Chapter1_symmetry and Point Group

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CBC 212 - Inorganic Chemistry

Asst. Prof. Chen Hongyu and So Cheuk Wai

Textbook:“Inorganic Chemistry”, 4th Edition Gary L. Miessler and Donald A. Tarr, Pearson.

Suggested Readings:“Inorganic Chemistry”, (Textbook of CBC 212 last year), C. E. Housecroft, and A. G. Sharpe, 2nd Ed., Pearson Education. 2005.

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CBC 212 INORGANIC AND BIOINORGANIC CHEMISTRY

Author : Gary L. Miessler

Donald A. Tarr

Publisher : Pearson

ISBN : 978-0-13-615383-2

Inorganic Chemistry : International Edition 4th ed

Student Solution Manual Available!

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Asst. Prof. Hongyu Chen (陈虹宇)

B. Sc. 1998 University of Science and Technology of China (中国科技大学)Ph.D. 2004 Yale UniversityPost Doc. 2005-6 Cornell UniversityAsst. Prof. 2006- Nanyang Technological University

Contact:Office Phone 6316-8795; E-mail: [email protected]

About me

My office:SPMS-CBC-03-Room 02

Office hour: Please call me directly to arrange a time. Schedule a meeting would take 2-3 e-mails, which costs too much of my time.

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Tutorials

CBC212 tutorials are on week 3, 5, 7, 10 and 12

I will teach T1 and T4: Thur 10:30am-11:30pm and 1:30pm-2:30pm

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What is Inorganic Chemistry?

Inorganic Chemistry deals with everything that is not organic chemistry.Inorganic Chemistry deals with everything that is not organic chemistry.

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What is Inorganic Chemistry?

Inorganic Chemistry deals with everything that is not organic chemistry.Inorganic Chemistry deals with everything that is not organic chemistry.

Organometallic Chemistry: metal-carbon bonds

Coordination Chemistry

Chemistry deals with gases, metals, salts, metal ions, ect.

Bioinorganic Chemistry: Metal ions that are

important in biology

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What will we cover?

Chapter 1 ⎯ Simple Bonding Theory (brief overview)Chapter 2 ⎯ Symmetry and Group TheoryChapter 3 ⎯ Molecular Orbitals

Chapter 6 ⎯ Coordination Chemistry - Structures and IsomersChapter 7 ⎯ Coordination Chemistry - BondingChapter 8 ⎯ Coordination Chemistry – Electronic SpectraChapter 9 ⎯ Coordination Chemistry – Reactions & Mechanism

Chapter 10 ⎯ Bioinorganic Chemistry

Organometallic Chemistry CBC312

Solid State Chemistry CBC951

Next week

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What will we cover?

Chapter 1 ⎯ Symmetry and Group TheoryChapter 2 ⎯ Molecular Orbitals

Chapter 3 ⎯ Coordination Chemistry - Structures and IsomersChapter 4 ⎯ Coordination Chemistry - BondingChapter 5 ⎯ Coordination Chemistry – Electronic SpectraChapter 6 ⎯ Coordination Chemistry – Reactions & Mechanism

Chapter 7 ⎯ Bioinorganic Chemistry (maybe)

Organometallic Chemistry CBC312

Solid State Chemistry CBC951

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Take note.

Weight of final mark.

⎯⎯ Homework 10%

⎯⎯ Mid-term (~week 8 or 9) 30%

⎯⎯ Final exam 60%

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Summary

Chapter 1 ⎯ Symmetry and Group Theory• Symmetry Elements and Operation• Point Groups

Chapter 3 ⎯ Molecular Orbitals

11

Symmetry ⎯⎯ Nomenclature

Symmetry Elements

Geometric entities that are used to manipulate molecules so as to transform the molecules, such as points, lines (axis) and planes

Symmetry Operation

The operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration.

Symmetry Group• Point group ⎯ a group of symmetry elements that leave at least one

common point unchanged (a single object).• Space group ⎯ translation through space included.

Point Symmetry The symmetry possessed by a single object that describes the repetition of identical parts of the object.

Space Symmetry The symmetry describing the relationship of different objects to each other in space

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Axes of Symmetry

C2 axis is the symmetry element (no movement is involved)Symmetry operation is the to rotate the molecule according to C2 axis.

