Matrices and Symmetry
A Simple Example
The algebra of matrices is ideal for describing
the symmetry elements of molecules.
Matrices can be used with varying degrees of
sophistication – the simplest is to use them to
operate on atomic labels.
A
B
B
 
C 
 A
 
 A
 
 B
C 
 
Combining Operations – R then σ
B
B
R120
C
C
A
A
Apply R120 then σv
A
C
B
A
σv
C
C
C
R120
B
B
A
Apply σv then R120
 A   1 0 0  0 1 0  A 
  

 
σ A R120  B  =  0 0 1  0 0 1  B 
C  0
  
0

= 1
0

Combining Operations - σ then R
B
σA
B
C
C
 A
 A   1 0 0  A 
 
 
  
 C  =  0 0 1  B  = σ A  B 
 B   0 1 0  C 
C 
  
 
 
 A
 B   0 1 0  A 
  
 
 
C
=
0
0
1
B
=
R
  
 
120  B 
C 
 A   1 0 0  C 
 
  
 
A
A
A
C
C
C
Reflections / Mirror Planes
B
B
B
Any symmetry operation can be characterised
by its effect on the column matrix – and thus
can be represented as a (3x3) matrix
Rotation Clockwise by 120º
A
A
 A
 
B
C 
 
The configuration of this
triangular molecule can be
represented by a column
matrix
1 0  1 0 0  C 
1 0  A   B 
   
0 0  B  =  A  = σ C
0 1  C   C 
 A   0 1 0  1 0 0  A 
  

 
R120σ A  B  =  0 0 1  0 0 1  B 
 C   1 0 0  0 1 0  C 
  

 
 0 0 1  A   C 

   
=  0 1 0  B  =  B  = σ B
 1 0 0  C   A 

   
1
Summary
General Rotations
The column vector conveniently represents
the essentials of the molecular topology.
• In the normal X,Y
system P is (x,y).
• In the rotated X′, Y′
system P is (x′,y′)
Symmetry operations can be represented
by matrices.
The normal rules of matrix multiplication
reproduce application of multiple symmops
x' = x cos(α ) + y sin(α )
y ' = − x sin(α ) + y cos(α )
A non-commutative algebra.
Rotation Matrices – 2D
Rewriting as a matrix equation
 x'   cos(α ) sin(α )  x 
 x
  = 
  = Rα  
 y '   − sin(α ) cos(α )  y 
 y
The operation “rotate the coordinate system
anti-clockwise by α”, is identical to “rotate
objects clockwise by α”
The operation is implemented by multiplying
the matrix onto the coordinates of any object.
Rotation Matrices – 3D
Rotation about the z-axis
 x
 x'   cos(α ) sin(α ) 0  x 
 
 
  
'
sin(
α
)
cos(
α
)
0
=
(
α
)
y
=
−
y
R
 
 y
  
z
 z'   0



 z
0
1  z 
  
 
Or the x axis…
0
0  x 
 x
 x'   1
 
 
  
'
0
cos(
α
)
sin(
α
)
=
(
α
)
y
=
y
R
 
 y
  
x
 z '   0 − sin(α ) cos(α )  z 
z
  
 
 
2