1. CH4 – four-fold improper rotation

2. Special Features of Improper Rotations
Certain improper rotations are equivalent to other symmetry operations.
Improper rotations are considered to be the lowest possible symmetry operations so when one of them is equivalent to another symmetry operation the symbol for the other symmetry operation is used. i
Since S2 is equivalent to the inversion operation, i, it is always designated “i”, not S2. y z x
C2(z)
x,y
S2 = x,yC2(z)

3.  C2  E Improper Rotations – A Combination of Rotation and Reflectionh 1 2 C4 h 1 2
 C2 C4 h 1 2 C4 h 1 2
 E

4. Consider the operation Snm for even values of m.
The reflection operation in S is always done an even number of times so Snm = Cnm E = Cnm when m is even.
Consider the operation Snn for odd values of n.
In this specific case where n = m, the rotation operation has been carried through an angle of 2.
Since n is odd, the reflection operation is carried out an odd number of times so n = .
The result is that in the specific case where n is odd, Snn = .
Consider Snm generally when n is odd.
When n is odd, Snn =  and Snn+1 = Cn.
When n is even, Snn = E and Snn+1 = Sn and cannot be reduced except for S2 which is the same as the inversion.
We conclude that when n is odd, the existence of a Sn axis requires the existence of both a Cn axis and a  plane.

5. Point Groups and Multiplication Tables

6. Point groups
So called because all the symmetry elements pass through one common point
It is useful to be able to classify the molecular point group so that we can easily identify all the symmetry elements
Symmetry classification can be used to discuss molecular properties.
We can use symmetry transformations of orbitals to decide which atomic orbitals contribute to the formation of molecular orbitals, and select linear combinations of atomic orbitals that match the symmetry of the molecule.

7. C1
If only the identity element is present, a molecule is in the C1 point group.

8. Ci
If the only additional element is inversion, the point group is Ci. An example is meso-tartaric acid.

9. Cs
Molecules in the Cs group, e.g., fluoroethane, have only one symmetry element other than E - a mirror plane.

10. Cn
If the only symmetry element other than E is an n -fold axis, the point group is Cn . For example, H2O2 is C2 .

11. Cnv
If in addition there are n vertical mirror planes, it belongs to the group Cnv . We saw that water has a C2 axis and two sv planes, so its point group is C2v .

12. Cv
All heteronuclear diatomics and linear molecules with different atoms on the ends, are symmetrical for any rotations around and reflections across the nuclear axis, so are Cv .

13. Cnh
If there is a horizontal mirror plane, the point group is Cnh . For C2h there is also an implied center of inversion.

14. Dn
The Dn group has the symmetry elements of the Cn group, as well as n C2 axes perpendicular to the principal axis.
Gauche Ethane, neither staggered nor eclipsed, is D3

15. Dnh
If in additional there is a horizontal mirror plane, the group is Dnh

16. Dh
All homonuclear diatomics and linear molecules which are symmetrical about the center point, have symmetry elements for any rotation about the nuclear axis and for end-to-end rotation and end-to-end reflection.

17. Dnd
If in addition to the elements of Dn there are n dihedral mirror planes the point group is Dnd . An example is staggered ethane, which is D3d .

18. Sn
Molecules which do not fit one of the above classifications, but which possess one Sn axis, belong to the Sn group.
n is a multiple of two, but S2 is equivalent to Ci , and the latter designation takes precedence.
Members of the Sn group also have a Cn/2 axis; e.g., an S4 molecule will have a C2 axis.

19. The cubic groups
So far we have seen molecules with one principal axis (if any).
Some highly symmetrical molecules have more than one principal axis, and most of these belong to the cubic groups.

20. Td
Molecules in the shape of a regular tetrahedron, e.g., CH4, are in the group Td

21. Th
If in addition to the symmetry of T there is an inversion center, the group is Th

22. Oh Molecules with a regular octahedron shape are in the group Oh
An example is SF6

23. Ih
Icosahedral (20-faced) molecules with the maximum symmetry for that arrangement belong to the point group Ih . Examples are some of the larger boranes and C60.

24. Td Oh Ih

25. Linear molecules

26. R3
An atom or a sphere has an infinite number of rotation axes in three dimensions and for all possible values of n. In these cases the point group is R3

27. Td, Oh, or Ih D∞h Start C∞v Cs Ci C1 Dnh Dnd Dn Cnh Cnv S2n Cn
yes
Td, Oh, or Ih
yes
D∞h
Start
yes
Is there a center of inversion (symmetry)? no
Has the molecule
Td, Oh, Ih symmetry?
Is the molecule linear? no
C∞v no
Is there a principal Cn axis? no
Is there a mirror plane?
yes Cs no
Is there a center of
inversion?
yes
Are there n C2 axes
perpendicular to the
Cn axis?
yes Ci no C1
yes
Is there a σh plane
(perpendicular to the
Cn axis)? no
Are there n σv planes
(containing the Cn axis)?
(These σv planes are of
the σd type.)
Is there a σh plane
(perpendicular to the
Cn axis)? no
yes
Dnh
yes
Dnd no Dn
yes
Cnh no
Are there n σv planes
(containing the Cn axis)?
yes
Cnv
yes
S2n
Is there an
S2n improper rotation axis? no Cn no