theories of intramolecular vibrational energy transfer · theories of intramolecular vibrational...

THEORIES OF INTRAMOLECULARVIBRATIONAL ENERGY TRANSFER

T. UZER

Schoolof Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430, USAand Bilkent University.Ankara, Turkey

with an appendix byW.H. MILLER

Departmentof Chemistry,and Materials and MolecularResearchDivision of theLawrenceBerkeleyLaboratory, Universityof California, Berkeley,California 94720, USA

INORTH-HOLLAND

PHYSICS REPORTS(Review Sectionof PhysicsLetters) 199, No. 2 (1991) 73—146. North-Holland

THEORIES OF INTRAMOLECULAR VIBRATIONAL ENERGY TRANSFER

T. UZERSchoolof Physics,Georgia Institute of Technology,Atlanta, Georgia30332-0430,USA

and Bilkent University, Ankara, Turkey

with an appendixbyW.H. MILLER

Departmentof Chemistry,and Materials and Molecular ResearchDivision of the LawrenceBerkeley Laboratory,University of California, Berkeley,California 94720, USA

Editor: E.W. McDaniel ReceivedJanuary1990

Contents:

1. Brief history and overview 75 4.3. The Onsetof stochasticity 1012. Definition and natureof intramolecularrelaxation 78 5. Classical microscopic theory of energy sharing; energy

2.1. General molecular model for intramolecularvibra- transferthroughnonlinearresonances 104tional energyredistribution (IVR) 80 5.1. Nonlinearresonancesand classical mechanicsof the

2.2. Criteria for IVR regimes 82 pendulum 1052.3. Dephasingversusrelaxationin isolatedmolecules 84 5.2. Vibrational energy transfer and overlapping reso-

3. “Phenomenological”theoriesof energysharing;approach nances 108of radiationlesstransitions 86 5.3. Generalizingthependulum 1123.1. Molecularmodel for fluorescence 87 5.4. Nonstatisticaleffects and phasespacestructures 1133.2. “Exact” moleculareigenstates 88 6. ‘Mesoscopic” description of intramolecular vibrational3.3. Long-time experiments 90 energytransfer 1163.4. Short-time experiments 91 6.1. Self-consistentdescriptionof quasiharmonicmode re-3.5. Quantumbeats 97 laxation 117

4. Vibrational energy transfer in model nonlinear oscillator 6.2. Relaxationof individual quasiharmonicmodesand itschains; the Fermi—Pasta—Ulamparadox and the Kol- spectralsignature 120mogorov—Arnol’d—Mosertheorem 98 Appendix. On the relation betweenabsorptionlinewidths,4.1. The Fermi—Pasta—Ulam(FPU) paradox 99 intramolecular vibrational energy redistribution4.2. The Kolmogorov—Arnol’d—Moser (KAM) theorem (IVR), and unimoleculardecayrates 124

and nonlinear resonances 100 References 129

Abstract:Intramolecularvibrationalenergytransfer is a processcentral to manyphysical andchemicalphenomenain molecules.Here, various theories

describingthe processare summarizedwith a special emphasison nonlinearresonances.A large bibliographysupplementsthe text.

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T. Uzer, Theories of intramolecularvibrational energy transfer 75

1. Brief history and overview

The flow of vibrational energyin excitedmoleculesis centralto chemicaland moleculardynamics.Various new developmentsin physics,chemistryandtechnologyof the 1970’sand 80’s havetendedtoconverge in intramolecular energy transfer (also known by the acronym IVR, for intramolecularvibrational energy redistribution): the increasinglypowerful and accuratemethodsof laserexcitationand analysis of molecularprocesseson the experimentalside, and the burgeoninginterest in theclassical-mechanicaland quantalbehaviorof nonlinearoscillator systemson the theoreticalside (forextensivereviews,seeChirikov [1979],Zaslavsky[1981,1985], Sagdeevetal. [1988],Eckhardt[1988]).Since its inception, intramolecularenergytransfer has been a field driven by experiment. This is anatural outcome of the usually overwhelming complexity of intramolecular interactions. Even asuperficial survey of the theoreticalliterature of the last thirty or so years revealsa few strandsoftheoretical thinking on IVR, each of which was suggestedto some degree by the experimentalachievementsat the time.

While the experimentalsituation has been reviewed many times (e.g., Parmenter[1982,1983],Smalley [1982],Bondybey[1984]),it seemsthat the various theoreticalviews of the processhavenotbeencollectedin one place before.This is the modestaim of this overview. During the “narration” Iwill occasionallyrefer to experiments,but thesereferencesare not intended to be comprehensivebecauseof spacerestrictions.Therefore,someslight to experimentalaccomplishmentsis inevitableandI apologizefor it at the outset.

Historically the mostwidelyknownstudiesof intramolecularvibrationalredistributionareassociatedwith thermalunimolecularreactions[Robinsonand Holbrook 1972; Forst 1973; Pritchard1984]. Thecurrentlyacceptedvarietiesof unimolecularreactionrate theoriesarosethroughthe testing of Slater’sdynamicaltheory [Slater 1959] and the statisticalRice—Ramsperger—Kassel—Marcus(RRKM) theories[Marcus 1952]. The key contentionin this debatewas the very existenceof intramolecularenergytransfer. Slater’s theory pictured the excitedmoleculeas an assemblyof harmonicoscillators.Withinthe Slater framework, vibrational energysharingbetweenmodesis forbidden, and the unimolecularreactionoccurswhen a reactionprogressvariable, theso-called“reaction coordinate”reachesa criticalextensionby the superpositionof variousharmonicmodedisplacements.In contrast,the RRKM theoryassumesthat excitationenergyrandomizesrapidly comparedto the reactionrate, and is distributedstatisticallyamongthe modesprior to reaction. (A similar formalism,albeit for the decayof excitednuclei, had beenintroducedby Bohr andWheeler[1939]independentlyfrom the developmentof thestatisticaltheoriesin chemicalphysics).The assumptionof randomization,which turns out to be widelyvalid, was neededvery early in the theoreticaldevelopmentbecausevery little was knownaboutenergysharingin moleculesat the time.

The statisticalapproachreducesthe (usuallyvery high) dimensionalityof the molecularproblemin amannersimilar to traditional “transition statetheory” [Glasstoneet al. 1941]. The phase-spacemotionof the system along the reactioncoordinateis assumedto be separablefrom its motion in all otherpossiblemodes,at least in the vicinity of a multidimensional surfacein phasespacethat separatesreagentsfrom products.This latter constructis called the “dividing surface” (for reviews, see Hase[1976],Truhlar et al. [1983], Hase [1983]), and the reaction rate calculation is thus reduced tocalculatingthe rate of passageof systemsacrossthe dividing surface,a “bottleneck” in onedimensiononly.

