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MOLECULAR SYMMETRY AND SPECTROSCOPY. [email protected] Download ppt file from. http://www.few.vu.nl/~rick. At bottom of page. We began by summarizing. Chapters 1 and 2. Spectroscopy and Quantum Mechanics. f. Absorption can only occur at resonance. h ν if = E f – E i = Δ E if.

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Molecular symmetry and spectroscopy

MOLECULAR SYMMETRY

AND SPECTROSCOPY

[email protected]

Download ppt file from

http://www.few.vu.nl/~rick

At bottom of page


Molecular symmetry and spectroscopy

We began by summarizing

Chapters 1 and 2. Spectroscopy and Quantum Mechanics

f

Absorption can only occur at resonance

hνif = Ef – Ei = ΔEif

νif

i

M

Integrated absorption coefficient (i.e. intensity) for a line is:

______

8π3 Na

~

~

~

ε(ν)dν

=

I(f ← i) = ∫

F(Ei )

νif

Rstim(f→i)

S(f ← i)

(4πε0)3hc

line

Use Q. Mech. to calculate:

ODME of H

μfi = ∫ (Ψf )* μAΨi dτ

ODME of μA


Molecular symmetry and spectroscopy

P. R. Bunker and Per Jensen:

Fundamentals of Molecular Symmetry,

Taylor and Francis, 2004.

The first 47 pages:

Chapter 1 (Spectroscopy)

Chapter 2 (Quantum Mechanics) and

Section 3.1 (The breakdown of the BO Approx.)

P. R. Bunker and Per Jensen:

Molecular Symmetry and Spectroscopy,

2nd Edition, 3rd Printing,

NRC Research Press, Ottawa, 2012.

Download pdf file from

To buy it go to:

http://www.crcpress.com

www.chem.uni-wuppertal.de/prb


Molecular symmetry and spectroscopy

P. R. Bunker and Per Jensen:

Fundamentals of Molecular Symmetry,

Taylor and Francis, 2004.

The first 47 pages:

Chapter 1 (Spectroscopy)

Chapter 2 (Quantum Mechanics) and

Section 3.1 (The breakdown of the BO Approx.)

P. R. Bunker and Per Jensen:

Molecular Symmetry and Spectroscopy,

2nd Edition, 3rd Printing,

NRC Research Press, Ottawa, 2012.

Download pdf file from

To buy it go to:

http://www.crcpress.com

www.chem.uni-wuppertal.de/prb

We then proceeded to discuss Group

Theory and Point Groups


Molecular symmetry and spectroscopy

Definitions for groups and point groups:

“Group”

A set of operations that is closed wrt “multiplication”

“Point Group”

All rotation, reflection and rotation-reflection operations

that leave the molecule (in its equilibrium configuration)

“looking” the same.

“Matrix group”

A set of matrices that forms a group.

“Representation”

A matrix group having the same shaped multiplication

table as the group it represents.

“Irreducible representation”

A representation that cannot be written as the sum

of smaller dimensioned representations.

“Character table”

A tabulation of the characters of the irreducible representations.


Molecular symmetry and spectroscopy

Character table for the point group C3v

E C3σ1

C32σ2

σ3

Two 1D

irreducible

representations

of the C3v group

The 2D representation M = {M1, M2, M3, ....., M6}

of C3v is the irreducible representation E. In this

table we give the characters of the matrices.

Elements in the same class have the same characters

3 classes and 3 irreducible representations


Molecular symmetry and spectroscopy

Character table for the point group C2v

x

EC2σyzσxy

(+y)

z

4 classes and 4 irreducible representations


Molecular symmetry and spectroscopy

E C3σ1

C32σ2

σ3

C3v

3

2

1

Spectroscopy

M

f

hνif = Ef – Ei = ΔEif

MMMM

i

S(f ← i) = ∑A | ∫ Φf* μAΦi dτ |2

Quantum Mechanics

ODME of H and μA

μfi =

∫ Φf* μAΦi dτ

Group Theory and Point Groups

(Character Tables and Irreducible Representations)

PH3

8


Molecular symmetry and spectroscopy

Point Group symmetry is based on

the geometrical symmetry of the

equilibrium structure.

Point group symmetry not appropriate

when there is rotation or tunneling

Use energy invariance symmetry

instead. We start by using inversion

symmetry and identical nuclear

permutation symmetry.


