1. Molecular Spectroscopy
박영동 교수

2. 5 Rotational Spectroscopy
For a linear molecule,
For a symmetric rotor, or symmetric top
Cn axis with n>2 or an S4 axis
prolate symmetric rotor.
oblate symmetric rotor
A spherical rotor

3. An asymmetric rotor
a prolate near-symmetric rotor
an oblate near-symmetric rotor

4. Rotational IR Spectroscopy Linear Molecules, D∞h or C∞v
𝐸 𝑟 =𝐹 𝐽 =𝐵𝐽 𝐽+1 −𝐷 𝐽 2 𝐽 …
𝐷= 4 𝐵 2 𝜔 2
For diatomic molecules(and XY2 molecules) with vibrational frequency ω,
For molecules of symmetry D∞h which have no dipole moment do not exhibit a (dipole) rotation spectrum in the infrared.
1. 𝜇 ≠0,
2. ∆𝐽=±1,
3. ∆ 𝑀 𝐽 =0, ±1

5. Figure 5.2 Rotational term values F(J) (horizontal lines): relative populations NJ /N0 (calculated from Equation 5.15) and transition wavenumbers 𝜈 (for the transitions indicated by the vertical arrows) for CO

6. Figure 5.3 Far-infrared spectrum of CO showing transitions with J” = 3 to 9. (Reproduced, with permission, from Fleming, J. W. and Chamberlain, J., Infrared Phys., 14, 277, Copyright 1974 Pergamon Press)

7. 5.2.2 Symmetric rotor molecules
a prolate symmetric rotor
K = 0, 1, 2, , J .
1. ∆𝐽=±1,
3. ∆𝐾=0
Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where Pa is the component along the a axis
an oblate symmetric rotor

8. 5.2.4 Asymmetric rotor molecules
For near prolate molecules,
𝐹 𝐽, 𝐾 ≅𝐵𝐽 𝐽+1 +(𝐴−𝐵) 𝐾 2 + …
For near oblate molecules,
𝐹 𝐽, 𝐾 ≅𝐵𝐽 𝐽+1 +(𝐶−𝐵) 𝐾 2 + …

9. 5.2.4 Asymmetric rotor molecules
s-cis
s-trans
Figure 5.9 Part of the microwave spectrum of crotonic acid. (Reproduced, with permission, from Scharpen, L. H. and Laurie, V. W., Analyt. Chem., 44, 378R, Copyright 1972 American Chemical Society)
Figure 5.8 The s-trans and s-cis isomers of crotonic acid

10. 5.2.5 Spherical rotor molecules
Figure 5.10 Part of the far-infrared spectrum of silane. (Reproduced, with permission, from Rosenberg, A. and Ozier, I., Can. J. Phys., 52, 575, 1974)

11. H2O rotational spectrum
Asymmetric top
(J, K-1, K+1)

13. Rotational Raman Spectroscopy Linear Molecules
1. 𝜇=𝑜𝑟 ≠0,
2. ∆𝐽=0, ±2.
𝐸 𝑟 =𝐹 𝐽 =𝐵𝐽 𝐽+1 −𝐷 𝐽 2 𝐽 …

14. Raman spectroscopy Detector
Figure 5.13 Experimental arrangement for gas phase Raman spectroscopy.

15. Rotational Raman Spectroscopy symmetric rotor molecules
∆𝐽=0, ±1,±2,
∆𝐾=0.

16. 5.3.2 Theory of rotational Raman scattering
Figure 5.14 The polarizability ellipsoid

17. 5.3.3 Rotational Raman spectra of diatomic and linear polyatomic molecules
Figure 5.15 Rotational Raman spectrum of a diatomic or linear polyatomic molecule

18. Rotational Raman spectrum of 15N2
obtained with nm radiation from an argon ion laser.
B0 : cm-1
Figure 5.17 Rotational Raman spectrum of 15N2 (the lines marked with a cross are grating ‘ghosts’ and not part of the spectrum)

19. 5.3.4 Nuclear spin statistical weights for a symmetrical (D∞h) diatomic or linear polyatomic molecules
𝜓𝑒: electronic wave function,
𝜓𝑣 : vibrational wave function,
𝜓𝑟 : rotational wave function,
𝜓𝑛𝑠 : nuclear spin wave function.
𝜓𝑒: ground electronic wave function: sym
𝜓𝑣 : vibrational wave function: sym
𝜓𝑟 : rotational wave function: sym for enen J, antisym for odd J
𝜓𝑛𝑠 : nuclear spin wave function: ?
𝜓𝑛𝑠 : nuclear spin wave function: (2I+1)(I+1) symmetric, and (2I+1)I antisymmetric
𝜓 : sym for integer I, follows Bose-Einstein statistics.
antisym for half-integer I, follows Fermi-Dirac statistics.