The first demonstrationthat IVR is a real physical phenomenon(Butler and Kistiakowski [1960])concernedmoleculesin the groundelectronicstatewith the highvibrationalexcitationcharacteristicof

76 T. Uzer, Theoriesof intramolecularvibrational energy transfer

reacting species.The kinetic and spectroscopicevidence for intramolecular relaxationprocessesinpolyatomicmoleculeshasbeendiscussedcritically by Quack[1983],who concludedthat,on the basisoftheir 0.1—10ps timescale,suchprocessesareslowcomparedwith vibrationalperiodsbut fastcomparedwith reactive andoptical processes.This picosecondtime scaleturnsout to betypical (seeKuip et al.[1985],Bagratashviliet al. [1986]).The “chemicaltiming” experimentsof the Parmentergroup(for anoverview,seeParmenter[1983])as well asthe experimentsof FelkerandZewail [1984,1985a—d]revealthe IVR processby providingstriking imagesof its temporalprogress.

Detailed experimentalstudiesperformedin the early 1960’s (reviewedby Oref and Rabinovitch[1979])producedresultsin harmonywith the statisticaltheory.The wide successof the predictionsofRRKM theory havebeentaken as proof of vibrational energyrandomization.However, theseearlyexperimentsare not of a dynamicalnature. Direct observationsrequire a dynamicalexperimentinwhicha very well specifiedinitial state(or a superpositionof states)is prepared.This statedecaysandameasurementis madeat a later time to demonstratethat the systemis in a statewhich differs from theinitial one.

The natureof the initial statein manyof theseearlyexperimentsis a statisticalsuperpositionbecauseof the collisionalpreparationof the initial state.Could it be thestatisticalpreparationof the initial statewhich gives such good agreementwith experiments?In fact, the RRKM theoryhasbeenreformulatedby assumingthat the vibrational relaxationis slow on the scalesof moleculardecomposition[Freed1979]. Thesetwo formulationsof the statisticaltheory take differentviewpoints aboutthe meaningofIVR. The randomizingapproachconsidersthe vibrational statesin terms of a zeroth-orderharmonicdescription,whereasthenonstatisticaltheory considersthe vibrationaleigenstateswhichdiagonalizethefull molecularHamiltonian.The questionarises,then: what is the properdescriptionof the initial state?

At about the same time as the last reformulation of the statisticaltheory by Marcus [19521,acalculation was being performedby Fermi, Pastaand Ulam [19551which was to inspire physiciststorethink their ideason energyrelaxationin coupledoscillator systems.The purposeof this calculationwas to see,in detail, the approachto ergodicity in a chain of couplednonlinearoscillators.The result,which hasbecomeknownas the Fermi—Pasta—Ulam(FPU)paradox,surprisedresearchersby showingvery little relaxation, thus demonstratingthat dynamicsof energysharingevenin a simple coupledoscillatorsystemcan be far from straightforwardand neednot be statisticalat all. This disturbinglackof ergodicity was explainedby Ford [19611by showing that the FPU systemlacked an essentialingredient for energy sharing, namely active nonlinear resonances.Also known as “internal” or“Fermi” resonances,theseintegerratiosbetweenfrequenciesare the unifying themeof this overview.The by then extant but little known Kolmogorov—Arnol’d—Moser (KAM) theorem revealed theconnectionbetweentheseresonancesandergodicity (for reviews,see Ford [1975],Berry [1978]).

Thesetwin developmentsof thefifties hada considerableimpacton IVR research.While, on the onehand,statisticaltheory affirmedthe existenceof redistributionfor real moleculesandthe conclusionsofexperimentswere in harmony with statistical assumptions,on the other hand paradigmatic,simplecoupledoscillator systemswere sometimesfound to relaxnonstatistically,andsometimesnot at all. Inthis contextit is worth noting that statisticaltheoriesof chemicalreactionshavealwaysallowedfor thepresenceof nonstatisticaleffects, by, for instance,excludingcertain“inactive” modesfrom considera-tion becausethey arepoorly coupledto the rest of the molecule [RobinsonandHolbrook 1972; Forst1973]. Notwithstandingthe many couplings and resonancesin even small molecules, the possibleexistenceof FPU-Iikebehaviorin molecularsystemshasbeenan intriguing andpersistentquestion,andone that will be addressedin this overview.

T. Uzer, Theoriesof intramolecularvibrational energytransfer 77

The planof this article is as follows. After generalremarksabout the natureof IVR, the quantumprocessis describedin sections2 and 3 by making relatively few assumptionsabout the detailedcoupling andenergytransferdynamics.This statistical-mechanicaldescriptionis basedon the theory ofradiationlesstransitions,where the central issue is to connectmolecularenergy-levelstructureto thedynamics of energyflow [Hunt et al. 1962; Robinsonand Frosch1962; Byrne et al. 1965; Robinson1967; Bixon and Jortner 1968, 1969a,b]. The “intermediate” version of this theory [Lahmani et al.1974;Freed1976;FreedandNitzan 1980]turns out to be applicableto commonexperimentalscenarios.We shall refer to this descriptionas the “phenomenological”theory, to indicateits frequentuse ofaveraged(over a large numberof states)propertieswhich can be determinedfrom experimentaldata(e.g., Matsumoto et al. [1983]). Section 2 refers specifically to IVR in a single Born—Oppenheimerelectronicstate,whereasin section3 the emphasisis on molecularfluorescenceas a well-establisheddiagnosticof IVR.

A contrastto this macroscopicapproachshall be demonstratedin section4 whenwe discussthe placeof the FPU paradoxandthe KAM theory in the developmentof the field. This sectionwill bring to thefore the central role nonlinearresonancesplay in intramolecularenergysharing. Subsequently,insection 5, we focus on energytransferthrough isolatedand interactingnonlinearresonances[OxtobyandRice 1976; Kay 1980; JafféandBrumer 1980;Sibertet al. 1982a,b;Sibert Ct al. 1984a,b],wheretheindividual couplingsare examinedrather than representativeaveragecouplings. In that section,wealso summarize the treatment of long-time correlations in intramolecular dynamics by means of“bottlenecks” in phasespace [Davis 1985]. In section 6, we review work that connectsexplicitlynonlinear oscillations to the experimentally accessible manifestationsof IVR [Kuzmin et al.1986a; Stuchebryukhov1986] using density matrix techniques [Faid and Fox 1987, 1988]. Theappendixexaminesthe connectionbetweenabsorption line widths, IVR and unimolecular decayrates.

