Chem 516, Spring 2013 - University Of Illinoisxuv.scs.illinois.edu/516/lectures/chem516.01.pdf ·...

1

© K. S. Suslick, 2014

Chem 516, Spring 2013

1


Chem 516, Spring 2013

2

2


Assignment of Point Groups

1. You should be completely familiar with and comfortable in assigning point groups to any molecule. I assume this as background knowledge.

2. If you are NOT comfortable with point group assignments and flow charts:

Check out www.scs.uiuc.edu/chem312/and specifically:

https://www.scs.illinois.edu/chem312/lectures/chem312.point groups.pdf

https://www.scs.illinois.edu/chem312/pointgrouppractice.htm

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

3


Point Group Flow Chart: Shriver

High

Sym

?

LowSymmetry ?

D groups C groups Low Sym.groups

High Sym.groups

Lineargroups

4

3


Flow Charts

Lots and lots of different flow charts exist.

Doesn’t matter which one you use,you’d better get the same answer!

We’ll give you Shriver’s on exams.

We use “Schoenflies” notation to designatePoint Group names.

Crystallographers use “Hermann-Mauguin”.Mathematicians use “Orbifold”.

5


Care and Feeding of Character Tables0. Molecular Symmetry: operations (movements) of a molecule to an

new orientation that is indistinguishable from the starting position. No bonds are broken. The molecule is not distorted in any way.

1. All symmetry operations use a geometric entity(e.g., point, line, plane) called a symmetry element.

2. Five classes of symmetry operations and elements: • Proper axis or rotation. Symbol Cn for rotation by 360º/n with n>1• Plane of reflection (mirror plane). Symbol: • Inversion symmetry, inversion center. Symbol: i• Improper axis of rotation. Symbol: Sn -- reflection followed by

a rotation by 360º/n• Identity. Symbol E. Equivalent to rotation of 360º C1

3. The collection of symmetry operations for any molecule iscalled a “Point Groups”. Combinations of symmetry operations are also symmetry operations.

4. The properties of these point groups are representedin a simple tabular format: the “Character Tables”.

6

4


Point Groups

Point Group:The collection of ALL symmetry operations that can be carried

out on a molecule (or any other object) of a specific symmetry.These will have a common intersection, generally a point.

All Point Groups behave as any algebraic group: completeness, identity element, associate law, reciprocals

Order of a point group:Total number of symmetry operations in a point group

(from 1 to 120 to ∞ )

7


Combinations of Symmetry Ops

Consecutive application of two symmetry operationsis itself a symmetry operation:

A x B do B and then do A

8

5


General Multiplicative Relationships

9


Point Groups

Similar Elements:Within a given point group, those elements that can be

taken into one another by some other symmetry operation.

Class of Symmetry Operations:Within a given point group, all symmetry operations whose elements are “similar” (i.e., transformable into one another) form a CLASS within the group.

E and i always form solo classes.

The columns in character tables (next) represent classes of symmetry operations.

Flow charts to assign point groups deal with symmetric elements, not classes of operations.

10

6


Similar Elements & Operationsconsider square planar CuCl42-

Cl Cl

Cu

Cl Cl

Cu

C4 (perpendicular)

C4 x C2′ = C2′′

C4 x C2′ = C2′′

Therefore, C2′ and C2′′ are “similar” elements or “equivalent”

C2′

C2′′

11


Combinations of Symmetry OpsGroup Multiplication Tables:

Within a symmetry group, all of the symmetry operations’combinations can be summarized in a table so thateach row and column will have one and only one occurrenceof each symmetry operation.

v v 2

E

12

7


Group Multiplication and Subgroups

Subgroup = a subset of symmetry operations within a group that is itself a group.

e.g., C2 is a subgroup of C2v C2 T C2v

13


Group Multiplication and Subgroups

Groups whose multiplication tables are diagonally symmetric have operations that commute (i.e., Op1 x Op2 = Op2 x Op1 ) i.e., “Abelian” (most pt. gr. are NOT Abelian).Abelian groups are C1, C2, Ci, Cs, C2v, C2h, D2, and D2h

14

8


Commutation

In general terms, A x B does not necessary = B x Afor symmetry operations!

The following sym. ops. do commute:

15


Irreducible Representations (IRR or )

IRR are the basis set vectors for the symmetry operators of a specific point group;

are n-dimensional vectors where n = order of group

can be used to represent ANY set of functionsthat represent a molecule: e.g., atomic orbitals,molecular orbitals,bond vectors, atomic coordinates,vibrations, bends, rotations, …

can be singly degenerate (A or B), or depending on the pt. group

doubly degenerate (E)or triply degenerate (T)or quadruply degenerate (G)or quintuply degenerate (H)

16

9


Irreducible Representations (IRR)

Characters ( χ )show how IRRs behave under particular sym. op.

17


The Character Tables: C2v

O

H H

z

y

x

Every point group is completely represented by a character table.The characters represent all possible responses to symmetry elements

for any function of any molecule with that point symmetry.

(xz) (yz)

18

10


The Components of Character Tables: C2v

O

H H

z

y

x

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

22

2221

22

Name of the point group

Symmetryoperations

Usually characters ( ) are 1, -1, 0, 2, -2, 3, -3.1 means symmetric, -1 means anti-symmetric, 0 means moved off original location.(In some groups, the characters can be trig. values or even imaginary.)

