MOLECULAR SYMMETRY AND SPECTROSCOPY

MOLECULAR SYMMETRYAND SPECTROSCOPY

[email protected]

P. R. Bunker and Per Jensen:

Molecular Symmetry and Spectroscopy,

2nd Edition, 3rd Printing,

NRC Research Press, Ottawa, 2012.


Fundamentals of Molecular Symmetry,

Taylor and Francis, 2004.

Learning about molecular symmetry from books

C$ 23.95








Learning about molecular symmetry from books

C$ 23.95

Chapter 1 in here

I0(ν) molecules Itr(ν)

Absorption spectrum: a plot of transmittance vs. ν

frequency

Transmittance = Itr(ν)/I0(ν)

(number of waves per cm)frequency




f

iM

hνif = Ef – Ei = ΔEif

νif

Absorption can only occur at resonance

E = Einternal

= Erve-spin

EACH “LINE’’ SIGNALS A RESONANCE;

Molecule undergoes a “transition.’’




f

iM


νif


E = Einternal

= Erve-spin

EACH “LINE’’ SIGNALS A RESONANCE;

Molecule undergoes a “transition.’’

[“Weak radiation”: Transmittance ≠ f(I0)]


Absorption spectrum: a plot of transmittance vs. ν, ν or λ ~

~ ~

ν = 1/ λ ~ = ν /c

frequency wavenumber wavelength

cm-1

Transmittance = Itr(ν)/I0(ν) ~ ~

(number of waves per cm)




f

iM


νif


~ ~

~ ~

E = Einternal

= Erve-spin

hνif = Ef – Ei = ΔEif ν = 1/ λ ~ = ν /ccm-1




f

iM


νif


~ ~

~ ~

E = Einternal

= Erve-spin

hνif = Ef – Ei = ΔEif ν = 1/ λ ~ = ν /ccm-1

νif /c = Ef /hc - Ei /hc = ΔEif /hc

Divide by hc

wavenumber

Term values

Term value difference

cm-1

cm-1

cm-1

I0(ν) moleculesconc = c* Itr(ν) ~ ~

Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]~ ~ ~

Absorption coefficient

hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f

iM

νif

Pierre_Bouguer

Beer-Lambert Law (weak radiation):

ℓ

http://en.wikipedia.org/wiki/Pierre_Bouguer






iM

νif

Pierre_Bouguer

Beer-Lambert Law (weak radiation):

ℓ

http://www.google.com/

(youtube feynman names)










iM

νif

Pierre_Bouguer

The exponential attenuation law (weak radn):

ℓ






I(f ← i) = ∫8π3 Na______

(4πε0)3hcF(Ei ) S(f ← i) = νif

~line

ε(ν)dν~ ~

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


iM

νif

Pierre_Bouguer

ℓ

Frequency Factor[1 – exp(-hυif /kT)]

Boltzmann Factor Line Strength

Stimulated Emission Factor

Rstim(f→i)

| ∫ Φf* μA Φi dτ |2 ∑A=X,Y,Z

gi exp [-Ei /kT ]

∑j gj exp [-Ej /kT ]

The exponential attenuation law (weak radn):



This depends on the wavefunctions that describe eachof the levels involved in the transition. They occur in theelectric dipole transition moment integral.

μA is the component of the molecular electric dipolemoment along the space fixed A (= X, Y or Z) axis:

μA = Σ Cre Arr

Charge onparticle r

A coordinateof particle r

S(f ← i) = | ∫ Φf* μA Φi dτ |2 ∑A=X,Y,Z

The Line Strength

Oscillating E and B fields contribute to the energy of emradiation. Above we have given the intensity of resonantlyabsorbed electric field energy in terms of the electric dipoletransition moment.

Molecules also absorb magnetic field energy; there is amagnetic dipole transition moment. But typically mdtm ≈ 10-5 edtm. Usually ignored, just as we ignore the electric quadrupole tm.

However sometimes we need the mdtm or the eqtm.ESR and NMR are magnetic dipole transitions. Electric field energy absorbed by low pressure H2 gas is elec. quadrupole.