Thereare manyfundamentalandexciting developmentsin theoreticalphysicsand chemistrywhichare intimately connectedwith the IVR problem. Restrictions of space preclude their thoroughdiscussionandthe explicit tracingof the connections.Amongtheseare:modespecificchemistry[Thieleet al. 1980a,b;Bloembergenand Zewail 1984; Miller 1987]; (for recentexperimentalexamples,seeButler et al. [1986a,1987]); quantumtransitionstatetheory[Tromp andMiller 1986], photodissociation[Simons1984;Brumer andShapiro1985]; electronicrelaxation[Boeglinet al. 1983;Amirav andJortner1987], chemiluminescentprocessesand ultrafast energy transfer amongelectronically excitedstates[Gelbart and Freed 1973; Bogdanov 1980; Gole 1985], and radiationlessprocessesin general [Freed1976; Ranfagniet al. 1984]; and the vast subjectof uñimolecularand intramoleculardynamics[Hase1976; Kay 1976, 1978; Brumer and Shapiro1980; Heller 1980; Hase1981; Marcus 1983; Heller 1983;Peres1984;StechelandHeller 1984;Marcus1988; HuberandHeller 1988; Huberet al. 1988;Ito 1988],andits relation to chaos[Kay 1983; StechelandHeller 1984; Ramachandranand Kay 1985; Farantosand Tennyson1987; Kay and Ramachandran1988; Shapiroet al. 1988], as well as localization [Kay1980; Heller 1987; O’Connorand Heller 1988; Dumont andPechukas1988; Parrisand Phillips 1988],and the spectral signature of chaos [Dai et al. 1985a,b;Hamilton et al. 1986; Pique et a!. 1988;Lombardi etal. 1988; Xie 1988]. For a “super-review” that includesmostintramolecularprocesses,seeJortnerandLevine [1981].I mustalsoleaveout energytransferon surfaces[Tully 1985; Micklavc 1987;Gadzuk 1987, 1988; Zangwill 1988] in spite of significant developmentsin the microscopicprobing ofvibrational energy transfer betweenabsorbatesand surfaces [Heidberget al. 1985; Jedrzejek1985;Heidberget al. 1987; Heilweil et al. 1988a,b;Harris and Levinos 1989].

78 T. Uzer, Theoriesof intramolecularvibrational energytransfer

2. Definition and nature of intramolecular relaxation

Collisional relaxationof polyatomicmoleculesin a thermal environmentis a well-defined process(among standardreviews are Quack and Troe [1977],Kneba and Wolfrum [1980]; for some newdevelopmentsemphasizingthe connectionto theintramolecularrelaxationprocess,seeRice andCerjan[1983],LawranceandKnight [1983],Villalonga and Micha [1983],NalewajskiandWyatt [1983,19841,Gilbert [19841,Lim andGilbert [1986],Orr andSmith [1987],HaubandOrr [1987],Koshi etal. [1987],Kable and Knight [1987],Bruehl and Schatz [1988],D.J. Muller et a!. [1988],Kable et a!. [1988],Rainbirdet al. [1988],Rock etal. [1988],Gordon[19881,Parson[1989,1990]; for competitionbetweeninter- andintramolecularenergytransfer,seeStraubandBerne[1986]).The samecannotbesaidaboutthe intramolecular vibrational relaxation of isolated molecules. A process can be consideredin-tramolecularif thereis negligible collisional andradiativeinteractionwith the environmentduring thetime scale of interest, which is less than a nanosecond.Vibrational relaxationcan be defined asirreversibledecay (on the time scale of interest) of a localizedvibrational excitation in a molecule[Quack 1983].

Dynamicalprocessesin isolatedmoleculesaregovernedby the time-dependentSchrödingerequation

H~1’=(H0+W)1P=ih8~PI8t. (2.1)

In this section,we will take Ii = 1. Neglectingradiative decay (or working in a basis that includesthestatesof the radiationfield) the solutionsof (2.1) areoscillatory. Assumingthat eq. (2.1) is rewrittenintermsof the amplitudesof the eigenstatesof H0 (the “unperturbed”,diagonalpart of H), andthat thecoupling W is off-diagonal in this basis,

ib’ = (H0 + W)b. (2.2)

For a time-independentH, the solution of (2.2) can be written as

b(t) = U(t, t0)b(t0) , (2.3)

U(t, to) = exp[—iH(t — t0)] . (2.4)

The time-evolutionmatrix solvesthe Liouville—von Neumannequationfor the time-dependentdensitymatrix r,

o’(t) = U(t, t0)ff(t0)U~(t,t0) , (2.5)

aswell as the Heisenbergequationof motion for the representationsof anyobservable,Q (in particularthe coordinatesand momentaof the atoms),

Q(t) = U~(t,t0)Q(t0)U(t, t0). (2.6)

WhenH is known, eqs. (2.3)—(2.6) constitutethe mostgeneralsolutionof awell-definedmathemati-cal formulationof thephysical problemof intramolecularmotion. While the solution is straightforwardin principle, both the size of the matrices and the possible intricacies of the couplings make the


emergenceof a relaxationfrom theseequationssomewhatsubtle.The very large size of thesecoupledequationshave naturally invited many ingenious approximationschemes(like those of Bixon andJortner[1968],Lahmani et al. [1974],Kay [1974,1976,1978], Kay et al. [1981],Milonni et al. [1983],andNadler andMarcus [1987]).By usingappropriatereducedHamiltonians,the numberof equationscrucial for the time developmentof nonstationarystatescan be decreased(seeMukamel [1981],andalsothe “adiabaticallyreducedequations”methoddue to Voth and Marcus[1986],Klippensteinet al.[1986],and Voth [1987]). The use of artificial intelligence methods for IVR problems has beenadvocatedby Ledermanet al. [1988],andLedermanand Marcus [19881.For other procedures,seeHeller [1981a,b],Scheckand Wyatt [1987]and Lopez et al. [1988].Gradet al. [19871haveusedasemiclassicalreductionprocedure.Classicaland semiclassicalaspectsof the intramolecular relaxationprocesshavebeenexaminedby Jaffé andBrumer[1984,1985], Parsonand Heller [1986a,blandParson[1988].