BF3

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Axes of Symmetry

Point symmetry

Space symmetry

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Symmetry ⎯ identity operator

Identity E

This operator leaves the object unchanged.

All molecules possess this symmetry element.

Identity operation.

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Axes of Symmetry

n-fold axis of symmetry Cn

After rotation of 360°/n, the object returns to its original configuration.

Principal axis, or highest order rotation axis, is defined as the Cnaxis having the largest n.

Rotation operation.

BF3

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Axes of Symmetry

× No axis of symmetry.

C2

C2

C2

C2

C3

C3

C3C3

C2

C2

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Axes of Symmetry

C3

C3

Chiral molecule

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Axes of Symmetry

C4

Octahedral center

C2

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Axes of Symmetry

C6

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Plane of Symmetry, XeF4

Plane of symmetry σv, σh, or σd

After reflection through a plane, the object returns to its original configuration.

σh means horizontal plane(relative to principle axis)

σv means vertical plane(relative to principle axis)

σd means dihedral plane

reflection operation.

The angle between two intersecting planes is known as the dihedral angle.

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Plane of Symmetry, XeF4

σv means vertical planeContains the principle axis,usually along bonds

σd means dihedral planeContains the principle axis,and bisects the angle between two adjacent C2 axes.Usually bisect bonds

The angle between two intersecting planes is known as the dihedral angle.

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Symmetry - Examples

C6H6

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Symmetry - Examples

σv

σv’

σv or σh?

σv or σh?

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Center of Symmetry

Center of symmetry i

After reflection through the center of the object, the object returns to its original configuration.

Inversion operation.

indistinguishable and superimposable

indistinguishable but NOT superimposable

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CO2 SF6 C6H6

H2S/H2O cis-N2F2 CH4/SiH4

√

×

Center of Symmetry

Center of symmetry i Inversion operation.

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Symmetry

Successive symmetry operation:

C32 = C3 × C3; C3

3 = C3 × C3 × C3

C32 is a symmetry operation, but not a symmetry element.

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Improper Axis of Symmetry

Improper axis of symmetry Sn

Successive symmetry operation: After rotation by 360°/n, followed by reflection through a plane that is perpendicular to the rotational axis, the object returns to its original configuration ( Sn ).After rotation by 360°/n, followed by inversion through the center of symmetry, the object returns to its original configuration ( n ).

Rotation-reflection operation(Improper rotation)

CF4/CH4/SiH4

S4 = C4 ×σh: Individually Cn and σh are not symmetry operation for these molecules.

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Symmetry - Examples

• S1 is equivalent to σ• S2 is equivalent to i

CF4/CH4/SiH4 Staggered form of C2H6

Ferrocene

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Summary of Point Symmetry

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How the related symmetry operation worksSymmetry element & its symbol

(x,y,z) (xcosα-ysinα,xsinα+ycosα,-z); α=2mπ/n Improper rotational axis, Sn (z axis)(x,y,z) (xcosα-ysinα,xsinα+ycosα,z); α=2mπ/n Proper rotational axis, Cn (z axis)(x,y,z) (x,y,-z)Mirror plane, σ (case of σxy)(x,y,z) (-x,-y,-z)Center of symmetry, i(x,y,z) (x,y,z)Identity, E

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Schoenflies Notation vs Hermann Mauguin Notation

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Symmetry - Examples

A chiral molecule does not have an improper rotation axis, Sn


Mirror

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[Ru(NH2CH2CH2NH2)3]2+

Symmetry - Examples



34


Symmetry - Examples



35

Symmetry - Examples



1,1'-bi-2-naphthol (BINOL)

trans-cyclooctene

cis-cyclooctene

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Symmetry - Examples



37

Symmetry - Examples



An exception:

tetrafluoro-spiropentane

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Summary

Chapter 1 ⎯ Symmetry and Group Theory• Symmetry Elements and Operation• Point Groups


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Point Groups

Symmetry Group ⎯ all the symmetry elements that describe a molecule.• Point group ⎯ a group of symmetry elements that leave at least one

common point unchanged (a single object).• Space group ⎯ translation through space included.

D6h

C6h

Six C2 axes perpendicular to C6

D3hC3v

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Point Groups

Method 1: list all symmetry elements and match with list characteristic of point group.

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Point Groups

Method 2: Use a flow chart.

