Vibrational Spectroscopy & Intramolecular Vibrational Redistribution (IVR) 1
The Role of Vibrational Energy in Chemical Reactions Unimolecular reactions: The Rice-Rampsberger-Kassel-Marcus (RRKM) theory of unimolecular reaction rate assumes that vibrational energy is randomized (IVR) quickly compared to the rate of the reaction. k 1 A + M A* k -1 collisional activation / deactivation k 2 A* products reaction of energized molecule - Works well in most but not all cases. - The alternative, Slater Theory, was found was found to be unsatisfactory 2
Rabinovitch: Chemical Activation (I) and (II) were detected by photolysis at 280 nm in CO/O 2 In mass spec (I) gives m/e = 95 and (II), m/e = 97. The low-pressure data give a small, collisionally noninterceptable fraction of nonrandomized product. The effective rate of intramolecular energy relaxation is 1.1 x 10 12 sec -1. 3 J. D. Rynbrandt and B. S. Rabinovitch, J. Chem. Phys. 54, 2275 (1971).
The Role of Vibrational Energy in Chemical Reactions Bimolecular reactions: Bond-selective chemistry has be demonstrated for simple reactions Only the excited bond was broken to form products. The distribution of vibrational amplitude in a molecule has a dramatic effect on its chemical reactivity. If the vibrational energy was randomized between the two bonds (IVR), then both products would be observed in comparable amounts. 4 Crim, F. F. Acc. Chem. Res. 1999, 32, 877.
5 IVR: Concepts from Classical Mechanics If the 3N-6 mass-weighted Cartesian coordinates are q = { q i } then the potential is V ( q ) = V e + 1 f 2 i, j q i q j + 1 (3) f 6 i, j,k q i q j q k + 1 (4 ) f 24 i, j,k,l q i q j q k q l + i, j In the Wilson GF matrix approach to normal modes, we used the approximation that all vibrational amplitudes were small enough that the 3 rd and higher order terms could be neglected. Now we will allow large amplitudes and examine the effects of the the higher order terms. The kinetic energy is T = 1 2 i, j,k i Using Newton s laws of motion, F = ma, and suitable choices of initial conditions (q i and q i for all atoms at t = 0), the classical equations of motion and be integrated. The result is a set of classical trajectories, one for each set of initial conditions. Equivalently, Hamilton s or Lagrange s formulations of classical mechanics may also be used. q i 2 This can also be done in internal coordinates, where the potential energy is simpler, but the kinetic energy is more complicated (i.e. the Watson Hamiltonian). i, j,k,l
Regular Motion in Classical Mechanics H 2 O trajectories for (a) normal modes, and (b) local modes. C. Jaffé & Paul Brumer, J. Chem. Phys. 73, 5646 (1980). Even for just the two OH stretches in water, we need 4 dimensions to represent the coordinates and momenta. Such 4-D trajectories lie on a hypertorus: See also J. Ford, Adv. Chem. Phys. 24, 155, (1973). 6
Poincaré Surfaces of Section Put a dot on the page whenever the trajectory crosses through the plane of the page. Examples for OH stretches of water: LEFT: normal modes at low energy. S=symmetric stretch; A=asymmetric RIGHT: local modes at higher energy. S becomes unstable. 7 C. Jaffé & Paul Brumer, J. Chem. Phys. 73, 5646 (1980).
Bifurcations in Acetylene Michael Kellman website: http://pages.uoregon.edu/meklab/migration2/ Normal Modes at low energy New modes are born at higher energies. Trajectories still trace out invariant tori, but their topology is different. 8
Classical chaos Poincaré surfaces of section develop some irregular speckled regions. The distance between neighboring trajectories increases exponentially with time until, there is no correlation between them. http://pages.physics.cornell.edu/sethna/teaching/sss/jupiter/web/chaos.htm 9
IVR: Quantum Concepts There is no analogous definition of chaos in quantum mechanics because following trajectories would violate the uncertainty principle. The Gaussian Orthogonal Ensemble (GOE) represents a universal strong coupling limit for quantum problems. Set up a Hamiltonian matrix in which all of the entries are chosen from a Gaussian distribution, but the diagonal elements have twice the variance of the rest. Create an ensemble of such matrices; the statistical properties of the ensemble are invariant to any orthogonal transformation. The GOE eigenstates have certain properties: Levels repel a picket fence-like (Wigner) distribution of level spacings Spectral rigidity a long distance correlation between levels A Porter-Thomas distribution of intensities with a few intense lines and a much larger number of very weak lines. 10
IVR and Spectroscopy For a single bright state, 1= s, a coherent excitation of the spectrum yields n Ψ( t) = c 1k φ k exp ie t k P j k=1 ( t) = φ j Ψ t ( ) 2 zeroth order states bright dark j vsj spectrum of eigenstates n ψ k Intensities I k Measures of IVR: φ d = Dilution factor n k=1 n 4 c 1k = I k 2 k=1 n k=1 I k 2 1 φ d = lim t t ( ) P 1 t τ IVR is the time for P 1 (t) to decay to 1/e. t 0 dt s h vij 1 Ps(t) The bright state i=1 or s 0 0 0 t / ns 1 11 McDonald, J. D. Annu. Rev. Phys. Chem. 1979, 30, 29.