An interestingway of seeingthe emergenceof relaxationin a systemthat containsonly infinitelylong-lived eigenstatesis to consider“coarse-grained”,i.e., averagedor reducedquantities[Zwanzig1960; Mon 1965; Kay 1974, 1976, 1978; Nordholm and Zwanzig1975; Garcia-Cohnand del Rio 1977;Ramaswamyet al. 1978; Alhassid andLevine 1979; Kay et al. 1981; Quack 1978; 1981a,1983; Lupoand Quack 1987]. Theseare quantitiesthat do not contain full mathematicalinformation about thesystem;usually simpler mathematicalstructuresarisefrom coarse-grainedquantities.In what follows,we will usethe argumentandnotationof Quack[1981a,1983] who concentrateson coarse-grainedlevelpopulationsPK for approximatelyisoenergeticstatescharacterizedby the samequantumnumberK in aparticular local vibration (i.e., separablein H0) but different quantumnumbersfor otherdegreesoffreedom,

x+N*

PK = L Pk(K) = L1 bk(K)bk(K). (2.7)k kx+1

Here ~‘ indicates sumsover the statesin one level, capital lettersbeing reservedfor levels, and xdenotesstateswhich are not included in the summation.Using eqs. (2.3) and (2.4), oneobtains

PK(t) = ~‘ ~ ~‘ U~1~b1(0)j~+ ~ UkJUlbJ(0)b~(0). (2.8)k J j k !j�l

The secondsum vanishesif the termsin it haveuncorrelatedphases,leavingthe coarse-grainedlevelpopulationas

PK(t) = ~ ~‘ ~‘ U~1~p1(0). (2.9)

If only the coarse-grainedlevel populationsP~,= ~ P1 are measurable,the same “random phase”argumentas before allows

PK(t) = ~ PJ(0)YKJ(t), YKJ(t) = NJ1~ ~ IUI~JI2, (2.lOa,b)

whereN~is the numberof statesin one level. This matrix Y, underfairly generalconditions,can be

expressed[Quack 1979a] as an exponentialinvolving a time-independentmatrix K [Quack 1978],

Y(t) = exp[K(t — ta)], (2.11)

80 T. Uzer. Theoriesof intramolecular vibrational energy transfer

and p(t) = Y(t)p(0). This can be reduced[Quack 1978, 1981a] to the differential form of the Pauhimasterequation[Pauli 1928],

Pz~Kp. (2.12)

In contrastto the oscillatory solutionsof eqs. (2.1)—(2.4),theseequations,(2.11) and(2.12), havedecayingsolutions,denoting relaxation. For sufficiently short times the exponentialfunction can beexpandedandone obtains[Quack1978]

Y(t) = 1 + K(t — t01) + ... , KKJ = 2WKJ~

2//~K, (2.13a,b)

where WKJI2 standsfor the averagesquarecouplingelementbetweenthe statesin levels K andJ, and

~K is the averagefrequencyseparationof statesin level K (11~K needsto bereplacedby the densityPkin the caseof the continuum). The precedingdevelopmentmakesclear the factorsthat need to beconsideredcarefully when studyingintramolecularrelaxation: the definition of what is beingobserved(e.g., what is relaxing into what), the time scale over which the observationis taking place,whichHamiltonian and which basis set is being used for the description.Moreover, the precedingdevelop-ment is valuablein connectingGolden-Ruletypeof first-orderperturbationtheory prescriptions,whichare widely used in IVR, to a fundamentalview of the process[Voth 1987].

Of course,intramolecularrelaxation is a more generalphenomenonthan the somewhatrestrictedPauhimasterequationwould suggest,i.e., thereareselectionsof course-grainingwhichmakeoscillatorybehaviorpossible[Quack 1981a],which are, however,the exceptionratherthanthe rule. We will seeexamplesof suchexponentialbehavioraswell as coherencesin the following sections.ThePauhimasterequationis merely one (and the earliest) attempt to describerelaxationphenomenain a quantum-mechanicalsetting [Van Hove 1962]; it proceedsby reducing the full density-matrix equationto asystemof coupledequationsfor the diagonaldensity-matrixelementsalone. The derivation,aswell asthe set of companionequationsfor the off-diagonal matrix elements,the “coherences”,have beencritically studiedby Fox [1978,1989]. The effect of thesecoherencesprovesvaluable,e.g.,in the theoryof spectral lineshapes[Faid and Fox 1988].

2.1. Generalmolecularmodelfor intramolecularvibrational energy redistribution ([VR)

Figure 1 shows schematicallythe vibrational energy levels in a polyatomic molecule relevanttointramolecularenergytransfer, as describedlucidly by Freed[1981],whosenotation will be adopted.The entire level in this figure belongsto either an excited electronicstate (experimentsreviewedbySmalley [1982],Parmenter[1982,1983]) or the ground electronicstate(e.g., Kim et al. [1987],andovertone experimentsreviewedby Crim [1984,19871). Here, 4~representsa vibrational level thatcarriesdipole oscillator strengthto someexcitedzero-ordervibrationallevel 4~.Thesezero-orderlevelsmay be local modes,or normal modeschosenfor reasonsof convenience,or a combination thereof(e.g., Quack[1981b],Ledermanet al. [1983],Child and Halonen[1984],Quack[1985],Perschet al.[1988]).The photophysicalexperimentbegins with the moleculebeing in somethermally accessiblevibrational—rotationallevel.

Isoenergeticwith the low-lying levels of ~ is a densemanifold of vibrational levels which do notcarry oscillator strengthfrom the ground vibrational level 4~or any of the otherthermally accessiblevibrational levels: dipole-allowedtransitions between thermally accessibleground-statevibrational

T. Uzer, Theories of intramolecularvibrational energytransfer 81

SI ~ {~}

Fig. 1. Molecularenergylevel diagram usedto discussradiationlessprocessesin polyatomic molecules.~ is theground electronicState,and ~OA

denotesa thermallyaccessiblecomponentof this state.Transitions from ~OA to the state~, are allowed while thoseto { ~,}areforbidden, ~,

designatesa componentof 4,, and4i,, is a componentof 4i~.(Adaptedfrom Freed [1981]).

levels, ~, andthesehigh-lying states{cb1} arenot possibledue to eitherunfavorableFranck—Condonfactors or becauseof symmetry considerations.The zeroth-orderfunctions {~,~} are not exacteigenfunctionsof the molecularvibrationalHamiltonianbecauseof the presenceof anharmonicitiesandCoriohis couplingsin the normal-modedescription,or, if a local modedescriptionis used,becauseoflocal mode—localmodecouplingsandanharmonicities.