AN ASIDE

Here ends the summary of Ch.1 15

Proportional to S(f ← i)

hνif = Ef – Ei = ΔEif POSITION: At a resonancef

iM

νif

INTENSITY: Line strength

μA = Σ Cre ArrS(f ← i) = | ∫ Φf* μA Φi dτ |2

~

∑A=X,Y,Z

CAN SIMULATE A SPECTRUM BY

CALC OF E and Φ

Quantum mechanics

Ch.2

Quantum mechanics

Schrödinger equation

For discussion of full Hincluding spin terms see pages 126-131

For spin-free H see pages 26-29 and 32-33

Q.M. Wave

Quantum mechanics

For discussion of full Hincluding spin terms see pages 126-131

For spin-free H see pages 26-28 and 32-33

Spin-free H is Hrve (rovibronic H)

The spin-free (rovibronic) Hamiltonian

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 thenselectively correct for the approximations made.

(after separating translation)

Off-diagonal matrix elements (ODME)

An pxp matrix A has elements Amn

A11 A12 A13 … A1p

A21 A22 A23 … A2p

A31 …………… A3p

A41

.

.Ap1 …………… App

Diagonalelements

Off-diagonalelements

A matrix is ‘diagonal’ if all off-diagonal elements are zero

Quantum mechanics and off-diagonal matrix elements

(ODME) Schrödinger equation

Eigenvalues and eigenfunctions are found by diagonalization of a matrix with elements

Eigenfunctions of an approximate H are “basis” functions

Exact Ej and ψj are obtained from ψn0 by

p 21-26

ODME has m ≠ n; perturbation Problem 2.5on page 41

Ch 2

0 0 0 0

ψm0

ψn0

Em0

En0

APPROXIMATE OR ZEROTHORDER SITUATION

Δmn0 = Em

0 - En0

Effect of nonvanishing ODME

ψm0

ψn0

Δmn0 = Em

0 - En0

Em

En

- - - - -

- - - - -

+δ

-δ

δ ~ Hmn2 / Δmn

0

,ψm

,ψn

ψm ~ ψm0 + [Hmn /Δmn

0 ] ψn0

Em0

En0

Using PerturbationTheory

ψelec ψvib ψrot ψns

Born-Oppapprox

NeglectCent. DistCoriolis

NeglectOrtho-para

mixing

MolecularOrbitals

Harmonicoscillator

Rigidrotor

Uncoupledspins

Ψe-v-r-ns0

(espin and Slater determinants)

ODME of H

μfi = ∫ (Ψf )* μA Ψi dτ

ODME of μA

To simulate a spectrum we need energies and line strengths i.e. weneed to calculate:

The Role of ApproximationsTo solve the Sch. Eq. we make approximations and then correct for them. In so doing we introduce concepts thatenable us to UNDERSTAND molecules.Key approximationis the BORN-OPPENHEIMER APPROXIMATION

Concepts that come about because we make approximations:

Electronic state, molecular orbital, electronic configuration,potential energy surface, molecular structure, force constants,electronic angular momentum, vibrational state, vibrational angular momentum, rotational state, Coriolis coupling, centrifugal distortion,…

We could call the approach that aims to solve the Sch. Eq. numerically without approximations using a gigantic computer “The few-concepts big-computer” approach.

BO Approx. Pages 43-47

When I am grumpy and sarcastic I like to call it “The small-brain big-computer approach.”

I(f ← i) = ∫8π3 Na______

(4πε0)3hcF(Ei ) S(f ← i) Rstim(f→i) = νif

~line

ε(ν)dν~ ~

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


iM

νif

ODME of H

μfi = ∫ (Ψf )* μA Ψi dτODME of μA

Use Q. Mech. to calculate:

Here ends the summary of Chapters 1 and 2








To buy it go to:

http://www.crcpress.comDownload pdf file from

www.chem.uni-wuppertal.de/prb

Chapter 1 (Spectroscopy)Chapter 2 (Quantum Mechanics) andSection 3.1 (The breakdown of the BO Approx.)

The first 47 pages:

28

http://www.crcpress.com/

http://www.crcpress.com/

http://www.chem.uni-wuppertal.de/prb

Now for Molecular Symmetry


Wikipedia molecular symmetry




Now for Molecular Symmetry


Wikipedia molecular symmetry

No discussion of the fundamentals of how or why the geometrical symmetry of the equilibrium structureof a moleule in a particular electronic state allows oneto do the things it says you can do. No discussion ofwhat the “rotations” and “reflection” symmetryoperations do to molecular coordinates.




