Symmetry and Spectroscopy

1. Symmetry and Spectroscopy

2. P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy,2nd Edition, 2nd Printing, NRC Research Press, Ottawa, 2006 (ISBN 0-660-19628-X). $49.95 for 747 pages. paperback. BJ1 P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry, IOP Publishing, Bristol, 2004 (ISBN 0-7503-0941-5). $57.95 paperback. BJ2

3. Examples of point group symmetry H2O C2v C3H4 D2d CH3F C3v C60 Ih

4. Examples of point group symmetry H2O C2v C3H4 D2d CH3F C3v C60 Ih

5. Point group symmetry of H2O y (-x) z The point group C2v consists of the four operations E, C2y, yz, and xy The word ´group´ is loaded. To see how we do two operations in succession

6. Point groups: Number of rotation axes and reflection planes.

7. y y σyz (-x) (-x) z z 2 2 1 1 C2 y (-x) z

8. σxy = C2σyz y y σyz (-x) (-x) z z 2 2 1 1 C2 y σxy (-x) z

9. Multiplication Table for H2O Point group y C2v = {E, C2, yz, xy } (-x) z Multiplication table (=RrowRcolumn, in succession) σxy = C2σyz Use multiplication table to prove that it is a “group.”

10. {E, C2, yz, xy } forms a “group“ if it obeys the following GROUP AXIOMS : • All possible products RS = T belong to the group • Group contains identity E (which does nothing) • The inverse of each operation R1 (R1R =RR1 =E ) is in the group • Associative law (AB )C = A(BC ) holds C2v

11. {E, C2, yz, xy } forms a “group“ if it obeys the following GROUP AXIOMS : • All possible products RS = T belong to the group • Group contains identity E (which does nothing) • The inverse of each operation R1 (R1R =RR1 =E ) is in the group • Associative law (AB )C = A(BC ) holds Fermi: C2v ‘‘Group theory is just a bunch of definitions‘‘

12. All possible products RS = T belong to the group • Group contains identity E (which does nothing) • The inverse of each operation R1 (R1R =RR1 =E ) is in the group • Associative law (AB )C = A(BC ) holds Not a GROUP

13. All possible products RS = T belong to the group • Group contains identity E (which does nothing) • The inverse of each operation R1 (R1R =RR1 =E ) is in the group • Associative law (AB )C = A(BC ) holds Is a GROUP (subgroup of C2v) Rotational subgroup

14. PH3 at equilibrium Symmetry elements: C3, 1, 2, 3 C3 Rotation axis kReflection plane Symmetry operations: C3v = {E,C3, C32, 1, 2, 3 }

15. Multiplying C3v symmetry operations Reflection Rotation Reflection C32 = 2 1

16. Multiplication table forC3v C32 = σ2σ1

17. Multiplication table forC3v C3 = σ1σ2 Note that C32 = σ2σ1

18. Multiplication table forC3v Rotational subgroup

19. Multiplication table forC3v 3 classes

20. A matrix group M4 = ´M1 = ´M2 = ´ M5 = ´M3 = ´ M6 =

21. Multiplication table forthe matrix group Products are Mrow Mcolumn

22. Multiplication tables have the ‘same shape’ E C3 C32 σ1 σ2 σ3 This matrix group forms a “representation” of the C3v group These two groups are isomorphic.

23. Irreducible Representations The matrix group we have just introduced is an irreducible representation of the C3v point group. The sum of the diagonal elements (character) of each matrix in an irreducible representation is tabulated in the character table of the point group.

24. The characters of this irreducible rep. C3v C3v 2 0 M4 = 1 ´M1 = E 0 -1 ´M2 = C3 ´ M5 = 2 -1 0 ´M3 = C32 ´ M6 = 3

25. The characters of this irreducible rep. E C3σ1 C32σ2 σ3 3 classes 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

26. 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

27. The matrices of the E irreducible rep. C3v C3v M4 = 1 ´M1 = E ´M2 = C3 ´ M5 = 2 ´M3 = C32 ´ M6 = 3

28. The matrices of the A1 + E reducible rep. C3v C3v 1 0 0 0 1 0 0 0 1 1 0 0 0 0 ‘ ‘ M4 = 1 ´M1 = E • 0 0 • 0 • 0 1 0 0 0 0 ‘ ‘ ´M2 = C3 ´ M5 = 2 1 0 0 0 0 1 0 0 0 0 ‘ ‘ ´M3 = C32 ´ M6 = 3

29. The matrices of the A2 + E reducible rep. C3v C3v ‘ 1 0 0 0 1 0 0 0 1 -1 0 0 0 0 ‘ ‘ ‘ ‘ M4 = 1 ´M1 = E • 0 0 • 0 • 0 -1 0 0 0 0 ‘ ‘ ‘ ‘ ´M2 = C3 ´ M5 = 2 1 0 0 0 0 -1 0 0 0 0 ‘ ‘ ‘ ‘ ´M3 = C32 ´ M6 = 3

