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MOLECULAR SYMMETRY AND SPECTROSCOPY

MOLECULAR SYMMETRY AND SPECTROSCOPY. Philip.Bunker@nrc.ca. Download ppt file from. http://www.few.vu.nl/~rick. At bottom of page. We began by summarizing. Chapters 1 and 2. Spectroscopy and Quantum Mechanics. f. Absorption can only occur at resonance. h ν if = E f – E i = Δ E if.

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MOLECULAR SYMMETRY AND SPECTROSCOPY

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  1. MOLECULAR SYMMETRY AND SPECTROSCOPY Philip.Bunker@nrc.ca Download ppt file from http://www.few.vu.nl/~rick At bottom of page

  2. We began by summarizing Chapters 1 and 2. Spectroscopy and Quantum Mechanics f Absorption can only occur at resonance hνif = Ef – Ei = ΔEif νif i M Integrated absorption coefficient (i.e. intensity) for a line is: ______ 8π3 Na ~ ~ ~ ε(ν)dν = I(f ← i) = ∫ F(Ei ) νif Rstim(f→i) S(f ← i) (4πε0)3hc line Use Q. Mech. to calculate: ODME of H μfi = ∫ (Ψf )* μAΨi dτ ODME of μA

  3. P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry, Taylor and Francis, 2004. The first 47 pages: Chapter 1 (Spectroscopy) Chapter 2 (Quantum Mechanics) and Section 3.1 (The breakdown of the BO Approx.) P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy, 2nd Edition, 3rd Printing, NRC Research Press, Ottawa, 2012. Download pdf file from To buy it go to: http://www.crcpress.com www.chem.uni-wuppertal.de/prb

  4. P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry, Taylor and Francis, 2004. The first 47 pages: Chapter 1 (Spectroscopy) Chapter 2 (Quantum Mechanics) and Section 3.1 (The breakdown of the BO Approx.) P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy, 2nd Edition, 3rd Printing, NRC Research Press, Ottawa, 2012. Download pdf file from To buy it go to: http://www.crcpress.com www.chem.uni-wuppertal.de/prb We then proceeded to discuss Group Theory and Point Groups

  5. Definitions for groups and point groups: “Group” A set of operations that is closed wrt “multiplication” “Point Group” All rotation, reflection and rotation-reflection operations that leave the molecule (in its equilibrium configuration) “looking” the same. “Matrix group” A set of matrices that forms a group. “Representation” A matrix group having the same shaped multiplication table as the group it represents. “Irreducible representation” A representation that cannot be written as the sum of smaller dimensioned representations. “Character table” A tabulation of the characters of the irreducible representations.

  6. Character table for the point group C3v E C3σ1 C32σ2 σ3 Two 1D irreducible representations of the C3v group The 2D representation M = {M1, M2, M3, ....., M6} of C3v is the irreducible representation E. In this table we give the characters of the matrices. Elements in the same class have the same characters 3 classes and 3 irreducible representations

  7. Character table for the point group C2v x EC2σyzσxy (+y) z 4 classes and 4 irreducible representations

  8. E C3σ1 C32σ2 σ3 C3v 3 2 1 Spectroscopy M f hνif = Ef – Ei = ΔEif MMMM i S(f ← i) = ∑A | ∫ Φf* μAΦi dτ |2 Quantum Mechanics ODME of H and μA μfi = ∫ Φf* μAΦi dτ Group Theory and Point Groups (Character Tables and Irreducible Representations) PH3 8

  9. Point Group symmetry is based on the geometrical symmetry of the equilibrium structure. Point group symmetry not appropriate when there is rotation or tunneling Use energy invariance symmetry instead. We start by using inversion symmetry and identical nuclear permutation symmetry.

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

  11. The spin-free (rovibronic) Hamiltonian (after separating translation) 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 then selectively correct for the approximations made.

  12. 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)*} + + e e H2 H1 O O E* (12) - e O H2 H1 H2 H1 (12)* Nuclear permutations permute nuclei (coordinates and spins). Do not change electron coordinates E* Inverts coordinates of nuclei and electrons. Does not change spins. Same CNPI group for CO2, H2, H2CO, HOOD, HDCCl2,…

  13. H H N1N2N3 1 C1 F 3 C2 2 I C3 O F D 2 O O 1 3 1 3 H H 12C 13C D H 2 1 2 H 1 + H H 3 2 3 GCNPI = {E, (12), (13), (23), (123), (132)}x {E, E*} = GCNP x {E, E*}

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

  15. H5 C1 H4 C2 I C3 D The CNPI Group of C3H2ID GCNPI = {E, (12), (13), (23), (123), (132)} x{E, (45)}x {E, E*} = {E, (12), (13), (23), (123), (132), (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

  16. H5 C1 H4 C2 I C3 D Number of elements = 3! x 2! x 2 = 6 x 2 x 2 = 24 If there are n1 nuclei of type 1, n2 of type 2, n3 of type 3, etc then the total number of elements in the CNPI group is n1! x n2! x n3!... x 2.

