Molecular Spectra and Molecular Structure - Lec 23S

Page 1 of 6 pages Lecture #23: Pure Rotation Spectra of Polyatomic Molecules Last time: Asymmetric tops ⎡ ⎛ A − B⎞ 2 ⎤ H ROT = ⎢BJ(J + 1) + ⎜ ⎟ K ⎥ JK JK ⎝ C − B⎠ ⎦ ⎣ ⎛B − C⎞ 1 1/2 +⎜ ⎟ [(J mK)(J mK −1)(J ± K + 2)(J ± K + 1)] JK JK ± 2 ⎝ A − B⎠ 4 asymmetry parameter κ ≡ 2B – A – C A–C prolate B=C κ = –1 oblate A=B A+C B= 2 κ = +1 most asymmetric Perturbation Theory: κ=0 H′JK, JK±2 1 = f(J, K) κ ² E° Factor HROT into four blocks — can label these blocks according to whether Kp and Ko (or Ka and Kc) are even or odd e,e e,o o,e o,o correspond to rotational symmetry species — good quantum numbers [not related to geometric symmetry operations on rigid molecule]. TODAY pure rotation spectra SYMMETRIC TOP → Mx = My = 0, ∆K = 0 like diatomic. Only information about B ASYMMETRIC TOP → M can have 1, 2, or 3 non-zero components Transition ∆Kp ∆Ko a-type even odd b-type odd odd c-type odd even NOTATION/ASSIGNMENT Pif ∝ ∑ εS vi  M b,if (Q) vf Ωi J i M i αSb Ωf J f M f S,b pure rotation Mb(Qe) BODY COMPONENT b → (a, b, or c) Hougen, p. 31 f(Ji,Jf)g(JiΩi;JfΩf)h(JiMi;JfMf) body polarization K↔Ω lab polarization 2 Page 2 of 6 pages Symmetric Tops - can’t have nonzero dipole ⊥ to unique axis: z z → a prolate → c oblate   ∆K = 0, ∆J = ±1 transitions only if there were a ⊥ µ component, it would have to point in same direction after a Cn(z) or Sn(z) operation. µ⊥ = 0 for n ≥ 2.  ⎛ A − B⎞ 2 E ROT = BJ(J +1) + ⎜ ⎟K ⎝ C − B⎠ (∆J = 0 weak, except for J = K) identical BJ(J + 1) stacks shifted by A - BC - B K2 no inter-stack transitions no information about other rotational constants ∆EJK↔J–1K = 2BJ (small splittings due to centrifugal distortion o EJK = EJK − DJ J 2 ( J + 1) 2 − DJK J ( J + 1) K 2 − DK K 4 ) but these give no information about A or C! ⎧other rotational constants ⎧from IR and UV spectra ⎨ Get information about ⎨equilibrium structure isotopomers 14 NH3 15 NH3 14 ⎩ ND3 etc. ⎩ and non-symmetric species (non-symmetric tops!) Asymmetric Tops - more than 1 component of M can be nonzero. K ≠ 0 levels split by asymmetry splitting. Same ∆J type transitions from different K’s not superimposed. No resemblance to diatomic molecule spectrum! ∆K symmetry label - C2 about specified axis prolate axis oblate axis e or o , e or o Kp Ko J K p ,K o Ma (e, o) Mb (o, o) Mc (o, e) K′p J′K p′ K o′ M a J′′K p′′ K o′′ b c e×e×e=e K′o Kp Ko K ′′p K ′′o ⎛a e o⎞ ⎛ e e ⎞⎜ ⎟⎛ e e ⎞ →⎜ ⎟ ⎟⎜ b o o ⎟⎜ ⎝ o o ⎠⎜ ⎟⎝ o o ⎠ ⎝c o e⎠ o × o × e = e etc. Page 3 of 6 pages integrand must be (e, e) for integral to be ≠ 0  a-type selection rule ∆Ka = even b-type selection rule ∆Ka = odd c-type selection rule ∆Ka = odd ∆Kc = odd ∆Kc = odd ∆Kc = even Want ∆K to be as small as possible. 0 or ±1 for near symmetry axis, ±2 or ±3 possible for other axes. Kp and Ko are (related to) symmetry labels. Only one of these two quantum numbers is an almost good projection quantum number. ∆J = 0 strong or weak depending on whether ∆Knear sym = ±1 or 0. near prolate near oblate a || ∆J = 0 weak ∆Ka = 0 ⊥ ∆J = 0 strong ∆Kc = ±1 b ⊥ ∆J = 0 strong ∆Ka = ±1 ⊥ ∆J = 0 strong ∆Kc = ±1 c ⊥ ∆J = 0 strong ∆Ka = ±1 || ∆J = 0 weak ∆Kc = ±0 NOTATION examples ² Kp p larger splitting larger o or o ² J a or b or c near prolate 