Chemistry Reference
In-Depth Information
4.12
Summary
In this chapter, we have covered the idea of symmetry representations in terms of
characters and matrices. The highlights are:
1. A representation of a set of symmetry operations allows the operations to be manipu-
lated algebraically. The representation must give the same multiplication table as the
operations themselves.
2. Matrices allow the effect of operations on sets of basis vectors to be written down
algebraically and show exactly how basis vectors are transformed under each operation.
3. The character for a basis function under a given operation is a single number that
describes how much of the original function remains after the transformation. For a
single basis function it will be in the range
1.
4. Total characters for sets of basis functions are the trace of the corresponding matrix.
5. Character tables for the point groups give the sets of characters for the simplest
representations in the group. These are the irreducible representations.
6. For a basis set of our choosing we will usually arrive at a set of characters that are not
present in the character table. This is a reducible representation given the symbol
−
1to
+
,
as a sum of the irreducible representations.
7. Any reducible representation can be constructed as a linear sum of the standard irre-
ducible representations. For the correct linear combination, the sum of the characters
for the irreducible representations in each class will give the reducible character, i.e.
and it will always be possible to express
χ
(
C
)
=
n
i
χ
i
(
C
)
i
where the symbol
and
i
signify the
reducible representation and the
i
th irreducible representation for a given class
C
respec-
tively.
n
i
is a positive integer (which can be zero) that gives the number of times the
irreducible representation
i
is present in the reducible representation
χ
is used for a character and the subscripts
.
4.13
Completed Tables
Figure 4.12 gives the completed table for Problem 4.1.
4.14
Self-Test Questions
1. Derive a reducible representation for the four CN bonds in the
D
4h
complex [Ni(CN)
4
]
2
pictured in Figure 4.11. By comparing your results with possible combinations of the
irreducible representations given in Table 4.8 identify the set that are equivalent to your
reducible representation.
2. Write out matrices for all the operations in
C
3v
for the basis of N H bonds shown in
Figures 4.6 and 4.7. Using your matrices, derive a multiplication table for the operations
in the
C
3v
point group and check your results against operations carried out on a model
of the molecule. Remember that the symmetry elements are fixed in space and do not
move when operations are carried out.
