Lecture 4: Polyatomic Spectra

Transcription

1 Lecture 4: Polyatomic Spectra 1. From diatomic to polyatomic Ammonia molecule A-axis. Classification of polyatomic molecules 3. Rotational spectra of polyatomic molecules N 4. Vibrational bands, vibrational spectra H

2 1. From diatomic to polyatomic Rotation Diatomics Recall: For diatomic molecules Energy: 1 h Rotational constant: B, cm 8 Ic F Selection Rule: J ' 1 J, cm BJ J 1 DJ J 1 J" 1 J R. R. 1 Line position: 3 J " 1 J " B J" 1 4D J" 1 Centrifugal distortion constant Notes: 1. D v is small, i.e.,. E.g., for NO, D B D / B 4 NO B 4 e B / vib J=60, 310 What about polyatomics ( 3 atoms)? 6 D / B J ~ 0.01

3 1. From diatomic to polyatomic 3D-body rotation B C A Convention: A-axis is the unique or figure axis, along which lies the molecule s defining symmetry 3 principal axes (orthogonal): A, B, C 3 principal moments of inertia: I A, I B, I C Molecules are classified in terms of the relative values of I A, I B, I C 3

4 . Classification of polyatomic molecules Types of molecules Type Linear Molecules Symmetric Tops Spherical Tops Asymmetric Rotors Relative magnitudes of I A,B,C I B =I C ; I A 0* I B =I C I A I A 0 I A =I B =I C I A I B I C CO NH 3 CH 4 H O Examples C H Acetylene CH 3 F NO OCS Carbon oxysulfide BCl 3 Boron trichloride Relatively simple No dipole moment Largest category Not microwave active Most complex *Actually finite, but quantized momentum means it is in lowest state of rotation 4

5 . Classification of polyatomic molecules Linear molecules E.g., Carbon oxy-sulfide (OCS) B C A O C S r CO r CS A B r C r S Center of mass r CO = 1.165Å r CS = 1.558Å I B =I C ; I A 0 B, cm 1 h 8 I B c C 5

6 . Classification of polyatomic molecules Symmetric tops Prolate I B =I C I A ; I A 0 A, cm B, cm C, cm 1 1 h 8 I h 8 I h 8 I A B C c c c 1 H I A <I B =I C, A>B=C E.g., CH 3 F B C H C F A H C.M. Tripod-like (tetrahedral bonding) A B C 6

7 . Classification of polyatomic molecules Symmetric tops I B =I C I A ; I A 0 A, cm B, cm C, cm B h 8 I h 8 I h 8 I A A B C c c c C Oblate I A >I B =I C, A<B=C E.g., BCl 3 (Planar) B B Cl Cl C.M. Cl Planar view No elec. dipole mom. no QM selection rule C A A B C 7

8 . Classification of polyatomic molecules Spherical tops Asymmetric rotors C.M. I A =I B =I C E.g., CH 4 (methane) H H C H H Cube w/ C at center and H at diagonal corners Symmetric, but No dipole moment No rotational spectrum I A I B I C E.g., H O A O H H B o C.M. Complex and not addressed here C 8

9 3. Rotational spectra of polyatomic molecules Linear molecules (I B =I C ; I A 0) Examples OCS HCN CO C H HC Cl Symmetric, no dipole moment Must be asymmetric to have electric dipole moment (isotopic substitution doesn t change this as bond lengths remain fixed) Energies and line positions Can treat like diatomic (1 value of I) same spectrum FJ BJ J 1 DJ J 1 _ J BJ 1 4DJ 1 3 Rotational const. Centrifugal distortion const. Note: Larger I, smaller B (& line spacing) than diatomics ( is suppressed, i.e. J=J ) 9

10 3. Rotational spectra of polyatomic molecules Linear molecules (I B =I C ; I A 0) Bond lengths N atoms N-1 bond lengths to be found Abs./Emis. spectra B 1 value of I B Use N-1 isotopes N-1 values of I B Example: OCS (carbon oxy-sulfide) O 16 C 1 S Å 1.558Å O C S Use isotopes for equations: I I O O 1 1 C C 3 3 S S F F masses, r masses, r CO CO, r, r CS CS r CO r CS r C r S Center of mass Solve for r CO, r CS 10