Molecular Symmetry(as it should be explained)

Knowing the symmetry labels it is easy to determine which ODME of H

and have to be zero.

An important use of symmetry is to put “symmetry labels” on the zeroth

order energy levels.

μA

Molecular Symmetry(as it should be explained)

GROUP THEORYFERMI:“Just a bunch of

definitions”

andPOINT GROUPS

(uses geometrical symmetry)

The Point Group of H2O

z

x

(+y)

The point group of H2O consists of the four symmetry operations

E, C2x, xz, and xy

It is called the C2v point group

The identityoperation

Examples of point group symmetry

H2O

CH3F

C60

C3H4

C2v

C3v

D2d

Ih120 symmetry operations

z

x

(+y)σxz

C2x

z

x

(+y)

z

x

(+y)

+

+

-

σxy

Successive application = “multiplication”

σxy C2xσxz=

+ -

+

z

x

(+y)

+


σxy C2xσxz=

+

C2v multiplication table

E C2x σxz σxy

E E C2x σxz σxy

C2x C2x E σxy σxz

σxz σxz σxy E C2x

σxy σxy σxz C2x EA “Group” contains all

products of its membersand it contains E

z

x

(+y)

+


σxy C2xσxz=

+

E C2x σxz σxy

E E C2x σxz σxy

C2x C2x E σxy σxz


σxy σxy σxz C2x EIS NOT A GROUP

A “Group” contains allproducts of its members

and it contains E E C2x σxz{ }

z

x

(+y)

+


E C2x C2x=

+

C2 multiplication table

E C2x σxz σxy

E E C2x σxz σxy

C2x C2x E σxy σxz


σxy σxy σxz C2x ETHE C2 GROUP – A SUBGROUP OF C2v

A “Group” contains allproducts of its members

and it contains E

PH3 at equilibrium: The C3v point group

Symmetry operations:C3v = {E, C3, C3

2, 1, 2, 3 }

3

12

Multiplication table for C3v

C32 = σ2σ1

C3 = σ1σ2

Multiplication is not necessarily commutative

A matrix group

1 0

0 1

21

23

23

21

21

23

23

21

1 0

0 1

21

23

23

21

21

23

23

21

´M1 = M4 =

´ M5 =

´ M6 =

´M2 =

´M3 =

M1

M2

M3

M4

M5

M6

M1

M1

M2

M3

M4

M5

M6

M2

M2

M3

M1

M6

M4

M5

M3

M3

M1

M2

M5

M6

M4

M4

M4

M5

M6

M1

M2

M3

M5

M5

M6

M4

M3

M1

M2

M6

M6

M4

M5

M2

M3

M1

Multiplication table for the matrix group

M3 = M5 M4

E C3 C32 σ1 σ2 σ3

This matrix group forms a “representation” of the C3v group

Multiplication tableshave the ‘same shape’

M1 M2 M3 M4 M5 M6

C32 = σ2σ1

M3 = M5 M4

Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations

E C3 C32 σ1 σ2 σ3



M1 M2 M3 M4 M5 M6

C32 = σ2σ1

M3 = M5 M4


E C3 C32 σ1 σ2 σ3



M1 M2 M3 M4 M5 M6

C32 = σ2σ1

M3 = M5 M4


E C3 C32 σ1 σ2 σ3

This matrix group forms an “irreducible representation” of the C3v group


M1 M2 M3 M4 M5 M6

C32 = σ2σ1

M3 = M5 M4

The characters of this irreducible rep.

1 0

0 1

21

23

23

21

21

23

23

21

1 0

0 1

21

23

23

21

21

23

23

21

´M1 = E M4 = 1

´ M5 = 2

´ M6 = 3

´M2 = C3

´M3 = C32

C3v C3v

2

-1

-1

0

0

0

Characters of the 2d matrix irrep of C3v

E (123) (12)

1 2 3

A1 1 1 1

A2 1 1 1

E 2 1 0

The 2D representation M = {M1, M2, M3, ....., M6}of C3v is the irreducible representation E. In thistable we give the characters of the matrices.

