bonding notes3

7/28/2019 Bonding Notes3

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Chem 59-250Symmetry and Introduction to Group Theory

Symmetry is all around us and is a fundamental property of nature.


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Chem 59-250Symmetry and Introduction to Group Theory

The term symmetry is derived from the Greek word symmetria whichmeans measured together. An object is symmetric if one part (e.g. oneside) of it is the same* as all of the other parts. You know intuitively if something is symmetric but we require a precise method to describe howan object or molecule is symmetric.

Group theory is a very powerful mathematical tool that allows us torationalize and simplify many problems in Chemistry. A group consists of a set of symmetry elements (and associated symmetry operations) that

completely describe the symmetry of an object.

We will use some aspects of group theory to help us understand thebonding and spectroscopic features of molecules.


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Chem 59-250

No symmetry CHFClBr

Some symmetry CHFCl 2

More symmetry CH 2Cl2

More symmetry ? CHCl 3

We need to be able to specify the symmetry of molecules clearly.

F

BrCl

H

F

ClCl

H

Cl

H H

Cl Cl

Cl

H

Cl

What about ?

Point groups provide us with a way to indicate the symmetry unambiguously.


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Chem 59-250 Symmetry and Point Groups

Symmetry elements are geometric entities about which a symmetryoperation can be performed. In a point group, all symmetry elementsmust pass through the center of mass (the point). A symmetry operationis the action that produces an object identical to the initial object.

Point groups have symmetry about a single point at the center of mass of

the system.

The symmetry elements and related operations that we will find inmolecules are:

Identity, E

InversionCenter of symmetry, iReflectionPlane of symmetry, n-fold improper rotationImproper rotation axis, S n

n-fold rotationRotation axis, C n

OperationElement

The Identity operation does nothing to the object it is necessary formathematical completeness, as we will see later.


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Chem 59-250n-fold rotation - a rotation of 360/n about the C n axis (n = 1 to )

H(2)

O(1)

H(3) H(3)

O(1)

H(2)

In water there is a C 2 axis so we can perform a 2-fold (180) rotation to getthe identical arrangement of atoms.

H(2)

H(3)H(4)

N(1)

H(2)

H(3)

H(4)

N(1)

In ammonia there is a C 3 axis so we can perform 3-fold (120) rotations toget identical arrangement of atoms.

H(2)H(3)

H(4)

N(1)

120 120

180


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Chem 59-250

H(2)

H(3)H(4)

N(1)

H(2)

H(3)

H(4)

N(1)

Notes about rotation operations:- Rotations are considered positive in the counter-clockwise direction.- Each possible rotation operation is assigned using a superscript integer mof the form Cnm.- The rotation C nn is equivalent to the identity operation (nothing is moved).

H(2)H(3)

H(4)

N(1)

C31 C32

C33 = E

H(2)

H(3)H(4)

N(1)


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Chem 59-250

Cl(4)

Cl(3)

Ni(1)

Cl(2)

Cl(5)

C42 = C 21C41

C43

Cl(4)

Cl(2) Ni(1) Cl(3)

Cl(5)

Cl(4)

Cl(2)

Ni(1)

Cl(3)

Cl(5)

Cl(4)

Cl(3) Ni(1) Cl(2)

Cl(5)

Notes about rotation operations, Cnm:- If n/m is an integer, then that rotation operation is equivalent to an n/m -fold rotation.e.g. C 42 = C 21, C 62 = C 31, C 63 = C 21, etc. (identical to simplifying fractions)


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Chem 59-250Notes about rotation operations, Cnm:- Linear molecules have an infinite number of rotation axes C because anyrotation on the molecular axis will give the same arrangement.

O(2)C(1)

C(1)O(2)

O(3) C(1) O(2)

N(2)N(1)N(1)N(2)


9/15

Chem 59-250 The Principal axis in an object is the highest order rotation axis. Itis usually easy to identify the principle axis and this is typicallyassigned to the z-axis if we are using Cartesian coordinates.

Ethane, C 2H6 Benzene, C 6H6

The principal axis is the three-foldaxis containing the C-C bond.

The principal axis is the six-fold axisthrough the center of the ring.

The principal axis in a tetrahedron isa three-fold axis going through onevertex and the center of the object.


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Chem 59-250Reflection across a plane of symmetry, (mirror plane)

H(2)

O(1)

H(3) H(3)

O(1)

H(2)

v

H(2)

O(1)

H(3)

v

H(2)

O(1)

H(3)

These mirror planes arecalled vertical mirrorplanes, v, because theycontain the principal axis.

The reflection illustratedin the top diagram is

through a mirror planeperpendicular to theplane of the watermolecule. The plane

shown on the bottom is inthe same plane as thewater molecule.Handedness is changed by reflection!


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Chem 59-250

A horizontal mirror plane, h, isperpendicular to the principal axis.

This must be the xy-plane if the z-axis is the principal axis.

In benzene, the h is in the planeof the molecule it reflectseachatom onto itself.

Notes about reflection operations:- A reflection operation exchanges one half of the object with the reflection of the other half.- Reflection planes may be vertical, horizontal or dihedral (more on d later).- Two successive reflections are equivalent to the identity operation (nothing ismoved).

h

v v

Vertical and dihedral mirrorplanes of geometric shapes.

d d

h


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Chem 59-250Inversion and centers of symmetry, i (inversion centers)

In this operation, every part of the object is reflected through theinversion center, which must be at the center of mass of the object.

i

Br

Cl

F

F

Cl

Br

Br

Cl

F

F

Cl

Br

1

1

1 1

1

12

2

2 2

2

2

2 2 11

[x, y, z]i

[-x, -y, -z]

We will not consider the matrix approach to each of the symmetry operations in thiscourse but it is particularly helpful for understanding what the inversion operationdoes. The inversion operation takes a point or object at [x, y, z] to [-x, -y, -z].


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Chem 59-250n-fold improper rotation , S nm (associated with an improper rotation axisor a rotation-reflection axis) This operation involves a rotation of 360/n

followed by a reflection perpendicular to the axis. It is a single operationand is labeled in the same manner as properrotations.F1

F2

F3

F4 S 41

90

F4

F1

F2

F3

h

F1

F2

F3

F4

Note that: S 1 = , S 2 = i, and sometimes S 2n = C n (e.g. in box) this makesmore sense when you examine the matrices that describe the operations.

C

H1

H3

H2H4

C

H2

H4

H3H1

C

H1

H3

H2 H4

S 41

S 42

C21

h d f


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Chem 59-250

We can use a flow chart such as this

one to determine the point group of any object. The steps in this processare:

1. Determine the symmetry is special(e.g. octahedral).

2. Determine if there is a principalrotation axis.

3. Determine if there are rotation axesperpendicular to the principal axis.

4. Determine if there are mirror planes.

5. Assign point group.

Identifying point groups

Ch 59 250 Id if i i


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Chem 59-250 Identifying point groups

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