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 μ fi = Spectroscopy Quantum Mechanics f i MMMM M ∫ Φ f * μ A Φ i dτ Group Theory and Point Groups C 2v = {E,C 2x,σ xz,σ xy } z x E C 2x σ xz σ xy The Complete Nuclear Permutation Inversion Group CNPI Group of C 3 H 4 has 3! x 4! x 2 = 288 elements allene +y

(12) E* E1111E1111 (12)* 1 1 A1A2B1B2A1A2B1B2 R  = ±  A 2 x B 1 = B 2, B 1 x B 2 = A 2, B 1 x A 2 x B 2 = A 1 ∫ Ψ a HΨ b dτ = 0 if symmetries of Ψ a and Ψ b are different. ∫ Ψ a μΨ b dτ = 0 if symmetry of product is not A 1 ψ is a wavefunction of A 2 symmetry ψ generates the A 2 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 R 2 = E

(12) E* E1111E1111 (12)* 1 1 A1A2B1B2A1A2B1B2 Symmetry of μ A ∫ Ψ a * μ A Ψ b dτ = 0 if symmetry of ψ a *μ A ψ b is not A 1, i.e., if the symmetry of the product Ψ a Ψ b is not = symm of μ A Symmetry of H Using symmetry labels and the vanishing integral theorem we deduce that: ∫ Ψ a *HΨ b dτ = 0 if symmetry of Ψ a *HΨ b is not A 1, i.e., if the symmetry of Ψ a is not the same as Ψ b The Vanishing Integral Theorem ∫f(τ)dτ = 0 if symmetry of f(τ) is not A 1

ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8 Ψ1Ψ2Ψ3Ψ4Ψ5Ψ6Ψ7ψ8Ψ1Ψ2Ψ3Ψ4Ψ5Ψ6Ψ7ψ8 A 1 A 2 B 1 A1 A2 B1A1 A2 B

ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8 Ψ1Ψ2Ψ3Ψ4Ψ5Ψ6Ψ7ψ8Ψ1Ψ2Ψ3Ψ4Ψ5Ψ6Ψ7ψ8 A 1 A 2 B 1 A1 A2 B1A1 A2 B Allowed Transitions A 1 ↔ A 2 B 1 ↔ B 2 A1A1 A2A2 B1B1 Connected by A 2

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 Group for any molecule BUT there is one big problem: Superfluous symmetry labels

Character Table of CNPI group of CH 3 F 12 elements 6 classes 6 irred. reps G CNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)* G CNPI Using the CNPI Group approach for CH 3 F μAμA A1”A1” HA1’A1’

A1’A1’ A 1 ’’ A2’A2’ A 2 ’’ E’ E’’ E’ + E’’ A 1 ’’ + A 2 ’ A 1 ’ + A 2 ’’ The CNPI Group approach CH 3 F Allowed transitions connected by A 1 ” Block diagonal H-matrix E’ + E’’

A1’A1’ A 1 ’’ A2’A2’ A 2 ’’ E’ E’’ A 1 ’’ + A 2 ’ A 1 ’ + A 2 ’’ CH 3 F Allows for tunneling between two VERSIONS The CNPI Group approach E’ + E’’

Very, very high potential barrier No observed tunneling through barrier CH 3 F F F TWO VERSIONS

The number of versions of the minimum is given by: (order of CNPI group)/(order of point group) For CH 3 F this is 12/6 = 2 For benzene C 6 H 6 this is /24 = 43200, and using the CNPI group each energy level would get as symmetry label the sum of irreps. Clearly using the CNPI group gives very unwieldy symmetry labels. 3! x 2C 3v group has 6 elements

Very, very high potential barrier No observed tunneling through barrier Only NPI OPERATIONS FROM IN HERE CH 3 F (12) superfluous E* superfluous (123),(12)* useful F F

G CNPI ={ E, (12), (13), (23), (123), (132), E*, ( 12)*, (13)*,(23)*, (123)*, (132)* } For CH 3 F: G MS ={ E, (123), (132), ( 12)*, (13)*,(23)* } The six feasible elements are superfluous useful 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.

