MOLECULAR SYMMETRY AND SPECTROSCOPY

1. MOLECULAR SYMMETRY AND SPECTROSCOPY Philip.Bunker@nrc.ca

2. Learning about molecular symmetry from books 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. C$ 23.95

3. Learning about molecular symmetry from books P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry, Taylor and Francis, 2004. Chapter 1 in here P. R. Bunker and Per Jensen: Molecular Symmetry and Spectroscopy, 2nd Edition, 3rd Printing, NRC Research Press, Ottawa, 2012. C$ 23.95

4. ν Absorption spectrum: a plot of transmittance vs. molecules I0(ν) Itr(ν) frequency Transmittance = Itr(ν)/I0(ν) frequency (number of waves per cm)

5. ν Absorption spectrum: a plot of transmittance vs. f molecules I0(ν) Itr(ν) νif i M Absorption can only occur at resonance Transmittance = Itr(ν)/I0(ν) hνif = Ef – Ei = ΔEif E = Einternal = Erve-spin EACH “LINE’’ SIGNALS A RESONANCE; Molecule undergoes a “transition.’’

6. ν Absorption spectrum: a plot of transmittance vs. f molecules I0(ν) Itr(ν) νif i M Absorption can only occur at resonance Transmittance = Itr(ν)/I0(ν) [“Weak radiation”: Transmittance ≠ f(I0)] hνif = Ef – Ei = ΔEif E = Einternal = Erve-spin EACH “LINE’’ SIGNALS A RESONANCE; Molecule undergoes a “transition.’’

7. ~ ν, ν or λ Absorption spectrum: a plot of transmittance vs. ~ ~ molecules I0(ν) Itr(ν) frequency wavenumber wavelength ~ ~ Transmittance = Itr(ν)/I0(ν) ~ ν= 1/ λ = ν/c cm-1 (number of waves per cm)

8. ~ ν, ν or λ Absorption spectrum: a plot of transmittance vs. f ~ ~ molecules I0(ν) Itr(ν) νif i M ~ ~ Absorption can only occur at resonance Transmittance = Itr(ν)/I0(ν) hνif = Ef – Ei = ΔEif ~ ν= 1/ λ = ν/c hνif = Ef – Ei = ΔEif cm-1 E = Einternal = Erve-spin

9. ~ ν, ν or λ Absorption spectrum: a plot of transmittance vs. f ~ ~ molecules I0(ν) Itr(ν) νif i M ~ ~ Absorption can only occur at resonance Transmittance = Itr(ν)/I0(ν) hνif = Ef – Ei = ΔEif ~ ν= 1/ λ = ν/c hνif = Ef – Ei = ΔEif cm-1 E = Einternal = Erve-spin Divide by hc νif /c = Ef /hc - Ei /hc = ΔEif /hc Term value difference cm-1 wavenumber cm-1 Term values cm-1

10. f Absorption can only occur at resonance hνif = Ef – Ei = ΔEif νif i M ℓ Absorption coefficient ~ molecules conc = c* ~ I0(ν) Itr(ν) Pierre_ Bouguer Beer-Lambert Law (weak radiation): ~ ~ ~ Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]

11. f Absorption can only occur at resonance hνif = Ef – Ei = ΔEif νif i M ℓ Absorption coefficient ~ molecules conc = c* ~ I0(ν) Itr(ν) Pierre_ Bouguer Beer-Lambert Law (weak radiation): ~ ~ ~ Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)] http://www.google.com/ (youtube feynman names)

12. f Absorption can only occur at resonance hνif = Ef – Ei = ΔEif νif i M ℓ Absorption coefficient ~ molecules conc = c* ~ I0(ν) Itr(ν) Pierre_ Bouguer The exponential attenuation law (weak radn): ~ ~ ~ Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]