20. I = ½
I = 1
I = 0
Figure 5.18 Nuclear spin statistical weights (ns stat wts) of rotational states of various diatomic
molecules; a, antisymmetric; s, symmetric; o, ortho; p, para; 𝜓𝑟 ; 𝜓𝑛𝑠 and 𝜓𝑒 ; rotational, nuclear spin
and electronic wave functions, respectively

21. Rotational Raman spectrum of 14N2

22. Rotational Raman spectrum of 16O2

23. Statistical weight factors of symmetric and antisymmetric (even and odd) rotational levels of some linear molecules.*
Molecule
Resultant statistics
Statistical weight factors
Symmetric (even) levels
Antisymmetric (odd) levels
12C16O2, 13C16O2, 12C18O2
Bose 1
0**
12C2H2
Fermi 3
12C2D2 6
13C2H2 10
13C3D2 15 21
12C214N2
12C215N2
12C316O2, 12C318O2
12C13C216O2, 13C316O2
3**
12C4H2
13C4H2 28 36
12C4D2
12C213C2H2
13C4D2 78 66
12C(14NH)2
12C(14ND)2 45
13C(15NH)2
* The following values for the nuclear spins have been assumed; I(H, 13C, 15N, 19F) =1/2, I(12C, 16O, 18O) =0, I(D, 14N) = 1, I(35Cl) = 3/2, I(37Cl) = 3/2.
** If 13C instead of 12C is at the center the statistical weights have to be multiplied by 2 [assuming I(13C) = 1/2].
(Table from Herzberg, vol II, p18, Table 2.)

24. 5.3.5 Rotational Raman spectra of symmetric and asymmetric rotor molecules
For a symmetric rotor molecule:
For asymmetric rotors
too complex for us!

25. 5.4 Structure determination from rotational constants

26. 5.4 Structure determination from rotational constants

27. 6. Vibrational Spectroscopy
6.1 Diatomic spectropscopy
1 aJ Å-2 = 1 mdyne Å-1 = 100 N m-1

28. 6.1.1 Infrared spectra

30. 6.1.2 Raman spectra

31. 6.1.3 Anharmonicity
De: dissociation energy
D0: zero-point energy

34. 35Cl: 75.76
37Cl: 24.24

35. HCl isotopes, from SJ. Chem. Educ. 40, 245(1963)

38. IR OH stretching vibration of Phenol
in diethylether has a wavenumber of 3344 cm-1.
O-H stretching vibration of phenol in solution in hexane has a wavenumber of 3622 cm-1.

40. 24B=67 cm-1
𝐸 𝑟 = ℏ 2 2𝐼 𝐽 𝐽+1 =𝐵𝐽 𝐽+1

48. Exam 2
1. ethylene(C2H4) 분자의 진동모드
H H C x y z

49. C=C str
CH2 s-str
CH2 twis
CH2 rock
H H C x y z
CH2 wag
CH2 rock
CH2 a-str
CH2 twis
CH2 a-str
CH2 sci
CH2 s-str
CH2 wag

50. m=0, mxy=0, mxz=0, myz=1, m2x=0, m2y=0, m2z=1, m0=0.
Hollas p.423
Hollas p.423
m=0, mxy=0, mxz=0, myz=1, m2x=0, m2y=0, m2z=1, m0=0.
Hollas p.416

51. 3ag+au+2b1u+b2g+2b2u+2b3g+b3u
Hollas p.423
m=0, mxy=0, mxz=0, myz=1, m2x=0, m2y=0, m2z=1, m0=0.
ag = = 3.
au = = 1.
b1g = – 1 = 0.
b1u = – 1 = 2.
b2g = = 1.
b2u = – 1 = 2.
b3g = – 1 = 2.
b3u = – 1 = 1.
3ag+au+2b1u+b2g+2b2u+2b3g+b3u

52. H H C x y z ag
b1u
b2u au
b3g
b3u
b2u ag ag
b2g
b1u
b3g

53. CH2=CH2 # 대칭종 cm-1 (*) 1 CH2 s-str (A1)ag 3022 2 C=C str 1625 3
CH2 i-p sci
1343 4
CH2 o-p twis
(A2)au
1026 5
CH2 o-p wag
(B1)b2g
940 6
(A1)b1u
2989 7
CH2 a-str
(B2)b2u
3105 8
CH2 i-p rock
826 9
1444 10
(B2)b3g
1222 11
3083 12
(B1)b3u
949
위 표와 그림에서 B1과 B2는 서로 바뀌어져 있음