Character table of the MS group of CH 3 F E (123) (12)* (132) (13)* (23)* μAμA A2A2 HA1A1 (12),(13),(23) E*,(123)*,(132)* are unfeasible elements of the CNPI Group Use this group to block-diagonalize H and to determine which transitions are forbidden

Using CNPIG versus MSG for PH 3 E A2A2 A1A1 Can use either to determine if an ODME vanishes. But clearly it is easier to use the MSG. E’ + E’’ A 1 ’’ + A 2 ’ A 1 ’ + A 2 ’’ CNPIGMSG E’ + E’’E

The subgroup of feasible elements forms a group called THE MOLECULAR SYMMETRY GROUP (MS GROUP) Unfeasible elements of the CNPI group interconvert versions that are separated by an insuperable energy barrier Superfluous useful End of Lecture Two

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 Groups of these molecules. We will examine this “resemblance” and show how it helps us to understand Point Groups.

C 2v {E C 2x σ xz σ xy } C 2v (M) elements: {E, (12), E*, (12)*} z x (+y) 12 H2OH2O elements:

C 2v E C 2x σ xz σ xy C 2v (M) z x (+y) 12 H2OH2O The C 2v and C 2v (M) character tables

E (123) (12)* (132) (13)* (23)* E C 3 σ 1 C 3 2 σ 2 σ 3 C 3v (M) C 3v CH 3 F The C 3v and C 3v (M) character tables F

H N 1 N 2 N 3 MSG is {E,E*} E E* A’ 1 1 A” 1 -1 All P and P* are unfeasible (superfluous) G CNPI ={ E, (12), (13), (23), (123), (132), E*, ( 12)*, (13)*,(23)*, (123)*, (132)* } HN 3 has 6 versions 12/2 = 6 3!x2CsCs

H N 1 N 2 N 3 MSG is {E,E*} E E* A’ 1 1 A” 1 -1 All P and P* are unfeasible (superfluous) G CNPI ={ E, (12), (13), (23), (123), (132), E*, ( 12)*, (13)*,(23)*, (123)*, (132)* } HN 3 has 6 versions 12/2 = 6 3!x2CsCs PG is {E,σ} E σ A’ 1 1 A” 1 -1

H3+H3+ Point group also has 12 elements: D 3h G CNPI ={ E, (12), (13), (23), (123), (132), E*, ( 12)*, (13)*,(23)*, (123)*, (132)* } Therefore only 1 version and MSG = CNPIG +

Character Table of MS group H elements 6 classes 6 irred. reps G CNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)* G CNPI

Character table for D 3h point group E2C 3 3C' 2 σhσh 2S 3 3σ v linear, rotations quadratic A' x 2 +y 2, z 2 A' RzRz E'202 0(x, y)(x 2 -y 2, xy) A'' A'' z E''20-210(R x, R y )(xz, yz)

STOP

Number of elements in CNPIG = 3! x 4! x 2 = 288 H5H5 C1C1 C2C2 C3C3 H4H4 H7H7 H6H6 The MSG of allene is: {E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*, (4657)(13)*, (4756)(13)*} The allene molecule C 3 H 4 Point group is D 2d has 8 elements Thus there are 288/8 = 36 versions

Number of elements in CNPIG = 3! x 4! x 2 = 288 H5H5 C1C1 C2C2 C3C3 H4H4 H7H7 H6H6 The MSG of allene and PG are: E C 2 C 2 ’ C 2 ’ σ d σ d {E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*, S 4 S 4 (4657)(13)*, (4756)(13)*} The allene molecule C 3 H 4 Point group is D 2d has 8 elements Thus there are 288/8 = 36 versions

