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

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Motivation: Unimolecular Reaction Rates

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A⇔k−1

k1

A * ⇒k2

productsRRKM 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-Er) 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.

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k E( ) =1

h

G E‡( )

N E − Er( )

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

Approach

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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:

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N E( ) ≈Ω E( )ΔE

Acetylene HamiltonianBadr Amyay

Normal mode basis set: (v1 v2 v3 v4 v5, l4 l5 ) with e/f g/u symmetries

Polyad numbers: Nr = 5v1 + 3v2 + 5v3 + v4 + v5 conserved Ns = v1 + v2 + v3 not conserved

Vibrational angular momentum: k = l4 + l5 not conserved

Four coupling types: Vibrational l-resonance: Δvn = 0, Δl4 = ±2, Δl5 = ∓2, Δk = 0 Anharmonic (e.g., DD4455): Δv4 = ±2, Δv5 = ∓2, Δk = 0 Rotational l-resonance: Δk = ±2, ±4, ~J 2

Coriolis: Δk = ±1, ΔNs = ±1, ~J

Fit included 19,582 lines up to 13,000 cm-1, ~150 off-diagonal parameters

Polyads studied in this work: {Nr, 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!

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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

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Measures of Phase Space Explored

Participation number

The Shannon entropy

Gruebele’s dispersion

Acetylene Phase Space Exploration

Polyad {16, e, g}

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Normal mode bright states

(Bends)

(CH) (CH) (CH)

(Bends)

(Bends)

Polyad {16, e, g}

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Normal mode bright states

(Bends)

(CH) (CH) (CH)

(Bends)

(Bends)

Acetylene Phase Space ExplorationPolyad {16, e, g}

Normal mode bright states

Acetylene Phase Space Explored

Acetylene Phase Space Explored

Polyad {16, e, g}

Acetylene Phase Space Explored

Polyad {16, e, g}

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 NS

– slower and less complete exploration

Strong rotational dependence: 16 360 states in Nr = 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

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NS = 0 NS = 1 NS = 2 3 4

Coriolis

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

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Acetylene

Vinylidene

NS = 0 NS = 1 NS = 2 3 4

Coriolis