As such, why not simply use the number 1 to represent the effect ofEon that
coordinate?* In this case, that would work fine. However, for the other sym-
metry operations in the above example, expressions equivalent to equation
13.3 do not exist for all coordinates. For example, in the C 2 operation, the xH1
coordinate becomes the xH2coordinate in the opposite direction, not a con-
stant times the original xcoordinate. Thus, imposing the C 2 operation on H 2 O
moves the xcoordinate of H1 to the xcoordinate of H2, implying a 0 in the 9
9 matrix in the appropriate diagonal position and a 1 in the 9 9 matrix
in an off-diagonal position. The behavior of the other coordinates upon oper-
ation ofC 2 , or any other symmetry operation, can be analyzed similarly.
Therefore, finding a simpler representation than a 9 9 matrix is not as sim-
ple as defining an equation like 13.3 for all symmetry operations.
An inspection of the 9 9 matrices in equation 13.2 does show a pattern,
however. In each 9 9 matrix is a repeating 3 3 “submatrix” that has a cer-
tain set of diagonal elements. In the cases ofEand (yz), these 3 3 “sub-
matrices” lie on the main diagonal of the original matrix. In the cases ofC 2
and (xz), some of these 3 3 matrices do not lie along the main diagonal of
the original matrix. But at least we have identified some characteristic of each
9 9 matrix. Recognizing this pattern gives us a way to simplify the matrix
representation of the symmetry operation.
To simplify the representation, one needs to identify the smallest square
“submatrix” patternthat occurs in the same place for allmatrix representations
of the symmetry operations. The numbers will be on the main diagonal of
these smaller submatrices. The blocks may have nonzero off-diagonal elements
or sometimes zeros for diagonal elements; that doesn’t matter. In the example
above, a 3 3 section in the middle of each matrix, in the same position in
each matrix, will serve. In the case ofE, the block is outlined as follows:
(13.4)
The corresponding sections of all 9 9 representations above contain the
characteristic 3 3 submatrix for each particular symmetry operation.
Consider the set of numbers along the main diagonal for one coordinate
for all the symmetry operations. (For example, consider the set of numbers
corresponding to the behavior of the xcoordinate for each of the four sym-
metry operations.) What we find is that only a certain number of possible sets
of numbers for any system has this point group symmetry. Consider the
symmetry operations for C2v[E,C 2 ,v(xz),v(yz)] from the representations
above. We find the following possible sets of numbers: (1,1, 1,1), (1,1,
1, 1), and (1, 1, 1, 1) for any coordinate. These are the only sets of numbers
represented in the 9 9 matrix representation for the symmetry operations
of H 2 O. The C2vpoint group has one more possible set of numbers, which
000
000
000
000
000
000
100
010
001
000
000
000
100
010
001
000
000
000
100
010
001
000
000
000
000
000
000
13.5 Character Tables 433*It might help to think of equation 13.3 as an eigenvalue equation, with 1 being the
eigenvalue of the operator E. Although this is a good analogy, it applies strictly only for
groups in which the “eigenvalue” for Eis 1.