gtlectures 03 basis functions

8/11/2019 GTlectures 03 Basis Functions

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Introduction to Group Theory

Basis Functions

Eduardo Bed Barros

Universidade Federal do Cear


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Group Theory and Quantum Mechanics:

If the Hamiltonian is invariant regarding a certain

symmetry operation OR.

The wavefunctions can then be labelled according to theIRs

The dimension of the IRs corresponds to the number of

degenerate states.

nnn EH nnRnR EH OO

)(O)(O nRnnR EH

'O

O

nnR

nnR

symmetric upon OR

n and n are

degenerate states


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Basis Functions

if we have n functions (=1,2,... n) such that

Defining a n-dimensional space

We can always find a representation of any groupG=(E,R1,R2,R3...) in terms of these functions

)(xf

dxxfxf )()(

orthogonal


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Basis Functions

We define the wigner operator for an operationR of G as the equivalent of this operation in the

coordinate space (x) such that

Thus...

)()()( xfaxgxfOR

RO

)()()( xfRDxfOR

If we do this for all R in G, well have a representation of the gr


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Basis Functions

If D is not an IR. it can always be written in terms ofa sum of irreducible representations.

We can thus find that

j

j

j

jj RaRa j )()(D

)(D

1DD' MM

0

00

0

00

0

000

00

D' j

i


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Basis Functions

This transformation is equivalent to finding a set oforthogonal functions such that ofr each IR i(of

dimension li), component of D, there is a subset of li

functions

with =1,2,..., li such that

we can use the notation

i

)()()( xRxO i

i

i

li

R

ixi )(

Partner functions


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Projection Operator

Is an operator which transforms a basis vector of into one of its partners

We can write P as a linear combination of the wigner

operators OR:

iiiP

R

RORaP i )(


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Projection Operators

We can easily find that

so that

Pkeeps the basis function invariant

)()( Rh

lRa ii

R

R

ii ORh

lP i )(


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Projection Operators

We can also define a space projetor as

Which projects any combination of basis functions ofthe iirreducible representation onto itself

The advantage of this projection operator is the factthat it can be obtained directly from the CharacterTable of the group

R

Ri OR

h

lPP

i

ii )(


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Using the projection operators

We can use the projection operators to find the basisfunctions of the IRs

Ex.: D3

czbyaxf

The Projection operators are:

2

3

1

3

222

2

3

1

3

22223

13

26

26

161

2

1

CCEP

CCCCCEP

CCCCCEP

E

A

A

If we want to obtain a linear basis, we can start by applying this PO on an

arbitrary linear function of the coordinates:


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Let us first see what each operation doesto the coordinates:

xxOE

C3

C2

C2

C2 `x

y

z

yxyO

yxxO

zzO

C

C

C

23

21

23

21

13

13

13

yxyO

yxxO

zzO

C

C

C

23

21

23

21

23

23

23

yyOE zzOE

Wigner Operator:

Operates on the

coordinate system


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xxOE

C3

C2

C2

C2 `x

y

z

yxyO

yxxO

zzO

C

C

C

21

23

23

21

13

13

13

yxyO

yxxO

zzO

C

C

C

21

23

23

21

23

23

23

yyOE zzOE

Operation on the

coordinate system


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xxOC 2

C3

C2

C2

C2 `x

y

z

yxyO

yxxO

zzO

C

C

C

2123

23

21

2

2

2

zzOC 2yyOC 2

yxyO

yxxO

zzO

C

C

C

2123

23

21

2

2

2


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C3

C2

C2

C2 `x

y

z

222

23

13

2

6

1CCCCCE

AOOOOOOP

06

1

06

1

)()()(

6

1

21

23

21

23

21

23

21

23

23

21

23

21

23

21

23

21

2

2

2

yxyxyyxyxyyP

yxyxxyxyxxxP

zzzzzzzzP

A

A

A

czfPA 2

The operator for the A2 IR projects the

function fonto the function z

This indicates that the function z is a

linear basis function of the A2IR of the

D3group

C


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C3

C2

C2

C2 `x

y

z

23

13

26

2CCE

E OOOzP

yyyxyxyyP

xxyxyxxxP

zzzzP

E

E

E

)3(6

2)()(

6

2

)3(6

2)()(2

6

2

02

6

2

21

23

21

23

23

21

23

21

byaxfPE

The operator for the EIR projects the

function fonto both the x and y functions

Since, the IR is bidimensional, this

indicates that the x and y funcitons form a

linear basis of the E IR of the D3group.

To check that they are partners, it is

necessary to use to partner projection

l


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Using the partner projection operator

Rember the E irreducible representation of the D3group:

The projection operator

Projects an arbitrary function on one of the first

partner

2/12/32/32/1

2/12/32/32/1

10

01

2/12/32/32/1

2/12/32/32/1

10

01

2

1

222

2

3

1

33

111111

111111

E

A

A

CCCCCED

222

23

13

21

21

21

21

11

6

2CCCCCE

E OOOOOOP

R

R

ii ORh

lP i )(

axfPzPyPxxP EEEE 11111111 0,0,It is easy to check that:

Soxis one of the partners of the E irreducible representatio

l


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Using the partner projection operator

Rember the E irreducible representation of the D3group:

The projection operator

Projects an one partner onto the other

2/12/32/32/1

2/12/32/32/1

10

01

2/12/32/32/1

2/12/32/32/1

10

01

2

1

222

2

3

1

33

111111

111111

E

A

A

CCCCCED

22

23

13

23

23

23

23

12

6

2CCCC

EOOOOP

R

R

ii ORh

lP i )(

yxPE 12It is easy to check that:

So yis the partner ofxforming

the basis of the E IR yx,


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Complete Character Tables for Point Groups

These tables are all indexed!


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gtlectures 03 basis functions

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