chem516.02.origins of character tablesxuv.scs.illinois.edu › 516 › lectures ›...

1

© K. S. Suslick, 2013

Origin of Character Tables

• Symmetry operations obey the rules of group theory.• Great! ? So we can use the mathematics of group theory:

completeness, associative law, reciprocals

• Any symmetry operation can be represented by a matrix operating on some basis set (e.g., axes) describing the molecule.

Symmetry operations combine according the associative law of multiplication:

(RS)T = R(ST)

If R and S are symmetry operations of an object,then RS is also a symmetry operation.

The inverse of each symmetry operation is also a symmetry operation:

R-1R= RR-1 =E


Group Multiplication Tables

These show the fundamental rules of Group Theory(completeness, reciprocals, associative law)

i.e., they show the outcome of the different products in a group

The number of elements = Group order (h)

here h = 4

C2v E C2 v v’

E E C2 v v’C2 C2 E v’ v

v (xz) v v’ E C2

v’(yz) v’ v C2 E

2


Matrix Representations

For any symmetry operation, we are just taking (x, y, z) to some new (x’, y’, z’)(well, xyz for each atom of the molecule)

(x,y,z) is a vector, we’ll call it r .

The “taking” is called a transformation: for us, a sym op.

That transformation can be represented as a matrix.

The matrices of the sym ops must/will fulfillthe multiplication table of that point group.

So, what do these matrices look like?



3


Matrix Representations of Sym Ops

Identity Element, E

z

y

x

z

y

x

100

010

001

E r = r

z

y

x

z

y

x

100

010

001

i r = -r

Inversion, i



σh (σxy i.e., x,y fixed, z to -z)

z

y

x

z

y

x

100

010

001

Mirror Plane, h

More generally:

4



C2 rotation (z-axis)

x -> -x

y -> -y

z -> z

Cn In general for rotations of 2/n

z

y

x

z

y

x

100

010

001

2

2

1

1

cossin

sincos

y

x

y

x



Sn improper rotation ( Cn followed by σh )

5



Matrix transformations will match the multiplication tables of any group.



• Matrix representations of symmetry operations can often be reduced into block matrices. Similarity transformations may help to reduce representations further. The goal is to find the irreducible representation, the only representation that can not be reduced further.

• The same ”type” of operations (rotations, reflections, etc.) belong to the same class. Formally R and R’ belong to the same class if there is a symmetry operation S such that R’=S-1RS. Symmetry operations of the same class will always have the same character.

6


Characters of Representation...

Even if the matrices have different basis (e.g., axes),ICBST, the trace of the matrices are the same.

Character (trace) of the matrices

The symmetry operations gives various classes that can bedistinguished by their characters.

In addition, there are only a certain number of distinct waysin which a function can behave when subjectedto the symmetry operations of a particular point group.

These possibilities are called the irreducible representations:the characters for each possible irr rep under each sym opmakes up the character table for each point group.

)()( RDRi

ii 3

100

010

001

tr


Characters / Traces

4.) tr(A•B•C) = tr(C•A•B) = tr(B•C•A) cyclic permutation

7


Characters and Classes

Definition:

R and R’ belongs to the same class if

there is some sym op (S) such that

R’ = S-1 R S

This implies that

tr(R’) = tr(R)

(R’) = (R)

)()( RDRi

ii

3

100

010

001

tr


00

00

000

000

00

00

00''0

000''

00

00

00'0

000'

B

A

B

A

B

A

C’ C’’ C

Block Matrices

A’A’’ = AB’B’’ = BC’C’’ = C

Block matrices are good!

icbst:

That is, if two matrices are block diagonal, the corresponding blocks of identical order can be multiplied individually.

8


• If a matrix representing a symmetry operation is transformed into block diagonal form then each little block is also a representation of the operation since they obey the same multiplication laws.

• When a matrix can not be reduced further we have reached the irreducible representation. The number of reducible representations of symmetry operations is infinite but there is a small finite number of irreducible representations.

• The number of irreducible representations is always equal to the number of classes of the symmetry point group.

Block Matrices


• As stated before, all representations of a certain symmetry operation have the same character (trace of their matrix representation). So we will work with them rather than the matrices themselves.

• The characters of different irreducible representations of point groups are found in character tables. Character tables can easily be found in textbooks.

• Reducing big matrices to block diagonal form is always possible but not easy. Fortunately we do not have to do this ourselves.

Block Matrices

9


An Example…

C3v in the basis {SN, S1, S2, S3} (S=sphere)

i.e., a NH3 molecule

E, C3+, C3

- 3 v are the sym ops. 1 23

N

)3()1()4( DDD D(n) = blocks of n dim

0

0

0

0001SN

S1

S2

S3

SN S1 S2 S3

10

1

1

01

E =4

0010

1000

0100

0001

C3 =1

0010

1000

0010

0001

v =2

0100

0010

1000

0001

C32 =1

0010

0100

1000

0001

v =2 v =2

1000

0010

0100

0001

matrix representations:


Irreducible representations

In principle, we can take any red. rep. and reduce it to a block diagonal through similarity transformations.

In this example, we already have two blocks:

0

0

0

0001

0010

0100

1000

0001

10


An Example…

4-D basis {SN, S1, S2, S3} (S=sphere)

reduced to 1-D and 3-D

ICBST the3-D can be further reduced(via a similarity transform. that separates z from x&y)

No further reduction is possible because the x & y axes transform into one another.D(2) (the orange block) is now an irreducible representation

of the molecular point group.