Molecular symmetry and spectroscopy

The Complete Nuclear Permutation

Inversion (CNPI) Group

Contains all possible permutations

of identical nuclei including E. It also

contains the inversion operation E*

and all possible products of E* with

the identical nuclear permutations.

GCNPI = GCNP x {E,E*}


Molecular symmetry and spectroscopy

The spin-free (rovibronic) Hamiltonian

(after separating translation)

Vee + VNN + VNe

THE GLUE

In a world of infinitely powerful computers we could solve the Sch. equation numerically and that would be that. However, we usually have to start by making approximations. We then

selectively correct for the approximations made.


The cnpi group for the water molecule

The CNPI Group for the Water Molecule

The Complete Nuclear Permutation Inversion (CNPI) group

for the water molecule is {E, (12)} x {E,E*} = {E, (12), E*, (12)*}

+

+

e

e

H2

H1

O

O

E*

(12)

-

e

O

H2

H1

H2

H1

(12)*

Nuclear permutations permute nuclei (coordinates and spins).

Do not change electron coordinates

E* Inverts coordinates of nuclei and electrons.

Does not change spins.

Same CNPI group for CO2, H2, H2CO, HOOD, HDCCl2,…


Molecular symmetry and spectroscopy

H

H

N1N2N3

1

C1

F

3

C2

2

I

C3

O

F

D

2

O

O

1

3

1

3

H H

12C 13C

D H

2

1

2

H

1

+

H

H

3

2

3

GCNPI = {E, (12), (13), (23), (123), (132)}x {E, E*}

= GCNP x {E, E*}


Molecular symmetry and spectroscopy

GCNPI = {E, (12), (13), (23), (123), (132)}x {E, E*}

GCNPI={E, (12), (13), (23), (123), (132),

E*, (12)*, (13)*,(23)*, (123)*, (132)*}

Number of elements = 3! x 2 = 6 x 2 = 12

Number of ways of permuting

three identical nuclei


Molecular symmetry and spectroscopy

H5

C1

H4

C2

I

C3

D

The CNPI Group of C3H2ID

GCNPI = {E, (12), (13), (23), (123), (132)}

x{E, (45)}x {E, E*}

= {E, (12), (13), (23), (123), (132),

(45),(12)(45), (13)(45), (23)(45), (123)(45), (132)(45),

E*, (12)*, (13)*, (23)*, (123)*, (132)*,

(45)*,(12)(45)*, (13)(45)*, (23)(45)*, (123)(45)*,

(132)(45)*}

Number of elements = 3! x 2! x 2 = 6 x 2 x 2 = 24


Molecular symmetry and spectroscopy

H5

C1

H4

C2

I

C3

D

Number of elements

= 3! x 2! x 2 = 6 x 2 x 2 = 24

If there are n1 nuclei of type 1, n2 of

type 2, n3 of type 3, etc then the total

number of elements in the CNPI group

is n1! x n2! x n3!... x 2.


Molecular symmetry and spectroscopy

The CNPI group of allene

H5

The

Allene molecule

C1

H4

C3H4

C2

H7

C3

H6

Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288


Molecular symmetry and spectroscopy

The CNPI group of allene

H5

The

Allene molecule

C1

H4

C3H4

C2

H7

C3

H6

Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288

Sample elements: (456), (12)(567), (4567), (45)(67)(123)


Molecular symmetry and spectroscopy

The CNPI group of allene

H5

00H

The

Allene molecule

C1

H4

C3H4

C2

H7

C3

H6

00H

Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288

How many elements?

C3H4O4


Molecular symmetry and spectroscopy

The CNPI group of allene

H5

00H

The

Allene molecule

C1

H4

C3H4

C2

H7

C3

H6

00H

Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288

C3H4O4

3! x 4! x 4! x 2 = 6912


Molecular symmetry and spectroscopy

The size of the CNPI group depends

only on the chemical formula

Number of elements in the CNPI groups of various

molecules

(C6H6)2 12! x 12! x 2 ≈ 4.6 x 1017

Just need the chemical formula to

determine the CNPI group. Can be BIG


Molecular symmetry and spectroscopy

An important number

Molecule PG h(PG) h(CNPIG) h(CNPIG)/h(PG)

H2O C2v 4 2!x2=4 1

PH3 C3v 6 3!x2=12 2

Allene D2d 8 4!x3!x2=288 36

C3H4

Benzene D6h 24 6!x6!x2=1036800 43200

C6H6

This number means something!

