Juggling with representations Matrix representation of Symmetry - - PowerPoint PPT Presentation

Juggling with representations

O Ha Hb

C2 σv σv’

How many such representations?

Representation Irreducible

Matrix representation of Symmetry Point Groups

C2v

O a b Basis E C2 (z) σv(zx) σv’(yz)

1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0

Two dimensional representation

Reducible?

Function space

X (e1) Y (e2) Z (e3) p1 p4 p3 p2

• fi + fj = fk

Collection of functions f1, f2, …, fi, …, f n

• n fm = fn
• Σ ai fi = fq
• (fi , fj)= ∫ fi* fi dτ
• If n of the functions are linearly

independent, then any of the other functions can be represented as a linear combination of these n functions. The space is n-dimensional Linear independence: Σ αι φι = 0 if and only if ai = 0 for all values of i

• Orthonormal basis functions

3

Transformation operators, OR

for a function space made up of n linearly independent basis functions, fj

4

Transformation operators leave the scalar product

• f two functions unchanged

d d d’ d

5

Transformation operators are linear

(a) If f and g are functions, a is a number and g = a f (b) If f, g, h are functions, h = f + g

=

6

Transformation operators produce a unitary representation if orthonormal basis functions are used

δij ⇒ *

∫

διϕ *

∫

* kl διϕ * διϕ *

D(R)†D(R) = E

8

Transformation operators produce a unitary representation if orthonormal basis functions are used

⇒ διϕ *

D(R)†D(R) = E

9

How does one switch to an orthonormal basis??

Similarity Transformation

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

P a g e 3 3 7

10

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

A

11

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B

12

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B B = A -1

13

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B

Let A = A ‘ and B = B ‘

14

B = A -1

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B

Let A = A ‘ and B = B ‘ ⇒ B = A-1

15

B = A -1

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B

Let A = A ‘ and B = B ‘ ⇒ B = A-1

16

fk = gl

Σ

Alk

l = 1 n

B = A -1

Switching bases

Let f1, f2, …., fn and g1, g2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1, p-1, p0 orbitals

=

B

Let A = A ‘ and B = B ‘ ⇒ B = A-1 gj = fi

Σ

Bij

i = 1 n

17

fk = gl

Σ

Alk

l = 1 n

B = A -1

Now what?

gj = fi

Σ

Bij

i = 1 n

18

fk = gl

Σ

Alk

l = 1 n

19

gj = fi

Σ

Bij

i = 1 n

fk = gl

Σ

Alk

l = 1 n

Transformation operators

20

gj = fi

Σ

Bij

i = 1 n

fk = gl

Σ

Alk

l = 1 n

Transformation operators

21

gj = fi

Σ

Bij

i = 1 n

fk = gl

Σ

Alk

l = 1 n

Transformation operators

Transformation operators

22

gj = fi

Σ

Bij

i = 1 n

fk = gl

Σ

Alk

l = 1 n

23

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

fk = gl

Σ

Alk

l = 1 n

Transformation operators

24

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

=

since OR is a linear operator fk = gl

Σ

Alk

l = 1 n

Transformation operators

25

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

=

fk = gl

Σ

Alk

l = 1 n

Transformation operators

26

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

fk = gl

Σ

Alk

l = 1 n

Transformation operators

27

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

fk = gl

Σ

Alk

l = 1 n

Transformation operators

28

gj = fi

Σ

Bij

i = 1 n

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

fk = gl

Σ

Alk

l = 1 n

Transformation operators

29

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl

Transformation operators

30

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk

Transformation operators

31

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk

Transformation operators

32

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk

Transformation operators

33

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk

Dg(R) = A Df(R) B

A relationship between Df(R) and Dg(R)

34

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk But B = A-1

Dg(R) = A Df(R) B

A relationship between Df(R) and Dg(R)

Similarity transformation switches the bases

35

fi

Σ

Bij

i = 1 n

i

Σ

Bij

i = 1 n

=

gl

Σ

Alk

l = 1 n

= Σ

l = 1 n

gl Σ

Bij

i = 1 n

Alk

Dg(R) = A Df(R) A-1

36

• Transformation matrices for two linearly idependent bases are conjugate

to each other

Dg(R) = A Df(R) A-1

Similarity transformation switches the bases

37

• The similarity transformation is achieved by the matrices that relate the

bases linearly

Dg(R) = A Df(R) A-1

Similarity transformation switches the bases

• Transformation matrices for two linearly idependent bases are conjugate

to each other

38

• The similarity transformation is achieved by the matrices that relate the

bases linearly

• Change of basis does not affect the multiplication rules

Dg(R) = A Df(R) A-1

Similarity transformation switches the bases

• Transformation matrices for two linearly idependent bases are conjugate

to each other

39

• The similarity transformation is achieved by the matrices that relate the

bases linearly

• Change of basis does not affect the multiplication rules

If Df(SR) = Df(S) Df(R), then Dg(SR) = Dg(S) Dg(R)

Dg(R) = A Df(R) A-1

Similarity transformation switches the bases

• Transformation matrices for two linearly idependent bases are conjugate

to each other

Equivalent representations

40

• The similarity transformation is achieved by the matrices that relate the

bases linearly

• Change of basis does not affect the multiplication rules

If Df(SR) = Df(S) Df(R), then Dg(SR) = Dg(S) Dg(R)

Two representations of a point group are EQUIVALENT if, for every symmetry operation R, Dg(R) = A Df(R) A-1 using the same pair of matrices A and A-1 Dg(R) = A Df(R) A-1

• Transformation matrices for two linearly idependent bases are conjugate

to each other

Homework Problem

41

• Consider the two sets of p orbitals: px, py, pz and p+1, p-1, p0
• Work out the transformation matrices for both the sets, for all the

symmetry elements of point group C3v

• With the help of the matrices that correlate px, py, pz with p+1, p-1, p0,

prove the equivalence of the two representations thus generated P a g e 3 3 7