While the zero-ordereigenstatesbasedon the separabilityof vibrations leadsoneto expectasharpabsorptionspectrum peakedat the energiesof 4~,the presenceof {~}alters this situation. Thealterationsdependpartly on the densityof levelsin this manifold,p,. When the manifold {4~}is sparse(asmight beexpectedin smallmolecules),perturbationsin thespectramayarisedueto the presenceofthe manifold. This small-moleculelimit of low densitiesis an ideal scenariofor the assignmentsandanalysisof spectrallines.

When, on the other hand, the manifold {4~} is dense,it can behavelike a continuum on thetimescaleof the experiment,and produce irreversible relaxation from ç~to {~}.Betweentheseextremeslies an intermediatecasewhich displays some characteristicsof both the small and largemolecule limits becauseof either strong variations in the 4~—{4~}couplings (due to symmetryconsiderations,say) or when the densityp1 is too high to allow the resolution or the assignmentofindividual levelsbut still not high enoughto leadto irreversiblerelaxation.The intermediatecasehasbeen termed the “too-many-level small-moleculecase” [Freed 1981] and is widely encounteredincurrentexperiments,aswe will seein detail in the next section.New insight into the intermediatecaseof radiationlesstransitionshasbeen provided by the experimentsof Van der Meer et al. [1982b]on


pyrazine.This work has beenreviewedlucidly by Kommandeuret a!. [19871.Formorerecentwork, seeTomer et al. [19881,Konings et a!. [1988],Siebrandet al. [1989].

2.2. Criteria for IVR regimes

IVR phenomenaaregovernedby the detailsof energylevel density,Pt’ by the decayrates(I~)dueto infrared emissionof the zero-orderlevels { ~}, and the s—I couplingsbetweenmanifolds. Anotherimportantconsiderationis the nature of the initially preparedstate.For instance,moleculesare nowcommonlypreparedwith substantialamountsof energyin local modes[Swoffordet al. 1976; Bray andBerry 1979], for recentreviews,seeCrim [1984],ReislerandWittig [1986].Experimentsareunderwayto studythe subsequenttime evolution, which will, onehopes,leadto unambiguousconclusionsaboutthe natureandrate of intramolecularprocessesfrom direct time-resolvedmeasurements[Schereret a!.1986; Schererand Zewaih 19871,ratherthanon the basisof opticalhinewidths[Rizzoet al. 1984;Ticichet a!. 1986; Butler et al. 1986b; Luo et a!. 1988].

On the other hand, if it were technically possible to excite an eigenstateof the molecularHamiltonian,thereshouldbe very little (if any) nontrivial time evolution. To quantifythe criteria thatcharacterizethe rangeof possible behavior,Freed[1981]definesthe parameter

X1 = , (2.14)

whereh1~representsthe energywidth of a level with a decayrate of 1, ande1 is the averagespacing

betweenthe { ~ levels. The conditionfor the small-moleculelimit isX1<<1 , (2.15)

whereasfor the large moleculeit is

X1~1. (2.16)

In the small-moleculelimit, the situationcorrespondsto the spectroscopist’sdescriptionof perturba-tions in the spectraof small molecules.The molecularHamiltonian is representedin the basis set ofzero-ordervibrational eigenfunctions{ ~, /~}. Couplingamongonly a few levelsneedsto be consid-ered,namely, thosewhich are nearly resonantand which haveappreciablecoupling matrix elements.These arelevels for which

— E1~~ 1, (2.17)

whereE5 and E1 are the zero-ordervibrationalenergiesof 4, and ~, and v~1arethe coupling matrixelements

v~~=~~lHI~1). (2.18)

This is the action of the Fermi resonancefamiliar to spectroscopists[Fermi 1931]. The partial, “local”diagonahizationof the molecular Hamiltonian within the resonantlycoupled states leads to themoleculareigenstates{ t/i~}, which are linear superpositionsof the basis states,viz.


(2.19)

wherej countsthe çb~states(N in number) includedin the superposition.Assumingthat 4~and4~donot radiateto any set of commonlevels, it is possibleto evaluatethe radiative decay ratesof themoleculareigenstates{t/i~} as

= ja~j~~ a11~~2F

1. (2.20)

For manycasesof interestthe zeroth-orderradiativedecayrateof 4~is muchgreaterthan that of the

{4~}•So, under the condition that

(2.21)

the radiative decayratesof the moleculareigenstatesare equalto

1~—Ha5~~2I~. (2.22)

The conditionthat the original zerothorderstates,~, be distributedamongstthe moleculareigenstatesimplies the normalizationcondition

,~a5~V= 1. (2.23)

It follows from this that if more thanone of the a,~are nonzero,we must have

(2.24)

Equations(2.22) and (2.24) yield the result

[,<1 for I~F1, (2.25)

which impliesthat the radiative decayratesof the mixed, moleculareigenstates{ t/i~} are less thantheradiativedecayrate of the parentzeroth-orderstateç~carrying all of the original oscillator strength.

When the spacingbetweenmoleculareigenstatesis large comparedwith the uncertaintywidths oftheselevels, i.e.,

minfE~— ~ ~ ~h(f~+ 1,), (2.26)

monochromaticexcitationcan only leadto the preparationof individual moleculareigenstates.Pulsedexcitation, on the other hand, can leadto the coherentexcitationof a superpositionof a numberofnearlymoleculareigenstates.However, when the pulse durationr satisfiesin addition to (2.26) theinequality

minIE~— ~ >IiIr, (2.27)

84 T. Uzer. Theoriesof intramolecularvibrational energy transfer

this pulsedexcitationcannotexcitemorethanoneof the individual moleculareigenstates.If the pulseissufficiently short or the spacingbetweenmoleculareigenstatesis sufficiently small that the inequality(2.27) and/or(2.26) is violated,then a coherentsuperpositionof thesenearbymoleculareigenstatesispossible (for the subsequentdynamics,see,e.g., Taylor and Brumer [19831).