End of Review of Lecture One

22

ANY QUESTIONS OR COMMENTS?


Cnpi group symmetry is based on energy invariance

CNPI group symmetry is based on energy invariance

Symmetry operations are operations that leave the energy of the system (a molecule in our case) unchanged.

Using quantum mechanics:

A symmetry operation is an operation that

commutes with the Hamiltonian:

RHn = HRn


Molecular symmetry and spectroscopy

The character table of the CNPI

group of the water molecule

(12) E* 1 1

1 -1

-1 -1

-1 1

(12)*

1

-1

1

-1

E 1

1

1

1

A1

A2

B1

B2

It is called C2v(M)


Molecular symmetry and spectroscopy

The character table of the CNPI

group of the water molecule

(12) E* 1 1

1 -1

-1 -1

-1 1

(12)*

1

-1

1

-1

E 1

1

1

1

A1

A2

B1

B2

It is called C2v(M)

Now to explain how we label

energy levels using

irreducible representations


Labelling energy levels

Labelling energy levels

For the water molecule (no degeneracies, and R2 = identity for all R) :

H = E

RH = RE

Since RH = HR and E is a number, this leads to HR = ER.

H(R) = E(R)

E is nondegenerate. Thus RΨ = cΨ.

But R2 = identity. Thus c2 = 1, so c = ±1 and R = ±

R = (12), E* or (12)*

The eigenfunctions have symmetry


Molecular symmetry and spectroscopy

R = E*

+ Parity

- Parity

Ψ1+(x)

Ψ2-(x)

x

x

Ψ-(-x) = -Ψ-(x)

Ψ3+(x)

Eigenfunctions of H

must satisfy

E*Ψ = ±Ψ

x

Ψ+(-x) = Ψ+(x)


Molecular symmetry and spectroscopy

E*ψ(xi) = ψE*(xi), a new function.

ψE*(xi) = ψ(E*xi) = ψ(-xi) = ±ψ(xi)

Since E*ψ(xi) can only be ±ψ(xi)

This is different from Wigner’s approach

See PRB and Howard (1983)


Molecular symmetry and spectroscopy

+ Parity

- Parity

Ψ1+(x)

Ψ2-(x)

x

x

Ψ-(-x) = -Ψ-(x)

Ψ3+(x)

Eigenfunctions of H

must satisfy

E*Ψ = ±Ψ

x

Ψ+(-x) = Ψ+(x)

and (12)ψ = ±ψ


There are four symmetry types of h 2 o wavefunction

There are four symmetry types of H2O wavefunction

(12) E* 1 1

1 -1

-1 -1

-1 1

E 1

1

1

1

(12)*

1

-1

1

-1

R = ±

A1

A2

B1

B2

A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1


The symmetry labels of the cnpi group of h 2 o

The Symmetry Labels of the CNPI Group of H2O

(12) E* 1 1

1 -1

-1 -1

-1 1

E 1

1

1

1

(12)*

1

-1

1

-1

A1

A2

B1

B2

We are labelling the states using

the irreps of the CNPI group

∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1


The symmetry labels of the cnpi group of h 2 o1

The Symmetry Labels of the CNPI Group of H2O

(12) E* 1 1

1 -1

-1 -1

-1 1

E 1

1

1

1

(12)*

1

-1

1

-1

A1

A2

B1

B2

Thus, for example, a wavefunction of “A2 symmetry”

will “generate” the A2 representation:

∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

Eψ=+1ψ(12)ψ=+1ψE*ψ=-1ψ(12)*ψ=-1ψ


The symmetry labels of the cnpi group of h 2 o2

The Symmetry Labels of the CNPI Group of H2O

(12) E* 1 1

1 -1

-1 -1

-1 1

E 1

1

1

1

(12)*

1

-1

1

-1

A1

A2

B1

B2

Thus, for example, a wavefunction of “A2 symmetry”

will “generate” the A2 representation:

∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

Eψ=+1ψ(12)ψ=+1ψE*ψ=-1ψ(12)*ψ=-1ψ

For the water molecule we can, therefore label

the energy levels as being A1, A2, B1 or B2

using the irreps of the CNPI group.


