Reaction Dynamics Lecture Notes 2012

8/12/2019 Reaction Dynamics Lecture Notes 2012

1/38

1

1

MolecularReactionDynamics

Lectures14

ClaireVallance

INTRODUCTION

Bynowyoushouldbewellandtrulyusedtotheconcepts involved inreactionkineticsonamacroscopic

scale. Youwillhaveencountereda rangeofexperimental techniques formeasuring rate constantsand

determining reactionorders,andavarietyof theoretical tools simplecollision theory, the steady state

approximation,transitionstatetheory,tonameafew thatcanbeusedto interpretorevenpredictthe

experimentalresults.

Arateconstantdeterminedinatypicalkineticsstudyisahighlyaveragedquantity,theaggregateoutcome

of an unimaginably large number of individual collisions between reactant molecules. In contrast, a

chemicalreactiondynamicsstudyaimstounderstandchemistryatanexquisite levelofdetailbyprobing

chemical reactions on the scale of single reactive collisions betweenmolecules. Aswe shall see, it is

possible to control thevelocities,quantum states,andeven the geometricalorientationof the colliding

moleculesandtomeasurethesamepropertiesintheproductsinordertounderstandtheeffectofeachof

thesevariablesonthereactionunderstudy. Whenwestudychemistryatthesinglecollisionlevel,wefind

that every chemical reaction bears itsown unique fingerprint, embodied in the kinetic energy, angular

distribution, and rotational and vibrational motion of the newly formed reaction products. These

quantitiesreflecttheforcesactingduringthereaction,particularlyinthetransitionstateregion,andtheir

measurement often provides unparalleled insight into the basic physics underlying chemical reactivity.

Gaininganunderstandingofchemistryatthesinglecollisionlevelistheaimofthislecturecourse.

Thefirstfourlecturesofthiscoursewillintroducemostofthekeyconceptsinreactiondynamics,including

adetaileddescriptionofcollisions,theconceptofreactionsoccurringoverpotentialenergysurfaces,and

anoverviewofexperimentaltechniquesusedtoprobesinglecollisionevents. Thefinalfourlectures,which

will be given byMark Brouard, will develop these ideas inmore detail and use them to explain the

observeddynamicsinawiderangeofchemicalsystems.

1. COLLISIONS

Collisionslieattheveryheartofchemistry. Inthegasorliquidphase,atypicalmoleculeundergoesbillions

of collisions every second. In this course, we will learn about the basic physics describing collisions,

experimentsandcalculationsthatcanbecarriedouttostudythem,andways inwhichtheresultscanbe

interpretedinordertolearnaboutreactionmechanisms.

Collisionsmaybecategorisedintothreetypes:

(i) Elasticcollisionsarecollisionsinwhichkineticenergyisconserved Thecollisionpartnershavethesame

total translational kinetic energy after the collision as they did before the collision, and no energy is


2/38

2

2

transferredintointernaldegreesoffreedom(rotation,vibrationetc). Notethatthekineticenergyofeach

individualcollisionpartnermaychange,butthetotalkineticenergyremainsconstant.

(ii) Inelastic collisions are collisions inwhich kinetic energy is not conserved, and energy is converted

betweendifferent formse.g.translationaltorotationalorvibrational. Reactivecollisionsareasubsetof

inelasticcollisions,butthetermisusuallyusedtodescribecollisionsinwhichtranslationalkineticenergyis

transformedintointernalexcitationofoneorbothofthecollisionpartners(orviceversa).

(iii) Areactivecollisionisacollisionleadingtoachemicalreactioni.e.acollisioninwhichchemicalbonds

aremadeorbroken,sothatthespeciesleavingthecollisionregionarechemicallydistinctfromthosethat

entered. The collisionenergymustbehighenough toovercome any activationbarrier associatedwith

reaction. Asnotedabove,reactivecollisionsareaspecialcaseofinelasticcollisions.

Reactivecollisionsare theprocesses inwhichchemicalchangeoccurs,while inelasticcollisionsareoften

themeans bywhichmolecules gain enoughenergy toovercome activationbarriers so that subsequent

collisionsmayleadtoreaction.

Allcollisionsmustobeycertainphysical laws,suchasconservationofenergyandboth linearandangular

momentum. There are also a number of concepts associatedwith collisions thatwill probably be less

familiar,andwhichwewillnowexplore.

1.1Relativevelocity

Ratherthanconsideringthe individualvelocities,v1andv2,ofthetwocollisionpartners, it isoftenmore

usefulto

consider

their

relative

velocity.

The

relative

velocity

issimply

the

difference

between

the

individual velocities, and corresponds to the apparent velocity of the second particle to an observer

travellingalongwiththefirstparticle(orviceversa).

vrel=v1v2 (1.1)

Analogy:imagineyouaresittingonamovingtrainandmeasuringtheapparentvelocityofasecondtrain.

Ifthesecondtrainistravellinginthesamedirectionasyourtrain(i.e.awayfromyourtrain),itwillappear

tohavea lowervelocitythan itstruevelocity,while if itistravellingintheoppositedirection(i.e.towards

yourtrain)itwillappeartohaveahighervelocity.

The relative velocity is very important in reaction dynamics, as it determines the collision energy, and

therefore the probability of overcoming any energy barrier to reaction. To extend the train analogy

above, the collisionenergy ismuch lowerwhen the two trainsare travelling in the samedirection (low

relativevelocity)thanwhentheysufferaheadoncollision(highrelativevelocity).

1.2Collisionenergy,totalkineticenergy,andconservationoflinearmomentum

The total kinetic energy of the colliding particles is simply equal to the sum of their individual kinetic

energies:

K = m1v12 + m2v2

2 (1.2)


3/38

3

3

Wherem1andm2arethemassesofthetwoparticles.Alternatively,thetotalkineticenergycanbebroken

downintoacontributionassociatedwiththevelocityofthecentreofmassofthecollisionpartners,anda

contributionassociatedwiththeirrelativevelocity.

K=MvCM2+vrel

2 (1.3)

whereM=m1+m2isthetotalmassandthereducedmassofthetwoparticles,vCMisthevelocityofthecentreofmassof the particles, and vrel is their relative velocity. The velocityof the centreofmass is

calculatedfrom

vCM =m1v1+m2v2

M (1.4)

Thetotallinearmomentumofthetwoparticlesmustbeconservedthroughoutthecollision,andisequalto

p = m1v1+m2v2 = MvCM (1.5)

(NotethatEquations1.4and1.5areequivalent). Sincethemomentumofthecentreofmassisconserved,

thekineticenergyassociatedwiththemotionofthecentreofmass,K=MvCM2=p

2/2M,mustalsobe

conserved throughout thecollision,and is thereforenotavailable toovercomeanyenergeticbarriers to

reaction. Only the kinetic energy contribution arising from the relative velocity of the two particles is

availableforreaction. Thisenergyisknownasthecollisionenergy.

Ecoll=vrel2 (1.6)

Exercise: ProvethatEquations(1.2)and(1.3)areequivalent.

1.3 Conservationofenergyandenergyavailabletotheproducts.

Thetotalenergymustbeconserved inanycollision. Beforethecollision,thetotalenergy is thesumof

contributions from the kinetic energies of the collision partners (or, equivalently, the kinetic energies

associatedwith their relativemotion and themotion of their centre ofmass) and any internal energy

associated with rotation, vibration, or electronic excitation of the reactants. The reaction itselfmay

consumeenergy(anendoergicreaction,withapositiverH)orreleaseenergy(anexoergicreaction,witha

negativerH),andthismustbetakenintoaccountwhenconsideringtheamountofenergyavailabletotheproducts.

Eavail=Ecoll+Eint(reactants) rH0 (1.7)

NotethatDrH0denotestheenthalpyofreactionat0Kelvin(i.e.forgroundstatereactantsreactingtoform

ground stateproducts). Theenergyavailable to theproductsmaybedistributedbetween translational

andinternalenergyoftheproducts.


4/38

4

4

1.4 Impactparameter,b,andopacityfunction,P(b)

Theimpactparameterquantifiestheinitialperpendicularseparationofthepathsofthecollisionpartners.

Essentially,thisisthedistancebywhichthecollidingpairwouldmisseachotheriftheydidnotinteractin

anyway, and canbe found by extrapolating the initial straightline trajectoriesof theparticles at large

separationstothedistanceofclosestapproach.

Theprobabilityofreactionasafunctionofimpactparameter,P(b),iscalledtheopacityfunction. Opacity

functionswillbecoveredinmoredetaillaterinthecourse.

1.5 Collisioncrosssection,cYouhavecomeacrossthecollisioncrosssectionbefore,inthecontextofsimplecollisiontheory(firstand

secondyearkinetics courses). The collisioncross section isdefinedas the cross sectionalarea that the

centresof twoparticlesmust liewithin iftheyaretocollide. Wecanuseaverysimplemodeltogaina

conceptualunderstandingofthecollisioncrosssection. Considerthekineticgasmodel,inwhichthereare

no interactionsbetweenparticles,and theparticlesact likehardspheresor billiardballs. In the figure

below, imagine thatwe have frozen themotion of all of the particles apart from the darker coloured

particle

positioned

on

the

left

in

the

side

on

view,

and

in

the

foreground

in

the

head

on

view.

We

see

that

themovingparticlewillonlycollidewithotherparticleswhosecentresliewithinthecrosssectionalareac

=d2,wheredistheparticlediameter.Wedefinethequantitycasthecollisioncrosssection.

In themoregeneralcaseof interactingparticles,attractive interactionsmaymean that trajectorieswithimpact parameters larger than the particle diameter, i.e. b>d, can lead to collisions. In this case the

collision cross sectionwill be somewhat larger than the hardsphere collision cross section illustrated

above.

1.6 Reactioncrosssection,rYouhavealsocomeacrossthereactioncrosssectionbefore inthecontextofsimplecollisiontheory. In

thatcontext,thereactioncrosssectionisfoundbymultiplyingthecollisioncrosssectionbyastericfactor,


5/38

5

5

i.e. r = Pc,which accounts for the fact that geometrical requirementsmean that in general not allcollisionsleadtoreaction. Inthiscoursewewilltreatthereactioncrosssectionmorerigorously.

Thereactioncrosssection isthemicroscopicorsinglecollisionversionofarateconstant,andrepresents

thereactionprobabilityasaneffectivecrosssectionalareawithinwhichthetworeactantsmustcollidein

orderforreactiontooccur. Forareactionbetweentwospeciesthatcanbemodelledashardspheres,so

that the reactionprobability isuniform for impactparametersbetween zero and somemaximum value

bmax,thenthereactioncrosssectionis

r =bmax2 (1.8)

Moregenerally,thereactioncrosssection isgivenby integratingthereactionprobabilityasafunctionof

impactparameterandangle.

r = 0

2

0

bmax

P(b)bdbd = 2 0

bmax

P(b)bdb (1.9)

Thevolumeelement2bdbisillustratedbelow.

