# MOLECULAR SYMMETRY AND SPECTROSCOPY

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MOLECULAR SYMMETRYAND [email protected] ppt file from

http://www.few.vu.nl/~rickAt bottom of page

I(f i) = 83 Na______(40)3hcF(Ei )S(f i) Rstim(fi) =if~line()d~~Integrated absorption coefficient (i.e. intensity) for a line is:hif = Ef Ei = Eif Absorption can only occur at resonancefiMif ODME of Hfi = (f )* A i dODME of AUse Q. Mech. to calculate:Chapters 1 and 2. Spectroscopy and Quantum MechanicsWe began by summarizing

P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry,Taylor and Francis, 2004. P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy, 2nd Edition, 3rd Printing, NRC Research Press, Ottawa, 2012.To buy it go to:http://www.crcpress.comDownload pdf file from

www.chem.uni-wuppertal.de/prbChapter 1 (Spectroscopy)Chapter 2 (Quantum Mechanics) andSection 3.1 (The breakdown of the BO Approx.)The first 47 pages:

P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry,Taylor and Francis, 2004. P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy, 2nd Edition, 3rd Printing, NRC Research Press, Ottawa, 2012.To buy it go to:http://www.crcpress.comDownload pdf file from

www.chem.uni-wuppertal.de/prbChapter 1 (Spectroscopy)Chapter 2 (Quantum Mechanics) andSection 3.1 (The breakdown of the BO Approx.)The first 47 pages:We then proceeded to discuss GroupTheory and Point Groups

GroupA set of operations that is closed wrt multiplicationPoint GroupAll rotation, reflection and rotation-reflection operationsthat leave the molecule (in its equilibrium configuration)looking the same.Matrix groupA set of matrices that forms a group.Irreducible representationA representation that cannot be written as the sumof smaller dimensioned representations.Definitions for groups and point groups:RepresentationA matrix group having the same shaped multiplicationtable as the group it represents.Character tableA tabulation of the characters of the irreducible representations.

Character table for the point group C3vThe 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 1 C32 2 3Elements in the same class have the same charactersTwo 1Dirreduciblerepresentationsof the C3v group3 classes and 3 irreducible representations

E

(123)

(12)

1

2

3

A1

1

1

1

A2

1

1

(1

E

2

(1

0

Character table for the point group C2v

E C2 yz xy4 classes and 4 irreducible representationszx(+y)

E

(12)

E*

(12)*

A1

1

1

1

1

A2

1

1

(1

(1

B1

1

(1

(1

1

B2

1

(1

1

(1

hif = Ef Ei = Eif S(f i) = A | f* A i d |2 ODME of H and Afi = Spectroscopy Quantum MechanicsGroup Theory and Point GroupsPH3fiMMMMM f* A i d (Character Tables and Irreducible Representations)8

Point Group symmetry is based onthe geometrical symmetry of theequilibrium structure.Point group symmetry not appropriate when there is rotation or tunnelingUse energy invariance symmetryinstead. We start by using inversionsymmetry and identical nuclear permutation symmetry.

The Complete Nuclear Permutation Inversion (CNPI) GroupContains all possible permutationsof identical nuclei including E. It also contains the inversion operation E*and all possible products of E* withthe identical nuclear permutations.GCNPI = GCNP x {E,E*}

The spin-free (rovibronic) HamiltonianVee + VNN + VNeTHE GLUEIn 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)

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)*} Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinatesE* Inverts coordinates of nuclei and electrons.Does not change spins.Same CNPI group for CO2, H2, H2CO, HOOD, HDCCl2,

IFHDC1C2C3OOO132H H12C 13CD H

123132132FH N1N2N3H1H2H3+GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*} = GCNP x {E, E*}

GCNPI={E, (12), (13), (23), (123), (132), E*, (12)*, (13)*,(23)*, (123)*, (132)*}GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*}Number of elements = 3! x 2 = 6 x 2 = 12Number of ways of permutingthree identical nuclei

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 = 24The CNPI Group of C3H2ID = {E, (12), (13), (23), (123), (132),

Number of elements = 3! x 2! x 2 = 6 x 2 x 2 = 24If there are n1 nuclei of type 1, n2 oftype 2, n3 of type 3, etc then the totalnumber of elements in the CNPI groupis n1! x n2! x n3!... x 2.

H5TheAllene moleculeC1C2C3H4Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288H7H6C3H4The CNPI group of allene

H5TheAllene moleculeC1C2C3H4Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288H7H6C3H4The CNPI group of alleneSample elements: (456), (12)(567), (4567), (45)(67)(123)

H5TheAllene moleculeC1C2C3H4Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288H7H6C3H4The CNPI group of allene00H00HC3H4O4How many elements?

H5TheAllene moleculeC1C2C3H4Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288H7H6C3H4The CNPI group of allene00H00H3! x 4! x 4! x 2 = 6912 C3H4O4

Number of elements in the CNPI groups of variousmoleculesThe size of the CNPI group depends only on the chemical formula(C6H6)2 12! x 12! x 2 4.6 x 1017Just need the chemical formula todetermine the CNPI group. Can be BIG

An important numberMolecule 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 43200C6H622This number means something!End of Review of Lecture OneANY QUESTIONS OR COMMENTS?

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:

RHn = HRn

(12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2The character table of the CNPIgroup of the water moleculeIt is called C2v(M)

(12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2The character table of the CNPIgroup of the water moleculeIt is called C2v(M)Now to explain how we labelenergy levels using irreducible representations

Labelling energy levelsRH = RE Since RH = HR and E is a number, this leads to HR = ER. H = EE is nondegenerate. Thus R = c.

But R2 = identity. Thus c2 = 1, so c = 1 and R = For the water molecule (no degeneracies, and R2 = identity for all R) :H(R) = E(R)The eigenfunctions have symmetryR = (12), E* or (12)*

+ Parity- Parityx1+(x)x3+(x)x2-(x) +(-x) = +(x)-(-x) = --(x)Eigenfunctions of Hmust satisfyE* = R = E*

E*(xi) = E*(xi), a new function.

E*(xi) = (E*xi) = (-xi) = (xi)

Since E*(xi) can only be (xi)This is different from Wigners approachSee PRB and Howard (1983)

+ Parity- Parityx1+(x)x3+(x)x2-(x) +(-x) = +(x)-(-x) = --(x)Eigenfunctions of Hmust satisfyE* = and (12) =

There are four symmetry types of H2O wavefunction (12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2R = A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1

The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1aHbd = 0 if symmetries of a and b are different. abd = 0 if symmetry of product is not A1We are labelling the states using the irreps of the CNPI group

The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2aHbd = 0 if symmetries of a and b are different. abd = 0 if symmetry of product is not A1E=+1 (12) =+1 E*=-1 (12)*=-1Thus, for example, a wavefunction of A2 symmetry will generate the A2 representation:

The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1111(12)* 1 -1 1-1A1A2B1B2aHbd = 0 if symmetries of a and b are different. abd = 0 if symmetry of product is not A1E=+1 (12) =+1 E*=-1 (12)*=-1Thus, for example, a wavefunct