11 3. Rotational spectra of polyatomic molecules Symmetric tops (I B =I C I A ; I A 0) main directions of rotation quantum numbers J (total angular momentum): 0, 1,, K (angular momentum about A): J, J-1,, 1, 0, -1, -J + & - allowed, w/o change in energy Quantized angular momentum As before: I Plus new: A A Energy levels 1 E J, K Ii i I I I A A i B B K C J+1 possibilities of K for each J C J J 1 F J, K BJ J 1 A B K Note degeneracy, i.e., independent of sign of K J K 11

12 3. Rotational spectra of polyatomic molecules Symmetric tops (I B =I C I A ; I A 0) Q.M. Selection rules J = +1 Remember that J = J J K = 0 No dipole moment for rotation about A-axis No change in K will occur with abs./emis. Line positions J. K F 1 J 1, K FJ, K BJ J 1 cm Note: Independent of K for a rigid rotor Same as rigid diatomic! K-dependence introduced for non-rigid rotation 1

13 F 3. Rotational spectra of polyatomic molecules Symmetric tops (I B =I C I A ; I A 0) Non-rigid rotation Effect of extending bond lengths (w/ changes in K) Change energies of rotation Centrifugal distortion const. D J, D K, D JK J, K BJ J 1 A BK DJ J J 1 4 DJK J J 1K DK K J. K J 1B D J 1 D K 1 cm J Note: Each J has J+1 components, but only J+1 frequencies JK B 0.851cm D D J JK 10 E.g., CH 3 F, Methyl Fluoride 6 1 cm H 1 cm 1 H C H C.M. F If J 0, J 400, DJ 1.6x10-3, DJ /B.% J 1 3 ν,cm lines for J= K=,1,0 B ~1.7cm -1 K 1 0 D JK (J+1)K ~4x10-4 cm -1 A D JK (J+1)K ~10-4 cm -1 13

14 3. Rotational spectra of polyatomic molecules Symmetric tops (I B =I C I A ; I A 0) gets complex fast! F Prolate I A <I B =I C, A>B=C J, K BJ J 1 A BK A B h c I A I B F Oblate I A >I B =I C, A<B=C J, K BJ J 1 A BK A B h c I A I B Example energy levels 14

15 3. Rotational spectra of polyatomic molecules Rotational partition function Linear Symmetric top Spherical top Asymmetric rotor B=C; I A 0 B=C A; I A 0 A=B=C A B C Q rot kt hcb Q rot 1 AB kt hc 3 Q rot 1 3 B kt hc 3 Q rot 1 ABC kt hc 3 A, cm B, cm 1 1 h 8 I h 8 I A B c c σ molecule-dependent symmetry factor Molecule σ Molecule Type CO Linear NH 3 3 Symmetric Top C, cm 1 h 8 I C c CH 4 1 Spherical Top H O Asymmetric Rotor 15

16 3. Rotational spectra of polyatomic molecules: Summary Linear (diatomic & polyatomic) and symmetric top molecules give similar (equal spacing) spectra at rigid rotor level High resolution needed to detect corrections / splittings Spectra microscopic parameters (r e, angles) Isotopes useful for spectral studies 16

17 4. Vibrational Bands, Rovibrational Spectra 1. Number of vibrational modes. Types of bands Spectrum of bending mode of HCN Parallel and perpendicular Fundamental, overtones, combination and difference bands 3. Relative strengths 4. Rovibrational spectra of polyatomic molecules Linear molecules Symmetric tops 17

18 4.1. Number of vibrational modes N-atom molecule 3N dynamical coordinates needed to specify instantaneous location and orientation Total: Center of Mass: Rotation: 3N 3 coordinates (3 translational modes) Linear molecules Nonlinear molecules angular coordinates 3 angular coordinates (rot. modes) (rot. modes) Vibration: Linear molecules Nonlinear molecules 3N-5 vibrational coordinates 3N-6 angular coordinates (vib. modes) (vib. modes) 18

