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Part 2.2: Symmetry and Point Groups 1
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Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Dec 14, 2015

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Cheyanne Kettell

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Page 1: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

1

Part 2.2: Symmetry and Point Groups

Page 2: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

2

• VSEPR• Define Symmetry• Symmetry elements• Symmetry operations• Point groups• Assigning point groups• Matrix math

Outline

Page 3: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Valence shell electron pair repulsion (VSEPR) theory

VSEPR

Page 4: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

4

• You know intuitively if something is symmetric but we require a precise method to describe how an object or molecule is symmetric.

• Definitions:– The quality of something that has two sides or halves that are the

same or very close in size, shape, and position. (Webster)

– When one shape becomes exactly like another if you flip, slide or turn it. (mathisfun.com)

– The correspondence in size, form and arrangement of parts on opposites sides of a plane, line or point. (dictionary.com)

• Symmetry Elements

• Symmetry Operations

Symmetry

Page 5: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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• Symmetry Elements– a point, line or plane with respect to which the symmetry

operation is performed– is a point of reference about which symmetry operations can

take place• Symmetry Operations

– a process that when complete leaves the object appearing as if nothing has changed (even through portions of the object may have moved).

Symmetry

Do Something!

If A, B and C are equivalent.

Page 6: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

6

Symmetry Element: Inversion Center

Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center.

For any part of an object there exists an identical part diametrically opposite this center an equal distance from it.

Page 7: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Symmetry Element: Axis

A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis.

An axis around which a rotation by 360°/n results in an object indistinguishable from the original.

Page 8: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

8

Symmetry Element: Plane

A molecule can have more than one plane of symmetry.

A plane of reflection through which an identical copy of the original is given.

Page 9: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

9

Symmetry ElementsPoint = Center of Symmetry

= 2 dimensions

Line = Axis of Symmetry

Plane = Plane of Symmetry

= 0 dimension

= 1 dimension

Volume of symmetry?

Page 10: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

10

• Symmetry elements– Axis– Mirror plane– Center of inversion

• Symmetry Operations– Identity (E)– Proper Rotation (Cn)– Reflection (s)– Inversion (i) – Improper Rotation (Sn)

Elements and Operations

Page 11: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

11

• Leaves the entire molecule unchanged.

• All objects have identity.

• Abbreviated E, from the German 'Einheit' meaning unity.

E

Identity (E)

~ doing nothing!

Page 12: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

12

Proper Rotation (Cn)• Rotation about an axis by 360°/n

• C2 rotates 180°, C6 rotates 60°.

• Each rotation brings is indistinguishable from the original.

C5 C2

Cnm : apply a Cn rotation m times

Page 13: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

13

Rotations (Cn)Cn

m : apply a Cn rotation m times

360°/6 = 60°

Page 14: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

14

More than 1 Cn

BH3

Highest n = principle axis (z-direction)

benzene

C6 , C3 , C2

6 x C⊥ 2

C3 3 x C⊥ 2

Page 15: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Reflection (s)Take each atom in the molecule and move it toward the reflection plane along a line perpendicular to that plane. Continue moving the atom through the plane to a point equidistant from the plane on the opposite side of the plane

Page 16: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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sv = through atom vertical planesd = between atoms/bonds dihedral planessh = perpendicular to the primary rotation axis

Reflection (s)Multiple Reflection Planes

benzene

3 x sv

3 x sd

1 x sh

Page 17: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

17

Reflection (s)Multiple Reflection Planes

sv

or

sxz

C2

1 2

sv’

or

syz

Page 18: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

18

Inversion (i)Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center.

Page 19: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Inversion (i)• Does not require an atom at the center of inversion.

C60 Benzene

Page 20: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

20

• Rotation about an axis by 360°/n followed by reflection through the plane perpendicular to the rotation axis.

• Labeled as an Sn axis.