Character(general symbol: )

Character under E:Dimension of each of irreducible representations: 1: one dimension; 2: two dimension; 3: three dimension

An Irreducible representation:collection of characters of symmetry operations on various functions – like those

of orbitals; general symbol: (gamma)

Axes; Rotations;Quadratic Functions

19


cis-Mo(CO)4Cl2

z

y

x

Character Tables

Order? h = 4

# of Classes? 4

20

11


cis-Mo(CO)4Cl2

z

y

x

Character Tables

Order? h = 4

# of Classes? 4

21

1 1 1 1


000112

111111

111111

'''

3

2

1

2333

vvvv CCEC

Character Tables; C3v

# of Classes of the symmetry operation: (# of distinct symmetry operation)

Order (h) of the symmetry operation:(total # of symmetry operation)

3

6

22

12


z

y

x

Class and order

Order? 1+8+6+6+3+1+6+8+3+6= 48 = h

# of Classes? 10

23


Character Table Properties

24

Irreducible representations are orthonormal:

= 1 if a=b= 0 if ab

= 1 if a=b= 0 if ab

= "Kronecker's delta function"

ab

i

biaiigh

)()( *1

13


Orthonormality of IRR

order = h =6

Is the IRR E normalized? i.e., does E ● E = h ?

25


Orthonormality of IRR

order = h =6

Are A2 and E orthogonal? i.e., does A2 ● E = 0 ?

26

14


yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22

s orbital behaves as x2 + y2 + z2

(because r = (x2 + y2 + z2))

O

H H

z

y

x

Axes, Rotations, Quadratic Functions

IdentityC2

reflections+1 (symmetric)

for each operation

27


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

z z

EC2

v (xz)’v (yz) No change

symmetric All 1’s in table

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22


px, py, pz behave as x, y, z

O

H H

z

y

x

pz

28

15


A py orbital has the same symmetry as an arrow pointing along the y-axis.

E’v (yz)

No change symmetric 1’s in table

py

C2

v (xz)

Opposite anti-symmetric -1’s in table

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22


Self test:Confirm px orbital

O

H H

z

y

x

29


y

x

y

x

E C2

v (xz)’v (yz)


y

x

dz2

dx2 - y2

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22

Self test:Confirm dxy, dxz and dyz

• dxy, dxz, dyz, as xy, xz, yz• dx2-y2 behaves as (x2 – y2)• dz2 behaves as 2z2 - (x2 + y2)


O

H H

z

y

x

y

x

E C2

v (xz)’v (yz)


30

16


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

EC2


z

z z

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22

v (xz)’v (yz) Opposite

anti-symmetric -1’s in table

Self test:Confirm Ry and Rx


O

H H

z

y

x

31


C2V E C2 v (xz) ’v (yz)

A1 1 1 1 1 z x2,y2,z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yz

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

D3h E 2 C3 3 C2 h 2 S3 3 v

A′1 1 1 1 1 1 1 x2 + y2, z2

A′2 1 1 -1 1 1 -1 Rz

E′ 2 -1 0 2 -1 0 (x,y) (x2 - y2, xy)

A′′1 1 1 1 -1 -1 -1

A′′2 1 1 -1 -1 -1 1 z

E′′ 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)


32

17


“A” = symmetric for rotation about the principle axis (1); one-dimensional.“B” = anti-symmetric for rotation about the principle axis (-1); one-dim.“E” for dimension = 2,“T” for dimension = 3 and “G” for dimension = 4

(different from identity operation E).

Subscript 1 = symmetric for rotation the secondary axis (or v).Subscript 2 = antisymmetric for rotation the secondary axis (or v).

′ = symmetric for reflection along .′′ = antisymmetric for reflection along .

g = symmetric for inversion with center i (gerade = even)u = antisymmetric for inversion with center i (ungerade = odd)

Mulliken Symbols

D3h E 2 C3 3 C2 h 2 S3 3 v

A′1 1 1 1 1 1 1 x2 + y2, z2

A′2 1 1 -1 1 1 -1 Rz

E′ 2 -1 0 2 -1 0 (x,y) (x2 - y2, xy)

A′′1 1 1 1 -1 -1 -1

A′′2 1 1 -1 -1 -1 1 z

E′′ 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)

33


Mulliken SymbolsWe use “Schoenflies” notation to designate Point Group names and “Mulliken” symbols for the Irreducible Representations. Crystallographers use “Hermann-Mauguin”. Mathematicians use “Orbifold”.

more on next page34

18


Mulliken SymbolsWe use “Schoenflies” notation to designate Point Group names and “Mulliken” symbols for the Irreducible Representations. Crystallographers use “Hermann-Mauguin”. Mathematicians use “Orbifold”.

35


Si

Cl

z

y

x

2-Dim IRR

doubly degenerate: px, py would have the same energy

36

19


z

y

xFBF

F

doubly degenerate: dxz, dyz would have the same energy

doubly degenerate: Px, Py same energy; also dxy, dx2-y2 same energy

37

2-Dim IRR


Si

Cl

z

y

x

triply degenerate

3-Dim IRR

38

20


Mulliken Symbols in Linear Pt. Groups

39

Chem 516, Spring 2013 - University Of Illinoisxuv.scs.illinois.edu/516/lectures/chem516.01.pdf ·...

Documents

Transcript of Chem 516, Spring 2013 - University Of Illinoisxuv.scs.illinois.edu/516/lectures/chem516.01.pdf ·...