbitals are used to explain the trigonal
planar structure of molecules like BF 3.