E C3 σ1

C32 σ2

σ3

Elements in the same class have the same characters

Two 1Dirreduciblerepresentationsof the C3v group

3 classes and 3 irreducible representations

Character table for the point group C3v

E (123) (12)

1 2 3

A1 1 1 1

A2 1 1 1

E 2 1 0

The 2D representation M = {M1, M2, M3, ....., M6}of C3v is the irreducible representation E. In thistable we give the characters of the matrices.

E C3 σ1

C32 σ2

σ3

Elements in the same class have the same characters

Two 1Dirreduciblerepresentationsof the C3v group


The matrices of the A1 + E reducible rep.

21

23

23

21

21

23

23

21

1 0

0 1

21

23

23

21

21

23

23

21

´M1 = E M4 = 1

´ M5 = 2

´ M6 = 3

´M2 = C3

´M3 = C32

A1 + E C3v A1 + E C3v

1 0 00 1 00 0 1

1 0 00

0

1 0 00

0

1 0 00

0

1 0 000

1 0 00

0

‘

‘

‘

‘

‘

‘

A similarity transformation would make the ‘origin’ of each of these reducible reps lessobvious…..but the characters would give theirreduction away!

A-1Mi’A = Ni

(A-1Mi’A)(A-1Mj’A) = A-1Mi’(A A-1)Mj’A

= A-1Mi’Mj’A

NiNj =

The matrices N form an equivalent representationand the character of Ni is the same as that of Mi’Both representations reduce to A1 + E.

‘

The matrices of the A2 + E reducible rep.

21

23

23

21

21

23

23

21

1 0

0 1

21

23

23

21

21

23

23

21

´M1 = E M4 = 1

´ M5 = 2

´ M6 = 3

´M2 = C3

´M3 = C32

A2 + E C3v A2 + E C3v

1 0 00 1 00 0 1

-1 0 00

0

1 0 00

0

1 0 00

0

-1 0 000

-1 0 00

0

‘

‘

‘

‘

‘

‘

‘

‘

‘

‘

‘

‘

Irreducible representationsThe elements of irrep matrices satisfy the„Grand Orthogonality Theorem“ (GOT).

We do not discuss the GOT here, but we list threeconsequences of it: • Number of irreps = Number of classes in the group.

• Dimensions of the irreps, l1, l2, l3 … satisfy

l12 + l2

2 + l32 + … = h,

where h is the number of elements in the group.

• Orthogonality relation

Character table for the point group C2v

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 C2 σyz σxy


z

x

(+y)

“Group”

A set of operations that is closed wrt “multiplication”

“Point Group”

All rotation, reflection and rotation-reflection operationsthat leave the molecule (in its equilibrium configuration)“looking” the same.

“Matrix group”

A set of matrices that forms a group.

“Irreducible representation”

An irreducible matrix group that maps onto the group.

Definitions a la Fermi:

As we shall see later, these are used as ‘‘symmetry labels‘‘ on energy levels.

We can use these symmetry labels to determine which ODME of H and μA

are zero.

IRREDUCIBLE REPRESENTATIONS

The “why” and the “how” will be explained

That’s it for Point Groups

Ch.6(15 pages)

This is where chemistry courses describing the Fundamentals of molecular symmetry usually end


Ch.6(15 pages)


59

PAUSEGroups, Point Groups, Matrix Groups and Irreducible Representations.


Ch.6(15 pages)


THERE CAN BE PROBLEMS IF WE TRY TO USE POINT

GROUP SYMMETRY.

BUT

How do we use point group symmetry when a molecule rotates and distorts?

H3+

D3h


H3+

D3h C2v


H3+

D3h C2vCH4 also

2

3

113

2

How do we use point group symmetry when the molecule tunnels?

NH3

C3vD3h

What are the symmetriesof B(CH3)3, CH3.CC.CH3, (CO)2, (NH3)2,(C6H6)2,…?

Nonrigid molecules (i.e. moleculesthat tunnel) are a problemif we try to use a point group.

What should we do if we study transitions (or interactions) between electronic states that have different point group symmetries at equilibrium?

Also

Further

People claim to be able to use thepoint group (or its rotational subgroup) to determine nuclear spin statistical weights. Thisis not really possible since point group operations do not just permute nuclei.