1.7 Excitationfunction,r(Ecoll)Theexcitationfunctionquantifiesthedependenceofthereactioncrosssectiononcollisionenergy. Itisthe

microscopicequivalentofthetemperaturedependenceoftherateconstant. Broadlyspeaking,thereare

twodifferent typesofexcitation function,shownbelow,with the typeexhibitedbyaparticular reaction

dependingonwhetherornotthereactionhasanactivationbarrier.

1. Exoergicwithnobarrier

Thistypeofexcitationfunctionistypicalofmanyexoergic1processes,e.g.ionmoleculereactions. Atlow

energies there is enough time for longrange attractive interactions (van derWaals interactions, dipole

interactionsetc). toactontheparticlesandthecrosssection is large. Ifthe longrange interactionsare

1Whentalkingaboutsinglereactivecollisions,thetermsendoergicandexoergicareusedinplaceofendothermicand

exothermic.


6/38

6

6

strong,evenparticleswith large impactparameterscanbepulledtogetherbytheattractive interactions

andundergoacollision.Asthecollisionenergyincreases,theparticlesflypasteachotherfasterandtheir

trajectoriesaredeviatedbyasmalleramountby theattractive interactions. Thecrosssection therefore

reduces,untileventually there isno time foranyattractive forces toactand the socalled hardsphere

limitisreached. Atthispointthecrosssectionisgivenby=d2,whered=rA+rBisthesumoftheparticleradii. Formanyreactionsthehardspherelimitisverysmall,andtheexcitationfunctionessentiallytends

tozeroatveryhighenergies.

2. Endoergicorexoergicwithabarrier.

Whenthereactionhasabarrier,noreactionoccursuntilthereisenoughenergyavailabletosurmountthe

barrierandthecrosssection iszero. Thecrosssectionthen increasestoamaximum intheregionwhere

attractiveforcescanact(c.f.Arrheniusequationformacroscopicrateconstants),beforedecreasingagain

tothehardspherelimitathighenergiesforthesamereasonsasdescribedabove. Theexcitationfunction

istheproductofthedecreasing barrierlessexcitationfunctiondescribedaboveandanArrheniustype

functionwhichrisesoncetheactivationbarrierissurmounted.

Inthesecondpartofthiscourse (taughtbyProf.Brouard),youwillbe introducedtosimplemodelsthat

reproducethekeyfeaturesofbothtypesofexcitationfunction. Themodelsfocusonexplainingtheshape

of the lowenergypartof theexcitation functions,which is relevant to the typesofcollisionexperiment

mostcommonlycarriedoutinthelaboratory.

Onceweknowtheexcitationfunction,expressedasafunctionofcollisionenergyorrelativevelocity,the

thermal rate constantmay be recovered. From simple collision theory, the rate constant for a given

relativevelocity issimplyk(v) = vr(v). IntegratingoverthevelocitydistributionP(v)of themoleculespresentinthesampleyieldsthethermalrateconstant.

k(T) = 0

vr(v)P(v)dv (1.10)

1.8Scatteringintoasolidangle,

Anangleisoftendefinedintermsofanarcofacircle;wetalkabouttheanglesubtendedbyanarc,one

radianistheanglesubtendedbyanarcthesamelengthastheradius,andthereare2radiansinacircle.

Byanalogy,wecandefinethesolidanglesubtendedbyanareaonthesurfaceofasphere,asshowninthe

diagrambelow.

Solidangle

ismeasured

in

units

of

steradians.

One

steradian

isthe

solid

angle

subtended

by

an

area

r

2on

thesurfaceofthesphere,andthereare4steradiansinasphere.


7/38

7

7

In reaction dynamics,we need the concept of solid anglewhen considering the angular distributionof

scatteredproducts. The relativevelocityvectorof thereactants isanaxisofcylindricalsymmetry inthe

collision,withtheconsequencethatscatteringoccurssymmetricallyaboutthisdirection. Whenwereferto

productsbeingscatteredatagivenanglefromtherelativevelocityvector,theyareactuallyscatteredintoa

coneratherthanasingledirectioni.e.theyarescatteredintoasolidangleratherthanatasingleangle.

1.9 Differentialcrosssection,dd

Formally,thedifferentialcrosssection(DCS)describestheprobabilityofscatteringintoagivensolidangled=sinddbetweenand+dinthecentreofmassframe(seelater). However,asnotedabove,scatteringissymmetricalabouttherelativevelocityvectorofthecollisionpartners,andsosincethereisno

dependenceon,itisoftenreferredtoastheprobabilityofscatteringatagivenanglefromthereactantrelativevelocity vector. TheDCS therefore tellsus theextent towhich collisionproducts are scattered

forwards( =0),backwards( =180),orsideways( =90)relativetotheincomingcollisionpartners. Asweshallsee,theDCScontainsagreatdealof informationonreactionmechanisms,anddifferentialcross

sectionswillbeconsideredinsomedetaillateroninthecourse.

1.10 Orbitalangularmomentum,L,andconservationofangularmomentum

Inthecontextofacollision,theorbitalangularmomentum isanangularmomentumassociatedwiththe

relativemotionof thecollisionpartnersastheyapproachandcollide. It isnot tobeconfusedwiththe

quantummechanical orbital angularmomentum of an electron in an atomic orbital. Even for two

particlestravellingincompletelystraightlines,thereisanassociatedorbitalangularmomentumwhentheir

relativemotionisconsidered. Wecanillustratethisbylookingatthelineofcentresofthetwoparticlesat

variouspointsintheirtrajectory.

Wesee thateven though theparticlesare travelling instraight lines, the lineofcentresof theparticles

rotates about their centre ofmass. Only head on collisions with an impact parameter b=0 have no

associatedorbitalangularmomentum.

Mathematically,theorbitalangularmomentumforacollidingpairofparticlesisgivenby

L=rxp (1.11)


8/38

8

8

where r is the (vector) separation of the particles andp is their relative linearmomentum. We can

thereforefindthemagnitudeofLfrom

|L| =|rxp| = |rxvrel|= vrelrsin (1.12)

whereistheanglebetweenrandvrel. Atlargeseparations,rsinissimplyequaltotheimpactparameter,b,giving

|L|=vrelb (1.13)

Because the total angularmomentum (the sumof theorbital angularmomentum L and any rotational

angularmomentumJofthecollisionpartners)mustbeconservedthroughoutthecollision,thisistrueright

upuntilthepointthattheparticlescollide,assumingtherotationalstatesoftheparticlesdonotchange.

Intheproducts,thetotalangularmomentum,L+J,mustbethesameasbeforethecollision,butangular

momentumcanbeexchangedbetweenLandJduringthecollision.

2. POTENTIALENERGYSURFACES

The previous section provided an introduction to some of the basic physics of atomic andmolecular

collisions. However, itgaveusno insight intowhysomecollisionssimply leadtoelasticscattering,while

othersleadtoenergytransferorevenchemicalreaction. Wheninelasticandreactivecollisionsarestudied

inevenmoredetail,we findthattheyoften leadtopreferredscattering inaparticulardirection,orthat

theypopulate specific rotational,vibrational,andelectronicquantum statesof theproducts,or that the

outcome of the collision depends strongly on the initial kinetic energies, quantum states, or relative

orientationsof the reactants. Toexplain allof theseobservations,weneed the conceptof apotential

energysurface. Wewillstartbyconsideringasimpleonedimensionalpotentialenergysurface,namely

theinteractionpotentialbetweentwoatoms.

2.1Theinteractionpotentialanditseffectonthecollisioncrosssection

We have discussed the effect of atomic and molecular interactions on reaction cross sections in a

qualitativeway inSection1.7, in thecontextofexcitation functions. Therewe introduced the idea that

attractive interactions between the colliding particles could increase the cross section abovewhatwewouldexpect fora hardspherecollision. Hereweprovideasomewhatmorequantitativetreatment in

thecontextofcollisioncrosssections.

Thepotentialenergyofacollidingpairofatomsdependsonasinglecoordinate,namelytheirseparation.

The simplestmodel is thehard spherepotential, inwhich there isno interactionuntil the two spheres

touch. Themaximum impactparameter isbmax=d=r1+r2, thesumoftheradii,and thecollisioncross

section isc=bmax2. Thecrosssection isnotenergydependent. The interactionpotential,anexample

collisiontrajectory,andthecrosssectionforsuchasystemareshownbelow.


9/38

9

9

In contrast, a realistic interatomic potential has a longrange attractive component and a shortrange

repulsive component. Often, the LennardJones potential is used, which has a repulsive component

proportionalto1/r12andanattractivecomponentproportionalto1/r

6.

V(r) =

r0r

12

2r0r

6

(2.1)

Hereisthedepthofthepotentialwellattheminimum,andr0istheparticleseparationattheminimum.

Thepotentialisshownbelow,togetherwithanexamplecollisiontrajectoryandcollisioncrosssection. In

contrast to the hard sphere potential, the impact parameter can now exceed d due to the attractive

interactionbetweentheparticles. Thetwoparticlesareattractedtoeachotherastheyapproach,butare

thenscatteredbeforeactuallytouchingbytherepulsivepartofthepotential.

Landau,LifshitzandSchiffproposedanexpressiontorelatethecollisioncrosssectiontotheLennardJones

parametersandr0andtotherelativevelocityvofthecollidingparticles.

c(v) = 8.083

2r06

v

2/5

(2.2)

Notethatthe1/v2/5

dependenceyieldsafunctionsimilartothat introduced intheexcitationfunctionfor

reactionswithoutabarrier,whichwemetinSection1.7.

Weseealreadythatthecollisioncrosssectiondependsontheinteractionsbetweenthecollidingparticles,

and therefore that measurement of the collision cross section can provide information about these

interactions. Asweshalldiscoverinthefollowingsections,theseinteractionsinfactcompletelydetermine

everyaspectofthecollisiondynamics.