Molecular symmetry and spectroscopy

The labelling business

The vibrational wavefunction for the v3 = 1

state of the water molecule can be written

approximately as ψ = N(Δr1 – Δr2).

Eψ=+1ψ(12)ψ=-1ψE*ψ=+1ψ(12)*ψ=-1ψ

This would be labelled as B2.


Molecular symmetry and spectroscopy

c = ε, ε2 (=ε*), or ε3 (=1)

If n = 3,

Suppose Rn = E where n > 2.

We still have RΨ = cΨ for nondegenerate Ψ, but now RnΨ = Ψ.

Thus cn = 1 and c = n√1, i.e. c = [ei2π/n]a where a = 1,2,…,n.

where  = ei2/3

C3 C32

eiπ = -1

ei2π = 1


For nondegenerate states we had this as the effect of a symmetry operation on an eigenfunction

For nondegenerate states we hadthis as the effect of a symmetry operation on an eigenfunction:

For the water molecule ( nondegenerate) :

H = E

RH = RE

HR = ER

Thus R = c since E is nondegenerate.

What about degenerate states?


Molecular symmetry and spectroscopy

ℓ-fold

degenerate energy level with energy En

RΨnk=D[R ]k1Ψn1 +D[R ]k2Ψn2 +D[R ]k3Ψn3 +…+D[R ]kℓΨnℓ

For each relevant symmetry operation R, the constants

D[R ]kp form the elements of an ℓℓ matrix D[R ].

ForT = RS it is straightforward to show that

D[T ] = D[R ] D[S ]

The matrices D[T ],D[R ], D[S ]….. form an ℓ-dimensional representation that is generated by the ℓ functions Ψnk

The ℓ functions Ψnktransform according to this representation


Labelling energy levels using the cnpi group

Labelling energy levels using the CNPI Group

We can label energy levels using

the irreps of the CNPI group for any molecule

∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

Pages 143-149

Pages 99-101

34


Molecular symmetry and spectroscopy

Determining symmetry and

reducing a representation

Example of using the symmetry operation (12):

(12)

r1´

r2´

´

H1

H2

We have (12) (r1, r2, ) = (r1´, r2´, ´)

We see that (r1´, r2´, ´) = (r2, r1, )


Molecular symmetry and spectroscopy

r1´

r2´

r2

r2

r2

r2

r1

r1

r1

r1

r2´

r1´

´

´

3

3

3

3

3

1

2

2

2

2

2

1

1

1

1

3

1

2

2

1

3

2

1

3

E

(12)

´

E*

r1´

r2´

(12)*

´

r2´

r1´


Molecular symmetry and spectroscopy

R a = a´ = D[R] a

E

 = 3

 = 1

(12)

 = 3

E*

(12)*

 = 1


Molecular symmetry and spectroscopy

aA1 = ( 13 + 11 + 13 + 11) = 2

aA2 = ( 13 + 11  13  11) = 0

aB1 = ( 13  11  13 + 11) = 0

aB2 = ( 13  11 + 13  11) = 1

Γ = Σ aiΓi

i

A reducible representation

 = 2A1 B2


Molecular symmetry and spectroscopy

aA1 = ( 13 + 11 + 13 + 11) = 2

aA2 = ( 13 + 11  13  11) = 0

aB1 = ( 13  11  13 + 11) = 0

aB2 = ( 13  11 + 13  11) = 1

Γ = Σ aiΓi

i

i

A reducible representation

 = 2A1 B2


Molecular symmetry and spectroscopy

aA1 = ( 13 + 11 + 13 + 11) = 2

aA2 = ( 13 + 11  13  11) = 0

aB1 = ( 13  11  13 + 11) = 0

aB2 = ( 13  11 + 13  11) = 1

Γ = Σ aiΓi

i

i

A reducible representation

 = 2A1 B2


Molecular symmetry and spectroscopy

aA1 = ( 13 + 11 + 13 + 11) = 2

aA2 = ( 13 + 11  13  11) = 0

aB1 = ( 13  11  13 + 11) = 0

aB2 = ( 13  11 + 13  11) = 1

Γ = Σ aiΓi

i

i

A reducible representation

 = 2A1 B2


Molecular symmetry and spectroscopy

We know now that r1, r2, andgenerate the

representation 2A1 B2

Consequently, we can generate from r1, r2, and three „symmetrized“ coordinates:

