S. Wiggins, Peter Collins, Frederic Mauguiere,University of Bristol

Gregory S. Ezra, Zeb C.Kramer, Cornell University

Stavros Farantos, University of Crete

Barry Carpenter, Cardiff University

Roaming: Dynamical Reaction Pathways in Phase Space

We gratefully acknowledge funding by the ONR, NSF, EPSRC, the Leverhulme Trust, and the University of Bristol

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Plan for the talk

•Explanation, and brief background, of “roaming”, with a focus on formaldehyde •The potential for formaldehyde (thanks to Joel Bowman). •Set-up for the Analysis—Define reactive events and construct phase space dividing surfaces •Trajectory behaviour. •The main result—The answer to the question “Why do trajectories “roam” and not dissociate through the radical channel? The phase space mechanism.” (And the implications of the answer to this question.) •Final remarks—generality of our approach

Phase Space Structures Explain Hydrogen Atom Roaming in Formaldehyde Decomposition. Frédéric A. L. Mauguière, Peter Collins, Zeb C. Kramer, Barry K. Carpenter,Gregory S. Ezra,  Stavros C. Farantos, and Stephen Wiggins. J. Phys. Chem. Lett., 2015, 6 (20), pp 4123–4128

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But this is a mathematics talk. Some general, underlying themes of the talk.

•Where is the mathematics?

•Can mathematicians make a contribution to chemistry?

•What “skills and knowledge” are required?

•A change in our understanding of “the form” of ODEs (the “rise of data”—very similar to the program on “Dynamical Systems Analysis of Lagrangian Transport in Oceanic Flows”.)

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What is “Roaming”?

J.M.Bowman and A. G. Suits, Roaming Reactions :The Third Way, Physics Today, 64(11), 33 (2011) .

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6 “Roaming—sketch”

H CO2Formaldehyde dissociation

roaming

7 “Roaming”

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Roaming—a little background Formaldehyde—the prototypical “roaming reaction”

D.Townsend,S.A.Lahankar,S.K.Lee,S.D.Chambreau,A.G.Suits,Z.Zhang,J.Rheinecker,L.B.Harding,andJ.M.Bowman.TheRoamingAtom:StrayingfromtheReactionPathinFormaldehydeDecomposition.Science,306,1158–1161,(2004).

J.M.Bowman and A. G. Suits, Roaming Reactions :The Third Way, Physics Today, 64(11), 33 (2011) .

Following this work, “roaming” has been found in many reactionsFor example…

L.B.Harding,S.J.Klippenstein,andA.W.Jasper[2012]SeparabilityofTightandRoamingPathwaysinMolecularDecomposition.J.Phys.Chem.A,116,6967-6982.

S.J.Klippenstein,Y.Georgievskii,andL.B.Harding[2011]Statisticaltheoryforthekineticsanddynamicsofroamingreactions.J.Phys.Chem.A,115,14370-14381.

Also

Li,J.Li,andH.Guo[2013]QuantummanifestationofroaminginH+MgHMg+H2.Thebirthofroamingresonances.J.Phys.Chem.A,117,5052-5060.

M.P.Grubb,M.L.Warter,H.Xiao,S.Maeda,K.Morokuma,andS.W.North[2012]NoStraightPath:MultistateRoamingastheOnlyRoutefortheNO3 →NO+O2 Reaction. Science, 355,1075.

Ketene Isomerization

I.G.Ulusoy,J.F.Stanton,andR.Hernandez[2013]EffectsofRoamingTrajectoriesontheTransitionStateTheoryRatesofaReduced-DimensionalModelofKeteneIsomerization.J.Phys.Chem.A,117,7553-7560.

I.G.Ulusoy,J.F.Stanton,andR.Hernandez[2013]Correctionto“EffectsofRoamingTrajectoriesontheTransitionStateTheoryRatesofaReduced-DimensionalModelofKeteneIsomerization”.J.Phys.Chem.A,117,10567-10568.

I.G.UlusoyandR.Hernandez[2014]Revisitingroamingtrajectoriesinketeneisomerization.Theor.Chem.Acc.133,1528.

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Roaming—Precursor Work

J.M.BowmanandB.C.Shepler[2011]RoamingRadicals.Annu.Rev.Phys.Chem.62,531-553.

J.M.Bowman[2014]Roaming.Mol.Phys.112,2516-2528.

Review papers

Ion-molecule reactions

Large body of work on reactions that avoid the MEP, e.g.,

Two interesting, and relevant, papers

J.Zhang,J.Mikosch,S.Trippel,R.Otto,M.Weidemüller,R.WesterandW.L.Hase[2010]F- +CH3I→FCH3 +I- ReactionDynamics.NontraditionalAtomisticMechanismsandFormationofaHydrogen-BondedComplex.JournalofPhysicalChemistryLetters 1,2747-2752.