  17. The CNPI group of allene H5 The Allene molecule C1 H4 C3H4 C2 H7 C3 H6 Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288

  18. The CNPI group of allene H5 The Allene molecule C1 H4 C3H4 C2 H7 C3 H6 Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288 Sample elements: (456), (12)(567), (4567), (45)(67)(123)

  19. The CNPI group of allene H5 00H The Allene molecule C1 H4 C3H4 C2 H7 C3 H6 00H Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288 How many elements? C3H4O4

  20. The CNPI group of allene H5 00H The Allene molecule C1 H4 C3H4 C2 H7 C3 H6 00H Number of elements = 3! x 4! x 2 = 6 x 24 x 2 = 288 C3H4O4 3! x 4! x 4! x 2 = 6912

  21. The size of the CNPI group depends only on the chemical formula Number of elements in the CNPI groups of various molecules (C6H6)2 12! x 12! x 2 ≈ 4.6 x 1017 Just need the chemical formula to determine the CNPI group. Can be BIG

  22. 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 43200 C6H6 This number means something! End of Review of Lecture One 22 ANY QUESTIONS OR COMMENTS?

  23. 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: RHn = HRn

  24. The character table of the CNPI group of the water molecule (12) E* 1 1 1 -1 -1 -1 -1 1 (12)* 1 -1 1 -1 E 1 1 1 1 A1 A2 B1 B2 It is called C2v(M)

  25. The character table of the CNPI group of the water molecule (12) E* 1 1 1 -1 -1 -1 -1 1 (12)* 1 -1 1 -1 E 1 1 1 1 A1 A2 B1 B2 It is called C2v(M) Now to explain how we label energy levels using irreducible representations

  26. Labelling energy levels For the water molecule (no degeneracies, and R2 = identity for all R) : H = E RH = RE Since RH = HR and E is a number, this leads to HR = ER. H(R) = E(R) E is nondegenerate. Thus RΨ = cΨ. But R2 = identity. Thus c2 = 1, so c = ±1 and R = ± R = (12), E* or (12)* The eigenfunctions have symmetry

  27. R = E* + Parity - Parity Ψ1+(x) Ψ2-(x) x x Ψ-(-x) = -Ψ-(x) Ψ3+(x) Eigenfunctions of H must satisfy E*Ψ = ±Ψ x Ψ+(-x) = Ψ+(x)

  28. E*ψ(xi) = ψE*(xi), a new function. ψE*(xi) = ψ(E*xi) = ψ(-xi) = ±ψ(xi) Since E*ψ(xi) can only be ±ψ(xi) This is different from Wigner’s approach See PRB and Howard (1983)

  29. + Parity - Parity Ψ1+(x) Ψ2-(x) x x Ψ-(-x) = -Ψ-(x) Ψ3+(x) Eigenfunctions of H must satisfy E*Ψ = ±Ψ x Ψ+(-x) = Ψ+(x) and (12)ψ = ±ψ

  30. There are four symmetry types of H2O wavefunction (12) E* 1 1 1 -1 -1 -1 -1 1 E 1 1 1 1 (12)* 1 -1 1 -1 R = ± A1 A2 B1 B2 A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1

  31. The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1 1 1 1 (12)* 1 -1 1 -1 A1 A2 B1 B2 We are labelling the states using the irreps of the CNPI group ∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different. ∫ΨaμΨbdτ = 0 if symmetry of product is not A1 A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1

  32. The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1 1 1 1 (12)* 1 -1 1 -1 A1 A2 B1 B2 Thus, for example, a wavefunction of “A2 symmetry” will “generate” the A2 representation: ∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different. ∫ΨaμΨbdτ = 0 if symmetry of product is not A1 Eψ=+1ψ(12)ψ=+1ψE*ψ=-1ψ(12)*ψ=-1ψ

  33. The Symmetry Labels of the CNPI Group of H2O (12) E* 1 1 1 -1 -1 -1 -1 1 E 1 1 1 1 (12)* 1 -1 1 -1 A1 A2 B1 B2 Thus, for example, a wavefunction of “A2 symmetry” will “generate” the A2 representation: ∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different. ∫ΨaμΨbdτ = 0 if symmetry of product is not A1 Eψ=+1ψ(12)ψ=+1ψE*ψ=-1ψ(12)*ψ=-1ψ For the water molecule we can, therefore label the energy levels as being A1, A2, B1 or B2 using the irreps of the CNPI group.

  34. The labelling business The vibrational wavefunction for the v3 = 1 state of the water molecule can be written approximately as ψ = N(Δr1 – Δr2). Eψ=+1ψ(12)ψ=-1ψE*ψ=+1ψ(12)*ψ=-1ψ This would be labelled as B2.

