5.80 Small-Molecule Spectroscopy and Dynamics

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1 MIT OpenCourseWare Small-Molecule Spectroscopy and Dynamics Fall 008 For information about citing these materials or our Terms of Use, visit:

2 5.76 Lecture #10 Fall, 008 Page 1 Last Time oscillating electric field Lecture #10: Transitions II Pif ε Mb,if Ω i J i M i α Sb Ω f J f M f θ,φ Intensity all electric dipole transition probabilities i e r f r S,b unique radial universal angular factor factor * polarization: S, M * band type: b, Ω * branch type J pure rotation i = f b = z for diatomic (µ along z) = σ(xz) and σ(yz) symmetry b = z Ω = 0 v M z,ii () v = M z,ii ( e ) + dm 1 d M Q vv + dq Q=0 dq if homonuclear =0 n Q vv matrix elements in Harmonic Oscillator Basis Set (Q = e ) Q=0 Q vv P if [ const. + small v term ] µ (dµ/d) ΩJ i M α Zz ΩJ f M Today: finish pure rotation spectrum Hönl-London Factors rotation-vibration spectrum v = ±1 propensity rule dm/d 0 anharmonic and centrifugal correction terms [PETUBATION THEOY] rotation-vibration-electric spectrum all v Franck Condon factors -centroid approximation stationary phase approximation Final factor is ΩJ i M α Zz ΩJ f M Each J consists of J + 1 degenerate M-components.

3 5.76 Lecture #10 Fall, 008 Page direction cosine matrix elements sum over M Hönl-London rotational linestrength [can't do this sum so simply for OOD because initial M s are not equally populated] factors see Hougen page 39, Table 7 M ΩJ i M α Zz ΩJ f M θφ Herzberg Diatomics, page 08 S Ji J f ΩΩ 3 sum rule (J f + Ω +1)(J f Ω +1) J f = ~ 3( J f +1) 3 common final state J i = J f + 1 ( or P) J f +1 g el f 3 useful for checking calculations Ω (J f +1) Ω = ~ 3J f 3J f (J f + Ω)(J f Ω ) J = ~ f 3J f 3 J i = J f (Q branch weak at high J) J i = J f 1 (P or ) The increase with J is due to J + 1 degeneracy factor being included. These formulas for a common final state do not depend on whether J i or J f is upper or lower state. Similar set of formulas for transitions out of common initial state. These formulas cannot depend on our choice of quantization axis. If we sum over equal X, Y, Z polarized absorption or emission, the factor of 3 must go away because of the isotropy of space and the equivalence of X, Y, Z. Next case: i = f otation-vibration Spectra (Diatomic or Linear Molecule) - still have Ω i = Ω f α Zz

4 5.76 Lecture #10 Fall, 008 Page 3 P if I M z,ii () S ΩΩ Ji J f exactly the same as for pure rotation spectra Here the absolute intensity factor is slightly different from that for pure rotation do the same power series expansion in Q about e (i.e. Q = 0). M z,ii () = M z,ii ( e ) + dm z,ii dq Q=0 = 0 when by orthogonality 1/ for harmonic oscillator v Q v 1 = v 1/ v = ±1 propensity rule µω e amplitude increases v 1/ requires dm 0 P if v dq Q=0 Other contributors to vibrational intensities: Q + 1 d M z,ii dq Q=0 Q * vibrational anharmonicity: perturbation theory mixes harmonic = ± 1 character into real ±1 levels * electrical anharmonicity: next term d M z,ii 0 v = ± dq Q=0 * rotational effects. for J is not orthogonal to for J ± 1! Then permanent dipole moment, M z,ii ( e ), can contribute to P if for v = ±1 transition. B() v ±1 J(J +1) v ±1 = v + v ±1 G vj = v0 + v Q B() = B( e ) 1+ e ( ( v0 B()J(J +1) v 0 v 0 o o E v E v = B e ) 1 Q e ) + v,0 Q v ±1,0 vj v0 B ( e ) J(J +1) e ω M v ±1J M() vj = v + M ( e )B e J(J +1) etc. Q v ±1,0 +

