Environmental Engineering Reference
In-Depth Information
These details are more important than one might expect. The same pattern, the
same angular dependences for the
p
and
d
wavefunctions, are duplicated for higher
values of principal quantum number
n
beyond
n¼
2.
3.5.1
Connection with Directed or Covalent Bonds
We can use these functions in turn to form other hybrid states. For example, the sp
3
set of wavefunctions that point to the corners of a tetrahedron occur in molecular
CCl
4
, and in crystalline germanium, silicon, and diamond.
A tetrahedron has four vertices at equal radius from the center, and this geomet-
rical
figure has both threefold (through each vertex) and twofold (through face
centers) rotation axes through the origin (cube center). If we think of a cube of side
L¼
2, then four diagonally related corners, out of the eight corners of the cube, are
vertices of a tetrahedron. These points have radius
p
3 from the center of the cube of
side 2, and the spacing between adjacent vertices is 2
p
2, which de
nes the bond
angle
b
. The law of cosines (
a
2
¼ b
2
þ c
2
2
bc
cos
b
) then gives cos
b¼
1/3 or
b¼
109.47
for the tetrahedral bond angle. This illustrates a method that can be
applied to other bonding geometries. Accurate measurement of bond angles and
bond lengths is a basic tool of the chemist and solid-state physicist.
To represent a linear combination of directed wavefunctions more simply, call it
r ¼ l
i þ m
j þ n
k
, where
l
,
m
, and
n
are direction cosines, which give the projection
r
of
along each of the axes. To
find the combinations that form the tetrahedron,
imagine vectors from the center of a cube of edge 2, to four diagonally related corners,
and represent these vectors by
l
,
m
, and
n
values. Taking the origin at the center of the
cube, a suitable set of linear combinations of
p
x
,
p
y
, and
p
z
wavefunctions (see
Table 3.1) is easily seen to be represented (by
l
,
m
, and
n
values) as (1,
1,1), (
1,1,1),
(1,1,
1), and (
1,
1,
1).
3.5.2
Bond Angle
A useful formula for the angle
H
between two lines described, respectively, by
l
,
m
,
and
n
values, is
H ¼ðl
1
l
2
þm
1
m
2
þn
1
n
2
Þ=½ðl
1
þm
1
þn
1
Þ
1
=
2
ðl
2
þm
2
þn
2
Þ
1
=
2
:
ð
3
:
46
Þ
cos
If we apply this to
find the angle, for example, between radius vectors (1,
1,1) and
(
1,1,1), we have
1
=
2
1
=
2
47
cos
H ¼ð
1
1
þ
1
Þ=½ð
3
Þ
ð
3
Þ
¼
1
=
3
:
So
H ¼
109
:
to verify the tetrahedral bonding angle we found for the diamond structures that
include silicon. Again, the directed wavefunctions
p
x
,
p
y
, and
p
z
have been repre-
sented by unit vectors
i
j
k
. We have also identi
ed a directed wavefunction with
a covalent bond, which will be further discussed.
,
, and