W.J.ChesnavichandM.T.Bowers[1982]TheoryofIon-NeutralInteractions:ApplicationofTransitionStateTheoryConceptstoBothCollisionalandReactivePropertiesofSimpleSystems.PergamonPress.

W.J.Chesnavich[1986]Multipletransitionstatesinunimolecularreactions.J.Chem.Phys.,84,2615.

W.H.Miller[1976]Uniliedstatisticalmodelfor’’complex’’and’’direct’’reactionmechanisms.J.Chem.Phys.65,2216.

K.RyneforsandN.Markovic.[1985]Dynamicsofcentrifugalbarriercomplexesclosetoorbiting.Chem.Phys.92,327-336.

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The Model: 2 DoF Model (thanks to Joel Bowman for providing the potential)

Rθ#

O# C#

H#

H#H2CO#well#

H+HCO#Radical#channel#

V#(×10

4 #cm

91)#

OC…H2#well######H2+CO##

CMHCO#

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Identify the “Reactive Events” of Interest, and Construct Dividing Surfaces the Crossing of which Defines these Reactive Events

We consider dynamics on a fixed energy surface—200 cm-1 above dissociation (33558 cm-1)

12 But what are these blue curves? The Nature and Structure of the DSs

Analysis and computation in phase space—“the setting for dynamics”

DS is a 2-sphere in the 3D energy surface

It is divided into two hemispheres by an unstable periodic orbit (PO)

These hemispheres control entrance and exit to the well They have the “no-recrossing” property and “minimal flux”

Periodic orbit dividing surface (“PODS”)—Work of Pollak, Pechukas, and Child from the late 70s. early 80’s (e.g. E. Pollak and M. S. Child [1980] Classical mechanics of a collinear exchange reaction:A direct evaluation of the reaction probability and product distribution. J. Chem. Phys. 73, 4373).

Generalized to more than 2 DoF in recent years using the idea of a normally hyperbolic invariant manifold (NHIM).

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−0.50

0.51

1.52 −15

−10−5

05

1015

−2

−1

0

1

2

3

pθ

θ

p R

(a)$ (b)$

(c)$ (d)$

Panel (a)—initial conditions

669 red—H’…HCO well (roaming—panel (b))

9155 blue—H + HCO (dissociating—panel (d))

22 green—return to H2CO well (panel (c))

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Why do trajectories “roam”, rather than dissociate?

There is an unstable PO in the “roaming region”.

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−0.50

0.51

1.52 −15

−10−5

05

1015

−2

−1

0

1

2

3

pθ

θ

p R(a)$ (b)$

(c)$ (d)$

Projection of the unstable PO in the roaming region into configuration space

Its effect on the dynamics is understood in phase space

16A “short” digression into some phase space notions—Poincare maps, and stable and unstable manifolds of unstable POs

p

q

H > 0

H > 0

H < 0H < 0

1 DoF saddle (H=p2-q2)

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172 DoF—Unstable POs, their stable and unstable manifolds, and Poincare maps

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Poincare maps associated with the unstable PO in the roaming region

19The roaming trajectories in relation to this structure: Shepherding Mechanism

20 Summary—For the 2 degree-of-freedom model we have…

But what about more than 2 degrees-of-freedom?

• Identified the “reactive events of interest”

• Constructed the phase space dividing surfaces (with “nice properties”) whose crossing defines these reactive events

• This allowed a sampling of the dividing surfaces in order to determine the nature of trajectories “encountering” the different

reactive events

• Determined the phase space mechanism that “decides” whether trajectories roam or dissociate

21 “The key idea”—Find a “saddle-like structure” on which to construct a dividing surface (with “properties”), and that has stable and unstable manifolds with one less dimension than the energy surface

For a n degree-of freedom Hamiltonian system..

• 2n dimensional phase space • 2n-1 dimensional energy surface

The key “building block” (a phase space generalisation of a “saddle point” with the “right dimensions” Normally hyperbolic invariant spheres of dimension 2n-3, S2n-3

These are examples of normally hyperbolic invariant manifolds (NHIMs)

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NHIMs generalize unstable periodic orbits to more the 2 DoF

Recall from previous slide Normally hyperbolic invariant spheres of dimension 2n-3, S2n-3

For 2 DoF, n=2, 2n-3=1, S1 —a periodic orbit

If you look at things “in the right way” similar ideas have been “almost there” in the chemistry literature (just lacking the necessary mathematics in order to elucidate their properties and develop related computational methods).

e.g.