S1 with A1 symmetry

S2 with A1 symmetry

S3 with B2 symmetry

For this, we need projection operators


Molecular symmetry and spectroscopy

Projection operators:

General for li-dimensional irrep i

Diagonal element of representation matrix

Symmetry operation

Simpler for 1-dimensional irrep i

Character

1


Molecular symmetry and spectroscopy

Projection operators:

General for li-dimensional irrep i

Simpler for 1-dimensional irrep i

Character

Diagonal element of representation matrix

1

Symmetry operation


Molecular symmetry and spectroscopy

Projection operators:

General for li-dimensional irrep i

Simpler for 1-dimensional irrep i

Character

Diagonal element of representation matrix

1

Symmetry operation

E (12) E* (12)*

A1 1 1 1 1

PA1 = (1/4) [ E + (12) + E* + (12)* ]


Molecular symmetry and spectroscopy

S1 = P11A1r1= [ E + (12) + E*+ (12)* ]r1

S3 = P11B2r1= [ E  (12) + E* (12)*] r1

= [ r1  r2 + r1 r2 ] = [ r1  r2 ]

= [ r1 + r2 + r1 + r2 ] = [ r1 + r2 ]

S2 = P11A1= [ E + (12) + E*+ (12)* ]

P11B2= [ E  (12) + E* (12)* ]

= [  +  + +  ] = 

= [   +   ] = 0

Projection operator for A1 acting on r1

PA1

PA1

PB2

PB2

 Is „annihilated“ by P11B2

PB2


Molecular symmetry and spectroscopy

S1 = P11A1r1= [ E + (12) + E*+ (12)* ]r1

S3 = P11B2r1= [ E  (12) + E* (12)*] r1

= [ r1  r2 + r1 r2 ] = [ r1  r2 ]

= [ r1 + r2 + r1 + r2 ] = [ r1 + r2 ]

S2 = P11A1= [ E + (12) + E*+ (12)* ]

P11B2= [ E  (12) + E* (12)* ]

= [  +  + +  ] = 

= [   +   ] = 0

Projection operators for A1 and B2

PA1

PA1

PB2

PB2

 Is „annihilated“ by P11B2

PB2


Molecular symmetry and spectroscopy

S1 = P11A1r1= [ E + (12) + E*+ (12)* ]r1

S3 = P11B2r1= [ E  (12) + E* (12)*] r1

= [ r1  r2 + r1 r2 ] = [ r1  r2 ]

= [ r1 + r2 + r1 + r2 ] = [ r1 + r2 ]

S2 = P11A1= [ E + (12) + E*+ (12)* ]

= [  +  + +  ] = 

Projection operators for A1 and B2

PA1

PA1

PB2

Aside: S1, S2 and S3 have the symmetry and form of the

normal coordinates.


Molecular symmetry and spectroscopy

A1

A1

The three

Normal modes

of the water

molecule

B2


Molecular symmetry and spectroscopy

Labeling is not just bureaucracy. It is useful.

PAUSE

50


Molecular symmetry and spectroscopy

Labeling is not just bureaucracy. It is useful.

The vanishing integral theorem

Pages 136-139

Pages 114-117

But first we look at

The symmetry of a product

Pages 109-114


Molecular symmetry and spectroscopy

(12) E* 1 1

1 -1

-1 -1

-1 1

THE SYMMETRY OF A PRODUCT

(12)*

1

-1

1

-1

E 1

1

1

1

A1

A2

B1

B2

A2

Eψ=+1ψ(12)ψ=+1ψE*ψ=-1ψ(12)*ψ=-1ψ

∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.

B1

Eφ=+1φ(12)φ=-1φE*φ=-1φ(12)*φ=+1φ

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

The symmetry of the product φψis B1 x A2 = B2.

A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1

B1 x B2, A1 x A2, B1 x A2, B2 x A2, B1 x B1,…

A2 A2 B2 B1 A1


Molecular symmetry and spectroscopy

Symmetry of a product. Example: C3v

A1 A1 = A1

A1 A2 = A2

A2 A2 = A1

A1 E= E

A2 E= E

E E= A1  A2  E

E E: 4 1 0

Reducible representation

Characters of the product representation are the products

of the characters of the representations being multiplied.