19 4.. Types of bands Numbering (identification) convention of vibrational modes Symmetry Decreasing energy Symmetric Declining frequency Asymmetric Declining frequency [cm -1 ] ν 1 ν ν i ν i+1 ν i+ Highest-frequency symmetric vibrational mode nd highest symmetric mode Lowest-frequency symmetric mode Highest-frequency asymmetric vibrational mode nd highest symmetric mode Exception: the perpendicular vibration for linear XY and XYZ molecules is always called ν 19

20 4.. Types of bands Parallel and perpendicular modes Examples: Parallel ( ) Perpendicular () Dipole changes are Dipole changes are to the main axis of symmetry to the main axis of symmetry H O (3x3-6=3 vib. modes) Symmetric stretch Symmetric bending Asymmetric stretch ν 1 =365cm -1 ν =1595cm -1 ν 3 =3756cm -1 CO (3x3-5=4 vib. modes) No dipole moment Not IR-active! Symmetric stretch Asymmetric stretch Symmetric bending ( degenerate) ν 1 =1330cm -1 ν 3 =349cm -1 ν =667cm -1 0

21 4.. Types of bands Parallel and perpendicular modes Symmetric molecules: vibrational modes are either IR-active or Raman-active (Chapter 6) Mode Frequency [cm -1 ] Vibrational modes of CO Type Description IR Raman ν Symmetric stretch Not active Active ν 667 Symmetric bend (Degenerate) Strong Not active ν Asymmetric stretch Very strong Not active Vibrational modes of HCN Mode Frequency [cm -1 ] Type Description IR Raman ν Symmetric stretch Strong Weak ν 715 Symmetric bend (Degenerate) Very strong Weak ν Asymmetric stretch Weak Strong 1

22 4.. Types of bands Terminology for different types of vibrational bands Fundamental Bands: ν i, the i th vibrational mode; v=v'-v"=1 for the i th mode 1 st Overtone: nd Overtone: ν i ; 3ν i ; v=v'-v"= for the i th mode v=v'-v"=3 for the i th mode Combination bands: Difference bands: Changes in multiple quantum numbers, e.g., ν 1 +ν ; v 1 = v =1, i.e., v 1 and v both increase by 1 for absorption or decrease by 1 for emission ν 1 +ν ; v 1 = and v =1 Quantum number changes with mixed sign ν 1 -ν ; v 1,final -v 1,initial = ±1 and v,final -v,initial = 1, i.e., a unit increase in v 1 is accompanied by a unit decrease in v, and vice-versa.

23 4.. Types of bands Vibrational partition function Q vib modes i 1 exp hc kt e, i g i E.g., NH 3 : 3N-6 = 6 vib. modes Degenerate Q vib 1 exp hc kt e,1 1 1 exp hc kt e, 1 1 exp hc kt e,3 1 exp hc kt e,4 Vibration Frequency [cm -1 ] Type Description ν Symmetric stretch ν 950 Symmetric bend ν Asymmetric stretch (Degenerate) ν Asymmetric bend (Degenerate) 3

24 4.3. Relative strength In general Fundamental bands are much stronger than combination, difference, and overtone bands Fairly harmonic molecules E.g., CO Relative strength between fundamental and overtones ~ 10 Closely SHO, overtone bands are nearly forbidden (low transition probabilities) Highly anharmonic molecules E.g., NH 3 Relative strength between fundamental and overtones 10 Overtone bands are less forbidden Exception Fermi resonance: Accidental degeneracies (i.e., near resonances) can strengthen weak processes. Two vib. Modes strongly coupled by radiative and collisional exchanges. E.g., ν CO (@ 1334cm -1 ) ν 1, CO 4

25 4.4. Rovibrational spectra of polyatomic molecules Linear polyatomic molecules (limit consideration to fundamental transitions) Energy: T v, J Gv FJ Case I: Parallel bands (symmetric and asymmetric stretch) i i Selection Rule: Absorption Spectrum: Example: v v 0, j i Note: No ν 1 parallel band for CO i J 1 ( R and P branches) j 1 P & R branches only HCN(ν 1, ν 3 ) Transition Probabilities v'=1 v"=0 P P branch R Null Gap P(1) R(0) J'=J"+1 J'= J" J'= J"-1 J"+1 J" R() R branch o B 5

