Molecular Symmetry
A set of nuclear coordinates defines a particular symmetry
for a molecule which is given by its symmetry operations.
These act to exchange equivalent atoms.
The symmetry operations present in a given molecule are
elements of a group.
There are only five types of symmetry operations possible:
E the do nothing operator - present in all groups
σ reflection through a plane
i inversion through a center point
Ci rotation about an axis
Si improper rotation about an axis
Water as an example
symmetry operations: E C2 σxz σ yz
Need to carefully define x, y, and z coordinates. There is a
convention which can be followed:
o z is defined to coincide with the principle axis of
rotation
o x is defined as the out-of-plane axis
A Gaussian calculation will reorient the molecule in this
standard orientation even if you do not initially provide it
as such. Results must be understood in terms of this
definition of coordinates. GaussView can quickly show the
Cartesian axes.
Now let's work out the symmetry operations present in each
of these:
Ethane Eclipsed
Ethane Staggered
Allene
The Group of Symmetry operations define a point group
for a molecule.
Point groups have properties that are contained in a
character table.
Use a flow diagram to identify the point group of the
following molecules. Then verify by examining the
character table for that point group.
eclipsed ethane
staggered ethane
allene
difluoro-bromomethane
methylene chloride
nitrogen monoxide
hydrogen peroxide (cis)
hydrogen peroxide (trans)
hydrogen peroxide (gauche)
phosphorus pentafluoride
The symmetry operations present in a molecule
define a group in the usual sense. Therefore, you
should be able to construct a group multiplication
table based on these elements.
Let’s try this for C2V water:
Fill in the blanks of this multiplication table:
E
σxz
C2
σyz
E
C2
σxz
σyz
Here is the Character table:
C2V
A1
A2
B1
B2
E
1
1
1
1
C2
1
1
-1
-1
σxz
1
-1
1
-1
σyz
1
-1
-1
1
z
Rz
x, Ry
y, Rx
Rows are the irreducible representations of the point
group. The last column shows how translations and
rotations transform as one of these irreducible
representations. Can you verify these using water?
Molecular Orbitals of a Molecule must Transform as
One of the Irreducible Representations of The Point
Group to Which the Molecule Belongs! DEMO
Here are some useful online links:
http://www.webqc.org/symmetry.php
http://www.colby.edu/chemistry/PChem/scripts/ABC.html
http://symmetry.otterbein.edu/
http://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html
You can reduce any representation into its irreducible parts by using:
mα =
1 N
∑ χ Γ (n) ⋅ h n ⋅ χα (n)
g n =1
HERE:
mα = # of irreducible representations of type α in the reducible representation Γ
g = order of the group
N = # classes in the group
hn = # of elements in the nth class
χ Γ (n) = character of the nth class in the reducible representation Γ
χ α (n) = character of the nth class in the irreducible representation α
For example, the cartesian (x,y,z) displacements of water can be
added up into one representation:
C2V
Γcart
E
9
C2
-1
σxz
3
σyz
1
Can you break this into the irreducible representations?
Which ones of these correspond to internal motions?
Which ones of the internal motions are IR active?
More detailed normal mode analysis of water can be found in this student
worksheet.
Try completing in the worksheet yourself. To check your work, here is a
copy of Roa Al-Qabbani’s notes from Fall 2009. Thanks Roa!