Character table for D 2d point group E2S 4 C 2 (z)2C' 2 2σ d linear, rotations quadratic A1A x 2 +y 2, z 2 A2A2 111 RzRz B1B x 2 -y 2 B2B2 11 1zxy E20-200(x, y) (R x, R y )(xz, yz) 29

USE OF GROUP THEORY AND SYMMETRY IN SPECTROSCOPY Point Group CNPI GroupMS Group Spoilt by rotation and tunneling Superfluous symmetry if many versions RH=HR, therefore can symmetry label energy levels Use molecular geometry Use energy invariance RH = HR

USE OF GROUP THEORY AND SYMMETRY IN SPECTROSCOPY Point Group CNPI GroupMS Group Spoilt by rotation and tunneling Superfluous symmetry if many versions RH=HR, therefore can symmetry label energy levels Use molecular geometry Use energy invariance RH = HR

USE OF GROUP THEORY AND SYMMETRY IN SPECTROSCOPY Point Group CNPI GroupMS Group Spoilt by rotation and tunneling Superfluous symmetry if many versions RH=HR, therefore can symmetry label energy levels Use molecular geometry Use energy invariance RH = HR For rigid (nontunneling) molecules MSG and PG are isomorphic. Leads to an understanding of PGs and how they can label energy levels.

Black are instantaneous positions in space. White are equilibrium positions. N.B. +z is 1 → 2. Then do (12). Note that axes have moved. Rotational coordinates are transformed by MS group.

Black are instantaneous positions in space. White are equilibrium positions. N.B. +z is 1 → 2. Then do (12). Note that axes have moved. Rotational coordinates are transformed by MS group.

Undo the permutation of the nuclear spins

Undo the permutation of the nuclear spins We next undo the Rotation of the axes

only ev coords

only ev coords only rot coords only nspin coords

MS Group and Point Group of H 2 O E = p 0 R 0 E (12) = p 12 R x  C 2x E* = p 0 R y   xz (12)* = p 12 R z   xy MS GroupPoint Group C 2v (M) C 2v

MS Group and Point Group of H 2 O E = p 0 R 0 E (12) = p 12 R x  C 2x E* = p 0 R y   xz (12)* = p 12 R z   xy MS GroupPoint Group C 2v (M) C 2v Jon Hougen and what the MSG is called by some people

The point group can be used to symmetry label the vibrational and electronic states of non-tunneling molecules. R PG H ev = H ev R PG Point group operations rotate/reflect electronic coords and vib displacements in a non-tunneling (rigid) molecule. 47 The rotational and nuclear spin coordinates are not transformed by the elements of a point group

The MSG can be used to classify The 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 Number of elements in CNPIG = 5! x 2! x 2 = 480 The MSG of internally rotating ethyl is the following group of order 12: {E, (123),(132),(12)*,(23)*,(31)*, (45),(123)(45),(132)(45),(12)(45)*,(23)(45)*,(31)(45)*} H 4 C b H 5 a The MSG of rigid (nontunneling) ethyl is {E,(23)*} There are still plenty of unfeasible elements such as (14), (23), (145), E*, etc.

MS group can change with resolution and temperature Recent theoretical calcs by Sousa-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 = cm -1 THE PH 3 MOLECULE AS AN EXAMPLE

7.2 cm cm MHz 2 MHz 240 kHz 15 kHz 600 Hz 12 Hz D 3h (M) C 3v (M) CNPIG

Tunneling splittings in NH 3 D 3h (M)

MS group can change with resolution and temperature For NH 3 MSG is usually D 3h (M) For PH 3 MSG is usually C 3v (M)

The subgroup of feasible elements form a group called THE MOLECULAR SYMMETRY GROUP Unfeasible elements of the CNPI group interconvert versions that are separated by an insuperable energy barrier MS group is used to symmetry label the rotational, ro-vibrational, rovibronic and nuclear spin states of any molecule. MS group can change with resolution and T SUMMARY 54