Notes on Symmetry and Point Groups on Advanced Inorganic Chemistry | CHE 3340, Study notes of Inorganic Chemistry

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Symmetry and Point

Groups

Chapter 4

Why study symmetry?

Principles of Symmetry and Group Theory

help us in following areas

1.Classifying molecular structures

2.Classifying molecular orbitals & constructing

them

3.Predicting splitting of electronic levels in

electric fields of various symmetries

4.Constructing Hybrid orbitals

5.Classifying electronic states of molecules

6.Classifying normal modes of vibrations

7.Predicting allowed transitions in spectra

Symmetry Element

 (^) Point, line or plane about which symmetry operation is performed Five Basic Symmetry Elements for Molecules

1. n-fold rotation axis, Cn

2. Mirror plane, 

3. Inversion Center, i

4. Improper Rotation Axis, Sn

5. Identity Operator, E

1. n-fold rotation axis (n-fold axis of symmetry ), C

n

.

 Rotation of an object by

360 °/n (where n = 2, 3, 4,

… an integer) about that

axis ( Cn ) gives an

equivalent position of the

object.

 (^) Symmetry operation  (^) Rotation  (^) Symmetry element  (^) axis

 Ex. H

2 O has 1^ C 2 axis

2. Plane of symmetry (mirror plane),   (^) Reflection of all parts of a molecule through a plane produces an indistinguishable configuration.

 Symmetry operation = reflection

 Symmetry element = plane

 (^) Ex. BF 3

 1  = plane of molecule

 3   to plane of molecule

 (^) each contains C 3 axis & one of the 3^ C 2 axes. F B F F C 3 C 2 C 2 C 2

Types of Reflection Planes

Ex. of mirror planes

3. Inversion Center, (Center of Symmetry) i  (^) Movement of every part of object in straight line through this center & equal distance on other side of center produces indistinguishable configuration.  (^) Every atom, when moved in straight line through this center &equal distance on other side of this center, encounters similar atom.  (^) (x, y, z)  (x, y, z)

 Symmetry operation = inversion

 Symmetry element = point

Inversion vs. C

Rotation

4. Improper Rotation Axis, ( n - fold Rotation – Reflection Axis), S

n

 Rotation by an angle

about an axis

 (^) (where n = 2, 3, 4, … an integer)

 Followed by reflection

through a plane  to that axis

yields an indistinguishable

configuration.

 (^) Symmetry operation

 Rotation – reflection

 (^) Symmetry element

 Improper rotation axis

n

360

5. Identity Operator, E

 (^) Do Nothing element  (^) Leaves the molecule unchanged.  (^) Every molecule has at least this.  (^) Simplest element/operator of symmetry  (^) Any combination of symmetry operations that takes you back to original orientation.

Successive Operations

 NH

3 possesses a^ C 3 rotation axis

 Tells us that rotation by 120° about this axis

gives us a molecular orientation that is

indistinguishable from initial one.

 However, it takes 3 such operations to get

back to exactly the first orientation.

N

H

H

H

C 3

N

H

H

H

C 3

C 3

N

H

H

H

C 3

N

H

H

H

120 o rotation 120 o rotation 120 o rotation

C 3

2

C 3

3

= E

Some Operations are

Equivalent to others

S 1 =  S 2 = i

Group Exercises

 List as many symmetry operations for each

molecule as you can find

O C O H Si H H H O C S S F F F F F F

B

Cl Cl

Cl

F P

F

F

F

F

Pt Cl Cl Cl Cl