2.2Atomicandmolecularinteractions

Morerealisticpotentialstakeintoaccountallofthevarioustypesofatomicandmolecularinteractionsthat

can contribute to the attractive part of the interaction potential. The various contributions can all be

writtenintheform


10/38

10

10

V(r) = a

rn (2.3)

whereaisaconstantthatdependsonquantitiessuchastheatomicormolecularcharge,dipolemoments,

polarisabilitiesetc,andndependsonthetypeofinteraction:

n=1 Ionioninteraction

n=2 Iondipoleinteraction

n=3 Dipoledipoleinteraction

n=4 Ioninduceddipoleinteraction

n=5 Dipoleinduceddipoleinteraction

n=6 Induced dipoleinduced dipole interaction (also known as van der Waals, London, or

dispersioninteraction)

2.3Thepotentialenergysurface(PES)forapolyatomicsystem

Foracollisionbetweentwoatoms,wehaveseenthatthepotentialenergydependsonasingleparameter,

the internuclear separation. When twomolecules collide, there aremanymore degrees of freedom,

associatedwiththepositionsofeachoftheindividualatoms,allofwhichmaybeimportantindetermining

theoutcomeofacollision. Thefigurebelowshowstheway inwhichthenumberofdegreesoffreedom

increaseswiththenumberofatomsinthesystem. Specifyingtherelativepositionoftwoatomsrequiresa

singlecoordinate;forthreeatomswerequiretwo interatomicdistancesandanangle;forfouratomswe

requirethree

interatomic

distances,

two

bond

angles,

and

adihedral

angle,

and

so

on

2

.

The result is that todescribe thepotentialenergy,wehave tomove fromapotentialenergycurve toa

potentialenergysurface. Thepotentialenergysurfacequantifiesthepotentialenergyofthesystemasa

functionofall (or some)of thenuclearcoordinatesof theatoms involved,or in termsofbond lengths,

bondanglesetc. Eachpointonthesurfacecorrespondstoaparticulargeometricalconfigurationofatoms,

andtheremaybeseveralpotentialenergysurfacescorrespondingtodifferentpossibleelectronicstatesof

thesystem. Thisisshownschematicallybelow.

2ThetotalnumberofcoordinatesrequiredtodescribethepositionsofNatomsis3N. However,theenergyofthe

systemdependsonlyontherelativecoordinatesoftheatoms. Thesearenotchangedbyatranslationofthewhole

systemor

by

rotation

of

the

whole

system,

each

of

which

require

3of

the

3N

degrees

of

freedom,

so

the

total

number

ofcoordinatesrequiredis3N6(or3N5foralinearsystem,inwhichonlytwodegreesoffreedomareneededto

describearotationofthewholesystem).


11/38

11

11

2.4. ThePESandthecollisiondynamics

IfyoucastyourmindbacktothefirstyearPhysicalBasisofChemistrylecturecourse,youmayrecallthat

ifthepotentialenergyofasystemisknown,theforceactingonthesystemmaybedeterminedfromthe

first derivative or gradient of the potential. For a potential energy function that depends on a single

coordinate,wehave

F(r)= dV(r)

dr (2.4)

whileforapotentialenergyfunctionthatdependsoncoordinatesr1,r2,r3etc,wehave3

F = V =

V

r1Vr2

Vr3... (2.5)

Thegradientofthepotentialenergysurfacethereforedeterminestheforcesactingonalloftheatoms,and

thereby determines their subsequentmotion. The potential energy surface (together with the initial

conditionswhereaboutsonthesurfacethesystemstarts,andhowmuchkineticenergyithas)thereforecompletelydefinesthereactiondynamics. Conceptually,wecanthinkofthedynamicsintermsofrollinga

balloverthepotentialenergysurface,asshownschematically inthefigureabove. Astheballrollsover

the surface, different atomic configurations are sampled, allowing us to reconstruct a movie of the

collisiondynamics. Wecanofcoursebemuchmorerigorousthanthis,andcancalculatethetrajectoryof

thesystemmathematically. Startingfromaparticularpointonthepotentialenergysurfacecorresponding

to the initialpositionsof the atoms,we calculate the gradientof the surface, and therefore the forces

actingontheatoms. WethensolveNewtonsthirdlawofmotiontodeterminethetrajectory,r(t),ofthe

systemi.e.thewayinwhichtheatomicpositionsevolveasafunctionoftime

F=ma

=m

d2r

dt2

(2.6)

Weallowthesystemtofollowthistrajectoryforashortperiodoftimeuntilithasreachedanewpointon

thesurface. Wemustthenfindthegradientagaintodetermineanuptodatesetofforces,solvetofind

thenewtrajectory,letthesystemfollowthistrajectoryforashortperiodoftimetoreachanewposition

on the surface,and soon. Thisprocedure isknownasaclassical trajectorycalculation,andwith some

modificationstotakeaccountofthequantisedenergylevelsofthesystem,theapproachmaybeusedto

3Applyingthegradientoperator,= r1 r2 r3... ,tothepotentialyieldstheforcevector,withcomponentsof

forcealongthecoordinatesr1,r2,r3etc.


12/38

12

12

predictvirtuallyalloftheexperimentalobservablesassociatedwithacollision,includingtheidentityofthe

products,theirspeedandangulardistributions,andtheirquantumstatepopulations.

Alternatively,wecancarryoutfullyquantummechanicalcalculationstostudythedynamics. Thisapproach

involvesconstructinganinitialwavefunctionforthesystemandpropagatingitacrossthepotentialenergy

surfacebysolvingthetimedependentSchrodingerequation.

2.5 ConstructionofthePES

Inordertocalculateapointonthepotentialenergysurface,weneedtosolvetheSchrodingerequationto

determine the energy at the atomic configuration of interest. While it is not possible to solve the

Schrodingerequationexactly, therearenumerouselectronicstructure softwarepackages (e.g.Gaussian,

Gamess,MolPro)whichcansolvetheequationnumericallytoahighdegreeofaccuracy. Inprinciple,to

construct thecompletePESwewouldneed tosolve theSchrodingerequation foreverypossibleatomic

configuration. Inpractice,wesolvetheSchrodingerequation forareasonablenumberofconfigurations

and then fit thedatapoints toa suitable function inorder togeneratea smooth surface. Though this

procedure is straightforward inprinciple, the constructionofan accuratepotentialenergy surface for a

chemicalsystemisachallengingproblem,whichquicklybecomesintractableasthenumberofatomsinthe

system, and therefore the number of degrees of freedom required, increases. Nonetheless, accurate

surfaces are available for a considerable number of reactive chemical systems, and comparing the

predictionsofthesesurfaceswiththeresultsofexperimentalmeasurementsprovidesastringenttestboth

for theory andexperiment. Note that the constructionofpotentialenergy surfaces assumes theBorn

Oppenheimerapproximation, i.e. that theelectronicandnuclearmotion canbedecoupled, so that it is

validtocalculatetheelectronicenergyataseriesoffixednucleargeometries.

Weshallnowlookatsomeexamplesofdifferenttypesofpotentialenergysurfacesandtheconsequences

forthecorrespondingscatteringprocesses.

2.6 Thepotentialenergysurfaceforalineartriatomicsystem

Considerareactingsystemofthreeatoms,i.e.A+BCAB+C. Forsimplicity,wewillconstraintheatomstolieinastraightlinethroughoutthereaction. Thisallowstherelativepositionsofallthreeatomstobe

defined by two distances, rAB and rBC, and the potential energy surface for the reaction is therefore afunctionof these two coordinates. Tobegin constructing thepotentialenergy surface,we consider the

interactionbetweentheatomsintheentrancechannelofthereactioni.e.beforethereactivecollision,as

theAatomapproaches theBCmolecule. Ifwe fix the rBCdistance, therewillbea longrangeattractive

interactionbetweenAandBC,andashortrangerepulsiveinteraction,andthepotentialenergycurvewill

looksomethinglikethatshowninthefigurebelow. Thefigurerepresentsacutthroughthefullpotential

energy surface,V(rAB,rBC) for a fixed valueof rBC. The picturewouldbe similar ifwe looked at theexit

channelof the reaction i.e. ifwe fixed theABdistanceandplotted the interactionpotentialenergyasa

functionofrBC.


13/38

13

13

By considering the energy of the system at various ABC separations, we now know enough about

molecularinteractionstobeabletopredicttheformofthecompletepotentialenergysurface. Theshort

range repulsive interactionsmean that the energywill be highwhen either or both of the rAB and rBC

distancesareverysmall,andtheenergywillbelowestwhenrABand/orrBCareattheirequilibriumdistance

(correspondingtotheminimuminthecurveabove). Theenergy increasesagainaswemoveoutto large

separationswheretherearenoattractiveinteractionsbetweentheatoms. Theresultingpotentialenergy

surfaceisshownbelowasacontourplot(ontheleft)andasa3Dplot(ontheright). Thesurfaceresembles

acurved halfpipe;keenskateboardersor snowboardersmightbe familiarwithanalogousgravitational

potentialenergysurfacesofthisform.

Theminimum energy path across the surface follows the base of the half pipe, and is known as the

reaction coordinate. Note that the transition state appears as a hump along the reaction coordinate,

which the reactantsmust surmount to formproducts. As shownbelow, ifweplot thepotentialenergy

alongthereactioncoordinate,werecoverthefamiliarreactionpotentialenergyprofile.


14/38

14

14

2.7 Reactiveandnonreactivetrajectoriesacrossthepotentialenergysurface

Asnotedearlier,theoutcomeofacollisiondependsbothontheshapeofthepotentialenergysurfaceand

ontheinitialvelocitiesandinternalstatesofthereactants. Inthissectionwewillconsiderafewexample

trajectoriesacrossasurfacesimilartothatconstructedinSection2.6above,againforanA+BC AB+C

reaction. Thefourtrajectorieswewillconsiderareshownbelow.

A. The reactantsare launchedonto the surfacewitha low relativevelocity. Theyhave insufficient

kinetic energy to surmount the activation barrier, and the trajectory turns round and returns

alongtheentrancechanneltothereactants. Suchatrajectorycorrespondstoelasticor inelastic

scattering,dependingonwhetherornotthere isachange inrotationalorvibrationalstateofthe

diatomicreactant.

B. The reactants are launchedonto the surfacewith a high relative velocity. Theyhave sufficient

energy to surmount theactivationbarrier,andat smallABdistances theyeventually rollup the

repulsivewallofthesurface,beforerollingbackdownagainandfollowinganoscillatingtrajectory

out intotheproductchannel. Note thatasatomCdeparts,theABdistanceperiodicallyextends

andcontracts,correspondingtovibrationoftheABproduct.

Fromthissimpleconsiderationoftwotrajectories,wehavethereforebeenabletopredictthatinorderfor

reaction to occur on this potential energy surface,we need a significant amount of energy in relative

translationalmotionofthereactants,andthatsomeofthisenergyisconvertedintovibrationalmotionof

theproducts. Thisisafairlycommonobservation,whichyouwillreturntointhesecondhalfofthislecture

courseundertheheadingofPolanyisrules.

TrajectoriesCandDcorrespondtothereversereaction,AB+C A+BC


15/38

15

15

C. Trajectory C is another unsuccessful trajectory. The reactants have a reasonably high relative

velocity in this case, but thepositionof the barrier in the exit channel for the reverse reaction

meansthatthetrajectorymerelyglancesoffthebarrierandreturnsalongtheentrancechannelto

reactants. Inthiscase,wecanseethatsomeofthetranslationalenergyoftheincomingreactants

isconverted intovibrationoftheABmoleculefollowingthecollision;thisisthereforean inelastic

scatteringtrajectory.