26 4.4. Rovibrational spectra of polyatomic molecules Linear polyatomic molecules Case I: Parallel band Energy = 1 Anti-symmetric Stretching Vibration = 0 Per cent Transmission _ 1 Example-: A parallel band of the linear molecule CO 6

27 (bra)4.4. Rovibrational spectra of polyatomic molecules Linear polyatomic molecules (limit consideration to fundamental transitions) Energy: Case II: Perpendicular bands Selection Rule: vi 1 J 1,0 RPandQsv j 0, j i If B'=B", all Q branch,1. lines occur at the same frequency 0. If B' B", Q J" 0 B' B" J" J" 1 Q branch degrades to lower frequencies (i.e., to the red in wavelength) nct v, J Gv FJ hei Transition Probabilities i v'=1 v"=0 P P branch Q Q branch J'=J"+1 J'= J" J'= J"-1 R J"+1 J" R branch o B 7

28 4.4. Rovibrational spectra of polyatomic molecules Linear polyatomic molecules Case II: Perpendicular bands Example: Spectrum of the bending mode of HCN, showing the PQR structure impurity _ 700 _

29 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules (e.g., CH 3 F, BCl 3 ) Recall: K quantum number for angular momentum around axis A Energy: T vi, J, K Gvi FJ, K i i i v 1/ x v 1/ BJ J 1 A BK i e e e i Case I: Parallel bands Tripod-like Selection Rule: v i K 1 J 1,0 ( P, Q, R branches) 0 B C 1. J+1 values of K (K=J, J-1,, 0,, -J). Intensity of Q branch is a function of (I A /I B ) 3. As (I A /I B ) 0 symmetric top linear molecule strength of Q branch 0 Planar 9

30 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules (e.g., CH 3 F, BCl 3 ) Case I: Parallel bands Note: 1. Splitting in P and R branch due to a difference in (A-B) in upper and lower vib. levels. Splitting in Q branch due to difference in B in upper and lower vib. levels 3. For K=0, spectrum reduces to that of linear molecules, no Q branch 4. K cannot exceed J Resolved components of a parallel band showing contributions from each of the K levels of the v=0 state 30

31 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules Case I: Parallel bands Per cent Transmission R-branch Q-branch P-branch _ 1 Example-1: A parallel absorption band of the symmetric top molecule CH 3 Br. The P branch is partly resolved, while only the contours of the R and Q branches is obtained 31

32 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules Case I: Parallel bands Absorbance _ 1 Example-: The parallel stretching vibration, centered at 151 cm -1, of the symmetric top molecule CH 3 I, showing the typical PQR contour. 3

33 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules (e.g., CH 3 F, BCl 3 ) Case II: Perpendicular bands Selection Rule: R Branch: P Branch: Q Branch: v i K 1 J 1,0 ( P, Q, R branches) J 1 1, K 1 J 1 A B1 K R o B J 1, K 1 A B1 K P o BJ J 0, K 1 A B1 K Q o Note: Two sets of R, P and Q branches for each lower state value of K 33

34 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules (e.g., CH 3 F, BCl 3 ) Case II: Perpendicular bands Energy levels of a symmetric top molecule showing transitions that are allowed for a perpendicular band Resulting spectrum, components of a perpendicular band showing the contributions from each K levels of the v=0 state 0 34

35 4.4. Rovibrational spectra of polyatomic molecules Symmetric top molecules Case II: Perpendicular bands Per cent Transmission Note: Spacing of the Q branch lines in a perpendicular band can be identified with (A-B), and hence are observable if A-B is large enough _ 1 Example: The Q-branch of a perpendicular band, for the symmetric top molecule CH 3 Cl 35

36 Next: Quantitative Emission/Absorption Spectral Absorptivity Eqn. of Radiative Transfer Einstein Coefficients/Theory Radiative Lifetime, Line Strength