13. f | ∫Φf* μAΦi dτ |2 ∑ A=X,Y,Z Absorption can only occur at resonance hνif = Ef – Ei = ΔEif νif i M ℓ Absorption coefficient ~ molecules conc = c* ~ I0(ν) Itr(ν) Pierre_ Bouguer The exponential attenuation law (weak radn): ~ ~ ~ Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)] Integrated absorption coefficient (i.e. intensity) for a line is: ______ 8π3 Na ~ ~ ~ ε(ν)dν = I(f ← i) = ∫ Rstim(f→i) F(Ei ) νif S(f ← i) (4πε0)3hc line Stimulated Emission Factor Frequency Factor [1 – exp(-hυif /kT)] Boltzmann Factor Line Strength giexp [-Ei /kT ] ∑jgjexp [-Ej /kT ]

14. The Line Strength S(f ← i) = | ∫Φf* μAΦi dτ |2 ∑ A=X,Y,Z This depends on the wavefunctions that describe each of the levels involved in the transition. They occur in the electric dipole transition moment integral. μA is the component of the molecular electric dipole moment along the space fixed A (= X, Y or Z) axis: μA = ΣCre Ar r Charge on particle r A coordinate of particle r

15. AN ASIDE Oscillating E and B fields contribute to the energy of em radiation. Above we have given the intensity of resonantly absorbed electric field energy in terms of the electric dipole transition moment. Molecules also absorb magnetic field energy; there is a magnetic dipole transition moment. But typically mdtm ≈ 10-5 edtm. Usually ignored, just as we ignore the electric quadrupole tm. However sometimes we need the mdtm or the eqtm. ESR and NMR are magnetic dipole transitions. Electric field energy absorbed by low pressure H2 gas is elec. quadrupole. Here ends the summary of Ch.1 15

16. Quantum mechanics Ch.2 CAN SIMULATE A SPECTRUM BY CALC OF E and Φ ~ f POSITION: At a resonance hνif = Ef – Ei = ΔEif νif i M INTENSITY: Proportional to Line strength S(f ← i) μA = ΣCre Ar ∑ S(f ← i) = | ∫Φf* μAΦi dτ |2 r A=X,Y,Z

17. For discussion of full H including spin terms see pages 126-131 For spin-free H see pages 26-29 and 32-33 Quantum mechanics Q.M. Wave Schrödinger equation

18. Quantum mechanics For discussion of full H including spin terms see pages 126-131 For spin-free H see pages 26-28 and 32-33 Spin-free H is Hrve (rovibronic H)

19. 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.

20. Off-diagonal matrix elements (ODME) An pxp matrix A has elements Amn A11 A12 A13… A1p A21A22 A23… A2p A31…………… A3p A41 . . Ap1……………App Off-diagonal elements Diagonal elements A matrix is ‘diagonal’ if all off-diagonal elements are zero

21. Quantum mechanics and off-diagonal matrix elements (ODME) Schrödinger equation Eigenfunctions of an approximate H are“basis” functions 0 0 0 0 Exact Ejandψjare obtained from ψn0by Eigenvalues and eigenfunctions are found by diagonalization of a matrix with elements Ch 2 p 21-26 Problem 2.5 on page 41 ODME has m ≠ n; perturbation

22. APPROXIMATE OR ZEROTH ORDER SITUATION ψm0 Em0 Δmn0 = Em0 - En0 En0 ψn0

23. Effect of nonvanishing ODME Em ,ψm ψm0 +δ Em0 - - - - - Δmn0 = Em0 - En0 En0 - - - - - -δ ψn0 En ,ψn δ ~ Hmn2 / Δmn0 Using Perturbation Theory ψm~ ψm0 + [Hmn /Δmn0 ]ψn0

24. Ψe-v-r-ns0 Neglect Ortho-para mixing Neglect Cent. Dist Coriolis Born-Opp approx ψelec ψvib ψrot ψns Harmonic oscillator Rigid rotor Uncoupled spins Molecular Orbitals (espin and Slater determinants)

25. To simulate a spectrum we need energies and line strengths i.e. we need to calculate: ODME of H ODME of μA μfi = ∫ (Ψf )* μAΨi dτ

26. The Role of Approximations To solve the Sch. Eq. we make approximations and then correct for them. In so doing we introduce concepts that enable us to UNDERSTAND molecules.Key approximation is the BORN-OPPENHEIMER APPROXIMATION Concepts that come about because we make approximations: Electronic state, molecular orbital, electronic configuration, potential energy surface, molecular structure, force constants, electronic angular momentum, vibrational state, vibrational angular momentum, rotational state, Coriolis coupling, centrifugal distortion,… We could call the approach that aims to solve the Sch. Eq. numerically without approximations using a gigantic computer “The few-concepts big-computer” approach. BO Approx. Pages 43-47 When I am grumpy and sarcastic I like to call it “The small-brain big-computer approach.”