D. Inthistrajectorythereactantsbeginwithamixtureoftranslationalandvibrationalkineticenergy,

and the trajectory successfully surmounts the barrier to form products with no vibrational

excitation.

2.8 Generalfeaturesofpotentialenergysurfaces

Thesimplepotentialenergysurfaceswehaveconsideredso farconsistof reactantandproduct valleys

separatedbyatransitionstate. Ingeneral,potentialenergysurfacescanbemuchmorecomplicatedthan

this, particularly for larger chemical systems. In the diagram below,we define a number of different

topologicalfeaturesthatmaybefoundonareactionpotentialenergysurface. Thediagramistakenfroma

chapteronGeometryOptimisationintheEncyclopediaofComputationalChemistry(JohnWileyandSons,

Chichester,1998,p11361142). Thereareseveralfeaturestonote:

(i) Stablespeciesappearasminimaonthe potential

energysurface,withthedeepestwellscorresponding

to themost stable species. Reactants and products

tend toappearas fairlydeepminimaon thesurface,

whileintermediateswillhaveshallowerwelldepths.

(ii) Transition states appear as saddlepointson the

surface. Asaddlepoint(sometimescalledafirstorder

saddle point) is amaximum inone dimension and a

minimum in all other dimensions, and as the name

suggests,issimilarinshapetoahorsessaddle.

(iii)Otherstructures,suchassecondordersaddlepoints(amaximumintwodimensionsandminimum in

allothers)andridges,canalsoappear,butwewillnotlookattheseinanydetailinthiscourse.

2.9 Examplesofpotentialenergysurfacesforrealchemicalsystems

Wewill now look at some examples of potential energy surfaces for a few bimolecular reactions and

photoinitiated events. Someof these processes involvemore than one electronic state, and therefore

morethanonepotentialenergysurface. Aswewillsee,populationcanbetransferredbetweendifferent

potential energy surfaces, opening up numerous possibilities for the reaction dynamics relative to

processesoccurringonasinglesurace.


16/38

16

16

(i)Thesimplestchemicalreaction:H+H2 H2+H

[FigureadaptedfromScience,331411412(2011)]

Thesimplestchemicalreaction,constrainedtoreactinalineargeometry, has a perfectly symmetrical potential energy

surface,withthetransitionstate in thecentreofthecurveof

thehalfpipe. Thefigureshowsthesurfacein3Dviewandin

contourview,withthereactioncoordinatesuperimposedonto

bothviews.

(ii)Photodissociation

of

NO2

[FigureadaptedfromScience,334208212(2011)]

The figure shows the ground and first electronically excited

states for theNO2molecule,plottedasa functionof thebond

angleandoneoftheNObond lengths (thesecondNObond is

held fixed). We can use the surfaces to understand the

dissociation dynamics following absorptionofaphoton. Note

that this process does not involve any collisions, though

photodissociation is sometimes referred toasa halfcollision.InNO2, theequilibriumgeometry for theexcitedstate ismuch

morebentthanforthegroundstate,sofollowingexcitationthe

moleculebends to reduce itsenergy. Theexcitedandground

statescrossataconicalintersection,allowingthewavefunction

tofunnelthroughtheconicalintersectionbacktohighlyexcitedvibrationallevelsofthegroundstate,from

whichitdissociates. Thebentgeometryfromwhichdissociationoccursleadstoahighdegreeofrotational

excitation in the diatomic fragment, since the N atom receives a strong kick when the NO bond

dissociates.

(iii)Photoinduced

cis

trans

isomerisation

in

retinal

[Figure adapted fromConical Intersections,editedbyW.Domcke,D.R.Yarkony, andH.Koppel, (World

Scientific,Singapore,2003)]

Rhodopsin is a protein that is intimately involved in human

visionandconsistsofaproteinmoeity,opsin,withacovalently

bound cofactor, 11cisretinal. Absorption of light by retinal

leadstorapidcistransisomerisation,inducingaconformational

changeintheopsin,whichtriggersasequenceofeventsleading

tovisualisationof the lightby thebrain. Thepotentialenergy

surfaces give us insight into the dynamics of the cistrans


17/38

17

17

isomerisation. Onabsorptionofaphoton, the11cis retinal isexcited toahigherlyingelectronicstate.

Theexcitedstatecanloweritsenergybyundergoing torsionaboutthedoublebond,withtheequilibrium

geometry lyingatanangleof90degrees. At thispointpopulationcan funnelbackdown to theground

statethroughaconicalintersectionandequilibratefurthertoyieldgroundstatealltransretinal.

(iv)H

+CO2

OH

+CO

[FigureadaptedfromAcc.Chem.Res.,431519(2010)]

In contrast to the systems we have considered so far, the

H+CO2reactionhasadeepwellonitspotentialenergysurface

corresponding to the relatively stablemoleculeHOCO. This

leads to very different collision dynamics. Unlike the direct

dynamics we have considered so far, in which molecules

collide, exchange atoms, and separate in essentially a single

concertedstep,whenadeepwellispresentonthesurfacethesystemcanbecometrappedinthewellforsignificantamounts

oftime. Asweshallseeinlaterlectures,directreactionsoften

leadtostronglyanisotropicangulardistributions. Incontrast,

when a longlived complex forms and survives for multiple

rotational periods, all memory of the initial approach

directionsofthereactantsare lost,andan isotropicangulardistributionresults. Theenergyavailableto

thesystemalsohastimetobecomedistributedstatisticallyamongstthevariousmodesofmotionofthe

complex,leadingtocharacteristicstatisticalpopulationdistributionsinthereactionproducts. Incontrast,

thenewlyformed(ornascent)productsofdirectreactionsoftenhavehighlynonstatisticalquantumstate

distributions.

(iv) OH+SO H+SO2

[FigureadaptedfromChem.Phys.Lett.,433(46)279286(2007)]

As we might expect, the situation is similar for the

H+SO2reaction, inwhichthecarbonatomhasbeenreplaced

bysulphur. Inthis3Dviewofthepotentialenergysurface,the

deepwell corresponding to theHOSO complex isvery clearly

seen.

2.10Massweightedcoordinatesandtheskewangle

Thepicturewehaveofrollingaballorpointmassacrossapotentialenergysurfacehassofarworkedvery

well ingivingusaconceptualunderstandingofthereactiondynamics. However,whenweexamine this

approachinalittlemoredetailwefindthatthisissomethingofanoversimplification. Wewilldemonstrate

theproblem(andhowtosolve it)usingthetriatomicA+BCsystemwehavealreadyconsidered insome


18/38

18

18

detail. For this typeof system it turnsout that theappropriatemass for the rollingball is simply the

reducedmassofthetworeactants.

=mAmBCmA+mBC

(2.7)

Ifwechooseacoordinatesystem(rAB,rBC,)todefinetherelativepositionsofA,BandC,andrestrictthe

atoms tobecolinear (=) forsimplicity, thenwecanwrite the total (kineticpluspotential)energyasfollows.

E =1

2

rAB2+2

mAmCM r

ABr

BC+'rBC

2+V(rAB,rBC, =) (2.8)

whereandarethereducedmassesofthereactantsandproducts,Misthetotalmass,andr=dr

dt.

Weimmediatelyseeaproblem. Usingthe(rAB,rBC)coordinatesystem,thekineticenergyhasacrosstermin

rABrBC,andtherearetwodifferentmassfactorsappearinginthetermsinr

AB2andrBC

2. Thekineticenergy

isnotofthesimpleformK=v2,andwecannotthinkofthemotionasthatofasingleparticleofmassmovingacrossthesurface. However,wecanrescuethesituation ifwedefineanewsetofcoordinates

calledJacobicoordinates,definedinthediagrambelow.

r=BCdistance

R=distancefromAtocentreofmassofBC

=anglebetweenrandR

Usingthesenewcoordinates,theexpressionforthetotalenergybecomes

E =1

2(R 2+1/2r2) +V(R,r,=0) (2.9)

where=BC/,withBC=mBmC/mBC.Thekineticenergynowdoeshavetherequiredsimpleform

K=(vx2+vy

2 )=v2,andwe findthat thepictureofaparticleofmassrollingoverthepotential

energy surface is in fact correct, so long as we plot the surface with the following mass weighted

coordinates.

x = R = rAB+mCmBC

rBC

y = 1/2r = 1/2rBC (2.10)

SincexandyaresimplylinearcombinationsofouroriginalaxesrABandrBC,thisturnsouttobeequivalent

toplottingthePESusingthe(rAB,rBC)axissystem,butwiththeaxesatanangleinsteadofat90degreesto

eachother.

The

angle

isknown

as

the

skew

angle,

and

isgiven

by


19/38

19

19

cos2 =

mAmCmABmBC

(2.11)

WheneitherAorCare lightatoms(i.e. lightattackingordepartingatom),cos2 issmall, iscloseto90

degrees,andthedynamicsareessentiallyasouroriginalsimplepicturewouldpredict. However,inother

casestheskewanglecanbequitesmall,leadingtosignificantlydifferentdynamics. Thefigurebelowshows

a skewedaxis system togetherwitha typical trajectory. The simplepicturedevelopedaboveofkinetic

energybeingreleasedintotranslationalorvibrationalmotionintheproductsbecomemorecomplicatedin

suchcases.

2.11 Orbitalangularmomentum,centrifugalbarriersandtheeffectivepotential

In Section 1.8we defined the orbital angularmomentum of a colliding system, and noted that orbital

angularmomentum

isconserved

during

acollision.

Consider

again

the

particle

trajectory

shown

in

Section

1.10andreproducedbelow.

Therelativekineticenergyofthetwoparticles,vrel2,canbeexpressedasasumoftranslationalkinetic

energydirectedalongthelineofcentresofthetwoparticlesandrotationalkineticenergyassociatedwith

theorbitalmotiondescribedabove,i.e.

Krel = vrel2 =

2+

L2

2r2 (2.12)

Onlythecomponentofthekineticenergyassociatedwithmotionalongthe lineofcentres isavailableto

helpovercomeanactivationbarriertoreaction. Thekineticenergyassociatedwiththeorbitalmotion is

notavailabletohelpsurmountanactivationbarrier,andbecauseithastheeffectofreducingtheavailable

energy, this term is often referred to as a centrifugal barrier. The centrifugal barrier term is often

combinedwiththepotentialenergysurfacetogiveaneffectivepotential.i.e.


20/38

20

20

Veff(r)=V(r)+L2

2r2 (2.13)

Asshowninthefigurebelow,thecentrifugalbarriercangiverisetoaneffectivebarriertoreaction,even

whenthepotentialenergysurfaceitselfhasnobarrier.

