1. POINT GROUPS
Molecular Symmetry
Symmetry element
Point Groups
LET’S GO

2. Molecular Symmetry
All molecules can be described in terms of their
symmetry
Symmetry operation  Reflection, rotation, or inversion
Symmetry elements such as  mirror, axes of
rotation, and inversion centers

3. There are two naming systems commonly used when describing
symmetry elements:
1. The Schoenflies notation used extensively by spectroscopists
2. The Hermann-Mauguin or international notation preferred by
crystallographers
Symmetry elements
Symmetry element Notation
Hermann-Manguin Schönflies
(crystallography) (spectroscopy)
Point Symmetry  Identity 1 for 1-fold rotation C
 Rotation axes n Cn
 Mirror planes m σh, σv, σd
 Centres of Ī i
inversion(centres
of symmetry) Sn
 Axes of rotary
inversion
(improper rotation)
Space symmetry  Glide plane n, d, a, b, c -
 Screw axis 21, 31, etc -

4. Symmetry Elements
Identitas (C ≡E atau 1)
1
Rotation axes (Cnatau n)
Centres of inversion (centre of

symmetry (i atau )
1
inversion axes (axes of rotary
inversion)

Mirror planes ( atau m)

5. 1. Identity (C1 ≡ E or 1)
 Rotasi dengan sudut putar
360° melalui sudut z sehingga
molekul kembali seperti posisi
semula.
 Putaran seperti ini diberi
simbol dengan C1 axis atau 1.
 Schoenflies: C1
 Hermann-Mauguin: 1 for 1-
fold rotation
 Operation: act of rotating
molecule through 360°
 Element: axis of symmetry
(i.e. the rotation axis).

6. 2. Rotation (Cn or n)
 Rotasi melalui sudut
selain 360°.
 Operation: act of
rotation
 Element: rotation axis
 Symbol untuk symmetry
element yang mana
rotasinya adalah rotasi
dari 360°/n
 Schoenflies: Cn
 Hermann–Mauguin: n.
Molekul mempunyai n-
fold axis dari symmetry.

7. a. Two-fold
rotation
A Symmetrical Pattern
= 360o/2 rotation
to reproduce a
motif in a
6
symmetrical
pattern
6

8. Operation
a. Two-fold
rotation
= 360o/2 rotation Motif
to reproduce a
motif in a
6
symmetrical Element
pattern
= the symbol for a two-fold
rotation
6

9. a. Two-fold
rotation
= 360o/2 rotation
to reproduce a
motif in a
6 first
operatio
n step
symmetrical
pattern
= the symbol for a two-fold
rotation
second 6
operatio
n step

10. b. Three-fold
rotation
= 360o/3 rotation
to reproduce a
motif in a
symmetrical
pattern

11. b. Three-fold
rotation
= 360o/3 rotation
to reproduce a
step 1
motif in a
symmetrical
pattern
step 3
step 2

12. Symmetry Elements
Rotation
6 6 6
6
6
6
6 6
1-fold 2-fold 3-fold 4-fold 6-fold
Objects with symmetry:
a
identity
Z t 9 d
5-fold and > 6-fold rotations will not work in combination with translations in
crystals (as we shall see later). Thus we will exclude them now.

13. Example:

14. 3. Inversion (i)
inversion through a
center to reproduce
a motif in a
symmetrical pattern
Operation:
inversion through this
6
point
Element: point
= symbol for an 6
inversion center

15. Example:

16. 4. Reflection (σ or m)
Reflection across a “mirror plane” reproduces
a motif
Mirror reflection through a plane.
Operation: act of reflection
Element: mirror plane
= symbol for a mirror plane

17. Schoenflies notation:
 Horizontal mirror plane ( σh): plane
perpendicular to the principal rotation
axis
 Vertical mirror plane ( σv): plane
includes principal rotation axis
Diagonal mirror plane ( σd): σd includes
the principle rotation axis, but lies
between C2 axes that are perpendicular to
the principle axis
σh
σh σv σdd
σ

18. Note inversion (i) and C2 are not equivalent

19. 5. Axes of rotary inversion (improper rotation Sn
or An improper rotation involves a combination of rotation and
n)
reflection
The operation is a combination of rotation by 360°/n (Cn) followed by
reflection in a plane normal ( σh) to the Sn axis
Molecule does not need to have either a Cn or a σh symmetry element

20. Combinations of symmetry elements are also possible
To create a complete analysis of symmetry about a point in
space, we must try all possible combinations of these
symmetry elements
In the interest of clarity and ease of illustration, we
continue to consider only 2-D examples

21. Try combining a 2-fold rotation axis with a mirror

22. Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)

23. Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)

24. Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required

25. Try combining a 2-fold rotation axis with a mirror
The result is Point Group 2mm
“2mm” indicates 2 mirrors

26. Now try combining a 4-fold rotation axis with a
mirror

27. Now try combining a 4-fold rotation axis with a
mirror
Step 1: reflect

28. Now try combining a 4-fold rotation axis with a
mirror
Step 1: reflect
Step 2: rotate 1

29. Now try combining a 4-fold rotation axis with a
mirror
Step 1: reflect
Step 2: rotate 2

30. Now try combining a 4-fold rotation axis with a
mirror
Step 1: reflect
Step 2: rotate 3

31. Now try combining a 4-fold rotation axis with a
mirror
Any other elements?

32. • Now try combining a 4-fold rotation axis with a
mirror
Any other elements?
Yes, two more mirrors
Point group name??
4mm

33. 3-fold rotation axis with a mirror creates point group
3m

34. 6-fold rotation axis with a mirror creates point
group 6mm

35. Point groups
Most molecules will possess more than one symmetry element.
All molecules characterised by 32 different combinations of symmetry
elements:
POINT GROUPS
There are symbols for each of the possible point groups
These symbols are often used to describe the symmetry of a molecule
For example: rather than saying water is bent, you can say that water has
C2v point symmetry

36. THE GROUPS
The groups C1, Ci and Cs
C1: no element other than the identity
Ci: identity and inversion alone
Cs:identity and a mirror plane alone
The groups Cn, Cnv and Cnh
Cn: n-fold rotation axis
Cnv: identity, Cn axis plus n vertical mirror
planes σv
Cnh: identity and an n-fold rotation
principal axis plus a horizontal mirror
plane σh
The groups Dn, Dnh and Dnd
Dn: n-fold principal axis and n two-fold
axes perpendicular to Cn
Dnh: molecule also possesses a horizontal
mirror plane
Dnd: in addition to the elements of Dn
possesses n dihedral
mirror planes σd

37. The groups Sn
Sn: Molecules not already classified
possessing one Sn axis
Molecules belonging to Sn with n > 4 are
rare
S2 ≡ Ci
The cubic groups
Td and Oh: groups of the regular
tetrahedron (e.g. CH4) and
regular octahedron (e.g. SF6), respectively.
T or O: object possesses the rotational
symmetry of the
tetrahedron or the octahedron, but none of
their planes of
reflection
Th: based on T but also contains a centre of
inversion
The full rotation group
R3: consists of an infinite number of
rotation axes with all
possible values of n. A sphere and an
atom belong to R3,
but no molecule does.

39. Examples:

40. Memiliki Cn yaitu C3
Tegak lurus dengan sumbu C2 ’ masuk grup D
Mempunyai σh  mencerminkan F atas dan F bawah
D3h