The observationof irreversibility for the large-moleculecase,eq. (2.16), has beenexplainedbyFreed[1981]from the radiationlesstransitionspoint of view. If such a moleculebeginsin state~ andcrossesover to the { 4~}manifold, thesefinal statesdecaywith ratesf beforethe moleculehasachanceto crossbackto the original state*1~.Thusit is the decayof the final manifold of levels { c/~} that drivesthe irreversible relaxationfrom 4~to {4,}. Thereare many situationsin which the inequality (2.21),generallytrue for condition (2.16), might seeminapplicableto such systems.However, in thesecasesone is primarily interestedin whetheror not the radiationlesstransition from ~ to {~~}appearsirreversibleon the timescaleof the experiment[Freed1976]. When the decaypropertiesof a systemona particulartime scale,r1, for an experimentareconsidered,the parameter(2.14) can be modified toread

X = h(I + r11)/Ej (2.28)

and thus characterizesthe decaycharacteristicsof a moleculeon the experimentaltimescale.When

X ~ 1 , (2.29)

small-moleculebehavioris manifeston the experimentaltimescale.Converselyif

X~1 (2.30)

is obeyed,the decaycharacteristicsof the moleculecorrespondto the statistical limit.

2.3. Dephasingversus relaxation in isolatedmolecules

It is alwayspossibleto view the time evolutionof nonstationarystatesof a moleculeas a dephasingprocessin the eigenstaterepresentation,sincethesestatesaremadeup of a superpositionof stationarystates,each of which evolves differently in time — any phaserelation establishedinitially tendsto bedestroyed.On the other hand, this descriptionof most processesas dephasingis neitherparticularlyusefulnor informativewhenwe concentrateon what happensto aparticularmodeor a groupof modes.In sucha case,the terms“pure dephasing”(alsocalled T

2 dephasingprocess)and “energyrelaxation”(T1 process)areusedto describedistinguishableappearancesof the generaldephasingprocess.In puredephasing,the energyof a mode(or a collectionof modes)doesnot change(e.g., Kay [1981],Stoneetal. [1981],Budimir andSkinner [1987]).This processis causedby energyexchangeamongothermodesin the molecule, which in turn causesthe effective frequencyof our chosenmodes to fluctuate[Mukamel 1978; Makarov and Tyakht 1982; and Stuchebrukhovet al. 1989]. This terminologyoriginatedin solid statephysics,whereone is frequentlyinterestedin phenomenain a subsystem;andmisleadinglyand inaccurately,often dephasingis used to meanpuredephasing.Onthe otherhand,inan energyrelaxationprocess,the energyof our subsystemdoeschange(see, e.g.,BloembergenandZewail [1984],Stuchebrukhovet a!. [19891).

For concreteness,let us returnto the “too-many-level small-molecule”case.As mentionedbefore,

T. Uzer, Theoriesof intramolecularvibrational energy transfer 85

broadbandexcitation can produce an initially preparednonstationarystateof the molecule whichcloselyapproximatesthe state‘/~at t = 0. The subsequenttime evolutionof the systemis governedbythe time evolution of the moleculareigenstates.The stateof the systemat time t in this limit can beshownto be (in the notation of Freed [1981])

~(t) = a~exp(—iE~tIh— ~f~t) ~. (2.31)

The probability of observingthe decaycharacteristicsof the state ~ at time t is proportional to theprobabilityof finding the moleculein the zeroth-orderstateç5~at t, which is given by (usingthe notationof Freedand Nitzan [1980])

P~(t) ( ~ ~(t)) 2 ~ a~j2exp(-iE~tIh- 1I~t)~. (2.32)

Note that the probability is a complicatedsum over moleculareigenstateswith both oscillatory anddampedtime dependences.The general behavior of (2.32) could, in general, be very involved.Howeverthe largenumberof contributingtermsprovidesa simplification. Again assumethevalidity of(2.21) and assumethat 1 is very small. However, on a short enoughtimescahe,the modified condition(2.30) may be satisfiedfor time scaleson the orderof or lessthan the radiativedecayrates,I~,of theindividual moleculareigenstates.Then, on the experimentaltimescaler~the moleculeappearsas if itconformsto the large-moleculestatisticallimit (2.16). Thus,as shownby Lahmanietal. [1974],P~(t)of(2.32) displaysan exponentialdecaywith a decayrate given by

= 1 + 4~, (2.33)

where

= ~ v~j~p11 (2.34)

is the rate at which the moleculeappearsto undergo decayfrom t~to {~}on the time scaler1.This short-time apparentdecay is an intramolecular“dephasing” process,which is causedby the

individual moleculareigenstatesin (2.32) all having slightly differentenergies{E~}(recently,Makarovand Tyakht [1987]have studied purely phaserelaxationand its effect on vibrational spectra.For aclassical-mechanical“dephasing” study, see Farantosand Flytzanis [1987]). When the molecule isinitially preparedin the nonstationarystate4~a coherentsuperpositionof all moleculareigenstateswith fixed relative phasesis constructed.However, becauseof the energydifferences,thesephasesevolveat different time rates.

The intramoleculardephasingof the different moleculareigenstatesis what leadsto the apparentexponentialdecayfor short timesgovernedby (2.30),makingit look as if the initial state/~is decayinginto the {çb1}. To put it morepicturesquely,undertheseconditions,this intermediate-casemoleculehasnot had enough time to realize that thereare only a finite numberof levels { ~}; it cannotresolveenergydifferencessmaller than lITe. Therefore,becauseof time—energyuncertainty,it appearsthat{ 4,} is a continuum,andexponentialdecayensues.This is an intramoleculardephasingprocesswhich,on this short timescale,is equivalentto IVR observedin the statistical limit (Carmeli et al. [1980],Gelbartet a!. [1975]).At longer times, when (2.29) holds, the moleculehashad time to find out that

86 T. Uzer. Theories of intramolecularvibrational energy transfer

the manifold {4,} is discrete, and it reverts to small-moleculebehavior (Heller [1981a,bJhasreformulatedthe time-domainbasis of spectroscopyin this fashion).Therefore,the initial exponentialdecay(known as “intramoleculardephasing”)is not atruly irreversiblerelaxationbecausein principle,a multiple pulsesequencecould beapplied to the systemto reconstitutethe state4~.The presenceof afinite numberof levels makes the dephasingreversible. In the statisticallimit, the condition (2.16)implies the participationof an effectively infinite numberof levels leading to irreversible relaxation.

On longer timescaleswhere(2.30) is violated, thesystemreturnsto thesmall-moleculelimit andthecomplicatedexpression (2.32) (for an experimentalstudy of dephasing in an intermediate-casemolecule,seeSmith etal. [1983]).Ordinarily, the long-timebehaviorof suchmoleculesis found to beadequatelyrepresentedby a simple exponentialdecaywith someaveragemoleculareigenstatedecayrate, (E,~).Betweenthe short-time exponentialdecayof (2.33) and the long-time averagemoleculareigenstateexponentialdecayof KI~),in principle,the decaygivenby (2.32)could bevery complicated,including, e.g., quantum interference effects (see next section). The decay propertiesof theseintermediate-casemoleculesin the too-many-level,small-moleculelimit, are well representedby abiexponentialdecayinvolving the short-timeapparentradiationlesstransitionandthe long-time decayof the individual moleculareigenstates.Muhlbach and Huber [1986]havereportedan experimentalobservationof such biexponentialdecay recently. For an experimentaldiscussionof biexponentialdecay,see Kommandeuret a!. [1987].