UsingL2=2vrel

2b2(seesection1.8),wecanrewritetheeffectivepotentialas

Veff(r) = V(r) +1

2vrel

2b2

r2 (2.14)

We see that the barrierwill be largest for heavy systems at large impact parameters. The centrifugal

barrier often has the effect of reducing the maximum impact parameter that can lead to successful

reaction,therebyreducingthereactioncrosssection.

3. EXPERIMENTALMETHODS

3.1 Experimentalaims

The ultimateexperiment in reactiondynamicswoulddetermine thecross section,velocitydistribution,

quantumstatedistribution,andanyangularmomentumpolarisationfortheproductsofacollisioninwhich

everypropertyof the reactants (velocity,quantum state,etc)wasspecified. Inpractice,weaim for the

mostdetailedobservationswecanmakeforcomparisonwiththeory.Measurablepropertiesinclude:

(i) Totalreactioncrosssectionsatspecifiedcollisionenergies;

(ii) Productvelocity(speedandangular)distributions;

(iii) Productinternalstatedistributions;

(iv) Productalignmentand/ororientationi.e.polarisationofangularmomentum

(v) Dependenceofthereactionontheorientationoralignmentofthereactants.

Inthefollowing,wewillexploresomeoftheexperimentaltechniquesthathavebeendevelopedtoachieve

thesegoals.


21/38

21

21

3.2 Isolatingsinglecollisionevents:temporalandspatialisolation

In order to obtain information about single collision events, the products must be detected before

undergoingfurthercollisions. Secondarycollisionscanchangethevelocities,internalstates,andeventhe

chemical identities of the collision partners, thereby destroying information on the properties of the

nascentproductmoleculeof interest. Eventually,secondarycollisions leadtocompletethermalisationof

thereactionproducts,asisthecaseinabulkkineticsexperiment.

Isolatingsinglecollisioneventscanbeachievedusingeitheroftwoapproaches,knownasspatialisolation

andtemporalisolation,respectively:

(i) Intemporalisolation,weensurethatthemeantimebetweencollisionsislongerthanthetimerequired

todetectthereactionproducts. Themeantimetbetweencollisions issimplythe inverseofthecollision

frequency,z(seefirstyearPropertiesofGasesandKineticscourses)

t =

1

z =

1

cvn (3.1)

wherec isthecollisioncrosssection,v isthemeanrelativevelocity,andn isthenumberdensityofthegas. Anexampleofanapproachtostudyingsinglecollisionsthatreliesontemporal isolation isthe laser

pumpprobeexperiment. Afirst,pump,laserpulseisusedtoinitiatereaction,andafterashortdelay(of

anything from tens of femtoseconds up to a few tens of nanoseconds, depending on the type of

experiment), a second probe laserpulse isused to characterise theproducts formed via anyoneof a

rangeofspectroscopictechniques. Themeasurementtimeisdeterminedbythepumpprobedelay. For

comparison, the time between collisions forN2 gas at atmospheric pressure and 298 K is around 1 ns.

Reducingthepressurereadilyallowstemporalisolationtobeachieved.

(ii) In spatial isolation,weensure that themean freepathof themolecules ismuch larger than thedistancetheproductsmusttravelbeforetheyaredetected. Spatialisolationisappropriateforexperiments

in which the products are detected through interaction with a physical detector, rather than

spectroscopicallythroughinteractionwithlight. Themeanfreepathisgivenby

= vt =1

cn (3.2)

Anexampleofanexperimentemployingspatialisolationisthecrossedmolecularbeamexperiment,which

wewillconsiderindetaillater. Crossedbeamexperimentsarecarriedoutunderhighvacuumconditions.In the classical crossed beam experiment, collisions occur at the intersection between twomolecular

beams, and the productsmust fly from the crossing region to the inlet of amass spectrometer to be

detected.

3.3 Controlofinitialvelocities

3.3.1. Molecularbeams

Using beams of molecules provides a sample with a welldefined velocity distribution, and allowsdirectionalpropertiesofchemicalprocessestobestudied. Therearetwotypesofmolecularbeamsources,


22/38

22

22

known as effusive and supersonic sources, respectively. Both types of sourcework by allowing gas to

escapefromahighpressureregionthroughasmallorificeintoavacuum. Theflowcharacteristicsofthe

sourcearedeterminedbytheratioofthemean freepath inthesource tothediameterof theorifice,a

quantityknownastheKnudsennumber.

Kn = (3.3)

Inaneffusivesourcethediameteroftheholeissmallerthanthemeanfreepathinthegas(Kn>1),andina

supersonicsourceitislarger(Kn


23/38

23

23

definitionthemolecules inasupersonicmolecularbeamareextremelycold. It isfairlystandardtoreach

temperaturesaslowas5Kbythisverysimpletechniqueofexpandingagasthroughasmallhole. Atthese

lowtemperatures,onlyafewlowenergyrotationalstatesareoccupied.

Themaximumorterminalvelocityofmoleculesinasupersonicbeamcanbedeterminedbyassumingthat

alloftheinternalenergyofthemoleculesinsidethesourceisconvertedintokineticenergydirectedalong

thebeamaxis,inanisenthalpicexpansion(i.e.H=0duringtheexpansion). Assumingidealgasbehaviour,wehave

1

2mu

2 =

Ts

Tf

CpdT (3.4)

wheremu2 isthekineticenergyofthebeammoleculesandCpdT=dH isthechange inenthalpyofthe

moleculeswithinthesourceastheyundergosupersonicexpansion. TsandTfarethetemperaturesofthe

source andmolecular beam, respectively. Since Tf is small relative to Ts,we have Ts Tf Ts. Also,

assumingCpisnottemperaturedependent,wecansetCp=R/(1),where=Cp/CV. Evaluating(3.4)thengives

1

2mu

2 =

R

1 Ts (3.5)

Rearrangingyieldsthebeamterminalvelocity.

u =

2RTs

m

(1)

1/2

(3.6)

WeseefromEquation(3.6)thattheterminalvelocityofasupersonicmolecularbeamdependsonthemass

ofthebeamgas,withlightgasesachievinghigherspeedsthanheaviergases. AbeamofHehasaterminal

velocityofaround2400ms1,whileforHClitismuchlower,ataround500ms

1. Higherbeamvelocitiesfor

heaviermoleculesmaybeachievedbyseedingtheheaviermoleculeinalightcarriergassoastoachieve

a lighter averagemass for the beammolecules; abeamof1%HCl inHe, for example,has a terminal

velocityofaround2100ms1.

Supersonicbeamsbydefinitiontravelfasterthanthelocalspeedofsound,whichresultsinacomplicated

shockwavestructureconsistingprimarilyofa barrelshockwavethatsurroundsthebeamanda Mach

disk

that

intersects

the

beam

axis.

Immediately

in

front

of

the

nozzle

orifice

is

a

zone

of

silence,

which

is

notdisturbedbyshockwaves. Byplacingaskimmer inthezoneofsilence,12cm fromthenozzle, it is

possible to create a collimatedbeamwhich isnotdisturbedby turbulence from the shockwaves. The

skimmer isshapedsoas topassmolecules from thecoreofthebeam,whiledeflecting thebarrelshock

waveandpreventingtheMachdiskfromforming.


24/38

24

24

3.3.2 Controllingthebeamvelocity

Intheprevioussectionwelearntthatthevelocityofamolecularbeamcouldbecontrolledbyseedingthe

moleculeofinterestinasuitablebuffergas. Dependingontheseedratio,beamvelocitiescanbeachieved

thatrangebetweentheterminalvelocityofthebuffergasandtheterminalvelocityoftheseedgas (see

Equation3.6). Arangeofothertechniqueshavebeendevelopedthatallowcontrolofthebeamvelocity,

many of which allow access to a range of beam velocities outside that determined simply by the

thermodynamicsofthesupersonicexpansion.

Afterseeding,thesimplestmethodforspeedinguporslowingdowna

molecular beam is tomount the source on the end of a rotating

assembly, as shown on the left. The labframe velocity of the

resultingbeam is then the sumof thebeamvelocity resulting from

the expansion and the velocity of the source. While this is a

completely general technique, it is technically quite difficult to

implement.

Recently, therehasbeenagreatdealof interest inusingmolecular

beams to create ultracold molecules in order to study quantum

mechanicaleffectsthatoccurwhenchemicalreactionsarecarriedoutatextremelylowtemperatures. The

molecules in a molecular beam are already very cold. The velocity distribution within a supersonic

molecularbeamtypicallycorrespondstoatemperatureoftheorderofafewKelvin;however,theentire

distribution ismoving at the velocity of themolecular beam. The problem therefore becomes one of

slowingthebeamvelocitydistributiondown. Variousmethodshavebeendeveloped inwhichelectricor

magneticfieldsareusedtoselectaslowsubsetofbeammolecules,and/ortoslowtheselectedmolecules

downtomuchlowervelocities. Wewilllookatoneexample,theStarkdecelerator,whichusestheStark

effecttoslowdownthebeammolecules.

Whenamoleculewithapermanentdipole issubjectedtoanexternal field, itsenergy is loweredby the

Starkeffectifitsdipoleisalignedparalleltothefield(alowerStarkstate),andraisedifitsdipoleisaligned

antiparalleltothefield(anupperStarkstate). Inan inhomogeneouselectricfield, lowerStarkstatescan

lowertheirenergybymovingtoregionsofhigherfield andforthisreasonaresometimesknownas`high

fieldseekingstates'whileupperStarkstatescan lowertheirenergybymovingtoregionsoflowerfield,

andarealsoknownas`lowfieldseekingstates'.


25/38

25

25

TheStarkdeceleratorusestimevaryingelectricfieldstodecelerateavelocityslicefromamolecularbeam

tozerovelocity. Thedeceleratorisshownschematicallyabove. Itconsistsofalternatepairsofelectrodes,

witheverysecondpairbiasedathighvoltageandtheremainingpairsgrounded. Whentravellingbetween

a grounded electrode and one at high potential, amolecule in a lowfieldseeking Stark statewill be

climbingapotentialgradientandwillbesloweddown. Ifthepotentialappliedtotherodswerestatic,once

themoleculehadpassedthehighfieldelectrodeitwouldthenacceleratedownthepotentialgradienton

theother sideof theelectrodeandconvert thepotentialenergy ithadgainedback intokineticenergy.

However, ifasthemoleculenearsthehighpotentialelectrode,theelectrodepotentialsareswitched (so

thatthoseathighpotentialarenowgrounded,andviceversa),themoleculeagainfinds itselfclimbinga

potentialgradient,andisslowedfurther,andsoon. Tensorevenhundredsofdecelerationstagescanbe

used,dependingon thedesired final velocity. Note thatonly a certain fractionofbeammolecules are

decelerated,whose initialvelocity,position,andquantum stateplace them inphasewith the switching

frequencyofthedecelerator.