Symmetry of triple product is obvious extension


Molecular symmetry and spectroscopy

+ Parity

- Parity

Ψ+(x)

Ψ-(x)

x

x

Ψ-(-x) = -Ψ-(x)

Ψ+(x)

∫Ψ+Ψ-Ψ+dx = 0

x

- parity

Ψ+(-x) = Ψ+(x)


Molecular symmetry and spectroscopy

+ Parity

- Parity

Ψ+(x)

Ψ-(x)

The vanishing integral theorem

x

x

Ψ-(-x) = -Ψ-(x)

Ψ+(x)

∫f(τ)dτ = 0 if symmetry of f(τ) is not A1

∫Ψ+Ψ-Ψ+dx = 0

x

- parity

Ψ+(-x) = Ψ+(x)


Molecular symmetry and spectroscopy

(12) E* 1 1

1 -1

-1 -1

-1 1

USING THE VANISHING

INTEGRAL THEOREM

E 1

1

1

1

(12)*

1

-1

1

-1

Symmetry of H

A1

A2

B1

B2

Using symmetry labels and the vanishing

integral theorem we deduce that:

∫Ψa*HΨbdτ = 0 if symmetry of Ψa*HΨb is not A1,

Integral vanishes if Ψa and Ψb have different symmetries


Molecular symmetry and spectroscopy

ODME of H

vanishes if symmetries not the same

This means that we can

“block-diagonalize” the

Hamiltonian matrix


Molecular symmetry and spectroscopy

A1 A2 B1

ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8

0

0

0

0

0

0

0

0

. . .

. . .

. . .

0

Ψ1

Ψ2

Ψ3

Ψ4

Ψ5

Ψ6

Ψ7

ψ8

0

0

A1

A2

B1

0

0

. . .

. . .

. . .

0

0

0

0

0

0

. .

. .

0

0

0

Symmetry is preserved on diagonalization


Molecular symmetry and spectroscopy

ODME of μA

μfi = ∫ (Ψf )* μAΨi dτ

Can use symmetry to determine if this ODME = 0

This ODME will vanish if the symmetry

of (Ψf)* μAΨi is not A1

0

0


Molecular symmetry and spectroscopy

(12) E* 1 1

1 -1

-1 -1

-1 1

What is the symmetry ofμA ?

(12)*

1

-1

1

-1

E 1

1

1

1

A1

A2

B1

B2

μA = ΣCre Ar

r

Charge on

particle r

A coordinate

of particle r

A = space-fixed X, Y or Z

EμA= ?μA (12)μA = ?μA E*μA= ?μA (12)*μA= ?μA


Molecular symmetry and spectroscopy

(12) E* 1 1

1 -1

-1 -1

-1 1

What is the symmetry ofμZ ?

(12)*

1

-1

1

-1

E 1

1

1

1

A1

A2

B1

B2

μA = ΣCre Ar

r

Charge on

particle r

A coordinate

of particle r

A = space-fixed X, Y or Z

EμA= +1μA (12)μA = +1μA E*μA= -1μA (12)*μA= -1μA

μA has symmetry A2


The symmetry labels of the cnpi group of h 2 o3

The Symmetry Labels of the CNPI Group of H2O

(12) E* 1 1

1 -1

-1 -1

-1 1

Γ(H) = A1

E 1

1

1

1

(12)*

1

-1

1

-1

Symmetry of H

R = ±

A1

A2

B1

B2

Symmetry of μA

Γ(μA) = A2

Using symmetry labels and the vanishing

integral theorem we deduce that:

∫Ψa*HΨbdτ = 0 if symmetry of Ψa*HΨb is not A1,

∫Ψa*μAΨbdτ = 0 if symmetry of ψa*μAψb is not A1,


The symmetry labels of the cnpi group of h 2 o4

The Symmetry Labels of the CNPI Group of H2O

(12) E* 1 1

1 -1

-1 -1

-1 1

Γ(H) = A1

E 1

1

1

1

(12)*

1

-1

1

-1

Symmetry of H

R = ±

A1

A2

B1

B2

Symmetry of μA

Γ(μA) = A2

Using symmetry labels and the vanishing

integral theorem we deduce that:

∫Ψa*HΨbdτ = 0 if symmetry of Ψa*HΨb is not A1,

that is, if the symmetry of

Ψa is not the same as Ψb

Pages 113-114

∫Ψa*μAΨbdτ = 0 if symmetry of ψa*μAψb is not A1,

that is, if the symmetry of the product ΨaΨbis not A2


Molecular symmetry and spectroscopy

Symmetry of rotational levels of H2O

b

JKaKc

a

c

Γrot

KaKc

e e A1

o o A2

e o B1

o e B2

Allowed transitions


Molecular symmetry and spectroscopy

  • So we can use the CNPI group to:

  • Symmetry label energy levels

  • and

  • 2. Determine which ODME vanish.

65

Ch. 7

Ch. 6


Molecular symmetry and spectroscopy

BUT BUT BUT...

There are problems with the CNPI Group

Number of elements in the CNPI groups of various

molecules

Huge groups. Size bears no relation to geometrical symmetry

C6H6, for example, has a 1036800-element CNPI group,

but a 24-element point group at equilibrium, D6h

Often gives SUPERFLUOUS multiple symmetry labels


Molecular symmetry and spectroscopy

PH3

3

2

1

There are two VERSIONS

of this molecule


Molecular symmetry and spectroscopy

PH3

TWO VERSIONS: Distinguished

by numbering the identical nuclei

Very, very

high

potential

barrier

2

~12000 cm-1

3

1

1

2

3

Bone et al., Mol. Phys., 72, 33 (1991)


Molecular symmetry and spectroscopy

The number of versions of the minimum is given by:

(order of CNPI group)/(order of point group)

For H2O this is 4/4 = 1

H

For H3+ this is 12/12= 1

C1

F

C2

For PH3 or CH3F this is 12/6 = 2

I

C3

For O3 this is 12/4 = 3

D

12/1 = 12

For HN3 this is 12/2 = 6


Molecular symmetry and spectroscopy

1 3 2

2 1 3

2 3 1

3 1 2

3 2 1

The six versions of HN3

H

N1 N2 N3

H

C1

F

C2

I

C3

D

12 versions


Molecular symmetry and spectroscopy

The number of versions of the minimum is given by:

(order of CNPI group)/(order of point group)

For H2O this is 4/4 = 1

H

For H3+ this is 12/12= 1

C1

F

C2

For PH3 or CH3F this is 12/6 = 2

I

C3

For O3 this is 12/4 = 3

D

12/1 = 12

For HN3 this is 12/2 = 6


Molecular symmetry and spectroscopy

The number of versions of the minimum is given by:

(order of CNPI group)/(order of point group)

For H2O this is 4/4 = 1

H

For H3+ this is 12/12= 1

C1

F

C2

For PH3 or CH3F this is 12/6 = 2

I

C3

For O3 this is 12/4 = 3

D

12/1 = 12

For HN3 this is 12/2 = 6

For C6H6 this is (6!x6!x2)/24

= 1036800/24 = 43200


Molecular symmetry and spectroscopy

Using the CNPI Group to symmetry

label the energy levels of a molecule

that has more than one version.

Character Table of CNPI group of PH3

E (123) (23) E* (123)* (23)*

(132) (31) (132)* (31)*

(23) (23)*

GCNPI

GCNPI

12 elements

6 classes

6 irred. reps


Molecular symmetry and spectroscopy

PH3 (or CH3F)

Using the CNPI Group

A1’

E’ + E’’

A1’’

A1’’ + A2’

A2’

A2’’

A1’ + A2’’

E’

E’’


Molecular symmetry and spectroscopy

PH3 (or CH3F)

Using the CNPI Group

A1’

E’ + E’’

A1’’

A1’’ + A2’

A2’

A2’’

A1’ + A2’’

E’

Why this

Degeneracy?

E’’


Molecular symmetry and spectroscopy

This double labeling results from the fact that there

are two versions of the PH3 molecule and the

tunneling splitting between these versions is not

observed.

For understanding the spectrum

this is a “superfluous” degeneracy

(particularly for CH3F)


Molecular symmetry and spectroscopy

The number of versions of the minimum is given by:

(order of CNPI group)/(order of point group)

For H2O this is 4/4 = 1

For PH3 or CH3F this is 12/6 = 2

For O3 this is 12/4 = 3

For C3H4 this is 288/8 = 36

For C6H6 this is 1036800/24 = 43200,

and using the CNPI group each energy level would

get as symmetry label the sum of 43200 irreps.

