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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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 = Spectroscopy Quantum Mechanics f i MMMM M Φ f * μ A Φ i Group Theory and Point Groups C 2v = {E,C 2x xz xy } z x E (12) E* (12)* A 1 1 1 1 1 A 2 1 1 1 1 B 1 1 1 1 1 B 2 1 1 1 1 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
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Transcript of 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...

  • Slide 1
  • h if = E f E i = E if S(f i) = A | f * A i d | 2 ODME of H and A fi = Spectroscopy Quantum Mechanics f i MMMM M f * A i d Group Theory and Point Groups C 2v = {E,C 2x, xz, xy } z x E C 2x xz xy The Complete Nuclear Permutation Inversion Group CNPI Group of C 3 H 4 has 3! x 4! x 2 = 288 elements allene +y
  • Slide 2
  • (12) E* 1 1 1 -1 -1 -1 -1 1 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 a H b d = 0 if symmetries of a and b are different. a b d = 0 if symmetry of product is not A 1 is a wavefunction of A 2 symmetry generates the A 2 representation E=+1 (12) =+1 E*=-1 (12)*=-1 RH = HR For example: EIGENFUNCTIONS TRANSFORM IRREDUCIBLY Labelling Energy Levels Using CNPIG Irreps R 2 = E
  • Slide 3
  • (12) E* 1 1 1 -1 -1 -1 -1 1 E1111E1111 (12)* 1 1 A1A2B1B2A1A2B1B2 Symmetry of A a * A b d = 0 if symmetry of a * A b is not A 1, i.e., if the symmetry of the product a b is not = symm of A Symmetry of H Using symmetry labels and the vanishing integral theorem we deduce that: a *H b d = 0 if symmetry of a *H b is not A 1, i.e., if the symmetry of a is not the same as b The Vanishing Integral Theorem f()d = 0 if symmetry of f() is not A 1
  • Slide 4
  • 1234567812345678 1234567812345678 A 1 A 2 B 1 A1 A2 B1A1 A2 B1 0 0 00 0 0 0 00 00000 0 0 0 0 0 0 0 0....
  • Slide 5
  • 1234567812345678 1234567812345678 A 1 A 2 B 1 A1 A2 B1A1 A2 B1 0 0 00 0 0 0 00 00000 0 0 0 0 0 0 0 0.... Allowed Transitions A 1 A 2 B 1 B 2 A1A1 A2A2 B1B1 Connected by A 2
  • Slide 6
  • 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
  • Slide 7
  • 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 AA A1A1 HA1A1
  • Slide 8
  • A1A1 A 1 A2A2 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
  • Slide 9
  • A1A1 A 1 A2A2 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
  • Slide 10
  • 2 3 1 1 3 2 Very, very high potential barrier No observed tunneling through barrier CH 3 F F F TWO VERSIONS
  • Slide 11
  • 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 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. 3! x 2C 3v group has 6 elements
  • Slide 12
  • 2 3 1 1 3 2 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
  • Slide 13
  • 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.
  • Slide 14
  • Character table of the MS group of CH 3 F E (123) (12)* (132) (13)* (23)* AA 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
  • Slide 15
  • 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 + EE
  • Slide 16
  • 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 ------- 16
  • Slide 17
  • 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.
  • Slide 18
  • C 2v {E C 2x xz xy } C 2v (M) elements: {E, (12), E*, (12)*} z x (+y) 12 H2OH2O elements:
  • Slide 19
  • C 2v E C 2x xz xy C 2v (M) z x (+y) 12 H2OH2O The C 2v and C 2v (M) character tables
  • Slide 20
  • E (123) (12)* (132) (13)* (23)* E C 3 1 C 3 2 2 3 C 3v (M) C 3v 1 2 3 CH 3 F The C 3v and C 3v (M) character tables F
  • Slide 21
  • 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
  • Slide 22
  • 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,} E A 1 1 A 1 -1
  • Slide 23
  • H3+H3+ Point group also has 12 elements: D 3h 3 1 2 G CNPI ={ E, (12), (13), (23), (123), (132), E*, ( 12)*, (13)*,(23)*, (123)*, (132)* } Therefore only 1 version and MSG = CNPIG +
  • Slide 24
  • Character Table of MS group H 3 + 12 elements 6 classes 6 irred. reps G CNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)* G CNPI
  • Slide 25
  • Character table for D 3h point group E2C 3 3C' 2 hh 2S 3 3 v linear, rotations quadratic A' 1 111111x 2 +y 2, z 2 A' 2 1111 RzRz E'202 0(x, y)(x 2 -y 2, xy) A'' 1 111 A'' 2 11 1z E''20-210(R x, R y )(xz, yz)
  • Slide 26
  • STOP
  • Slide 27
  • 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
  • Slide 29
  • Character table for D 2d point group E2S 4 C 2 (z)2C' 2 2 d linear, rotations quadratic A1A1 11111x 2 +y 2, z 2 A2A2 111 RzRz B1B1 1 11 x 2 -y 2 B2B2 11 1zxy E20-200(x, y) (R x, R y )(xz, yz) 29
  • Slide 30
  • 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
  • Slide 31
  • 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
  • Slide 32
  • 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.
  • Slide 33
  • 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.
  • Slide 34
  • 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.
  • Slide 35
  • Undo the permutation of the nuclear spins
  • Slide 36
  • Undo the permutation of the nuclear spins We next undo the Rotation of the axes
  • Slide 37
  • Slide 38
  • only ev coords
  • Slide 39
  • only ev coords only rot coords only nspin coords
  • Slide 40
  • Slide 41
  • Slide 42
  • Slide 43
  • Slide 44
  • Slide 45
  • 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
  • Slide 46
  • 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
  • Slide 47
  • 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
  • Slide 48
  • 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
  • Slide 49
  • 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.