3.3.3Polarisedlaserphotolysis

Radical species for reaction dynamics experiments are often generated through laser photolysis of a

suitableprecursormolecule. Ifapolarised laser isused tocarryout thephotodissociation, the radical is

often formed with a highly anisotropic velocity distribution, providing a reference direction for

measurementsofproductvelocitydistributions.

Thetransitiondipolevectoriffortheexcitationfromstateitostatefthatleadstodissociationisgivenbyif = f|d|i (3.7)

whered istheelectricdipoleoperator. Radicalprecursorstendtobeeitherdiatomicor lineartriatomic

molecules, and in these molecules the transition dipole vector will generally lie either parallel orperpendicular to thebond. Foranelectricdipole transition tooccur, the transitiondipolemust interact

withtheelectricvector ofthelaserpulse. ThetransitionprobabilityPifisthen

Pif = |if.|2 = if22cos2 (3.8)whereistheanglebetweenifand .Theprobabilityofamoleculeabsorbingaphotonanddissociatingisthereforestronglydependentontheanglebetweenthetransitiondipoleandtheelectricfieldvectorofthe

light. Whenlinearlypolarisedlightisincidentonarandomlyorientedsampleofgas,immediatelyfollowing

excitationthetransitiondipolesoftheexcitedstatemoleculeswillhaveacos2angulardistributionabout

thelaser

polarisation

vector.

Ifthe

transition

dipole

lies

parallel

to

the

breaking

bond,

this

will

translate

intoacos2distributionofthefragmentvelocitiesaboutthelaserpolarisationvector,whileifthetransition


26/38

26

26

dipoleliesperpendiculartothebreakingbondthenasin2distributionwillresult. Thisisillustratedbelow

for Cl atoms formed in the polarised laser dissociation of Cl2 via a perpendicular transition. The sin2

distribution isclearlyvisible,with signal intensitypeakingat rightangles to the laserpolarisationvector

(markedinwhiteontheimages).

3.5 Crosssectionmeasurements

Total cross sections can bemeasured bymeasuring the attenuation of amolecular beam as it passes

throughacollisionchambercontainingthescatteringgasofinterest,asshowninthefigurebelow.

Ifthepathlengththroughthescatteringchamberislandthenumberdensityofthescatteringgasisn,we

candeterminethemeanfreepathandthecollisioncrosssectionfromtheBeerLambertlaw.

I = I0exp(nl) = I0exp

l

with =1

n (3.9)

The beam intensity transmitted through the collision cell in the presence (I) and absence (I0) of the

scatteringgasaremeasured,thepath lengthl isknown,andthenumberdensityn iscalculatedfromthe

measuredpressureandtemperatureofthecollisioncell,assumingtheidealgaslaw.

Note that themeasuredattenuationcrosssection is the sumof thecross sections forallprocesses that

scattertheincidentbeam. Foranatomicbeamandscatteringgas,thisissimplytheelasticscatteringcross

sectionE,whileforanatomicormolecularbeamwithamolecularscatteringgas,thecrosssectionisthe

sumoftheelasticandinelastic(includingreactive)crosssectionsi.e.=E+IN.

3.6 Controlofinitialquantumstates

Therearenumerousways inwhichthe initialquantumstatesofthereactantmoleculescanbeselected.

Simplyexpanding themolecules inamolecularbeamusuallyyieldsmolecules intheirgroundvibrational

stateandlowrotationalstates,whileelectronicallyorvibrationallyexcitedstatesmaybepreparedbylaser

excitationfromthegroundstateimmediatelypriortothecollisionprocessofinterest.


27/38

27

27

Dipolarmolecules,suchasheteronucleardiatomicsorsymmetrictops,maybepreparedinsinglerotational

statesbyexploitingtheirtrajectoriesthroughaninhomogeneouselectricfield. InSection3.3.2wemetthe

Stark effect, and noted that in an inhomogenous electric field,molecules in lower Stark states (dipole

aligned parallel to the field) are highfield seeking, while those in upper Stark states (dipole aligned

antiparalleltothefield)arelowfieldseeking.

Stateselectionofdiatomicsislimitedtoafewmoleculeswithverylargedipolemoments,andwewillfocus

here on state selection of symmetric topmolecules using a hexapole field. A hexapole stateselector

consists of sixmetal rodsmounted on an inscribed radius r0,with positive and negative high voltages

appliedtoalternaterods,asshownbelow.

Therotationalstateofasymmetrictopmolecule isdefinedbythreequantumnumbers:thetotalangular

momentum,J,theprojectionK ofJontothemolecularaxis;andtheprojectionMJofJontoa labframe

axis, in this case the external field direction. The rotationalmotion is precessional: themolecular axis

precessesaboutthetotalangularmomentumvectorJ,which inturnprecessesaboutthe labframeaxis.

Solving the equations ofmotion for a symmetric topmolecule through a hexapole field yields simple

harmonicoscillatorsolutionsfortheradialcoordinate.Molecules in lowerStarkstatesfollowexponential

trajectoriesthatdivergefromthehexapoleaxis,andarelostfromthemolecularbeam. Conversely,upper

Stark states follow sinusoidal trajectories, r(t) sint,which focus back to the axis at a distance thatdependsontherotationalstate|JKMJandthepotentialV0appliedtothehexapolerods,asdeterminedby

theoscillationfrequency,

=

6V0d

mr03 |KMJ|

J(J+1)

1/2

(3.10)

wherem andd are themass anddipolemomentof themolecule, and r0 is the inscribed radiusof the

hexapolerods. Thedependenceof onbothV0andtherotationalstateofthemoleculemeansthatthe

hexapolepotentialcanbechosentofocusanydesiredrotationalstatewithKMJ0ontothebeamaxisat

thehexapoleexit. Othernonfocusingrotationalstatesmaybeeliminatedfromthebeambyusinganirisorpinholetoallowonlythefocusedstateofinteresttoexitthehexapole.

Aswellasbeingused forrotationalstateselection,thehexapoletechniquecanalsobeused tocreatea

beam of spatially oriented symmetric topmolecules. The upper Stark states transmitted through the

hexapoleareallalignedwiththeirdipolesopposingthelocalappliedelectricfield,butarenotyetuniformly

oriented in space, since thehexapole field is inhomogeneous. Inorder to spatiallyorient thebeam it is

allowed topassadiabatically (i.e.withoutchangingquantum state) intoa regionofweakhomogeneous

electric fieldmaintainedbetween apairofparallelplatespositioneddirectly after thehexapole.As the

beamleavesthehexapolethemoleculardipolesrotate inspacetolineupagainstthenewhomogeneous

fieldandorientationisachieved. Ahomogeneousfieldofonlyaround2Vcm1

ormoreisrequiredinordertoensurethattheupperStarkstatesremainwelldefinedandtherebytoachievesomedegreeofmolecular


28/38

28

28

orientation. However,considerablystrongerfieldsareoftenrequiredinordertoovercomecouplingofJto

anynonzeronuclearspinswithinthemoleculeinordertoachieveoptimumorientation.

Thedegreeoforientationisdeterminedbytherotationalstatesofthemoleculesinthebeam. Inthe

strongfieldcase,wherecouplingofJtotheelectricfieldismuchstrongerthancouplingtoanynonzero

nuclearspins,theaverageanglebetweentheelectricfieldvectorandthemoleculardipole(i.e.the

precessionangle)foragiven|JKMJstateisgivenby

cos =KMJ

J(J+1) (3.11)

Inthesecondhalfofthecourse,youwill lookanexample inwhichthehexapoleorientationmethodwas

usedtostudystericeffectsintheRb+CH3Ireaction.

Numerousothertechniquesareavailableforgeneratingspatiallyorientedmoleculesfordynamicsstudies.

Ifyouwouldliketoreadmoreaboutthesemethods,adetaileddiscussioncanbefoundinarecentreview

article4.

3.7Measurementoffinalvelocities

3.7.1 Crossedbeamexperimentwithrotatingmassspectrometer

One of the most commonly used methods for measuring the product velocity (speed and angular)

distribution isthecrossedmolecularbeamtechnique. Twomolecularbeamscontaining thereactantsof

interestarecrossed,usuallyatrightangles,andreactionproductsscatteroutfromthecrossingregion. A

rotatabledetector,normallyaquadrupolemassspectrometer,isthenusedtomeasuretheproductfluxas

afunctionofangle. Ateachangle,anarrivaltimedistributionisrecordedfortheproductofinterest,which

canbe transformed intoa speeddistribution forproducts scattered in thatdirection, since thedistance

from thecrossing region to thedetector isknown. By recordinga speeddistributionateachangle, the

complete velocity distribution can bemeasured. Note that this typeof detection is not product state

selective.

4 C. Vallance, Generation, characterisation, and applications of atomic and molecular alignment and

orientation, Phys. Chem. Chem. Phys., 13 14427-14441 (2011).


29/38

29

29

Somecrossedmolecularbeam instrumentsallowthecrossingangleofthebeamstobevariedtoprovide

controloverthecollisionenergy. Crossingthebeamsatasmalleranglereducesthecollisionenergy,while

movingtolargercrossingangles(i.e.closertoaheadon)geometryincreasesthecollisionenergy.

Wenoted inSection1.2thatthevelocityofthecentreofmass isconservedthroughoutacollision. We

shouldalsonotethatdifferentexperimentalsetupswillyielddifferentvelocitiesforthecentreofmass;for

example,asweincreasethebeamcrossingangleinacrossedmolecularbeamexperiment,thevelocityof

thecentreofmassbecomessmaller. Inordertocomparetheresultsofdifferentexperiments(measuredin

thesocalledlaboratoryframeorlabframeofreference),wenormallysubtractthevelocityofthecentre

ofmass from ourmeasured velocity distribution. The result is the velocity distribution thatwewould

observe ifweweretravellingalongwiththecentreofmass,andforthisreason it iscalledthecentreof

mass(CM)framevelocitydistribution. ThelabtoCMframetransformation,asthesubtraction iscalled,

will be covered in detail in Section 3.9. However, the transformation is shown in terms of aNewton

diagrambelow.

TheNewtondiagramontheleftaboveshowsboththelabframeandCMframevelocitiesofthereactants,

togetherwiththevelocityofthecentreofmass,vCM. The labframevelocitiesare labelledv1andv2,and

theCMframevelocitiesarelabelledu1andu2. NotethatintheCMframe,thereactantscollideheadon

along the relative velocity vector. As noted in Section 1.9, the relative velocity vector is an axis of

symmetry for the collision, and scattering is therefore symmetric about this direction. The reaction

productswillhavesomemaximumvelocityumax,determinedfromenergyconservationrequirements(see

Section1.3),andwillthereforescattersomewherewithinthecircleofradiusumaxcentredontheCMframe

origin. Wesee that in the labframe,scattering isonlypossibleoverarather restricted rangeofangles.