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Point Groups

There are infinitely many 3D point groups, however, in crystallography the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. There are only 32 crystallographic point groups. (see Wikipedia: Crystallographic restriction theorem)

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Point Groups

(a) The groups C1, Ci, Cs• C1 point group : E (identity) only• Ci point group : E and i• Cs point group : E and mirror plane σ

C1 point groupCi point group

Cs point group

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Point Groups

(b) The groups Cn, Cnv, and Cnh• C2 point group : E, C2 Ex) H2O2• C2v point group : E, C2, 2σv Ex) H2O• C3v point group : E, C3, 3σv Ex) NH3• C∞v point group : linear molecule, no inversion center, Ex) CO, HCN• C2h point group : E, C2, σh Ex) trans-CHCl=CHCl• C3h point group : E, C3, σh Ex) B(OH)3

C2 point group C2h point group C3h point group

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Point Groups

(c) The groups Dn, Dnh, and Dnd• Dn point group: n-fold principal axis Cn and n × C2 axis present in the group• Dnh point group: group element of Dn + σh

D6 point group D6h point group

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Point Groups

(c) The groups Dn, Dnh, and Dnd• Dn point group: n-fold principal axis Cn and n × C2 axis present in the group• Dnh point group: group element of Dn + σh• Dnd point group: Dn elements + n × σd• D∞h point group: linear molecule + inversion center, Ex) O2, C2H2

Allene, H2C=C=CH2 (D2d) Ethane, C2H6 (D3d)

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Viewing of C2 axis

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Viewing of C2 axis

x x

x

x

x

o

o

o

o

o

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Viewing of C2 axis

o

o o

x x

x

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Viewing of C2 axis

o

o o

x x

x

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Viewing of C2 axis

o

o

xx

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Point Groups

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Point Groups

(d) Highly symmetric groups Td and Oh

Td point group Oh point group

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Point Groups


Td point groupNOT Td point group

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Point Groups


56

Point Groups


NOT Oh point group

Oh point group

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Point Groups

(d) Highly symmetric groups Ih

Icosahedron

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Point Groups

(d) Highly symmetric groups Ih

Truncated IcosahedronIcosahedron

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Point Groups

60

Point Groups

61

Point Groups

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Point Groups

D3h point groupD3h point group

D2h point group

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Point Groups

D5d point group

D3 point groupFe(C5H5)2, staggered


D5h point group

Os(C5H5)2, eclipsed

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Common Molecular Symmetries

TdC3v

Oh D4hC3v

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D∞h C2v D3h

C2vC3vD4h


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C2v


C4v

D3h

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Coordination Chemistry ― Bonding

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Summary

Chapter 1 ⎯ Symmetry and Group Theory• Symmetry Elements and Operation• Point Groups• Character Table


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Character Table

A set of symmetry operations form a mathematical group(group theory) when:

• the result of combination of any two operations is also a memberof the group. This is called the multiplication rule: AB = C • the multiplication rule is also associative: A(BC) = AB(C) • the group contains a identity element such that AE = EA • the inverse element of an entity is also a member AA-1 = E

The order of a group is the number of symmetry operations for that group.

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Character Table

• We have learned symmetry elements and operations• We know how to assign an object to a point group

So, what can we do about this knowledge?• Mathematical treatment leads to character tables which summarize the symmetry characteristics of point groups. (read page 92~102)• The character tables can be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and toconstruct molecular orbitals.

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NR- Character Table

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

100010001

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

'''

zyx

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

zyx

Original coordinates

Transformation matrix

Results of transformation.

=

Each symmetry operation may be expressed as a transformation matrix:[New coordinates] = [transformation matrix][old coordinates]

×reflection in x = 0 plane.

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

100010001

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100010001

E

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

−

100010001

C2

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

100010001

σv (xz): σv’ (yz):

The matrices record how the x, y, z coordinates are modified as a result of an operation. For example, the C2v point group consists of the following operations:• E: do nothing. Unchanged.• C2: rotate 180 degrees about the z axis • σ(xz): y becomes –y• σ’(yz): x becomes -x The character is defined as the trace of a square matrix

3 -1 1 1A reducible representation:

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Character Table

Consider a function: f(x) = x2

σv’ (f(x)) = σv(x2) = (-x)2 = x2 = f(x)

f(x) is an eigenfunction of this reflection operator with an eigenvalue of +1. This is called a symmetric eigenfunction.