Improper Rotation (Sn)

Snm : apply a Cn rotation then reflection m times

Page 21: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

21

Improper Rotation (Sn)

Snm : apply a Cn rotation then reflection m times

Allene B2H4

Page 22: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

22

Improper Rotation (Sn)Sn

m : apply a Cn rotation then reflection m times

n = even

Page 23: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

23

Improper Rotation (Sn)Sn

m : apply a Cn rotation then reflection m times

n = odd

Page 24: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

24

– Identity (E)

– Proper Rotation (Cn)

– Reflection (s)

– Inversion (i)

– Improper Rotation (Sn)

Symmetry Operations

Page 25: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

25

Side note: Crystals

Molecular Symmetry Crystallographic Symmetry

Center of Mass Unchanged

IdentityRotation

ReflectionInversion

Additional Symmetry Operations

Page 26: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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• Screw Symmetry

• Glide-reflection SymmetryThe combination of reflection and translation.

The combination of rotation and translation.

Crystallographic Symmetry Operations

Page 27: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

27

Groups of Symmetry Operations• Many molecules have 2 or more

symmetry operations/elements.

• Not all symmetry operations occur independently.

• Molecules can be classified and grouped based on their symmetry.

• Molecules with similar symmetry elements and operations belong to the same point group.

• Called point group because symmetry operations are related to a fixed point in the molecule (the center of mass).

Page 28: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Point Groups

Infinite number of molecules, finite number of point groups.

Page 29: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

29

• Nonaxial (no rotation)- C1, Cs, Ci

• Cyclic (rotational)-Cn, Cnv, Cnh, Sn

• Dihedral ( C⊥ 2)

- Dn, Dnd, Dnh

• Polyhedral- T, Th, Td, O, Oh, I, Ih

• Linear- C∞v, D ∞h

Types of Point Groups

http://symmetry.jacobs-university.de/

http://symmetry.jacobs-university.de/
Page 30: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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There are no symmetry elements except E

Nonaxial: C1

Many molecules are C1

CH3

NH2

H

Page 31: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

31

Symmetry elements: E and s

Nonaxial: Cs

Page 32: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

32

Symmetry elements: E and i

Nonaxial: Ci

Page 33: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

33

Cyclic: Cn

Symmetry elements: E and Cn

C3C2

C6

[6]-rotane

Page 34: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

34

Cyclic: Cnv

Symmetry elements: E, sv/sdand Cn

C2v

C2 principle axis

sv contains C2 axis

sd contains C2 axis

C6v

C6 principle axis

3 x sv contains C6 axis

3 x sd contains C6 axis

gallium(h6-C6Me6)

Page 35: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

35

Cyclic: Cnh

Symmetry elements: E, sh, Sn and Cn

C3h

C3 principle axis

sh to C⊥ 3 axis

S3 axis

C6h

C6 principle axis

sh to C⊥ 6 axis

S6 axis

hexakis(Me2N)benzene

Page 36: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Cyclic: Cnh

Symmetry elements: E, sh, Sn and Cn

Page 37: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

37

Cyclic: Sn

Symmetry elements: E and Sn

S4

S4, C2 principle axis

S6

S6, C3 principle axis

tetrabromopentane 18-crown-6

Only S4, S6, S8, etc.

Page 38: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

38

Dihedral Point Groups

Cn

Dn DndDnh

C2 perpendicular to Cn

C2

Cn

C2

Cn

C2

sd

sh

Page 39: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

39

Dihedral: Dn

D3

Symmetry elements: E, Cn and C⊥ 2

D2(SCH2CH2)3

D3

Page 40: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

40

Dihedral: Dn

Symmetry elements: E, Cn and C⊥ 2

Ni(en)3 en = H2NCH2CH2NH2

1 x C3

3 x C⊥ 2

Page 41: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

41

Dihedral: Dnd

Symmetry elements: E, Cn, C⊥ 2 and sd

D2d

Allene

Page 42: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

42

Dihedra: Dnd

Symmetry elements: E, Cn, C⊥ 2 and sd

D2d

Mg AlFe

Staggard Sandwich complexes (D5d)

