GROUP THEORY ( SYMMETRY)

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Done by Shobana.N.S

SymmetryBY

SHOBANA.N.SQUEEN MARY’S COLLEGE

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Symmetry is present in nature and in human culture

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Understand what orbitals are used in bonding.

Predict optical activity of a molecule.

Predict IR and Raman spectral activity

Using symmetry in Chemistry

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A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged.

Symmetry operations

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There are 5 kinds of operations

1. Identity2. n-Fold Rotations3. Reflection4. Inversion5. Improper n-Fold Rotation

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IDENTITY

E (Identity Operation) = no change in the object.

Needed for mathematical completeness.

Every molecule has at least this symmetry operation.

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It is equal to rotation of the object 360/n degree about an axis .

The symmetry element is line.

Principle axis = axis with the largest possible n value.

Cnn Is equal to identity (E)

Cn Rotation (n-fold rotation)

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Symmetry element is plane.

Linear object has infinite σ.

σv- plane including principle axis

σh- plane perpendicular to principle axis.

σd- plane bisecting the dihedral angle between two σv plane.

REFLECTION OPERATION (σ)

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(x,y,z) --> (-x,-y,-z).

Symmetry element : point

Symmetry operation : inversion through a point.

i n is equal to identity (E)

INVERSION

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It is also known as ROTATION-REFLECTION AXIS.

Rotation followed by reflection.

Snn = E ( n= even number)

Sn2n = E ( n= odd number)

Improper axis of symmetry

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A symmetry element is a point of reference about which symmetry operations can take place

Symmetry elements can be 1. point 2. axis and 3. plane

Symmetry elements

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Symmetry element : point

Symmetry operation : inversion

Center of Symmetry

1,3-trans-disubstituted cyclobutane

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Symmetry element : plane

Symmetry operation : reflection

Plane of Symmetry

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Symmetry element : line

Symmetry operation : rotation

Axis of Symmetry

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element operation symbol

symmetry plane reflection through plane σ

inversion centerinversion: every point x,y,z translated to -x,-y,-z

i

proper axis rotation about axis by 360/n degrees Cn

improper axis

1. rotation by 360/n degrees2. reflection through plane perpendicular to rotation axis

Sn

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Point group

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The collection of symmetry elements present in a molecule forms a ‘group’, typically called a POINT GROUP.

The symmetry elements can combine only in a limited number of ways and these combinations are called the POINT GROUP.

WHY IS IT CALLED A “POINT GROUP”??

Because all the symmetry elements (points, lines, and planes) will intersect at a single point.

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Cyclic Symmetries

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Ci has 2 symmetry operations : E the identity operation

i point of inversion

Ci Group

C2H2F2Cl2

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It has two symmetry operations

E – identity operation

σ – reflection

Cs group

CH2BrCl

1- bromo, 2-chloro ethene

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Only one symmetry operation (E)

Molecules in this group have no symmetry

This means no symmetrical operations possible.

C1 axis of rotation

CHFBrClBromochlorofluoromethane

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C2 axis of rotationRotation of the molecule to 180 degree.

This point group contains only two symmetry operations:

E the identity operationC2 a twofold symmetry axis

Examples : water, chlorine trifluoride, hydrogen peroxide, formaldehyde

hydrazine

24(2R,3R)-tartaric acid D-mannitol

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C3 axis of rotationRotation of the molecule to 120 degree.

This point group contains only two symmetry operations:

E the identity operationC3 a three fold symmetry axis

Examples: ammonia, boron trifluoride, triphenyl phosphine

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9b H-Phenalene 3,7,11-trimethyl cyclo dodeca 1,5,9-triene

2,6,7-trimethyl-1-aza-bicyclo [2.2.2]octane

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This point group contains the following symmetry operations

E the identity operation Cn n-fold symmetry axis.

nσv n reflection operation

Cnv Point Group

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E the identity operation

C2 2-fold symmetry axis.