5 5.76 Lecture #10 Fall, 008 Page 4 Herman-Wallis effect. See 3 elegant papers (especially the first one) from David Nesbitt s group: D. Nelson, Jr., A. Schiffman, D. Yaron, and D. Nesbitt, Absolute Infrared Transition Moments for Open Shell Diatomics from J Dependence of Transition Intensities: Application to OH, J. Chem. Phys. 90, 5443 (1989); D. Nelson, Jr., A. Schiffman, and D. Nesbitt, The Dipole Moment Function and Vibrational Transition Intensities of OH, J. Chem. Phys. 90, 5455 (1989); D. Nelson, Jr., A. Schiffman, J. Orlando, J. Burkholder, and D. Nesbitt, H + O 3 Fourier- Transform Infrared Emission and Laser Absorption Studies of OH (X ) adical: An Experimental Dipole Moment Function and State-To-State Einstein A Coefficients, J. Chem. Phys. 93, 7003 (1990). cross terms give sign of dm/d with respect to M( e ). Add transition amplitudes before taking. Non-Lecture: Anharmonic Correction to vibrational wavefunction. v = v + c v,v±1 V() = kq / + aq 3 H v ±1 ω 3/ from Q 3 v aq 3 v ±1 = aω 5/ c = µ 3/ 1/ f(v) v 3/ v,v±1 ± ω Pure rotation requires M z ( e ) 0 see perturbation theory for formulas. Vibration-otation mostly due to dm z 0 dq Q=0 v = ±1 propensity, P if v but also vibrational and electronic anharmonicities and centrifugal distortion nearly perfect J-independence (except centrifugal effects - hydrides) M z 0 rotation M x = M y = 0 parallel type always dmz 0 rotation vibration (weak Q branches) dq (no such restriction to only M z 0 in polyatomic molecules) recall Hönl-London factor can have strong Q branches

6 5.76 Lecture #10 Fall, 008 Page 5 Big differences when we consider electronic transitions i f * not restricted to only M z () (x,y components also) * no simple vibrational selection or propensity rules as for the harmonic limit of vibration-rotation because { } is not orthogonal to { } * awkwardness about M b,if (). We would like to express vibrational matrix elements of M b,if as function of vi rather than [M b,if () ]. centroid approximation stationary phase, semi-classical Franck-Condon principle vibrational intensity distribution provides information about difference in structure P if I Z Ω i J i M α Zb b Ω f J f M M() b,if electronic transition moment expand M() about some arbitrary value since ei ef expand, take ME and divide through by M() b,if dm = M( ) b,if + + d = This looks exactly like ( ) n expansion if is replaced everywhere by the -centroid vi = provided that the -centroid approximation is valid. need to look at higher terms in expansion to see the necessity for -centroids

7 5.76 Lecture #10 Fall, 008 Page 6 v n n v f ( ) n i v f always true approximation (justification for typical type of spectroscopic simplification) Semi-classical Franck-Condon principle See: * J. Tellinghuisen, eflection and Interference Structure in Diatomic Franck-Condon Distributions, J. Mol. Spectrosc. 103, 455 (1984) * C. Noda and. N. Zare, elation Between Classical and Quantum Formulations of the Franck- Condon Principle: The Generalized r-centroid Approximation, J. Mol. Spectrosc. 95, 54 (198) This -centroid approximation is convenient because we can think of M() as a simple function of a single variable (which usually turns out to be a monotonic function of the wavelength of the transition, λ) In the -centroid approximation ( S Ω i Ω f Ji J f P Iq M ) ij vi v v f f b,if Franck-Condon factor (overlap squared) Now what is stationary phase approximation? How is it related to semi-classical F-C principle? vertical * = 0 no impulse * P = 0

8 5.76 Lecture #10 Fall, 008 Page 7 p-mismatched p-matched Transition is vertical ( = 0) and occurs at that where p upper = p lower. This means that wavefunctions are oscillating at the same spatial frequency. I() = χ * ( ) χ vf ( )d r < ( ) This shows how F C overlap integral accumulates. h Spatial oscillations of χ given by de Broglie λ = p. I() stationary phase All n integrals accumulate near stationary phase point.