POINT GROUPS

Molecular SymmetrySymmetry elementPoint Groups

LET’S GO

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

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 crystallographersSymmetry elementsSymmetry element Notation

Hermann-Manguin(crystallography)

Schönflies(spectroscopy)

Point Symmetry Identity Rotation axes Mirror planes Centres of inversion(centres of symmetry) Axes of rotary inversion (improper rotation)

1 for 1-fold rotationnmĪ

CCn

σh, σv, σd

i

Sn

Space symmetry Glide plane Screw axis

n, d, a, b, c21, 31, etc

--

Symmetry ElementsIdentitas (C1≡E atau 1)

Rotation axes (Cnatau n)

Centres of inversion (centre of symmetry (i atau )inversion axes (axes of rotary inversion)Mirror planes ( atau m)

1

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).

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.

a. Two-fold rotation

= 360o/2 rotation to reproduce a motif in a symmetrical pattern

A Symmetrical Pattern

6

6

a. Two-fold rotation

= 360o/2 rotation to reproduce a motif in a symmetrical pattern

Motif

Element

6

6

Operation

= the symbol for a two-fold rotation

a. Two-fold rotation

= 360o/2 rotation to reproduce a motif in a symmetrical pattern

6

6

first operation step

second operation step

= the symbol for a two-fold rotation

b. Three-fold rotation

= 360o/3 rotation to reproduce a motif in a symmetrical pattern 6

6

6

b. Three-fold rotation

= 360o/3 rotation to reproduce a motif in a symmetrical pattern

6

66

step 1

step 2

step 3

Symmetry ElementsRotation

6

6

6

6

6

66

6

6

6

6

6

6

6

6

6

1-fold 2-fold 3-fold 4-fold 6-fold

9t dZaidentity

Objects with symmetry:

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.

Example:

3. Inversion (i)

inversion through a center to reproduce a motif in a symmetrical pattern

Operation: inversion through this point Element: point

= symbol for an inversion center

6

6

Example:

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

σh σdσvσh

σd

Schoenflies notation: Horizontal mirror plane ( σh): plane

perpendicular to the principal rotation axis Vertical mirror plane ( σv): plane

includes principal rotation axisDiagonal mirror plane ( σd): σd

includes the principle rotation axis, but lies between C2 axes that are

perpendicular to the principle axis

Note inversion (i) and C2 are not equivalent

5. Axes of rotary inversion (improper rotation Sn or n)An improper rotation involves a combination of rotation

and reflectionThe 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

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

Try combining a 2-fold rotation axis with a mirror

Try combining a 2-fold rotation axis with a mirror

Step 1: reflect

(could do either step first)

Try combining a 2-fold rotation axis with a mirror

Step 1: reflect

Step 2: rotate (everything)

Try combining a 2-fold rotation axis with a mirror

Step 1: reflect

Step 2: rotate (everything)

No! A second mirror is required

Try combining a 2-fold rotation axis with a mirror

The result is Point Group 2mm

“2mm” indicates 2 mirrors

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

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

Step 1: reflect

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

Step 1: reflect

Step 2: rotate 1

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

Step 1: reflect

Step 2: rotate 2

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

Step 1: reflect

Step 2: rotate 3

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

Any other elements?

• Now try combining a 4-fold rotation axis with a mirror

Yes, two more mirrors

Any other elements?

4mm

Point group name??

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

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

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

The groups C1, Ci and Cs

C1: no element other than the identityCi: identity and inversion aloneCs:identity and a mirror plane alone

THE GROUPS

The groups Cn, Cnv and Cnh

Cn: n-fold rotation axisCnv: identity, Cn axis plus n vertical mirror planes σvCnh: 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 planeDnd: in addition to the elements of Dn possesses n dihedralmirror planes σd

The groups Sn

Sn: Molecules not already classified possessing one Sn axisMolecules belonging to Sn with n > 4 are rareS2 ≡ Ci The cubic groups

Td and Oh: groups of the regular tetrahedron (e.g. CH4) andregular octahedron (e.g. SF6), respectively.T or O: object possesses the rotational symmetry of thetetrahedron or the octahedron, but none of their planes ofreflectionTh: based on T but also contains a centre of inversion

The full rotation groupR3: consists of an infinite number of rotation axes with allpossible values of n. A sphere and an atom belong to R3,but no molecule does.

Examples:

Memiliki Cn yaitu C3

Tegak lurus dengan sumbu C2 ’ masuk grup D

Mempunyai σh mencerminkan F atas dan F bawah

D3h