Notes on Symmetry and Group Theory
Let's go back and review the characteristics of character tables, using the
C
2
v
point group as an example:
C
2
v
E
C
2
σ
v
(xz)
σ
v
(yz)
A
1
1
1
1
1
z
x
2
, y
2
, z
2
A
2
1
1
–1
–1
R
z
xy
B
1
1
–1
1
–1
x, R
y
xz
B
2
1
–1
–1
1
y, R
x
yz
II
I
III
IV
As we briefly mentioned on Friday, these tables compile data for all of the irreducible representations of a
given point group (
C
2
v
in this case).
There are 4 main areas of the table:
I:
Characters of irreducible representations (remember these indicate the "trace" or "character" of
the matrices describing each operation for each irreducible representation.
The value under
E
is always the "dimensionality" or "degeneracy" of the irreducible
representation, since
E
will always leave every element unchanged.
II:
Names of irreducible representations (Mulliken symbols).
I've included info about how these are
assigned, but the most important thing is that A/B are for singly degenerate irreducible
representations (meaning that each constituent matrix is 1×1), E (not the same as the symmetry
operation) is for doubly degenerate irreducible reps (come from 2×2 matrices), and T is for triply
degenerate irreducible reps (3×3 matrices).
III:
Symbols representing x, y, and z coordinates/axes and rotations about each axis
* These can be grouped together for higher-dimensional irreducible reps
if
symmetry operations
mix them
* R
x
, R
y
, and R
z
can be visualized as curved arrows around x, y, and z axes
IV:
Quadratics
→
squares and binary products of x, y, and z can also form bases for irreducible
representations
* We'll get to the utility of these (mostly because they relate to
d
orbitals) when we discuss MO
theory

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