27. 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 Here ends the summary of Chapters 1 and 2

28. 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 28

29. Now for Molecular Symmetry http://www.google.com/ Wikipedia molecular symmetry

30. Now for Molecular Symmetry http://www.google.com/ Wikipedia molecular symmetry No discussion of the fundamentals of how or why the geometrical symmetry of the equilibrium structure of a moleule in a particular electronic state allows one to do the things it says you can do. No discussion of what the “rotations” and “reflection” symmetry operations do to molecular coordinates.

31. Molecular Symmetry (as it should be explained)

32. Molecular Symmetry (as it should be explained) An important use of symmetry is to put “symmetry labels” on the zeroth order energy levels. Knowing the symmetry labels it is easy to determine which ODME of H and have to be zero. μA

33. GROUP THEORY “Just a bunch of definitions” FERMI: and POINT GROUPS (uses geometrical symmetry)

34. The Point Group of H2O x The identity operation (+y) z The point group of H2O consists of the four symmetry operations E, C2x, xz, and xy It is called the C2v point group

35. Examples of point group symmetry H2O C2v C3H4 D2d CH3F C3v C60 Ih 120 symmetry operations

36. x x (+y) (+y) z z Successive application = “multiplication” x σxz (+y) z - - + + C2x σxy σxz σxy C2x = + +

37. x (+y) z Successive application = “multiplication” C2v multiplication table E C2xσxzσxy E E C2xσxzσxy C2x C2x E σxy σxz σxzσxzσxy E C2x σxyσxyσxz C2x E + + σxz σxy C2x = A “Group” contains all products of its members and it contains E

38. x (+y) z Successive application = “multiplication” E C2xσxzσxy E E C2xσxzσxy C2x C2x E σxy σxz σxzσxzσxy E C2x σxyσxyσxz C2x E + + σxz σxy C2x = A “Group” contains all products of its members and it contains E { } E C2xσxz IS NOT A GROUP

39. x (+y) z Successive application = “multiplication” C2 multiplication table E C2xσxzσxy E E C2xσxzσxy C2x C2x E σxy σxz σxzσxzσxy E C2x σxyσxyσxz C2x E + + E C2x C2x = A “Group” contains all products of its members and it contains E THE C2 GROUP – A SUBGROUP OF C2v

40. PH3 at equilibrium: The C3v point group 3 2 1 Symmetry operations: C3v = {E,C3, C32, 1, 2, 3 }

41. Multiplication table forC3v Multiplication is not necessarily commutative C3 = σ1σ2 C32 = σ2σ1

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

43. Multiplication table forthe matrix group M3 = M5 M4

44. E C3 C32 σ1 σ2 σ3 M1 M2 M3 M4 M5 M6 C32 = σ2σ1 Multiplication tables have the ‘same shape’ M3 = M5 M4 This matrix group forms a “representation” of the C3v group Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations

45. E C3 C32 σ1 σ2 σ3 M1 M2 M3 M4 M5 M6 C32 = σ2σ1 Multiplication tables have the ‘same shape’ M3 = M5 M4 This matrix group forms a “representation” of the C3v group Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations

46. E C3 C32 σ1 σ2 σ3 M1 M2 M3 M4 M5 M6 C32 = σ2σ1 Multiplication tables have the ‘same shape’ M3 = M5 M4 This matrix group forms a “representation” of the C3v group Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations

47. E C3 C32 σ1 σ2 σ3 M1 M2 M3 M4 M5 M6 C32 = σ2σ1 Multiplication tables have the ‘same shape’ M3 = M5 M4 This matrix group forms an “irreducible representation” of the C3v group

48. 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

49. Characters of the 2d matrix irrep of C3v EC3σ1 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

50. 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