s orbital

Reflection

Get the same orbital back, multiplied by +1, The s orbital is an eigenfunction of the reflection, symmetric with respect to the reflection.

z

73

Character Table

Similarly for a function: f(x) = x3

σv’ (f(x)) = -1 * f(x)

f(x) is an eigenfunction of this reflection operator with an eigenvalue of -1. This is called a antisymmetric eigenfunction.

p orbital

Reflection

Get the same orbital back, multiplied by -1. The p orbital is an eigenfunction of the reflection, antisymmetric with respect to the reflection. z

74

Character Table

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

vvv

,1111,1111

1111,,1111

)(')(

2

1

22

2221

22

−−−−−−

σσCharacter Table of C2v

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Character Table

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

vvv

,1111,1111

1111,,1111

)(')(

2

1

22

2221

22

−−−−−−

σσCharacter Table of C2v

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Character Table

000112111111

111111'''

3

2

1

2333

−−Γ−−−Γ

Γvvvv CCEC σσσ

6 Symmetry operations3 classes: E, 2C3, 3σv

Irreducible representationsRepresentations are subsets of the complete point group – they indicate the characters of symmetry operations on various functions – like those of orbitals.

Character under E:Dimension of each of irreducible representations

Name of the point group

Character

Each point group has a complete set of possible symmetry operations that are conveniently listed as a matrix known as a Character Table.

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Character Table

000112111111

111111'''

3

2

1

2333

−−Γ−−−Γ

Γvvvv CCEC σσσ

Notes about symmetry labels (Mulliken Symbols) and characters:• “A” means symmetric with regard to rotation about the principle axis.• “B” means anti-symmetric with regard to rotation about the principle axis.• “E” for dimension = 2 and “T” for dimension = 3. • Subscript numbers are used to differentiate symmetry labels, if necessary.• “1” indicates that the operation leaves the function unchanged: it is called “symmetric”. • “-1” indicates that the operation changes the function’s sign: it is called “antisymmetric”.

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Character Table

Order = 1 + 2 + 3 = 6

• The total number of symmetry operation in the group is called order.• The number of irreducible representations equals the number of classes.• The sum of the square of the dimensions of each irreducible representation equals the order of the group. 12 + 12 + 22 = 6• Irreducible representations are orthogonal to each other.

For A2 and E 1(1)(2) + 2(1)(-1) + 3(-1)(0) = 0• A totally symmetric representation exist in the group, with characters of 1 for all operation. in this case A1

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Character Table

• dxy, dxz, dyz, as xy, xz, yz• dx2-y2 behaves as (x2 – y2)• dz2 behaves as 2z2 - (x2 + y2)• px, py, pz behave as x, y, z• s behaves as x2 + y2 + z2

(because r = √(x2 + y2 + z2)

80

Character Table

A set of symmetry operations form a mathematical group(group theory) when:

• the result of combination of any two operations is also a memberof the group. This is called the multiplication rule: AB = C • the multiplication rule is also associative: A(BC) = AB(C) • the group contains a identity element such that AE = EA • the inverse element of an entity is also a member AA-1 = E

The order of a group is the number of symmetry operations for that group.

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NR- Character Table

• AB = C • A(BC) = AB(C) • AE = EA • AA-1 = E

)()( '2 yzxzC vv σσ =×

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

222 CECCE =×=×

ECC =× 22 Evv =×σσ1 0 0

0 -1 0 0 0 1

1 0 0

0 -1 0 0 0 1

=1 0 0

0 1 0 0 0 1

For example in C2v group:

82

Character Table

z

y

x

z

y

x

A pz orbital has the same symmetry as an arrow pointing along the z-axis.

EC2σv (xz)σ’v (yz) No change

∴ symmetric∴ 1’s in table

≡

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

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Character Table

z

y

x

Rz is the vector of rotation around the z-axis (right-hand rule).

EC2

No change∴ symmetric∴ 1’s in table

z

y

x

z

y

xσv (xz)σ’v (yz)

Opposite∴ anti-symmetric∴ -1’s in table

z

y

x

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

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Character Table

z

y

x

A px orbital has the same symmetry as an arrow pointing along the x-axis.