M

View down the C5 axis

http://images.google.com/imgres?imgurl=www.grace-collection.com/images/tennis-ball.JPG&imgrefurl=http://www.grace-collection.com/sports-patterns.html&h=354&w=360&prev=/images?q=+tennis++ball&svnum=10&hl=en&sa=G
Page 43: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Dihedra: Dnh

Symmetry elements: E, Cn, C⊥ 2 and sh

Benzene

D6h

Page 44: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

44

Dihedra: Dnh

Symmetry elements: E, Cn, C⊥ 2 and sh

D2h

D3h

D5h D4h

D6h

Page 45: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

45

Dihedra: Dnh

Symmetry elements: E, Cn, C⊥ 2 and sh

XX

XX

Y

Y

trans-MY2X4

OsCl2(CO)4

PtCl4

Page 46: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Dihedral Point Groups

D5dD5h

Ferrocene (Fc)

D5

Page 47: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral Point GroupsMore than two high-order axes

Sometimes referred to as the Platonic solids.

Earth Air WaterFire

Page 48: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral Point GroupsMore than two high-order axes

I, O and T rarely foundNo mirror planes!

Point Groups• Tetrahedral

- T, Th, Td

• Octahedral- O, Oh

• Icosahedral- I, Ih

Page 49: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Tetrahedron

Point Groups

Has sdNo s

Has sh

Page 50: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: T

No s

Only Rotational Symmetry!

[Ca(THF)6]2+

Very rare!

Page 51: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Td

Rotational + sd

Most common tetrahedral point group

Page 52: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Th

Rotational + sh

Has sh

Page 53: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: OctahedronPoint Groups

Has sh

No s

Page 54: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Octahedron

No s

V6P8O24 -core Dodeka(ethylene)octamine

Very rare!

Page 55: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Octahedron

Page 56: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Octahedron

Page 57: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: IcosahedralPoint Groups

Has sh

No sI

snub dodecahedron

IhIcosahedron

E, 15C5, 12C52, 20C3, 15C2

E, 15C5, 12C52, 20C3, 15C2, i,

12S10, 12S103, 20S6, 15σ

I couldn’t find a molecule

Page 58: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral: Icosahedral

Has sh

IhIcosahedron

E, 15C5, 12C52, 20C3, 15C2, i,

12S10, 12S103, 20S6, 15σ

B12H12

Page 59: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Polyhedral Point GroupsMore than two high-order axes

I, O and T rarely foundNo mirror planes!

Point Groups• Tetrahedral

- T, Th, Td

• Octahedral- O, Oh

• Icosahedral- I, Ih

Page 60: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

60

High Symmetry

BxHx

Page 61: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Linear Point GroupsPoint Groups

Has sh

No sC∞v

D∞h

E, C∞, ∞σv

E, C∞, ∞σv, ∞C2, I

HCl, CO, NCNCS, HCN, HCCH

H2, N2, O2, F2, Cl2 CO2, BeH2, N3

Page 62: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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• Nonaxial (no rotation)- C1, Cs, Ci

• Cyclic (rotational)-Cn, Cnv, Cnh, Sn

• Dihedral ( C⊥ 2)

- Dn, Dnd, Dnh

• Polyhedral- T, Th, Td, O, Oh, I, Ih

• Linear- C∞v, D ∞h

Types of Point Groups

http://symmetry.jacobs-university.de/

http://symmetry.jacobs-university.de/
Page 63: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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• Memorize– Cs: E and S

– Oh: E, 3S4, 3C4, 6C2, 4S6, 4C3, 3σh, 6σd, i

• Flow chart

Assigning point groups

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Page 65: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Page 66: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Page 67: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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CO2

Page 68: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

68

∞ ∞

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69

Page 70: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Cotton’s “five-step” Procedure

Page 71: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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General Assignments

