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.
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
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
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
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.
