H ν if = E f – E i = ΔE if S(f ← i) = ∑ A | ∫ Φ f * μ A Φ i dτ | 2 ODME of H and μ A...

hνif = Ef – Ei = ΔEif

S(f ← i) = ∑A | ∫ Φf* μA Φi dτ |2

ODME of H and μA μfi =

Spectroscopy

Quantum Mechanics

f

iMMMM

M

∫ Φf* μA Φi dτ

Group Theory and Point Groups

C2v = {E,C2x,σxz,σxy}z

x E (12) E* (12)*

A1 1 1 1 1

A2 1 1 1 1

B1 1 1 1 1

B2 1 1 1 1

E C2x σxz σxy

The Complete Nuclear Permutation Inversion Group

CNPI Group of C3H4 has 3! x 4! x 2 = 288 elementsallene

+y

(12) E* 1 1

1 -1 -1 -1 -1 1

E 1111

(12)* 1 -1 1-1

A1

A2

B1

B2

R = ±

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

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

∫ΨaμΨbdτ = 0 if symmetry of product is not A1

ψ is a wavefunction of A2 symmetry ψ generates the A2 representation

Eψ=+1ψ (12)ψ =+1ψ E*ψ=-1ψ (12)*ψ=-1ψ

RH = HR

For example:

EIGENFUNCTIONS TRANSFORM IRREDUCIBLY

Labelling Energy Levels Using CNPIG Irreps

R2 = E

(12) E* 1 1

1 -1 -1 -1 -1 1

E 1111

(12)* 1 -1 1-1

A1

A2

B1

B2

Symmetry of μA

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

i.e., if the symmetry of the product ΨaΨb is not = symm of μA

Symmetry of H

Using symmetry labels and the vanishingintegral theorem we deduce that:

∫Ψa*HΨbdτ = 0 if symmetry of Ψa*HΨb is not A1,i.e., if the symmetry of Ψa is not the same as Ψb

The Vanishing Integral Theorem∫f(τ)dτ = 0 if symmetry of f(τ) is not A1

ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8

Ψ1

Ψ2

Ψ3

Ψ4

Ψ5

Ψ6

Ψ7

ψ8

A1 A2 B1

A1

A2

B1

0 0

000 0

0 0 0 0 0 0 0 0

0

0

0

0

0

0

0

0

. . .

. . .

. . .. . .. . .. . .

. .

. .

ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8

Ψ1

Ψ2

Ψ3

Ψ4

Ψ5

Ψ6

Ψ7

ψ8

A1 A2 B1

A1

A2

B1

0 0

000 0

0 0 0 0 0 0 0 0

0

0

0

0

0

0

0

0

. . .

. . .

. . .. . .. . .. . .

. .

. .

AllowedTransitions

A1 ↔ A2

B1 ↔ B2

A1

A2

B1Connectedby A2

Like using a big computer to calculate E from exact H, we can, in principle, use

the CNPI Group to label energy levels and to determine which ODME vanish

for any molecule

Using the CNPI Groupfor any molecule

BUT there is one big problem: Superfluous symmetry labels

Character Table of CNPI group of CH3F

12 elements 6 classes

6 irred. reps

GCNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)*

GCNPI

Using the CNPI Group approach for CH3F

μA A1”H A1’

A1’

A1’’

A2’

A2’’

E’

E’’

E’ + E’’

A1’’ + A2’

A1’ + A2’’

The CNPI Group approachCH3F

Allowedtransitionsconnectedby A1”Block diagonal H-matrix

E’ + E’’

A1’

A1’’

A2’

A2’’

E’

E’’

A1’’ + A2’

A1’ + A2’’

CH3F

Allows for tunnelingbetween two

VERSIONS

The CNPI Group approach

E’ + E’’

E’ + E’’

2

3113

2

Very, veryhighpotentialbarrier

No observed tunneling through barrier

CH3F

F F

TWO VERSIONS

The number of versions of the minimum is given by: (order of CNPI group)/(order of point group)

For CH3F this is 12/6 = 2

For benzene C6H6 this is 1036800/24 = 43200,and using the CNPI group each energy level wouldget as symmetry label the sum of 43200 irreps.

Clearly using the CNPI group givesvery unwieldy symmetry labels.