  35. c = ε, ε2 (=ε*), or ε3 (=1) If n = 3, Suppose Rn = E where n > 2. We still have RΨ = cΨ for nondegenerate Ψ, but now RnΨ = Ψ. Thus cn = 1 and c = n√1, i.e. c = [ei2π/n]a where a = 1,2,…,n. where  = ei2/3 C3 C32 eiπ = -1 ei2π = 1

  36. For nondegenerate states we hadthis as the effect of a symmetry operation on an eigenfunction:  For the water molecule ( nondegenerate) : H = E RH = RE HR = ER Thus R = c since E is nondegenerate. What about degenerate states?

  37. ℓ-fold degenerate energy level with energy En RΨnk=D[R ]k1Ψn1 +D[R ]k2Ψn2 +D[R ]k3Ψn3 +…+D[R ]kℓΨnℓ For each relevant symmetry operation R, the constants D[R ]kp form the elements of an ℓℓ matrix D[R ]. ForT = RS it is straightforward to show that D[T ] = D[R ] D[S ] The matrices D[T ],D[R ], D[S ]….. form an ℓ-dimensional representation that is generated by the ℓ functions Ψnk The ℓ functions Ψnktransform according to this representation

  38. Labelling energy levels using the CNPI Group We can label energy levels using the irreps of the CNPI group for any molecule ∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different. ∫ΨaμΨbdτ = 0 if symmetry of product is not A1 Pages 143-149 Pages 99-101 34

  39. Determining symmetry and reducing a representation Example of using the symmetry operation (12): (12) r1´  r2´ ´ H1 H2 We have (12) (r1, r2, ) = (r1´, r2´, ´) We see that (r1´, r2´, ´) = (r2, r1, )

  40. r1´ r2´ r2 r2 r2 r2 r1 r1 r1 r1 r2´ r1´     ´ ´ 3 3 3 3 3 1 2 2 2 2 2 1 1 1 1 3 1 2 2 1 3 2 1 3 E (12) ´ E* r1´ r2´ (12)* ´ r2´ r1´

  41. R a = a´ = D[R] a E  = 3  = 1 (12)  = 3 E* (12)*  = 1

  42. aA1 = ( 13 + 11 + 13 + 11) = 2 aA2 = ( 13 + 11  13  11) = 0 aB1 = ( 13  11  13 + 11) = 0 aB2 = ( 13  11 + 13  11) = 1 Γ = Σ aiΓi i A reducible representation  = 2A1 B2

  43. aA1 = ( 13 + 11 + 13 + 11) = 2 aA2 = ( 13 + 11  13  11) = 0 aB1 = ( 13  11  13 + 11) = 0 aB2 = ( 13  11 + 13  11) = 1 Γ = Σ aiΓi i i A reducible representation  = 2A1 B2

  44. aA1 = ( 13 + 11 + 13 + 11) = 2 aA2 = ( 13 + 11  13  11) = 0 aB1 = ( 13  11  13 + 11) = 0 aB2 = ( 13  11 + 13  11) = 1 Γ = Σ aiΓi i i A reducible representation  = 2A1 B2

  45. aA1 = ( 13 + 11 + 13 + 11) = 2 aA2 = ( 13 + 11  13  11) = 0 aB1 = ( 13  11  13 + 11) = 0 aB2 = ( 13  11 + 13  11) = 1 Γ = Σ aiΓi i i A reducible representation  = 2A1 B2

  46. We know now that r1, r2, andgenerate the representation 2A1 B2 Consequently, we can generate from r1, r2, and three „symmetrized“ coordinates: S1 with A1 symmetry S2 with A1 symmetry S3 with B2 symmetry For this, we need projection operators

  47. Projection operators: General for li-dimensional irrep i Diagonal element of representation matrix Symmetry operation Simpler for 1-dimensional irrep i Character 1

  48. Projection operators: General for li-dimensional irrep i Simpler for 1-dimensional irrep i Character Diagonal element of representation matrix 1 Symmetry operation

  49. Projection operators: General for li-dimensional irrep i Simpler for 1-dimensional irrep i Character Diagonal element of representation matrix 1 Symmetry operation E (12) E* (12)* A1 1 1 1 1 PA1 = (1/4) [ E + (12) + E* + (12)* ]

  50. S1 = P11A1r1= [ E + (12) + E*+ (12)* ]r1 S3 = P11B2r1= [ E  (12) + E* (12)*] r1 = [ r1  r2 + r1 r2 ] = [ r1  r2 ] = [ r1 + r2 + r1 + r2 ] = [ r1 + r2 ] S2 = P11A1= [ E + (12) + E*+ (12)* ] P11B2= [ E  (12) + E* (12)* ] = [  +  + +  ] =  = [   +   ] = 0 Projection operator for A1 acting on r1 PA1 PA1 PB2 PB2  Is „annihilated“ by P11B2 PB2

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