2σv reflection operation

C2v Point Group

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Examples:

1. Ozone2. Thiophene3. Furan4. Pyridine

Sulphur dioxide

Formaldehyde(Z)-1,2-DICHLORO ETHENE

30m-Xylene

Phenanthrene

O-dichloro benzenep-dichloro benzene

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Cyclohexane (boat) Water

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3σv reflection operation

C3v Point Group

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Examples:

Ammonia POCl3 Trichloro methane

Tert-butyl bromide

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C4 n-fold symmetry axis.

4σv n reflection operation

C4v Point Group

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EXAMPLES

Xenon oxytetrafluoride

Sulfur chloride pentafluorideBromine pentafluoride

Fluorine pentafluoride

Calix[4]arene

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C∞ ∞ -fold symmetry axis. ∞ σv n reflection operation

C∞v Point Group

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Linear Hetero nuclear Diatomic Molecule belongs to this category

These molecules don’t have centre of inversion.

Chloro ethyne

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Cn n-fold symmetry axis. σh n reflection operation

NOTE : If n is even ‘i’ is present.

Cnh Point Group

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C2 2-fold symmetry axis. σh reflection operation i inversion

C2h Point Group

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EXAMPLES

trans-1,2-dichloroethylene

Trans-1,3-butadiene

C2H2F2

N2F2

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1,4-dibromo-2,5-dichloro-benzene (E)-1,2-dichloro ethene

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C3 3-fold symmetry axis. σh reflection operation

S3 improper axis of symmetry

C3h Point Group

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Benzene-1,3,5-triol

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DIHEDRAL GROUP

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Cn n-fold symmetry axis.

nC2 2-fold symmetry axis. (perpendicular to Cn)

Dn Point Group

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2C2 2-fold symmetry axis.

D2 Point Group

twistane

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D3 Point Group This point group contains the following

symmetry operations




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Examples Ru(en)3

Perchlorotriphenylamine

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Tris(oxalato)iron111 Molecular knot

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nC2 2-fold symmetry axis.

nσd dihedral plane

Dnd Point Group

NOTE : ‘i’ is present when n is odd and S2n coincident to C2 axis

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2C2* 2-fold symmetry axis.

2σd dihedral plane

2S4 improper axis of symmetry

D2d Point Group

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allene (propa-1,2-diene) biphenyl

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COT1,3,5,7-

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3σd dihedral plane


D3d Point Group

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Cyclohexane chair form

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Ethane staggered form

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2C4 n-fold symmetry axis.


4σd dihedral plane



D4d Point Group

Mn2(CO)10

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4C5 n-fold symmetry axis.


5σd dihedral plane


i inversion

D5d Point Group

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nC2 2-fold symmetry axis.

σh horizontal plane

nσv vertical plane

Sn improper axis of symmetry

Dnh Point Group

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D2h Point Group This point group contains the following symmetry operations





2σv vertical plane


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M2F6

ETHENE

DIBORANE

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1,4-DICHLOROBENZENE

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[2,2] PARACYCLOPHANE

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2σv vertical plane


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cyclopropane

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4σv vertical plane


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Nickel tetracarbonyl

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[AlCl₄]− Xenon tetrafluoride

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5σv vertical plane


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6σv vertical plane


i inversion

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D∞h Point Group This point group contains the following symmetry operations


C ∞ 4-fold symmetry axis.

∞ C2 2-fold symmetry axis. σh horizontal plane ∞ σv vertical plane

S ∞ improper axis of symmetry

i inversion

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LINEAR HOMONUCLEAR DIATOMIC MOLECULE

POSSESS CENTER OF SYMMETRY

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Cubic Point Group

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Tetrahedral Point Group





6σd dihedral plane

3S4 improper axis of symmetry Total: 24 elements

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METHANE

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NEOPENTANE

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Octahedral Point Group





3σh dihedral plane


i inversion

C3 3-fold symmetry axis



6σd dihedral plane

TOTAL :48 elements

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Cr(CO)6[PtCl6]2-

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PF6- CUBANE

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SF6

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Icosohedral (Ih) This point group contains the following symmetry operations




15σh horizontal plane


i inversion

20C3 3-fold symmetry axis



12S10* improper axis of symmetry

TOTAL : 120 elements

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dodecahedran fullerenes

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THANK YOU !!!!

GROUP THEORY ( SYMMETRY)

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