VSEPR Geometry Point group 

Linear D∞h

Bent or V-shape  C2v 

Trigonal planar  D3h 

Trigonal pyramidal  C3v 

Trigonal bipyramidal  D5h 

Tetrahedral  Td 

Sawhorse or see-saw  C2v 

T-shape  C2v

Octahedral  Oh 

Square pyramidal C4v

Square planar  D4h 

Pentagonal bipyramidal  D5h

For MXn

M is a central atomX is a ligand

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http://symmetry.otterbein.edu/gallery/index.html

Webpage

http://symmetry.otterbein.edu/gallery/index.html
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• 2-D vs 3-D• Global vs. Local• Dynamic Molecules

Pitfalls in Assigning Symmetry

Page 74: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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2-D vs 3-D Molecules

D3h C3

Page 75: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

75

Global vs. Local Symmetry

vs.

C2v Cs

hn+

2 + 2 cycloaddition

Page 76: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Global vs. Local Symmetry

D3 Oh

A1 = z2

E = x2-y2, xy, xz, yzEg = z2, x2-y2

T2g = xy, xz, yz

Page 77: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Global vs. Local Symmetryhemoglobin

C2

heme

Cs

porphyrin

D4h

Page 78: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Global vs. Local Symmetry

No translation

Quasicrystals

Dan Shechtman 2011 Nobel Prize in Chemistry

Crystals

Page 79: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Dynamic MoleculesAmine Inversion

10^10 per second

Symmetry Through the Eyes of a Chemist

Changes in:

Symmetry

Dipole moment

Chirality

Vibrational motion

Page 80: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Dynamic Molecules

Page 81: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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a) NH3

b) SF6

c) CCl4

d) H2C=CH2

e) H2C=CF2

CO2

OCS

Assign the point group

Page 82: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Stereographic Projections

two-fold rotational axis

three-fold rotational axis

Solid line = sv

Dashed line = sd

Objection Position• = in planex = below planeo = above plane

Dashed circle = no sd

Solid circle = sh

Page 83: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Stereographic Projections

S4 axis

S4 and C2

C3, 3C2, S6, and 3sd

Page 84: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Stereographic Projections

What about Ih?

Page 85: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Stereographic Projections

6C5, 10C3, 15C2, 15σ

Page 86: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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What Point Group?

D4h C6hD3

Page 87: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Symmetry Operation Math

Easy Alternative: Represent symmetry operations with matrices.

sxz

O (xo, yo, zo) O (xo, -yo, zo)

H1 (xh1, yh1, zh1) H1 (xh1, -yh1, zh1)

H2 (xh2, yh2, zh2) H2 (xh2, -yh2, zh2)

C5

C1 (xc1, yc1, zc1)

C60 (xc60, yc60, zc60)

.

.

.

ObjectObject

after sxz=•Matrix for sxz

Page 88: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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Matrix Math Outline

• Matrices• Trace/character• Matrix algebra• Inverse Matrix• Block Diagonal• Direct Sum• Direct Product• Symmetry operation matrix

Page 89: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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A matrix is a rectangular array of numbers, or symbols for numbers. These elements are put between square brackets.

Generally a matrix has m rows and n columns:

m Rows

n columns

Square matrix, m=n

Matrices

Page 90: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

90Method to simplify a matrix.

The character of a matrix is the sum of its diagonal elements.

square matrix

Trace/Character

Page 91: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

91

Unit Matrix/Identity: All elements in the diagonal are 1 and the rest are zero.

Column Matrix: used to represent a point or a vector.

The Identity Matrix (E)

Special Matricies

Page 92: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

92

Addition: Subtraction:

Matrix AlgebraFor two matricies with equal dimensions:

Multiply by a constant:

Page 93: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

93

Matrix MultiplicationIf two matrices are to be multiplied together they must be conformable; i.e., the number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix.

To get the first member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding elements of the first column of the second matrix and the results are added. To get the second member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding members of the second column of the second matrix and the results are added, and so on.

Page 94: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

94

Matrix MultiplicationIf two matrices are to be multiplied together they must be conformable; i.e., the number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix.

