Symmetry &

Teory Group

OlehYateman Arryanto

The concept of symmetry is exist in an everyday sense.

If we say something is ‘symmetrical’, we usually mean it has mirror symmetry, or ‘left-right’ symmetry, and would look the same if viewed in a mirror.

Symmetry is also very important in chemistry.

Some molecules are clearly ‘more symmetrical’ than others, but what consequences does this have, if any?

Symmetry & Teory Group

The aim of this course is to provide a systematic treatment of symmetry in chemical systems within the mathematical framework known as group theory

Symmetry & Teory Group

The aim of applications of group theory are : Predicting whether a given molecule will

be chiral, or polar. Examining chemical bonding and

visualising molecular orbitals Chemical Properties

Predicting whether a molecule may absorb light, and which spectroscopic transitions occurs.

Investigating the vibrational motions of the molecule.

Symmetry, Art & Science

often used as an implied symbol

of the essential symmetry of the human body, and by

extension, to the universe as a whole

Symmetry & Nature

Symmetry & Nature

Symmetry & Nature

Apex episcopus Nautilus pompilus

Symmetry & Nature

Spiral Symmetry

Symmetry & Nature

Architecture (Spiral Symmetry)

Symmetry & Nature

(Bilateral / Mirror Sym)

Symmetry & Architecture

Taj Mahal (Agra, India) Giza Pyramid Field (Cairo, Egypt)

Symmetry & Architecture

Eiffel Tower (Paris)325 m, 10.1 tons)

Symmetry & Landscaping

Toukouji (Japan) Trocadero Palace (Paris) from Eiffel tower

Symmetry & Nature

Symmetry & Nature

Super-High Symmetry in Super-Low “Life Forms”

Super-High Symmetry in Super-Low “Life Forms”

Symmetry & Nature

Symmetry & Nature

Symmetry & Nature

Wallpaper pattern for Ontmoeting. From: M.C.Escher, 1944

A frieze pattern

Symmetry is BeautifulSymmetry is Living

Symmetry is chemistrySymmetry is Beautiful

Living

Be Symmetric and be best chemist

To make a beautiful life

The Icosahedron

The Icosahedron

The Dodecahedron

The Cube (Hexahedron)

The Oktahedron

The Tetrahedron

The Platonic Solids

The Cube The Methane Molecule

The Structure of Crystal

The Oktahedron

Simetri adalah PolaSimetri adalah KeteraturanSimetri adalah Keindahan

Belajar Simetri berarti belajar tentang pola, keteraturan,

keindahan sehingga dengan belajar simetri kita menjadi simetri dan mampu berperanan menciptakan

dunia yang teratur, simetri dan indah

(mamahayu ning bawono lan angkoso )

P. H. Walton “Beginning Group Theory for Chemistry”Oxford University Press Inc., New York, 1998ISBN 019855964

A.F.Cotton “ Chemical Applications of Group Theory”ISBN 0471510947

S.F.A Kettle “ Symmetry and Structure, Readable Group Theory for Chemist “, 2nd Edition, John Wiley & SonsISBN 0 471 9554 7

TEXT BOOKS

Symmetry Symmetry is the notion that

an object of study may look the same from different points of view.

Figure 1 : Bilateral and rotational symmetry. From: [8].

What is symmetry? An object displays symmetry if

you can “do something to it” (rotate it, reflect it, invert it etc.) without changing its appearance

The process of doing the rotation, reflection etc. Is referred to as a SYMMETRY OPERATION if it does not

change the appearance of the molecule

Symmetry elements

Symmetry element is the line about which a rotation occurs or the plane through which a reflection takes place etc,

Symmetry elements are labeled based on : Rotation axis, Cn ( The n indicates the

angle through which the rotation takes place. (Angle = 360/n )

Mirror plane, σh, σv, σd ( h=horizontal, v=vertical or d=dihedral mirror indicate orientation of plane with respect to any rotation axes )

Symmetry elements

Inversion center, i

Improper rotation axis, Sn (This involves a combination of rotation and reflection. The n indicates the angle through which the rotation takes place. Angle = 360/n

Identity, E (E is always a symmetry element as doing nothing to an object, always the same

Mirror planes

σv includes the principle rotation axis

σvσv

v = vertical

σh is perpendicular to the principle rotation axis

Mirror planes

σh

h = horizontal

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

Mirror planes

Mirror planes

Symmetry elements of benzene ringThere is one horizontal reflection plane σh and two sets of vertical reflection plane (σd) and (σh )

Mirror planes

Borazine

B B

Rotate 120O

F1 F1

F2F3

F3F2

Operation rotation by 360/3 around C3 axis (element)

Rotations 360/n where n is an integer

BF3

B B

Rotate 120O

F1 F1

F2F3

F3F2

Operation rotation by 360/3 around C3 axis (element)

Rotations 360/n where n is an integer

BF3

A threefold rotation and the corresponding C3 in NH3There are two rotation assosiated with this axis, one through 120o [ C3

+ ] and the other -120o [ C3

- ]

C3+ C3

+

C3-

Rotation axesRotasi positif

Rotasi negatif

An H2O molecule may be rotated through any angle about the bisector of the HOH bond angle, but only a rotation of 180o, C2 leaves it apparently unchanged

Rotation axes

Rotation axes examples 1

The two fold rotational axis in cis-dinitrogen di florida

Molecule having n fold axis(a) Amonia (b) penta carbonil iron, (c) hexa carbonil iron

(a) (b) (c)

C2 C2 C2

C2

C2

C2C2

C2

C2

C2, C3, C6

Symmetry of hexagonal planar

Rotation axes examples 1

Rotation axes examples 2

Molecules containing five-fold rotational axes(a)Eclipsed ferrocene side and top view

(b) Staggered ferrocene

Inversion center

The invertion operation and invertion center of SF6

Note inversion (i) and C2 are not equivalent

Inversion operation

Two fold rotation

Inversion center examples

The center of symmetry of 1,2-dimethyl-1,2 –diphenyldiphospine disulfide

(a) (b) (c)

(a) and (b) do not have inversion center

Improper rotation axes An improper rotation involves a combination of

rotation and reflection The rotation and the reflection do not have

to besymmetry operations of the molecule in their

own right

A fourfold axis of improper rotation S4 in the CH4 molecule