30. Character table for the point group C2v EC2σyzσxy Irreducible representations are “symmetry labels”

31. Some of Fermi’s definitions • Group • Subgroup • Multiplication table of group operations • Classes • Representations • Irreducible and reducible representations • Character table See, for example, pp 14-15 and Chapter 5 of BJ1

32. Some of Fermi’s definitions Group Subgroup Multiplication table of group operations Classes Representations Irreducible and reducible representations Character table See, for example, pp 14-15 and Chapter 5 of BJ1

33. Irreducible representations • The elements of irrep matrices satisfy the • „Grand Orthogonality Theorem“ (GOT). • We do not discuss the GOT here, but we list three • consequences of it: • Number of irreps = Number of classes in the group. • Dimensions of the irreps, l1, l2, l3 … satisfy • l12 + l22 + l32 + … = h, • where h is the number of elements in the group. • Orthogonality relation

34. These are used as ‘‘symmetry labels‘‘ on energy levels. Which energy levels can ‘‘interact‘‘ and which transitions can occur. • Irreducible and reducible representations Can also determine whether certain terms are in the Hamiltonian.

35. BUT IN SOME CIRCUMSTANCES THERE ARE PROBLEMS IF WE TRY TO USE POINT GROUP SYMMETRY TO DO THESE THINGS

36. How do we use point group symmetry if the molecule rotates and distorts? H3+ D3h C2v

37. 3 1 2 Or if tunnels? NH3 2 1 3 D3h C3v

38. What are the symmetries of B(CH3)3, CH3.CC.CH3, (CO)2, (NH3)2,…? Nonrigid molecules (i.e. molecules that tunnel) are a problem if we try to use a point group.

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

40. Point groups used for classifying: The electronic states for any molecule at a fixed nuclear geometry (see BJ2 Chapter 10), and The vibrational states for molecules, called “rigid” molecules, undergoing infinitesimal vibrations about a unique equilibrium structure (see BJ2 Pages 230-238).

41. To understand how we use symmetry labels and where the point group goes wrong we must study what we mean by “symmetry” Rotations and reflections Permutations and 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] See also BJ1 and BJ2

42. Centrifugal distortion eg. H3+ or CH4 dipole moment Nonrigid molecules: eg. ethane, ammonia, (H2O)2, (CO)2,… Breakdown of BOA: eg. HCCH* - H2CC Symmetry not from geometry since molecules are dynamic Also symmetry applies to atoms, nuclei and fundamental particles. Geometrical point group symmetry is not possible for them. We need a more general definition of symmetry

43. Symmetry Based on Energy Invariance Symmetry operations are operations that leave the energy of the system (a molecule in our case) unchanged. Using quantum mechanics we define a symmetry operation as follows: A symmetry operation is an operation that commutes with the Hamiltonian: (RH – HR)n = [R,H]n = 0

44. Symmetry Operations (energy invariance) • Uniform Space ----------Translation • Isotropic Space----------Rotation • Identical electrons------Permute electrons • Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q,s) (-p,-q,s)P(E*) • Reversal symmetry-----Time reversal (p,q,s) (-p,q,-s) T • Ch. conj. Symmetry-----Particle antiparticle C

45. Uniform Space ----------Translation Symmetry Operations (energy invariance) Separate translation… Translational momentum Ψtot = Ψtrans Ψint int = rot-vib-elec.orb-elec.spin-nuc.spin

46. Uniform Space ----------Translation Isotropic Space----------Rotation Identical electrons-------Permute electrons Identical nuclei-----------Permute identical nuclei Parity conservation-----Inversion (p,q) (-p,-q)P(E*) Reversal symmetry-----Time reversal (p,s) (-p,-s) T Ch. conj. Symmetry-----Particle antiparticle C Symmetry Operations (energy invariance) K(spatial) group, J, mJor F,mF labels

47. Uniform Space ----------Translation Isotropic Space----------Rotation Identical electrons-------Permute electrons Identical nuclei-----------Permute identical nuclei Parity conservation-----Inversion (p,q)(-p,-q)P(E*) Reversal symmetry-----Time reversal (p,s) (-p,-s) T Ch. conj. Symmetry-----Particle antiparticle C Symmetry Operations (energy invariance) Symmetric group Sn

48. For the BeH molecule (5 electrons) Ψorb-spin transforms as D(0) of S5 PEP Slater determinant ensures antisymmetry so do not need S5

49. Uniform Space ----------Translation Isotropic Space----------Rotation Identical electrons-------Permute electrons Identical nuclei-----------Permute identical nuclei Parity conservation-----Inversion (p,q) (-p,-q)P(E*) Reversal symmetry-----Time reversal (p,s) (-p,-s) T Ch. conj. Symmetry-----Particle antiparticle C Symmetry Operations (energy invariance)

50. CNPI group = Complete Nuclear Permutation Inversion Group Symmetry Operations (energy invariance) • Uniform Space ----------Translation • Isotropic Space----------Rotation • Identical electrons-------Permute electrons • Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q) (-p,-q)P(E*) • Reversal symmetry-----Time reversal (p,s) (-p,-s) T • Ch. conj. Symmetry-----Particle antiparticle C