Clearly using the CNPI group gives

very unwieldy symmetry labels.

77


Molecular symmetry and spectroscopy

The CNPI Group approach

works, in principle, and can be used

to determine which ODME vanish.

BUT IT IS OFTEN HOPELESSLY UNWIELDY

In 1963 Longuet-Higgins figured out how to set up a sub-group of the CNPI Group that achieves the same result

without superfluous degeneracies.

This subgroup is called

The Molecular Symmetry (MS) group.


Molecular symmetry and spectroscopy

PH3 (or CH3F)

Ab initio calc with neglect of tunneling

Ab Initio CALC IN HERE

Very, very

high

potential

barrier

2

3

1

1

2

3

It would be

superfluous

to calc points

in other min

No observed tunneling through barrier


Molecular symmetry and spectroscopy

PH3 (or CH3F)

Only NPI OPERATIONS FROM IN HERE

Very, very

high

potential

barrier

2

3

1

1

2

3

No observed tunneling through barrier


Molecular symmetry and spectroscopy

PH3 (or CH3F)

Only NPI OPERATIONS FROM IN HERE

Very, very

high

potential

barrier

2

3

1

1

2

3

(12) superfluous

E* superfluous

(123), (12)* useful

No observed tunneling through barrier


Molecular symmetry and spectroscopy

The six feasible elements are

GMS ={E, (123), (132), (12)*, (13)*,(23)*}

If we cannot see any effects

of the tunneling through the

barrier then we only need

NPI operations for one

version. Omit NPI elements

that connect versions since

they are not useful; they are

superfluous.

superfluous

useful

superfluous

GCNPI={E,(12), (13), (23), (123), (132),

E*, (12)*, (13)*,(23)*, (123)*, (132)*}

For CH3F:

useful


Molecular symmetry and spectroscopy

The six feasible elements are

GMS ={E, (123), (132), (12)*, (13)*,(23)*}

If we cannot see any effects

of the tunneling through the

barrier then we only need

NPI operations or one

version. Omit NPI elements

that take us between

versions since they are not

useful; they are unfeasible.

If we cannot see any effects

of the tunneling through the

barrier then we only need

NPI operations for one

version. Omit NPI elements

that connect versions since

they are not useful; they are

unfeasible.

superfluous

For PH3

or CH3F:

GCNPI={E,(12), (13), (23), (123), (132),

E*, (12)*, (13)*,(23)*, (123)*, (132)*}


Molecular symmetry and spectroscopy

Character Table of CNPI group of PH3 or CH3F

GCNPI={E, (12), (13), (23), (123), (132),

E*, (12)*, (13)*,(23)*, (123)*, (132)*}

H

A1’

E (123) (23) E* (123)* (23)*

(132) (31) (132)* (31)*

(23) (23)*

GCNPI

GCNPI

12 elements

μA

6 irred. reps

A1”


Molecular symmetry and spectroscopy

Using the CNPI Group

E’ + E’’

A1’

E’ + E’’

A1’’

A1’’ + A2’

A2’

A2’’

A1’ + A2’’

E’

E’’

2 versions → sum or two irrep labels on each level


Molecular symmetry and spectroscopy

Character table of the MS group of PH3 or CH3F

E (123) (12)*

(132) (13)*

(23)*

H

A1

μA

A2


Molecular symmetry and spectroscopy

Using the MS Group

E

E

A1

A2

A2

E

A1

The MS group gives all the information we need

as long as there is no observable tunneling, i.e.,

as long as the barrier is insuperable.


Molecular symmetry and spectroscopy

Using CNPIG versus MSG for PH3

E’ + E’’

E

E’ + E’’

E

A1’’ + A2’

A2

A1’ + A2’’

A1

CNPIG

MSG

Can use either to determine if an ODME vanishes.

But clearly it is easier to use the MSG.


Molecular symmetry and spectroscopy

superfluous

Unfeasible elements of the CNPI group

interconvert versions that are separated

by an insuperable energy barrier

useful

The subgroup of feasible elements forms a group called

THE MOLECULAR SYMMETRY GROUP

(MS GROUP)

89


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