Eσv (xz)


z

y

x≡z

y

xC2σ’v (yz)

Opposite∴ anti-symmetric∴ -1’s in table

z

y

x

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

85

Character Table

y

x

d orbital functions can also be treated in a similar way

EC2σv (xz)σ’v (yz)


y

x

The z axis is pointing out of the screen. So these are representations of the view of the dz2 orbital and dx2-y2 orbital down the z-axis.

y

x

y

x

dz2 dx

2 - y

2

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

86

Character Table

yzy, Rx1-1-11B2

xzxy

x2,y2,z2

x, Ry

Rz

z

-11-11B1

-1-111A2

1111A1

σ’v (yz)σv (xz)C2EC2V

Note that the representation of orbital depends on the point group (which must be correctly identified).

(xz, yz)(Rx, Ry)01-20-12E’’z1-1-1-111A’’2

-1-111

2 S3

10-11

3 C2

-1-111A’’1

(x2 - y2, xy)

x2 + y2, z2

(x,y)Rz

02-12E’-1111A’2

1111A’1

3 σvσh2 C3ED3h

87

Summary


1. σ-bonding in a C2v group2. σ-bonding in a Td group3. Molecular Vibration in C2v4. Selected vibrational modes in C2v


88

Cl

H H

Cl

Application of Character Table

σ bonding in CH2Cl2

2204Γσ


Γσ (reducible representation of the sigma bonding) is the characters of each C2v symmetry operation acting on the 4 σ bonds. A bond contributes 0 to the character if the symmetry operation moves it and 1 if it doesn’t. The sums of the 1’s and 0’s for each operation give the representation of the bonding.

The E operation leaves all 4 bonds unchanged, and the character is 4 (1+1+1+1).

The C2 operation moves all four bonds so the character is 0.

Each σv operation leaves two bonds unchanged and moves the other two bonds so the character is 2 (1+1+ 0 + 0).

Overall, the reducible representation is thus:

89

2204Γσ


This representation is not one of those in the point group, but can be reduced (factored) into some combination of those that are. This can be done either by inspection or by the use of a relationship illustrated on the next slide.

yzy, Rx1-1-11B2

xzxy

x2,y2,z2

x, Ry

Rz

z

-11-11B1

-1-111A2

1111A1


Because the character under E is 4, there must be a total of 4 symmetry representations (sometimes called basis functions) that combine to make Γσ. Since the character under C2 is 0, there must be two of A symmetry and two of B symmetry. The irreducible representation is (2A1 + B1 + B2), which corresponds to: s, pz, px, and py orbitals – the familiar sp3.

Note: highly symmetric spherical s orbitals always are unchanged- characters all +1.


90


yzy, Rx1-1-11B2

xzxy

x2,y2,z2

x, Ry

Rz

z

-11-11B1

-1-111A2

1111A1


The formula to figure out the number of symmetry representations of a given type is:

( ) ( )[ ]n 1order

# of operations in class (character of RR) character of XX = × ×∑

( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )[ ]n 14A1

= + + +1 4 1 1 0 1 1 2 1 1 2 1 ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )[ ]n 14B1

= + − + + −1 4 1 1 0 1 1 2 1 1 2 1

( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )[ ]n 14B2

= + − + − +1 4 1 1 0 1 1 2 1 1 2 1( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )[ ]n 14A2

= + + − + −1 4 1 1 0 1 1 2 1 1 2 1

Which gives: 2 A1’s, 0 A2’s, 1 B1 and 1 B2.

reducible representation 2204Γσ


91

Summary




92


x

y

z

C3 1→33 →44→12=2

C2 2↔31↔4

also S4

σd 2↔31,4 move

σ bonding for a tetrahedral molecule (CH4, ClO4-, etc..)

Character:

1

0

2

93


σ bonding for a tetrahedral molecule (CH4, ClO4-, etc.).

The point group is Td so we must use the appropriate character table to find the reducible representation of the sigma bonding, Γσ first, then we can go the representation of the π bonding, Γπ.

The E operation leaves everything where it is so all four bonds stay in the same place and the character is 4.

Each C3 operation moves three bonds leaves one where it was so the character is 1.

The C2 and S4 operations move all four bonds so their characters are 0.

Each σd operation leaves two bonds where they were and moves two bonds so the character is 2.

06 S4

03 C2

214Γσ

6 σd8 C3ETd

94


(xy, xz, yz)(x, y, z)1-1-103T2

10-11

6 S4

-1211

3 C2

(Rx, Ry, Rz)-103T1

(2z2 - x2 - y2, x2 - y2)

x2 + y2 + z2

0-12E-111A2

111A1

6 σd8 C3ETd

06 S4

03 C2

214Γσ

6 σd8 C3ETd

Work out by yourself the irreducible representation for the σ bonding.

