1. Phase Space Exploration in Acetylene at Energies up to 13,000 cm -1 Jonathan Martens Badr Amyay David S. Perry U.S. Department of Energy The University of Akron Université libre de Bruxelles Michel Herman 1

2. Motivation: Unimolecular Reaction Rates 2 RRKM Theory Assumptions: 1. All internal molecular states of A* at energy E are accessible and will ultimately lead to … products, and 2. vibrational energy redistribution [IVR] within the energized molecule is much faster than unimolecular reaction. Questions: - Which degrees of freedom are active? - How do we deal with partially active degrees of freedom? - Does N(E-E r ) depend on the time available before reaction? Steinfeld, J. I.; Francisco, J. S.; Hase, W. L. Chemical Kinetics and Dynamics; Prentice Hall: New Jersey, 1989. Sum of states for the active degrees of freedom at the transition state Vibrational density of states for the active degrees of freedom in the reactants

3. Approach 3 1. Use the spectroscopic Hamiltonian for ground state acetylene to compute the dynamics following a coherent excitation of certain bright states. 2. Evaluate the volume of phase space explored to estimate the density of active vibrational states:

4. Acetylene Hamiltonian Badr Amyay Normal mode basis set: (v 1 v 2 v 3 v 4 v 5, l 4 l 5 ) with e/f g/u symmetries Polyad numbers: N r = 5v 1 + 3v 2 + 5v 3 + v 4 + v 5 conserved N s = v 1 + v 2 + v 3 not conserved Vibrational angular momentum: k = l 4 + l 5 not conserved Four coupling types: Vibrational l-resonance: Δv n = 0, Δl 4 = ±  Δl 5 = ∓  Δk = 0 Anharmonic (e.g., DD4455): Δv 4 = ±  Δv 5 = ∓  Δk = 0 Rotational l-resonance: Δk = ±  ±  ~J 2 Coriolis: Δk = ±  ΔN s = ±  ~J Fit included 19,582 lines up to 13,000 cm -1, ~150 off-diagonal parameters Polyads studied in this work: {N r, e, g} … all below the vinylidene threshold { 8, e, g} 5076 – 5682 cm -1 74 states {12, e, g} 7760 – 8415 cm -1 295 states {16, e, g} 10,421 – 11,076 cm -1 897 states {18, e, g} 11,808 – 12,379 cm -1 1459 states B. Amyay, M. Herman, A. Fayt, L. Fusina, A. Predoi-Cross, Chem. Phys. Lett. 491, (2010) 17-19. + additional lines! 4

5. 5 Express zeroth order basis states in terms of eigenstates Bright state is zeroth order state j = 1 After a coherent excitation of a bright states, calculate the time dependence of the wavefunction. Project the time-dependent wavefunction onto various zeroth-order states to monitor its time evolution. Acetylene: n Coupled Levels

6. 6 Measures of Phase Space Explored Participation number The Shannon entropy Gruebele’s dispersion

7. Acetylene Phase Space Exploration Polyad {16, e, g} 7 Normal mode bright states (Bends) (CH) (Bends)

8. Polyad {16, e, g} 8 Normal mode bright states (Bends) (CH) (Bends) Acetylene Phase Space Exploration Polyad {16, e, g} Normal mode bright states

9. Acetylene Phase Space Explored

10. Polyad {16, e, g}

11. Acetylene Phase Space Explored Polyad {16, e, g}

12. Increases over 3 decades of time: 20 fs to 20 ps. - Vibrational coupling, then rotational l-resonance, then Coriolis - A polyad-breaking Hamiltonian might yield even slower stages Bottlenecks for changes in N S – slower and less complete exploration Strong rotational dependence: 16  360 states in N r = 16 polyad Qualitatively similar between polyads, increasing with energy Qualitative dependence on the nature of the bright state – stretch vs. bend; normal mode vs. local mode Summary of Phase Space Exploration in Acetylene 12 N S = 0 N S = 1N S = 234 Coriolis

13. Which is the best measure of the volume of phase space explored? Participation number The Shannon entropy Gruebele’s dispersion What is the best strategy for getting better unimolecular reaction rates? Density of coupled states ~ Phase space volume / interaction width Rotational dependence: Rotation couples reactive and unreactive phase space. Phase Space exploration in Acetylene: Remaining Questions 13 Acetylene Vinylidene N S = 0 N S = 1N S = 234 Coriolis