Chemistry Reference
In-Depth Information
From Figure 4.7, the reflection has swapped
b
1
and
b
2
but has left
b
3
unchanged. The
matrix corresponding to this operation can be deduced from this result:
σ
v
C
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
b
1
b
2
b
3
010
100
001
b
1
b
2
b
3
b
2
b
1
b
3
⎝
⎠
=
⎝
⎠
⎝
⎠
=
⎝
⎠
(4.14)
The other two vertical mirror planes have similar matrices:
σ
v
A
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
b
1
b
2
b
3
100
001
010
b
1
b
2
b
3
b
1
b
3
b
2
⎝
⎠
=
⎝
⎠
⎝
⎠
=
⎝
⎠
(4.15)
σ
v
B
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
b
1
b
2
b
3
001
010
100
b
1
b
2
b
3
b
3
b
2
b
1
⎝
⎠
=
⎝
⎠
⎝
⎠
=
⎝
⎠
Here,
σ
v
B
are the planes containing vectors
b
1
and
b
2
respectively in the original
configuration. In all three cases the matrices swap two of the basis vectors and leave one
unchanged. If we add up the diagonal elements of each matrix we find that the trace is 1 in
each case, and so the total character of the three N H bonds under any of the reflection
operations is 1. The three matrices differ in which of the diagonal elements is 1, but the
trace has the same value in each case; this is a consequence of the equivalence of the three
mirror planes and means that only one heading, 3
σ
v
A
and
σ
v
, needs to appear in the
C
3v
character
table for all three mirror planes (see Appendix 12).
Problem 4.7:
Using the basis of N H bonds defined in Figure 4.6, write down the
matrix representation for a
C
3
2
operation. Take the trace of this matrix and, hence,
show that
C
3
2
and
C
3
1
operations have the same character for this basis.
4.7
Noninteger Characters
The examples up to now have been chosen so that each operation causes a given basis
vector to be unaffected, reversed or transformed into a different basis function. In the
analysis, we have shown that the matrix representation allows us to describe each of these
transformations leading to the characters of
1 and 0 respectively. However, if the
basis vectors are not arranged to fit nicely with the symmetry elements, then the character
assignment may not be so straightforward.
+
1,
−
4.7.1 Boron Trifluoride, BF
3
,
D
3h
For example, Figure 4.8 shows the result of the
C
3
1
operation on a p
x
(B) orbital in the
BF
3
molecule. The orbital is no longer aligned with either the
X
or
Y
direction after the
operation, so assigning its character requires some additional attention. The new orbital
is in between the
X
and
Y
axes and so could also be constructed by some mixture of the
