Chemistry Reference
In-Depth Information
In general, the diagonal terms in a transformation matrix tell us how much of each
original basis function is retained. The off-diagonal terms give additional information on
exactly how to build the new functions from the old.
So, the character for a particular operation and basis function is just the associated diag-
onal element in the matrix. In the NH
3
example, the
C
3
1
rotation would have a character
of 0 for each of the basis vectors.
A zero character for a basis function simply implies that it has been transformed into
some combination of the other members of the basis set.
4.6
The Trace of a Matrix as the Character for an Operation
The character we assign for a particular basis vector has been linked to the diagonal ele-
ment in the operation's matrix. This can be generalized to say that the sum of the diagonal
elements of a matrix representing an operation on a particular basis is the sum of the
characters for the basis under that operation. In matrix algebra, the sum of the diagonal
elements of a matrix
A
is known as the trace of the matrix, Tr(
A
), i.e.
⎛
⎞
a
11
a
12
a
13
3
⎝
⎠
A
=
a
21
a
22
a
23
Tr(
A
)
=
a
ii
=
a
11
+
a
22
+
a
33
(4.13)
a
31
a
32
a
33
i
=
1
This example shows the trace for a 3
3 matrix, but the idea of taking the sum of diagonal
terms can be extended to matrices of any order.
Using the trace rather than picking out particular elements of the matrix allows a single
character to be assigned for the result of the operation on the entire set of basis func-
tions being considered. This will turn out to be a useful tactic in dealing with symmetry
problems, since it allows whole sets of basis vectors to be considered together.
For example, the ammonia molecule belongs to the point group
C
3v
and so, in addition to
the principal axis, there are three vertical mirror planes. The result of a reflection operation
is shown in Figure 4.7; to see the how the transformation affects the basis set, one of the
three degenerate mirror planes has been chosen as an example. The mirror contains the
b
3
vector, and so the superscript C has been added to the mirror plane symbol following
the convention employed in earlier chapters.
×
σ
v
C
b
1
'
σ
v
C
b
2
b
3
'
b
3
b
1
b
2
'
Figure 4.7
The effect of the
σ
v
C
plane reflection on the basis of N
H bonds on ammonia.
