Chemistry Reference
In-Depth Information
To recover the equations we can multiply the vector by the matrix. The multiplication is
carried out following the convention that the first element on the left is given by sum-
ming the products of the first row of the matrix with the column vector on the right-
hand side:
x
′
−
1
0
x
−
0
=
gives
x
′
=
−1
×
x
+ 0
×
y
(A5.3)
y
y
′
Similarly, the second element on the left-hand side is a sum of products of the second row
of the matrix and the column vector on the right-hand side:
x
′
y
′
x
y
−1
0
0
=
gives
y
′
=
0
×
x
+ −
1
×
y
(A5.4)
−
Since the matrix represents a symmetry operation, the act of multiplying the vec-
tor by the matrix is often referred to as an operation, i.e. the matrix operates on the
vector.
Larger systems of basis vectors can be dealt with in much the same way. For example, in
Section 4.5 we considered the effect of a
C
3
1
rotation on a basis of the three N H bonds
of ammonia in the
C
3v
point group. As before, to carry out the multiplication the column
b
1
,
b
2
,
b
3
(representing N H bonds 1-3) is multiplied by the top row of the matrix to give
the vector now in the
b
1
direction, then by the middle row to get the new term in
b
2
and
finally by the bottom row to get the new
b
3
:
b
1
b
3
0
0
1
b
2
b
3
1
0
0
=
giving
b
1
now replaced by
b
3
0
1
0
b
1
b
3
0
0
1
b
1
b
2
b
3
giving
b
2
now replaced by
b
1
1
0
0
=
0
1
0
b
1
b
3
0
0
1
b
2
b
3
b
1
b
2
giving
b
3
now replaced by
b
2
1
0
0
=
(A5.5)
0
1
0
A5.1.1 Products of Matrices
We can extend the algebra of matrices to include products formed by multiplying two
matrices together. The product of the matrices must give a third matrix, since we know,
for example, that the combined
C
2
σ
v
operation in the
C
2v
point group is equivalent to the
other vertical reflection
σ
v
. The multiplication of the two matrices can be carried out by
treating the columns of the second matrix as vectors and multiplying each one by the rows