42,3 (J Kp , J Ko ) near oblate 41,3 (other asymmetry component in respective limit) 41,4 32,2 smaller smaller q Ra(31,2) r Ra(31,2) 31,2 31,3 41,4 (level does not exist) 40,4 40,5 p 32,2 Rc(31,2) s Rc(31,2) weak ∆Ko = 2 31,2 31,3 Can decide whether the other asymmetry doublet transition is at higher or lower frequency. Asymmetry ν (q R a (31,2 ))> ν (q R a (31, 3 )). splitting is largest for largest J–K: B Page 4 of 6 pages A lot of information about expected patterns can be deduced from qualitative prolate↔oblate correlation diagram and simple transition intensity ideas. NON-LECTURE Table of transition intensities in Townes and Schawlow, pages 557-612 (similar to diatomic M-averaged rotational linestrength factors). M averaged x SJkl J mn ′ (κ) ≡ 10 (2J +1) J kl M x α xS J′mn 4 listed for κ = ±1, ±0.5, 0 x = a, b, c Kp: k, m Ko: l ,n Transitions are listed by sub-branch and labeled by lower state quantum numbers. J Kp Ko x R ² K p ² K o (J K p ,K o ) 2 left columns are for upper sign of κ 2 right columns are for lower sign of κ But we are modern and have computers. How to do it? * specify A, B, C * diagonalize H ROT ⎧⎪E (J K K ) p o ⎨ ⎩⎪ ψJKpKo * evaluate MbαbS matrix elements following Hougen, pp. 32-41 (especially 37-41) * sum over MJ components What does actual microwave spectrum look like? Complicated J → 2J + 1 sublevels for each J Many rotational branches No obvious pattern for Q branches Do H2CO instead Which is a, b, c oblate or prolate? (possibility of switch to oblate?) where is µ? what is type? selection rules? 2 Page 5 of 6 pages A0 = 8.0012 cm–1 B0 = 0.43364 C0 = 0.41040 κ = –0.8857 (near prolate) µ is in-plane ⊥ to O—O direction Example: NO2 N B κ≡ 2B − A − C A−C R B O θ O R0 = 1.1934Å θ0 = 134.1° left axis: a, || to O—O direction (through center of mass) heavy axis: c, ⊥ to plane ∴ µ is || b axis Special relationship for planar molecules 1 1 1 A+B=C B 1 1⎞ ⎛1 inertial defect: –⎜A + B – C ⎟ = +5.6 × 10–3 ⎝ e e e⎠ (not quite 0 because v = 0 not e constants used ∆Kp = odd, ∆Ko = odd, ∆J = 0 strong, ∆J = ±1 b-type transitions: p types of branches P downward p p Q downward r R downward at low J P upward at low J r Q upward ~3 upward transitions for each J K a K c ~3(2J + 1) upward transitions for each J How to assign? * Look for patterns series of lines spaced by ~2 (R-branches) also repeated K-stack spacings pairs of lines corresponding to asymmetry doublets (R-branches) size of asymmetry splitting - large for large J–K absence of asymmetry splitting : K = 0↔0, 1↔0, 0↔1 * T-dependence high EROT lines increase in intensity rapidly as T↑ low EROT lines decrease in intensity slightly as T↑ * guess structure and compute spectrum — frequencies and relative intensities r R upward Page 6 of 6 pages * tricks: Stark (ε-field) modulation picks out low-J lines where asymmetry splitting is small J ≈ K Double Resonance — bootstrap — transfer one assignment to neighboring linked transitions saturate modulate label Recall, for planar molecule there are only 2 independent bits of information. 2 correct assignments lead to accurate prediction of entire spectrum. * isotope shifts. Change mass of atom that lies on one a, b, or c axis - corresponding rotational constant unchanged, others affected!