1 23

N

00

00

0010

0001

)2()1()1()4( DDDD


The characters of each Irr Rep must also fulfill the multiplication table.

Characters and Group Multiplication Tables

Always must be true for thetotally symmetric Irr Rep!

11







E • σv = -1x1 = -1 ≠ σv

σv • C2 = 1x-1 = -1 ≠ σv'

12


Character Tables

There are a number of useful properties of character tables:

012

111

111

32

3

2

1

33

vv CEC

• The sum of the squares of the dimensionality of all the irr reps is equal to the order of the group. (normalization)

• The sum of (the squares of characters of any irr rep X the number of elements in each class) is equal to the order of the group.

• The sum of the products of the corresponding characters of any two different irr reps of the same group is zero. (orthogonality)

• The characters of all matrices belonging to the operations in the same class are identical in a given irreducible representation. (similarity)

• The number of irreducible representations in a group is equal to the number of classes of that group. (completeness)


Reducible to Irreducible RepresentationFrom the orthonormalization of the IRRs (also known as the ”Grand Orthogonalization Theorem”), we can see that

where (i.e., the red. rep. is a lin comb.)

where ai is the number of times the irreducible representation irr appears in red , h the order of the group, l an operation of the group, g(c) #of sym ops in class l,red the character of the operation l in the red. rep., and irr the character of l in the irreducible representation.

l

llredi cccg

ha )(*)()(

1 )()(

irri

irrirrred a

13


Reducible to Irreducible representation

1 23

NAs an example, in C3v (NH3), consider the following red. rep.

214

32 33

red

vv CEC

Here’s the character table of the irreducible representations:

By inspection we find red = 2A1 + EBUT WE USUALLY CAN’T DO IT BY INSPECTION!

012

111

111

32

2

1

33

E

A

A

CEC vv


Reducible to Irreducible Representation

Let’s go back to our example again.

10231122416

1

01231121416

1

21231121416

1

2

1

E

A

A

a

a

a

So once again we find red = 2A1 + E

012

111

111

214

32

2

1

33

E

A

A

CEC

red

vv

l

llredi cccg

ha )(*)()(

1 )()(

14


Direct Products

• In many cases, we will want to know how a setof product functions transform, already knowing how the individual functions transform, within a pt gr.

• We could, of course, construct matrix reps for the product basis functions under each sym op. Ugg.

• Easier: Products of functions transform as thedirect products of the irr rep of the initial functions!

• Direct products are simply the red. rep. generated by multiplying each symmetry’s characters of the initial functions.


Direct Products

• Consider C3v . Let’s find out how (x,y) (x,y) behaves.of course, in this case it’s obvious (x,y) (x,y) = combined (xy, yx, x2, y2)

• So, (x,y) (x,y) = E E =

4 1 0

l

llredi cccg

ha )(*)()(

1 )()(

10031122416

1

11031121416

1

11031121416

1

2

1

E

A

A

a

a

a So, = A1 + A2 + E

15


Direct Product Shortcut Rules


Projection Operators

Symmetry-adapted bases:

The projection operator takes a non-symmetry-adapted basis set of a representation and projects it along new directions,so that it belongs to a specific irr. rep. of the group.

R

ll RRh

P ˆ)(1ˆ )(

where P l is the projection operator of the irr rep l, (l) is the character of the operation R for the rep l, and R means application of R to our original basis component.

More later...

^

^

16


Integrals

A last example… (with more next time!)

Integrals of product functions often appear in for example quantum mechanics and symmetry analysis can be helpful with them to.

ki fOf |ˆ|

An integral will be non-zero only if the integrand belongs to the totally symmetric irr. rep. of the point group.

Off ki ˆ


Summary

• Molecules (and their electronic orbitals, vibrations etc) are invariant under certain symmetry operations.

• The symmetry operations can be described by a representation determined by the basis we choose to describe the molecule.

• The representation can be broken up into its symmetry species (irreducible representations).

• In character tables we find information about the symmetry properties of the irreducible representations.

17


Lagniappe: A Little ExtraSome Point Groups have complex characters:

specifically Cn and Cnh for n ≥ 3, S2n, T, and Th (all uncommon)


Lagniappe: A Little Extra

Realize, however, that the real number rep. is actually reducible!e.g., for C3, h = 3, so Σ(χE)2 = 3 = 12 + 12 +12 ≠ 12 + 22 !

for n=3, 2cos2π/3 = -1

= 2 -1 -1

18



Double Rotation Groups

Within Dirac Equations (relativistic wave functions), the functionsthat represent half-integer spins are treated as double valuedwithin a single 2 rotation (sorta like a Möbius strip), “spinors”.

So the usual identity element (C1) does not return the function, but rather the negative of the function!

Solution:In addition to the usual rotations, we add the rotation operation

R (= C2 = original E) and all the operations formed by taking the product of R with the original sym ops.

For angular momentum basis functions with half-integer values ONLY, these new sym ops, R (i.e., C2) can have characters different than the “old” ops.

Twice the number of sym ops, but not twice as many classes. This also give us a new identity element E = C4


Lagniappe: A Little ExtraNormal D4 Group

Double Rotation Group D4' adds the following'

19


Lagniappe: A Little ExtraDouble Rotation Group D4'

Adding R doubles the number of sym ops, but not the # of classes (7 vs. 5)

' or *



Double Group for Octahedral Field

chem516.02.origins of character tablesxuv.scs.illinois.edu › 516 › lectures ›...

Documents

Transcript of chem516.02.origins of character tablesxuv.scs.illinois.edu › 516 › lectures ›...