The rotatable detector in a crossed beam experiment therefore does not need to sample the full 360

degreesaboutthecrossingregion.

Thedata fromacrossedmolecularbeamexperiment isnormallyshown inthe formofaCMframeflux

velocityangle contour map, inwhich the radial coordinate corresponds to product speed, the angular

coordinatecorrespondstoscatteringangle,andthe intensityatagivenpointonthemapcorrespondsto

thefluxofproductsscatteredinthecorrespondingdirection. Someexamplesoffluxvelocityanglecontour

mapsareshownbelow,illustratingreactionswithvariousdifferentscatteringbehaviours. Theplotscanbe

analysedtoextractdifferentialcrosssections,productspeedand internalstatedistributions,allofwhich

provide detailed information on the reaction mechanism. The dynamics leading to these different

scatteringbehaviourswillbeexaminedinthesecondhalfofthecourse.


30/38

30

30

3.7.3 Velocitymapimaging

Thecrossedmolecularbeamexperimentsdescribedabovebuilduptheproductvelocitydistributionangle

byangle. This isatimeconsumingand labourintensiveprocess. Morerecently, laserbasedtechniques

havebeendevelopedwhichallowthecompleteproductscatteringdistributiontoberecordedinessentially

a single measurement, with the added capability of product quantum state selection. Velocitymap

imagingcanbecombinedeitherwithasinglemolecularbeam(forexample,inphotodissociationstudies),

orwithcrossedbeamexperiments. Typicalvelocitymapimagingsetupsforbothtypesofstudyareshown

below.

Theessentialelementsofthesetupareasetofelectrostatic lenses,aflighttube,andapositionsensitivedetector. Thechemicalprocessunderstudywillgenerallyyieldproducts flyingoutfromthe interaction

region (e.g. thepointwhere twomolecularbeams cross in the caseof a crossed beam experiment,or

where a laser andmolecular beam cross in the case of a photodissociation experiment) in a Newton

sphere inalldirections. Toperformavelocitymappingmeasurement,thereactionproductof interest is

ionized (stateselectively ifdesired)within theelectrostatic lens systembya laserpulse. The ionization

step usually employs a REMPI processof the type described in Section 3.8.3. As soon as the neutral

productsareionized,theyfeeltheelectricfieldmaintainedwithintheelectrostaticlenssystem. Thefield

playsthreesimultaneousroles. Itacceleratestheionsalongtheflighttubetothedetector,itmapsallions

with the same velocity component in the detector plane onto the same point on the detector, and it

compressestheNewtonsphereforagiven ionmass intoapancakeas itstrikesthedetector. Thefield

acceleratesallionstothesamekineticenergy,andsinceK=mv2,thismeansthationsofdifferentmasses


31/38

31

31

flytothedetectoratdifferentvelocities,asshownbelow,andarethereforeseparatedintimewhenthey

reach the detector. This is the principle underlying timeofflight mass spectrometry. In velocitymap

imagingexperiments,the ionof interestcaneasilybemassselectedbytimegatingthedetector(turning

thedetectoronbrieflyattheappropriatetime)sothatonlyionsofthedesiredmassareimaged.

Thedetectorconsistsofapairofmicrochannelplates (MCPs)coupled toaphosphorscreen. TheMCPs

converteachincidentionintoapulseofelectrons,andthephosphorconvertstheelectronsintoaflashof

lightwhich can be imaged on a CCD camera placed behind the screen. The resulting image is a two

dimensionalprojectionofthecompletethreedimensional(CMframe)velocitydistributionofthenascent

reactionproducts. Someexampleimagesrecordedforarangeofchemicalsystemsareshownbelow.

Inmany cases, the scatteringdistributionhasanaxisofcylindrical symmetry,and the full3D scattering

distributioncanbe recovered from the2Dprojectionusingamathematical transformcalled the inverse

Abel transform. In theexamplebelow, the imageon the left isa typical2D crushed image. TheAbel


32/38

32

32

inversion reinflatesthe imageandextractsthecentreslice (centre image). Rotatingthesliceaboutthe

axisofcylindricalsymmetryreturnsthe3Ddistribution(rightimage).

Atechniquecalledslice imagingreliesoncarefullytimegatingthedetector inordertoallowthecentral

sliceofthe3Ddistributiontoberecordeddirectly,eliminatingtheneedfortheAbelinversion.

3.7.3Dopplerprofilesandpumpprobeexperiments

Aswe shall see inSection3.8,various typesofabsorption spectroscopymaybeused todetectproductmolecules state selectively. The spectral linewidths of the rovibronic transitions excited in such

measurements are determined by the product velocity distribution throughDoppler broadening of the

peaks. AmovingmoleculehasaDopplershiftgivenby

=

1 vzc 0 (3.12)

where0istheabsorptionfrequencyofthemoleculeatrest,istheabsorptionfrequencyofthemoleculemovingwithavelocitycomponentvzalongthelaserpropagationdirection,andcisthespeedoflight.

Ifthefrequencybandwidthoftheprobe laser ismuchnarrowerthanthewidthofthespectral line,then

thespectral lineshapeconstitutesaonedimensionalprojectionof theproductvelocitydistributiononto

the laser propagation axis. If several onedimensional projections are recorded, enough information is

obtainedtoreconstructtheentirethreedimensionalvelocitydistribution.

Dopplerprofilemeasurementsareusuallycarriedoutas thedetection,or probe,step ina laserpump

probeexperiment. WefirstmetlaserpumpprobeexperimentsinSection3.2asanexampleofatechnique

for achieving temporal isolationof collisions. A simple example, shown above, is the dissociationof a

diatomicmoleculeviaaparalleltransition(i.e.thetransitiondipoleliesalongthebond). Whenthepump

laserispolarisedalongthetransitiondipole/bondaxis,themoleculedissociates,andtheatomicproductsfly in theup and downdirections, as shown. On the rightof the figurewe see the two very different

Doppler lineshapes recorded by a probe pulse travelling either parallel or perpendicular to the recoil


33/38

33

33

direction of the fragments. When the probe laser beam propagates along the recoil direction of the

photofragments,thelineprofileissplitintotwopeaks,correspondingtofragmentsflyingtowardsoraway

fromthe laserbeam. Dopplerprofilescanberecorded for individualproductquantumstates,simplyby

tuning the probe laserwavelength to a transition involving the stateof interest. This allows complete

productvelocitydistributionstoberecordedasafunctionofproductquantumstate.

Thepumpprobe/Dopplerprofiletechniquecanalsobeusedtostudyphotoinitiatedbimolecularreactions.

Thetechniquerequiresthatoneofthereactantscanbepreparedbylaserphotolysisofasuitableprecursor

molecule (for example, oxygen atoms can be prepared by dissociation of O2, N2O, and various other

precursors;chlorineatomscanbepreparedbyphotolysisofCl2,andsoon). Agasmixture isprepared,

eitherinabeamorabulb,thatcontainstheprecursormolecule(AX)andthesecondreactant(B),andthe

pumpprobesequenceisasfollows:

AX+h A+X Pump(photoinitiation)

A+B P Bimolecularreaction

P+h P* Probe(LIForREMPI)

Inthepumpstep,theprecursormolecule isdissociatedto formthereactantA. The tworeactantsthen

reacttoformproductsP,andintheprobestepeitherLIForREMPIisusedtorecordappropriateDoppler

profilesfortheproductquantumstatesofinterest. Thedetailsofextractingproductvelocitydistributions

from the measured Doppler profiles are somewhat complicated, but the basic principle is easy to

understand,andisshownintheFigurebelow. PhotolysisoftheprecursorformsreactantAexpandingina

Newton spherewith awelldefined velocitydistribution (shownon the leftbelow). Following reaction,

forwardscatteredproductslieonalargerNewtonspherethanthatforthereactantA(belowcentre),while

backwardscatteredproductslieonasmallersphere(belowright). ThiswillleadtoverydifferentDoppler

profiles for the two cases. Assuming theNewton sphere ofA has beenwell characterised in separate

experiments, thedatacanbeanalysed toextract thevelocitydistribution (and therefore the speedand

angulardistribution/differentialcrosssection)ofthereactionproducts.

3.8Measurementoffinalquantumstates

3.8.1 Infraredchemiluminescence

Chemical reactants often produce products with significant amounts of internal excitation.

Chemiluminescenceemploysaspectroscopicanalysisof lightemittedbytheseexcitedstatestomeasure

therelativepopulationsofthestates.


34/38

34

34

A chemiluminescence experiment is typically carried out in a fast flow system under steady state

conditions,usinganexperimentalsetupsimilartothatshown inthediagrambelow. Toavoidcollisional

relaxationofthenewlyformedreactionproducts,thecollisionchamberismaintainedatalowbackground

pressureof105mbaror less. Awatercooledor liquidnitrogencooledjacket stopsanyparticles thatstrike thewall, furtherhelping topreventsecondarycollisions. Duringanexperiment,twouncollimated

jets (or better, supersonic beams) of reactants intersect, and are pumped away as fast as possible.

Radiationemitted fromthereactionzone iscollectedand focussed intoan infraredspectrometer. From

intensitymeasurementsof theemission linescorresponding toassigned transitionsbetweenknown (v,J)

states, relativepopulationsof the statescanbe inferred. The techniqueemploys temporal isolation to

observethenascentreactionproducts,workingontheprinciplethatthetimebetweencollisionsisgreater

thantheradiativelifetimeforinfraredemission. Asanexample,inastudyofthereactionCl+HIHCl+I,

thetimebetweencollisionsinthecrossingregion(p101to102mbar)wasaround106s,comparedwith

aninfraredemissionlifetimeforHClofof109s.

Notethatsincethetechnique isatypeofemissionspectroscopy, itdoesnotgiveany informationabout

populationofthegroundstate. Chemiluminescencedatacanbeusedtodeterminetherelativeratesof

reactiontoproducedifferentproductquantumstates,butsinceabsoluteproductyieldsarenotrecorded,

itcannotgiveabsoluterates.

3.8.2Laserinducedfluorescence(LIF)

In

the

previous

section,

we

saw

how

emission

spectroscopy

could

be

used

to

determine

the

internal

state

distributions of newly formed reaction products. An alternative approach is to use absorption

spectroscopy,whichhas the added advantageofbeing sensitive to thepopulationof the ground state.

However,sinceatypicalpumpprobeexperimentmayproduceonlyafewhundredproductmoleculesper

lasercycle,andthesewillbepresentamongstmuchhigherconcentrationsofotherspecies,suchasunused

reactants and background gases, a typical BeerLambert type absorptionmeasurement is completely

inadequate. We requirea technique that isbothextremelysensitiveandhighlyspecific to thechemical

speciesof interest. Laserinduced fluorescence (LIF) isone technique thatmatches these requirements.