To get the first member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding elements of the first column of the second matrix and the results are added. To get the second member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding members of the second column of the second matrix and the results are added, and so on.

Page 95: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

95

Matrix Multiplication

Page 96: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

96

Non-square matrix-The number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix.

Matrix Multiplication

2 columns

2 Rows

columns of a= rows of b

3x2 times 2x3 = 3x3

Page 97: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

97

Non-square matrix-The number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix.

Matrix Multiplication

2 columns

2 Rows

columns of a= rows of b

3x2 times 2x3 = 3x3

Page 98: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

98

Matrix Multiplication: Vectors

Identity Matrix (E)

vector vector

Page 99: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

99

Inverse MatricesIf two square matrices A and B multiply together,

and A B = E

then B is said to the inverse of A (written A-1). B is also the inverse of A.

A B Eaka A-1

Rules that govern groups:4) For every element in the group there exists an inverse.

Page 100: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

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A block-diagonal matrix has nonzero values only in square blocks along the diagonal from the top left to the bottom right

Block-diagonal Matrix

Page 101: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

101

Block Diagonal Matrix

Row 1

and so on…

Full Matrix Multiplication Block Multiplication

Important in reducing matrix representations.

Page 102: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

102

Direct Sum

The direct sum is very different from ordinary matrix addition since it produces a matrix of higher dimensionality.

A direct sum of two matrices of orders n and m is performed by placing the matrices to be summed along the diagonal of a matrix of order n+m and filling in the remaining elements with zeroes

=⊕ ⊕

Γ(5)(g) = Γ(2)(g) ⊕ Γ(2)(g) ⊕ Γ(1)(g)

Page 103: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

103

Direct ProductThe direct product of two matrices (given the symbol ) is a ⊗special type of matrix product that generates a matrix of higher dimensionality.

2 x 2 2 x 2 = 4 x 4⊗

Page 104: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

104

Matrix Math in Group TheoryEasy Alternative: Represent symmetry operations with matrices.

ObjectObjectafter X=•

Matrix for X

X = Symmetry Operations– Identity (E)– Proper Rotation (Cn)– Reflection (s)– Inversion (i) – Improper Rotation (Sn)

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x,y or z stay the same = 1 on the diagonal

x,y or z shift negative = -1 on the diagonal

If a coordinate is transformed into another coordinate into the intersection position, respectively. These intersection positions will be off the matrix diagonal.

A symmetry operation can be represented by a matrix.

Symmetry Operation Matrix

Page 106: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

106

Reflection (s)

Page 107: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

107

Inversion (i)

Page 108: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

108

Rotation (Cn)

x and y are dependent

10000

z does not change

Trigonometry!

Page 109: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

109

Rotation (Cn)

sin a = O/Hcos a = A/H

x = Oy = Al = H

x1 = l cos ay1 = l sin a

x2 = l cos ( + a q)y2 = l sin ( + a q)

x1, y1 x2, y2?

Page 110: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

110

Rotation (Cn)General Matrix for Cn

C2 C3For C3 θ = 120°

cos θ = -1/2 and sin θ = √3/2 For C2 θ = 180°

cos θ = -1 and sin θ = 0

Page 111: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

111

Rotation (Cn)General Matrix for Cn

Page 112: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

112

Improper Rotation (Sn)

sh • Cn = Sn

• =

• =

Page 113: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

113

C2h Matrix MathSymmetry Operations: E, C2(z), i, sh

E

C2(z)

sh

i

Page 114: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

114

C3v Matrix MathSymmetry Operations: E, C3, C3

2, sv’,v”,v’’’

E

C31

sv’

C32

sv’’

sv’’’

sv’

sv’’’

sv’’

x

y

Page 115: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

115

Summary

Page 116: Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

116

• VSEPR• Define Symmetry• Symmetry elements• Symmetry operations• Point groups• Assigning point groups• Matrix math

Review


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