So, the obvious answer is A1 + T2.• VB theory of CH4 uses 4 sp3 hybrid C orbitals (although sp3 and sd3 is allowed!)• MO theory constructs orbitals from the C2s and C2p and proper combinationsof the orbitals from the 4 groups bonded to the central atom carbon

95


Carbon atomic valence orbitals

BONDING (in phase) Combinations of ligand (H) orbitals

96


MO with high C2s contribution

3 degenerate MO’s withhigh C2p contributiomn

4 H1s orbitals

8 atomic orbitals → 8 MO’s

MO Diagram for CH4

97


The photoelectron spectrum of CH4 support ionizations from two strongly bondingMO’s having different C2s and C2p characters, not from a single set of sp3 hybridorbitals. Two MO’s delocalized over all 5 nuclei result in 4 equivalent C-H bonds.

98

Summary




99

Introduction

Magnetic component (EPR or NMR)

Electric component (IR or Raman)

Selection rules: In order for matter to absorb photon:1.) Energy must match.2.) (For electric interactions) The energy transition must be accompanied by a change in the electrical center of the molecule so that electrical work can be done.

• Infrared absorption: there must be a change of dipole moment in the molecule as it vibrates.• Raman absorption: there must be a change in polarizability during the vibration.

Allowed transitions ↔ Forbidden transitions

100

Introduction

See: http://en.wikipedia.org/wiki/Infrared_spectroscopy

How do you describe the movement of a molecule: translation, rotation and vibration?

101


3339Example3 (H2O)

3N-6333NN (non-linear)

4239Example3 (HCN)

3N-5233NN (linear)

Vibrational modes

Rotationalmodes

Translationalmodes

degrees offreedom

# of atoms

In a water molecule, each atom can move in all three directions, leading to a total of nine transformations. We view the movement (translation, rotation and vibration) of the water molecule as the combination of those vectors.

We are interested in the symmetry of the transformations.

102


yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

How many vibrational modes belong to each irreducible representation?

You need the molecular geometry (point group) and the character table

Use the translation vectors of the atoms as the basis of a reducible representation.

Since you only need the trace recognize that only the vectors that are either unchanged or have become the negatives of themselves by a symmetry operation contribute to the character.

103


yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

9 -1 3 1Γ =Total of vectors

A reducible representation

104


Γ = 3A1 + A2 + 3B1 + 2B2

yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

Selection rules: In order for matter to absorb photon:1.) Energy must match.2.) (For electric interactions) The energy transition must be accompanied by a change in the electrical center of the molecule so that electrical work can be done.

• Infrared absorption: there must be a change of dipole moment in the molecule as it vibrates a normal mode must belong to one of the irreducible representations corresponding to the x, y, and z vectors.• Raman absorption: there must be a change in polarizability during the vibration.

a normal mode must belong to the same irreducible representations of the quadratic functions of x, y, and z vectors (x2, y2, z2, xy, yz, xz, x2-y2, etc.), or one of the rotational functions Rx, Ry, Rz

105


yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

Symmetric stretch

Antisymmetric stretch

Symmetric bendIR active

Raman active

106

Summary




107


ML C

CLO

OC2

E

1

2

ML C

CLO

O

1

2

ML C

CLO

O1

2

ML C

CLO

O

1

2

ML C

CLO

O1

2

C2 σv(xz) σv(yz)

ν(CO)

Find: # vectors remaining unchanged after operation.

2 0 2 0Γ

Example: C-O stretch in C2v (cis) complex.

and C-O stretch in D2h (trans) complex.

108


Example: C-O stretch in C2v (cis) complex.

and C-O stretch in D2h (trans) complex.

Isolated vibration Coupled vibrations

109


yzRyBxzRxBxyRA

zyxzAyzxzCEC

x

y

z

vvv

,1111,1111

1111,,1111

)(')(

2

1

2

2221

22

−−−−−−

σσ

2 0 2 0Γ

A1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1

A2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x-1) = 0

B1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1

B2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x1) = 0

is both IR and Raman active

110


2 0 20Γ 0 2 02

111

Summary




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Chapter1_symmetry and Point Group

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