Molecular Spectroscopy 박영동 교수

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

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

Rotational IR Spectroscopy Linear Molecules, D∞h or C∞v 𝐸 𝑟 =𝐹 𝐽 =𝐵𝐽 𝐽+1 −𝐷 𝐽 2 𝐽+1 2 + … 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

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

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, 1974. Copyright 1974 Pergamon Press)

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

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 + …

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, 1972. Copyright 1972 American Chemical Society) Figure 5.8 The s-trans and s-cis isomers of crotonic acid

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)

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

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

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

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

5.3.2 Theory of rotational Raman scattering Figure 5.14 The polarizability ellipsoid

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

Rotational Raman spectrum of 15N2 obtained with 476.5 nm radiation from an argon ion laser. B0 : 1.857 6720.000 027 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)

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.

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

Rotational Raman spectrum of 14N2

Rotational Raman spectrum of 16O2

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.)

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

5.4 Structure determination from rotational constants

5.4 Structure determination from rotational constants

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

6.1.1 Infrared spectra

6.1.2 Raman spectra

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

35Cl: 75.76 37Cl: 24.24

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

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.

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

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

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

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

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 = 0 + 0 + 0 + 2 + 0 + 0 + 1 = 3. au = 0 + 0 + 0 + 1 = 1. b1g = 0 + 0 + 0 + 1 + 0 + 0 – 1 = 0. b1u = 0 + 0 + 0 + 2 + 0 + 0 + 1 + 0 – 1 = 2. b2g = 0 + 0 + 0 + 1 + 0 +1 - 1 = 1. b2u = 0 + 0 + 0 + 2 + 0 + 0 + 1 + 0 – 1 = 2. b3g = 0 + 0 + 0 + 2 + 0 + 1 – 1 = 2. b3u = 0 + 0 + 0 + 1 + 0 + 0 + 1 + 0 – 1 = 1. 3ag+au+2b1u+b2g+2b2u+2b3g+b3u

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

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