3. “Phenomenological” theories of energysharing; approach of radiationless transitions

Therealwayshasbeena needfor direct, dynamicalIVR experiments[Mukameland Smalley 1980;Moore et a!. 1983; Imre et al. 1984; SundbergandHeller 1984; Felker andZewail 1984, 1985a—d;Daiet al. 1985a,b;Tannoret al. 1985; Heppeneret al. 1985; Zewail 1985; Hamilton et al. 1986; Makarov1987; Graner1988] so that the sometimespuzzlingoutcomesof indirect experimentscould be avoided.Oneclass of experimentalstudiesaims at detectingthe time evolution of molecularfluorescenceas adirect observationof IVR [Parmenter1983; Coveleskieet al. 1985a,b;Dolson et al. 1985; Holtzclawand Parmenter1986; Knight 1988a,b].Theseexperimentsobservethe gradualfilling-in of an initialsparsefluorescencespectrumbecauseinitially unexcitedstatesarebeing populatedby IVR. This timeevolutionhasbeenobservedin the “chemicaltiming” experimentsof the Parmentergroup, andprovidean approximatepicosecondtimescalefor IVR.

Briefly, a mode of a large moleculeis excitedand the fluorescenceis detected.The experimentalapparatuscontainssomequenchergas, which removesthe excitationenergyby meansof collisionalenergytransfer.For very largepressuresof quenchergas, the fluorescencespectrumis sparsebecausethe excitationenergyis removedbeforeothermodescan be populatedby intramolecularenergyflow.On the other hand, for very low gas pressuresenergyflow can proceedfor a longer time, and thefluorescencespectrumlooks much richer. Without quenchinggas, a smooth spectrumis obtained,indicating that all coupledstatesare populated.Oneof the drawbacksof theseproceduresis that therangeof observationtimes is limited to the fluorescencetimescale.

The following theoreticaldescriptionwill specifically refer to relaxationin an electronicallyexcitedstate,and how the progressof IVR affects the fluorescencespectrumWe choseto restatethe theory,which was explainedby Freedand Nitzan [1980,1983] for fluorescence,for two reasons:firstly, it isapplicableto mostclassicmanifestationsof IVR; andsecondly,while thistheory wasformulated,as hasbeentraditional and useful, for fluorescencedevelopingfrom IVR, it has some relevancefor novel

T. Uzer, Theoriesof intramolecular vibrational energytransfer 87

processeslike vibrationally mediatedphotodissociation[Ticich et a!. 1987; Sinha et al. 1987] whichprobe the interplay of IVR with photodissociation[Simons1984].

The state densities in these moleculesput them into an area where a theoretical frameworkanalogousto the intermediatecasein the theory of electronicrelaxationprocessesshould beapplicable :[Freed1976,Tric 1976]. As statedbefore,in this intermediatecasethe densityof final vibrational levelsis insufficient to drive purely irreversiblebehavioron the timescaleof the experiment(which is of theorder of the fluorescencelifetime). In the literatureof electronicrelaxation,the intermediatecaseisclearlydistinguishedfrom the statisticalcase[Nitzanet al. 1972; Fradet a!. 1974;Freed1976; MukamelandJortner1977]. But in the caseof IVR thereis a qualitativedifferencebetweenthesetwo scenarios:for instance,using the intermediatetheory, it is possibleto determinethe averagenumberof stronglycoupledlevels and the averagecoupling betweenthe zeroth-orderlevel and thesestrongly coupledlevels. The differencebetweenthe intermediatetheory,as formulatedfor electronicrelaxation,andtheversion of the theoryneededfor IVR, is that in the electronicallyexcitedcase,the densemanifold oflevels which is strongly coupledto the initially excitedone, is eithernonemittingor is emitting on atimescahemuchlonger than the experimentalone.The currenttheoryis characterizedby a manifold of“final” levelswhose emissionis monitoredby the experiment.

Thesegeneralizationshavebeenclearly delineatedin the work of FreedandNitzan [1980],and canbe used to distinguish between the intermediateand statistical behavior, and thus determinethethresholdfor irreversiblevibrationalrelaxation.While mostapplicationsof this theoryhavebeenmadeto IVR taking placein excitedelectronicstates,the experimentson relaxationin the groundelectronicstate(e.g., Minton and McDonald [1988])indicatethat thereis very little, if any, differencebetweenthe scenarios(Parmenter[1983]),when thereare no electroniccurve-crossings.

The reverseprocedureof deconvolutingspectrahas recently been addressedby Lawrance andKnight [1988]using a Greenfunction approach.This and similar work hasbeenreviewedby Knight[1988a,b].For a model Hamiltonian treatmentin the spirit of this section,see Kommandeuret al.[1988].

3.1. Molecularmodelfor fluorescence

The adiabaticBorn—Oppenheimerapproximationassumesthat the wavefunctionsof the systemcanbe written in termsof a product of electronic,vibrational, rotational,and spin wavefunctions.If thisapproximationwere exact,excitation and relaxationprocesseswould be straightforwardto describe.However,such a wavefunction is not an eigenfunctionof thefull molecularHamiltonian,andnumerousinteractions,which are ignored by the Born—Oppenheimerapproximation,can couple thesestates,leadingto the removalof degeneraciesandthe occurrenceof radiationless,e.g.,transitionprocesses.Inwhat follows, we haveadoptedthe reasoningand notationof Freedand Nitzan [1980].For a recentexposition,with detailedreferencesto current experiments,see the reviews by Knight [1988a,b].

The ideal experimentalscenariofinds the moleculeinitially in the groundvibrational—rotationallevelof the groundelectronicstate.Such ideal conditionscan be approximatedin supersonicjet experiments.For definiteness,supposethat the excitationis taking placeinto the molecularvibronic levelsof the firstexcited molecularsinglet, and ignore electronicradiationlessrelaxation.Conventionalspectroscopicalpracticeassignsthe spectrumin termsof a basis set of zero-orderlevelsbasedon the populationsof asetof zero-ordermodes(which maybenormal modesor symmetrizedlocal modes,to citetwo commonexamples).