3! x 2 C3v group has 6 elements

2

3113

2

Very, veryhighpotentialbarrier

No observed tunneling through barrier

Only NPI OPERATIONS FROM IN HERE

CH3F

(12) superfluousE* superfluous(123),(12)* useful

F F

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

E*, (12)*, (13)*,(23)*, (123)*, (132)*}For CH3F:

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

The six feasible elements are

superfluous

superfluous

useful

useful

If we cannot see any effectsof the tunneling through thebarrier then we only needNPI operations for one version. Omit NPI elementsthat connect versions since they are not useful; they are

superfluous.

Character table of the MS group of CH3F

E (123) (12)

1 2 3

A1 1 1 1

A2 1 1 1

E 2 1 0

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

μA A2

H A1

(12),(13),(23)E*,(123)*,(132)*are unfeasibleelements of theCNPI Group

Use this group to block-diagonalize Hand to determine which transitions areforbidden

Using CNPIG versus MSG for PH3

E

A2

A1

Can use either to determine if an ODME vanishes.But clearly it is easier to use the MSG.

E’ + E’’

A1’’ + A2’

A1’ + A2’’

CNPIG MSG

E’ + E’’ E

The subgroup of feasible elements forms a group called

THE MOLECULAR SYMMETRY GROUP(MS GROUP)

Unfeasible elements of the CNPI groupinterconvert versions that are separated

by an insuperable energy barrier

Superfluous

useful

----------- End of Lecture Two -------

16

We are first going to set up the Molecular Symmetry Group for several non-tunneling (or “rigid”)molecules. We will notice a strange“resemblance” to the Point Groupsof these molecules. We will examinethis “resemblance” and show how ithelps us to understand Point Groups.

C2v

{E C2x σxz σxy}

C2v(M) elements:

{E, (12), E*, (12)*}

z

x

(+y)

1 2

H2O

elements:

E (12) E* (12)*

A1 1 1 1 1

A2 1 1 1 1

B1 1 1 1 1

B2 1 1 1 1

C2v E C2x σxz σxy E (12) E* (12)*

A1 1 1 1 1

A2 1 1 1 1

B1 1 1 1 1

B2 1 1 1 1

C2v(M)

z

x

(+y)

1 2

H2O

The C2v and C2v(M) character tables

E (123) (12)

1 2 3

A1 1 1 1

A2 1 1 1

E 2 1 0

E (123) (12)

1 2 3

A1 1 1 1

A2 1 1 1

E 2 1 0

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

E C3 σ1

C32 σ2

σ3

C3v(M) C3v

1

2

3

CH3F

The C3v and C3v(M) character tablesF

H N1 N2 N3

MSG is {E,E*}

E E*A’ 1 1A” 1 -1

All P and P* are unfeasible (superfluous)

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

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

HN3 has6 versions12/2 = 6

3!x2 Cs

H N1 N2 N3

MSG is {E,E*}

E E*A’ 1 1A” 1 -1

All P and P* are unfeasible (superfluous)

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

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

HN3 has6 versions12/2 = 6

3!x2 Cs

PG is {E,σ}

E σA’ 1 1A” 1 -1

H3+

Point group also has 12 elements: D3h

3

1

2

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

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

Therefore only 1 version and MSG = CNPIG

+

Character Table of MS group H3+

12 elements 6 classes

6 irred. reps

GCNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)*

GCNPI

Character table for D3h point group

E 2C3 3C'2 σh 2S3 3σvlinear,

rotationsquadratic

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)

Number of elements in CNPIG = 3! x 4! x 2 = 288

H5

C1

C2

C3

H4

H7

H6

The MSG of allene is:

{E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*,

(4657)(13)*, (4756)(13)*}

The allene molecule C3H4

Point group is D2d has 8 elements

Thus there are 288/8 = 36 versions


H5

C1

C2

C3

H4

H7

H6

The MSG of allene and PG are: E C2 C2’ C2’ σd σd

{E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*, S4 S4

(4657)(13)*, (4756)(13)*}

The allene molecule C3H4

Point group is D2d has 8 elements

Thus there are 288/8 = 36 versions

Character table for D2d point group

E 2S4 C2 (z) 2C'2 2σdlinear,rotations

quadratic

A1 1 1 1 1 1 x2+y2, z2

A2 1 1 1 -1 -1 Rz

B1 1 -1 1 1 -1 x2-y2

B2 1 -1 1 -1 1 z xy

E 2 0 -2 0 0 (x, y) (Rx, Ry) (xz, yz)29

USE OF GROUP THEORY ANDSYMMETRY IN SPECTROSCOPY

Point Group

CNPI GroupMS Group

Spoilt by rotation and tunneling

Superfluoussymmetry ifmany versions

RH=HR, thereforecan symmetry labelenergy levels

Use moleculargeometry

Use energyinvariance RH = HR

USE OF GROUP THEORY ANDSYMMETRY IN SPECTROSCOPY

Point Group

CNPI GroupMS Group

Spoilt by rotation and tunneling

Superfluoussymmetry ifmany versions

RH=HR, thereforecan symmetry labelenergy levels

Use moleculargeometry

Use energyinvariance RH = HRFor rigid (nontunneling)

molecules MSG and PGare isomorphic. Leads toan understanding of PGsand how they can label energy levels.

Black are instantaneouspositions in space. White are equilibrium positions.N.B. +z is 1→2.

Then do (12). Note that axes have moved.Rotational coordinates aretransformed by MS group.

Undo the permutationof the nuclear spins

Undo the permutationof the nuclear spins

We next undo theRotation of the axes

only evcoords

only evcoords

only rotcoords

only nspincoords

MS Group and Point Group of H2O

E = p0 R0 E

(12) = p12 Rx C2x

E* = p0 Ry xz

(12)* = p12 Rz xy

MS Group Point GroupC2v(M)C2v(M) C2vC2v

MS Group and Point Group of H2O

E = p0 R0 E

(12) = p12 Rx C2x

E* = p0 Ry xz

(12)* = p12 Rz xy

MS Group Point GroupC2v(M)C2v(M) C2vC2v

Jon Hougenand what theMSG is calledby somepeople

The point group can be used to symmetrylabel the vibrational and electronic states of non-tunneling molecules.

RPG Hev = Hev RPG

Point group operations rotate/reflectelectronic coords and vib displacementsin a non-tunneling (rigid) molecule.

47

The rotational and nuclear spincoordinates are not transformedby the elements of a point group

The MSG can be used to classifyThe nspin, rotational, vibrational

and electronic states of any molecule, including those

molecules that exhibit tunneling splittings (“nonrigid” molecules).

WE USE THE ETHYL RADICAL AS AN EXAMPLE OF A NONRIGID MOLECULE

The ethyl radical


The MSG of internally rotating ethyl is thefollowing group of order 12:{E, (123),(132),(12)*,(23)*,(31)*,(45),(123)(45),(132)(45),(12)(45)*,(23)(45)*,(31)(45)*}

H4

Cb

H5

a

The MSG of rigid (nontunneling) ethyl is {E,(23)*}

There are still plenty of unfeasible elementssuch as (14), (23), (145), E*, etc.

MS group can change with resolution and temperature

Recent theoretical calcs bySousa-Silva, Polyanski, Yurchenko and Tennyson

From UCL, UK.Yield the tunneling splittings given on the next slide

Schwerdtfeger, Laakkonen and Pekka Pyykkö,J. Chem. Phys., 96, 6807 (1992)

Barrier height = 12300 cm-1

THE PH3 MOLECULE AS AN EXAMPLE

7.2 cm-1

0.7750.1450.0230.0017 cm-1

15 MHz 2 MHz240 kHz 15 kHz600 Hz 12 Hz

D3h(M)

C3v(M)

CNPIG

Tunneling splittings in NH3

D3h(M)

MS group can change with resolution

and temperature

For NH3 MSG is usually D3h(M)

For PH3 MSG is usually C3v(M)

The subgroup of feasible elements form a group calledTHE MOLECULAR SYMMETRY GROUP

Unfeasible elements of the CNPI groupinterconvert versions that are separated

by an insuperable energy barrier

MS group is used to symmetry label the rotational, ro-vibrational, rovibronic andnuclear spin states of any molecule.

MS group can change with resolution and T

SUMMARY 54

H ν if = E f – E i = ΔE if S(f ← i) = ∑ A | ∫ Φ f * μ A Φ i dτ | 2 ODME of H and μ A...

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