Point groups used for classifying:

The electronic states for any moleculeat a fixed nuclear geometry and

The vibrational states for molecules,called “rigid” molecules, undergoing infinitesimal vibrations about a unique equilibrium structure.

Rotations andreflections

Permutationsand the inversion

J.T.Hougen, JCP 37, 1422 (1962); ibid, 39, 358 (1963)H.C.Longuet-Higgins, Mol. Phys., 6, 445 (1963)P.R.B. and Per Jensen, JMS 228, 640 (2004) [historical introduction]

To understand how we use symmetrylabels and where the point groupgoes wrong we must studywhat we mean by “symmetry”

70

Symmetry not from geometry since molecules are dynamic

• Centrifugal distortion

eg. H3+ or CH4 dipole

moment

• Nonrigid molecules: eg. ethane, ammonia, (H2O)2, (CO)2,…

• Breakdown of BOA: eg. HCCH* - H2CC

Also symmetry appliesto atoms, nuclei and fundamental particles. Geometrical point group symmetry is not possible for them.

We need a more general definition of symmetry

We define symmetry operationsas being those operations that leave the

energy of the molecule unchanged.

Doing this allows us to treat molecular symmetryon the same footing as it is used elsewhere in Physics.It makes it easy to see how energy levels are labelled.

And finally it allows us to understand what Point Group symmetry operations really do to a molecule

and how they too also involve energy invariance.

72

• Uniform Space-----------Translation• Isotropic Space----------Rotation • Identical electrons------Permute electrons• Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q,s,I) (-p,-q,s,I)

Symmetry Operations (energy invariance)

We might say something about Charge Conjugation and Time Reversal (p,q,s,I) (-p,q,-s,-I)at the end

• Uniform Space-----------Translation• Isotropic Space----------Rotation • Identical electrons-------Permute electrons• Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q) (-p,-q) E*

Symmetry Operations (energy invariance)

CNPI group = Complete Nuclear Permutation Inversion Group

EXAMPLE:The CNPI group for H2O is {E, (12),E*,(12)*}

Permutation of protons inversion (12)* = (12)E* = E*(12)

The Complete Nuclear Permutation Inversion (CNPI) group

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

H1 H2

O e+

H2 H1

O e+ H1 H2

Oe-(12) E*

(12)*

Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinates

E* Inverts coordinates of nuclei and electrons.Does not change spins.

The CNPI Group for the Water Molecule



H1 H2

O e+

H2 H1

O e+ H1 H2

Oe-(12) E*

(12)*




CNPI group of OCO?



H1 H2

O e+

H2 H1

O e+ H1 H2

Oe-(12) E*

(12)*




CNPI group of H2?



H1 H2

O e+

H2 H1

O e+ H1 H2

Oe-(12) E*

(12)*




CNPI group of CH2D13CFClBr?

The CNPI group of PH3

1

2

3

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

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

N! ways of permuting N identical nuclei

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

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

1 is replaced by 22 is replaced by 33 is replaced by 1

The CNPI group of O3

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

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


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


The ozonemolecule

O OO

1 3

2

The CNPI group of HD12C=13CH2

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

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


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


H H12C 13C

D H

1 2

3

I

F

H

D

SubstitutedAllene molecule C1

C2

C3

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

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

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


The CNPI Group of a Substituted Allene Molecule

The same CNPI group as for PH3, O3, H3+ and HD12C=13CH2

GCNPI = {E, (12), (13), (23), (123), (132)} x{E, (45)} x {E, E*}

(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

The CNPI Group of a slightly less substituted Allene Molecule

I

H5

D

C1

C2

C3

H4

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

How many elements in the CNPI groups of SF6, C2H6, C6H6, C6H5CH3, C60

(use ! In answers)

I

H5

D

C1

C2

C3

H4

Number of elements in CNPI group = 3! x 2! x 2

CH3FNumber of elements in CNPI group = 3! x 2

C3H2ID

The number of elements in a group is called the “order” of the groupand is denoted by the letter “h”

CNPI GROUPDONE

Ch.7

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 43200C6H6

86This number means something!

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MOLECULAR SYMMETRY AND SPECTROSCOPY