Thetechniquerequiresthatthemoleculeunderstudyhasafluorescentexcitedstatethatisaccessiblefrom

theelectronicgroundstate,andthatthespectroscopyoftheelectronictransitioniswellunderstood.The

techniqueisillustratedschematicallybelowontheleft.


35/38

35

35

Nascentproductsareexcitedtotheupperstate,andthephotonsemittedastheexcitedstaterelaxesare

detected,with themeasured signalbeingproportional to theoriginalpopulationof the lowerstate. By

scanningtheexcitation laseroverthevariousaccessiblerovibronictransitions fromthegroundstate,the

rovibrational state populations of the nascent productsmay be determined. In order to compare the

signalsfromtwostates,theLIFsignalsmustbecorrectedtotakeaccountofthedifferingabsorptioncross

sectionsofthelowerstateandemissioncrosssectionsoftheupperstate(s)involvedinthetransitions. LIF

iswidelyused fordetectionof smallmolecules suchasOHandNOproduced inapumpprobe reaction

scheme. OntherightinthefigureaboveisshownaLIFspectrumforOHmoleculesformedintheinsertion

reactionO(1D)+H2OH+H.

3.8.3Resonanceenhancedmultiphotonionization(REMPI)

Asecondtechniquethatsatisfiesthesensitivityandselectivityrequirementsoutlinedaboveisresonance

enhancedmultiphotonionization(REMPI). Selective ionizationoftheproductmoleculeof interestmakes

detectionrelatively

straightforward.

Ions

can

be

steered

towards

adetector

by

means

of

an

appropriately

electricfield,andparticlemultiplierdetectorscapableofdetectingsingle ionshavebeenwidelyavailable

for several decades. Single photon ionization of most molecules is unfeasible, since their ionization

energies correspond to photons in the soft Xray region, which are not easy to generate in most

laboratories. Instead, ionization isusuallybroughtabout throughessentially simultaneousabsorptionof

twoormorephotons,asshowninthefigurebelow.

Thefirstphotonaccessesavirtualstate. SuchastateisnotasolutionofthemolecularHamiltonian,buta

veryshort livedstateofthe molecule+photonsystem. As longasasecondphotonarrivessufficiently

quickly,requiringhigh light intensitiessuchasthosefound ina laserpulse,thevirtualstatecanabsorba

photon to reacha realorvirtual stateofhigherenergy. Nonresonantmultiphoton ionization, inwhich

noneof

the

intermediate

levels

accessed

by

successive

photon

absorptions

correspond

to

real

levels

of

the

molecule, is not a particularly efficient process. However,when one of the intermediate states does


36/38

36

36

correspondtoanactualstateofthemolecule,the ionizationefficiency is increasedenormously,oftenby

orders of magnitude, and we have resonanceenhanced multiphoton ionization. REMPI schemes are

usuallylabelledintermsofthenumberofphotonsrequiredtoreachtheresonantstateandthenumberof

photonsrequiredtoionizefromtheresonantstate. So,forexample,ina(2+1)REMPIprocess,twophotons

areabsorbed to take themolecule from theground state to the intermediate state,anda thirdphoton

ionizesthemoleculefromthisstate.

EfficientREMPIschemesareavailableforawiderangeofmolecules,andforthisreasonthetechnique is

morewidelyusedthanLIFdetection. Thephotonenergyusedfortheinitialtransitiontotheresonantstate

can be tuned to ionize a specific quantum state of themolecule of interest, and scanning the laser

wavelength to record a REMPI absorption spectrum allows product quantum state distributions to be

determined.

TheresonantstateemployedinaREMPIschemeisoftenaRydbergstate(i.e.averyhighlyingstate)ofthe

molecule. Such states have low ionization potentials, and can befield ionised simply by exposing the

moleculetoasufficientlystrongelectric field. InanalternativedetectiontechniquetoREMPI,knownas

Rydbergtagging,alaserisusedtoexcitemoleculestoaRydbergstatewithoutionisingthem. Theneutral

Rydbergtaggedmoleculesarethenallowedtoflytothedetector,wheretheyare ionizedbyanelectric

fieldimmediatelypriortodetection.Rydbergtaggingisaveryusefuldetectiontechniqueforexperiments

inwhich creating ionsorusingelectric fields toextract ions in the interaction regionwoulddisturb the

collisionalprocessunderstudy. However,itisalsousedinothersituations. Someexampleswillbegivenin

thesecondhalfofthislecturecourse.

Both LIF and REMPI can be combinedwithDoppler resolvedmeasurements (Section 3.7.3) in order to

obtain informationontheproductvelocities. REMPI isalsocommonlycombinedwithtimeofflightmass

spectrometryorvelocitymap imaging toobtainvelocity information. A timeofflightprofileprovidesa

onedimensional projection of the velocity distribution, while velocitymap imaging yeilds a two

dimensionalprojection.

3.9 Comparisonofresults:thelabtocentreofmassframetransformation

Asnoted inSection3.7,theproductscatteringdistributionsrecorded ina reactiondynamicsexperiment

arerecorded insome fairlyarbitrary laboratory (lab)frameofreference,determinedbythegeometryof

theexperiment. Tocomparetheresultsofdifferentexperiments (andalso tocompareexperimentwith

theory)weneed to transform the results into a common frameof reference called the centre ofmass

frame. IntheCM frame,theobserver istravellingalongwiththecentreofmassofthesystem,so the

totalmomentum in this frame is zero, and the reactants appear to undergo a headon collision at the

positionoftheircentreofmass,asillustratedbelow.


37/38

37

37

The CM frame is independent of experimental geometry, allowing results from different types of

experiments tobe compared,andalsoprovidesamuchmore intuitivepictureof thecollisiondynamics

thanany lab frame. Asanexample, forwardand backwardscattering isalwaysdefinedrelativetothe

CMframe,whereasinthelabframethesetermscanbeambiguoustosaytheleast.

Analogy: Insomesense,choosing to transformalldatafrom reactiondynamicsexperiments into theCMframe isalittleliketransformingalldatafromNMRspectroscopyexperimentsintounitsofchemicalshift.

Thetransformationallowsdatafromdifferentdynamicsexperiments(c.f.differentspectrometers inNMR)

tobecompareddirectly.

3.9.1. CalculationsinvolvingtheCMandlabframes

WewillnowworkthroughalabtoCMframetransformationforanelasticcollision. Itisstraightforwardto

extendthefollowingtoinelasticorreactivecollisions;allyouhavetodoisaddorsubtracttheappropriate

energychange (rH in thecaseofareaction) from theenergyof thecollisionproducts. Whileworkingthroughthefollowinganalysis,notethatvelocityisavectorquantity.

Usually,we know the velocitiesof the twoparticlesbefore the collision, v1 and v2. These areour lab

framevelocities. ThekineticenergiesofthetwoparticlesareK1=m1v12andK2=m2v2

2,andthetotal

kineticenergyis

K = K1+K2 = m1v12+m2v2

2 (labframe) (3.13)

At some point we will need to determine the velocities in the CM frame call them u1 and u2 to

differentiate them fromthe lab framevelocities. This isasimplecalculation todo. Remember that thecentreofmassframeisjusttheframeinwhichwearetravellingalongwiththecentreofmass. Thismeans

thatallwehavetodotogofromlabframetoCMframevelocitiesissubtractthevelocityvCMofthecentre

ofmass.

u1=v1vCM

u2=v2vCM (3.14)

We candeterminevCMbynoting that the totalmomentumof the systemmaybewritteneither as the

momentumofthecentreofmass,MvCM,withMthetotalmassofthetwoparticles,orasthesumofthe

momentaofthetwoindividualparticles.

MvCM=m1v1+m2v2 vCM =m1v1+m2v2

M (3.15)

Thefactthatthetotalmomentum iszero intheCMframegivesusanotherapproachtofindingtheCM

framevelocities. Sincevrel=v1v2 = u1u2,andm1u1=m2u2,weobtain

u1 =m2M

vrel and u2 =m1M

vrel (3.16)

Inadditiontodefiningthemomentumassociatedwiththemotionofthecentreofmass,wecanalsodefinethekineticenergyassociatedwiththismotion.


38/38

38

KCM=MvCM2 (3.17)

Notethatbecausethetotalmomentumofthesystemhastobeconserved,thevelocity,momentum,and

kineticenergyof the centreofmassareconserved throughout the collision (this is true forany typeof

collision, including inelastic and reactiveones). As noted earlier, energy tied up in themotionof the

centreofmassisthereforenotavailableforthecollision. Forareactivecollision,thisenergydoesnothelp

overcomeanyactivationbarrierthatmightbepresent.

Having defined the total kinetic energy and the kinetic energy associatedwith the centreofmass, the

remainingkineticenergyisofcoursetheenergyassociatedwithrelativemotionofthetwoparticles. This

energy is available for the collision, and from Section 1.2we recognise it as the collision energy, also

sometimesreferredtoastheCMframekineticenergy.

Krel=vrel2 (3.18)

Wehavenowdeterminedalloftherelevantparametersinvolvingthereactants. Theproductvelocitesmay

bedeterminedbyrequiringthatmomentumandkineticenergy(ortotalenergy inthecaseofaninelastic

or reactive collision) are conserved during the collision. Usually it ismost straightforward to do this

calculation in theCM frame, though it shouldworkjust aswell in the lab frame, albeitwith somewhat

trickieralgebra.

m1u1+m2u2 = m1u1+m2u2 (momentumconservation)

m1u12+m2u2

2= m1u1

2+m2u2

2 (energyconservation) (3.19)

Thesetwoequationssimplyconstitutetwoequationsinthetwounknownsu1andu2(thefinalCMframe

velocities). For an inelasticor reactive collision, the secondequationwould include an additional termcontaining the change in internal energy for an inelastic collision, or reaction enthalpy for a reactive

collision. RecallthatintheCMframethetotalmomentumiszerobothbeforeandafterthecollision. This

simplifiesthesituationconsiderably,sincethefirstequationbecomessimply

m1u1+m2u2=0 (3.20)

OncewehavesolvedtheequationstodeterminetheCMframevelocitiesafterthecollision,wecanfind

theequivalentlabframevelocitiesforcomparisonwithexperimentsimplybyaddingonthevelocityofthe

CM(which,remember,staysconstantthroughoutthecollision).

v1=u1+vCM

v2=u2+vCM (3.21)

SUMMARY

Atthispoint,youshouldbewellequippedwithmanyofthebasicconceptsofmolecularreactiondynamics.

In the next four lectures these ideaswill be developed inmore detail and youwill be introduced to a

numberofexamplestoillustratethedynamicsofabroadrangeofdifferentchemicalsystems.

Documents

Reaction Dynamics Lecture Notes 2012