Low-Order Resonances and Tier Models 12 Divide bath states into tiers according to coupling order relative to the bright state. Density of resonances at 3 rd and 4 th order play a key role in determining the IVR rate. At right, substitution of Si for C increases the total density of states, but decreases the number of 3 rd and 4 th order resonances, and hence the IVR rate following excitation of the acetylenic CH stretch is slower. Useful reviews: Lehmann, K. K.; Scoles, G.; Pate, B. H. Intramolecular dynamics from infrared eigenstate-resolved spectra. Ann. Rev. Phys. Chem.; Annual Reviews, 1994; Vol. 45; pp 241. Nesbitt, D. J.; Field, R. W. J. Phys. Chem. 1996, 100, 12735. Stuchebrukhov, A. A.; Marcus, R. A. J. Chem. Phys. 1993, 98, 6044.
Eigenstate-Resolved Spectroscopy Often near-resonant clumps of eigenstates are observed at high resolution. Super-resonance is indirect (low-order) coupling via non-resonant doorway states. Propyne 2ν 1 band 13
State-Space Models of IVR with Scaling Gruebele, M. Advances in Chemical Physics 2001, 114, 193. Madsen, D.; Pearman, R.; Gruebele, M. J. Chem. Phys. 1997, 106, 5874. Pearman, R.; Gruebele, M. Z. Phys. Chem. (Muenchen) 2000, 214, 1439. 14 P 1 ( t) t δ σ / 2 SCCl 2 If low-order resonances dominate then small hops in in the vibrational quantum number state space are expected. In rigid molecules, the coupling matrix element declines by 5x to 20x for each additional coupling order. a small number of large matrix elements and a huge number of very small ones. higher order couplings in aggregate make a significant contribution. In the time domain, this means relaxation times on multiple timescales spread over orders of magnitude. Gives the overall form of power law decays extending to rather longer times than simple exponential decays.
Rotational Effects in IVR The form of the Coriolis matrix elements and scaling with J and K was established by E. B. Wilson (J. Chem. Phys. 1936, 4, 313). W. D. Lawrance & A. E. Knight (J. Phys. Chem. 1988, 92, 5900): Phase space volume / Number of states 400 300 200 100 0 acetylene J = 100 J = 30 J = 2 The acetylene vibration-rotation Hamiltonian is known with precision up to 13,000 cm-1, enabling detailed calculation of the rotational effects on the IVR dynamics. 15 4 kinds of coupling: vibrational l-resonance, anharmonic (Darling-Dennison, etc.), rotational l-resonance, and Coriolis. 10 1 10 2 10 3 10 4 10 5 t / fs Calculated volume of phase space explored for acetylene as a function of time following a coherent excitation of the bending vibration {v 4, v 5 } = {14, 2} at a total energy near 10,000 cm -1.
Role of Large-Amplitude Motion Gruebele found that the large-amplitude motion of an internal rotator causes coupling matrix elements to decline less steeply only half an order of magnitude per coupling order, increasing the relative importance of direct higher-order couplings. This can make the higher IVR faster for modes connected to the large-amplitude coordinate. Large-amplitude coordinates are in general much more anharmonic than the ordinary small amplitude vibrations. Pearman, R.; Gruebele, M. Z. Phys. Chem. (Muenchen) 2000, 214, 1439. 16 Perry, D. S.; Bethardy, G. A.; Go, J. Ber. Bunsen-Ges. Phys. Chem. 1995, 99, 530.
Rotational Spectroscopy for the Study of Vibrational (Isomerization) Rates k = 5.6 x 10 9 sec -1 Cyclopropane Carboxoaldehyde Dian, B. C.; Brown, G. G.; Douglass, K. O.; Pate, B. H. Science (Washington, DC, U. S.) 2008, 320, 924. 17
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