E.Thiele[1962]Comparisonofclassicaltheoriesofunimolecularreaction.J.Chem.Phys.36,1466.

N.B.Slater[1959]TheoryofUnimolecularReactions.CornellUniversityPress,Ithaca.

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TheValueofa“NewPointofView”

ChemicalandEngineeringNews Volume 90 Issue 12 | pp. 42-43 Issue Date: March 19, 2012

Reacting By Roaming Studies strengthen role for alternative to transition-state pathways

“We are indeed coming to the conclusion that these roaming mechanisms are quite common,” says Lawrence B. Harding, a research fellow at Argonne National Laboratory who has done several computational studies on roaming and is now investigating alkyl halides. He and others emphasize that the reason roaming mechanisms hadn’t been identified earlier is simply because people hadn’t looked in the parts of potential energy surfaces where the roaming pathways lie.

Adds Suits, “If we had just looked at and thought about this 50 years ago with an open mind, we could have anticipated roaming easily.”

Some references24

F. A. L. Mauguiere, P. Collins, G.S. Ezra, S. C. Farantos, and S. Wiggins, Multiple transition states and roaming in ion-molecule reactions: A phase space perspective, Chemical Physics Letters, 592, 282-287 (2014).

F. A. L. Mauguiere, P. Collins, G.S. Ezra, S. C. Farantos, and S. Wiggins, Roaming dynamics in ion-molecule reactions: Phase space reaction pathways and geometrical interpretation, Journal of Chemical Physics, 140, 134112 (2014).

F. A. L. Mauguiere, P. Collins, G.S. Ezra, S. C. Farantos, and S. Wiggins, Roaming dynamics in ketene isomerization, Theoretical Chemistry Accounts, 133, 1507 (2014).

F. A. L. Mauguiere, P. Collins, Z. C. Cramer, B. K. Carpenter, G.S. Ezra, S. C. Farantos, and S. Wiggins, Phase space barriers and dividing surfaces in the absence of critical points of the potential energy surface. Application to roaming in ozone. Journal of Chemical Physics, 144(5), 054107 (2016).

F. A. L. Mauguiere, P Collins, S. Stamatiadis, A. Li, G. S. Ezra, S. C. Farantos, Z. C. Kramer, B. K. Carpenter, S. Wiggins, and H. Guo. Toward Understanding the Roaming Mechanism in H + MgH → Mg + HH Reaction. J. Phys. Chem. A. (2016). DOI: 10.1021/acs.jpca.6b00682.

Extra slides25

2 4 6 8 10 12 14 160

1

2

3

4

5

6

R (bohr)

θ (r

ad)

EP1EP2

EP3

EP4EP5

EP6

EP7

EP8EP9

label index energy (cm-1) EP1 1 33500 EP2 1 33507 EP3 1 33482 EP4 2 33515 EP5 0 32960 EP6 0 33502 EP7 1 33449 EP8 1 33463 EP9 0 33451

Critical points on the PES26

Dissociation Energy

33556.2

33556.4

33556.6

33556.8

33557

33557.2

33557.4

33557.6

33557.8

33558

33558.2

33558.4

0 1 2 3 4 5 6 7

V (cm−

1 )

e (rad)

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Procedure for Computing the 2 DoF Model from Bowman’s Potential

1)Bowman’scodeworkswithcartesiancoordinatesforthe4atoms.Cartesiancoordinatesarein3-space(x,y,z).Toobtainaplanarmodelwejustsetz=0forallatoms.

2)WefixedtheorigintothecenterofmassoftheHCOfragment.Bowmangaveustheinter-parNcledistancesfortheminimumoftheformaldehydewell.Sowecanfixtheoriginofthecoordinateframeandcalculatethe(x,y,z)coordinatesofeachatom.Sincetheformaldehydeminimumisplanar,zcanbetakenequalto0.

3)WeorientedthexaxisparalleltotheCOaxis.

4)Wethencalculatedthe(x,y,z=0)coordinatesofthe3atomsoftheHCOfragmenttakenattheformaldehydeminimum.Thesecoordinatesdon’tchangefromnowon.

5)TheposiNonoftheremaininghydrogenisobtainedfromthepolarcoordinates(R,theta)andthenconvertedto(x,y,z=0)coordinates.

6)ThenBowman’scodehasasubrouNnewhereyouprovidethe(x,y,z)coordinatesofthe4atomsanditconvertsthemtointer-parNcledistancesandfromthesedistancesreturnsthevalueofthepotenNal.

7)Werecordedforagridin(R,theta)thevaluesofthepotenNalandthendidasplineinterpolaNon.

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