The zero-orderstatesmaybe divided into two groups,which aredenotedby a for active, and by bfor bath. The a modesare those whosepotential energysurfacesin the ground (g) andelectronic


excited(e) statesmay differ appreciablyfrom each other. The b modesdo not changeduring thiselectronic transition. The a modes are optically active, “bright” statesand thus correspondto thedifferent progressionsin the molecularabsorptionspectrum.The zero-ordermolecularvibronic levelsare denotedg, ~ga’ ~gb)’ and e, pea’ nCh~l,where~ia.th’ i = g, e denotethe populationsof the a andbmodesin the g and e electronicstates.It should be noted that thesen standfor the collectionof allmodesa andb in this system. In this notation,the initial statecan be abbreviatedas g, 0, 0~.

In monitoring molecularfluorescence,one in fact monitors the populationsof the a modes.Thepropermolecularmodel (fig. 2) is composedof manifolds of b statesseatedon the different statesofthe modes.All thesestatesare the eigenstatesof the zero-ordermolecularHamiltonian H~.The totalHamiltoniangoverningthe time evolution is

H—H0+~u.+W, H—H~+H~, (3.la,b)

whereH~is the Hamiltonian of the free radiationfield, ~ is the molecule—radiationfield interactionwhich couplesthe e andg statesand W is the intramolecularvibrationalinteractionwhich couplesthedifferent a andb states. IVR then is viewed as transitionsbetweenthe manifolds shown in fig. 2,inducedby the off-diagonalcouplingW. Sometimesit may be necessaryto havetwo varietiesof W, onethat couplesa andb modes,and the otherthe b modesamongthemselves.Obviously, the dynamicsisespeciallysensitiveto the first variety of coupling.

3.2. “Exact” molecular eigenstates

It is convenient(thoughnot widely customary)to analyzethe basicsof the IVR processnot in termsof “zero-order” states, i.e., the eigenstatesof H00, but in terms of the exact eigenstates,i.e., the

W2 __ __ __

e 2eaneb>

e ~eaneb>

e 050 fleb>

Fig. 2. The molecularmodelusedin thediscussionof intramolecularvibrationalrelaxation(IVR) in anexcitedelectronicstate(~e)).Eachmanifoldcorrespondsto a particular stateof the optically active (a) modesand is composedof levels associatedwith different statesof all other(bath. h)modes.W1 denotesan intramolecularcouplingbetweenthea modesandtheb modes(which leadsto processeswhich changethepopulationsof thea modes). W, denotesthecoupling betweenthe b modes.(From Freedand Nitzan [1980]).


eigenstatesof H~+ W + H~.These statescan be written as combinationsof the zero-orderstatesas

~J) = ~ Cna,nbie,na, nb) e, 1) (3.2)

(here, we have suppressedthe statesof the radiation field H~).A narrow pulse which induces atransitionfrom the ground state g, 0,0) to anotherparticularzero-orderlevel e, na,0) encompasses,in the exact moleculareigenstatespicture, thosemoleculareigenstatese, j) which contain contribu-tions from e, ~a’ 0),

Je, j) = Ai;n ~ ~a’ 0) + ~ ~ n~,n~). (3.3)

Assuming that the intramolecularcoupling W is much smaller than the energyspacingbetweentheprimaryzero-orderstatese, na,0) leadsto the following picture. Eachexact e, j) statecorrespondstoa single primary e, na, 0) state and a group of quasidegeneratee, n~,n~)stateswith n~< na. Thenumberof zero-orderstatescontributingto the right-handside of eq. (3.3) is of the orderof (IW~p)2,wherep is the densityof levelsin the {~e,n~,n~,)}manifold.

In the Condon approximation, the total radiative lifetimes of the different rovibronic statese, ~a’ nb) are the same for all levels. Therefore, the sameis also true for the levels e, j) which

diagonalizethe molecularHamiltonian.The total width of a zero-ordermolecularlevel e,n), wheren = (na,nb), is

R NRyfl—y +y, , (3.4)

where R is the radiative width and y~ is the nonradiativewidth associatedwith radiationlesselectronictransitions.

Let us begin by discussingthe IVR problemusing a much simplified scenario.If the moleculeisinitially cold, i.e., in the state g,0,0), an incident pulse which is extremelynarrow in energymayinduce transitionsto a single e, na, 0) level. This assumptionis valid if the spacingbetweene, na,0)levels is larger than the bandwidth of the radiation. Under theseconditions, the excitedzero-orderstateswhich participatein the absorption—emissionprocessarethe state e, na,0) anda groupof statese, n~,n~)which aredegeneratewith it to within anenergyrangeof orderW. The latter are radiativelycoupled to the ground levels (other than g, n~,0), which is coupled to e, na,0)), and relaxedfluorescencearisesfrom this coupling. Denote

je, na, 0) s), {~e,n~,n~)} {~e)}. (3.5)

Ground-statelevelsto which s) and 1) arecoupledradiativelyaredenotedby {~m)}.We sometimesuse m~)to denotea level (like g, 0,0)) which is coupledto s) but not to the levels {~l)}.Themodelis shownin fig. 3a. In the correspondingexactmolecularstatespicture, fig. 3b, the groupof levels Is),

{ I I)) is replacedby the group { j) } which areobtainedas a linear combinationof Is) and { 1)) stateswhich diagonalizethe hamiltonianH~+ W.

The mostsignificant differencebetweenthe modeldiscussedhereandsimilar modelsusedbefore forthe theory of electronicradiationlesstransitionsis that the underlyingcontinua(fig. 3) which providethe “bath” for the decayof the initially preparedlevel, areoptically active. The emissionfrom these

90 T. Uzer, Theories of intramolecularvibrational energytransfer

(a)

{is>} ____ {ID~ ~j>} ____

\ ~~~Im> } { rn> _____

rn5> rn5>

Fig. 3. A simplified molecularmodel, a. s) are zero-orderlevels belonging to theexcited electronicstatewhich are directly accessiblefrom theinitial groundstatelevel m,). The intramolecularzero-orderbathlevelsare { I)). b. The correspondingexactmolecularstatepicture. (FromFreedand Nitzan [1980]).

states constitutes the relaxed part of the fluorescenceas discussedabove. Now we explore theimplications of this emission processon the line shapeand time evolution of molecularfluorescencespectra.

3.3. Long-timeexperiments

In the intermediatelevel structure case, the decay widths y~(eq. 3.4) of the individual exactmolecularlevels are smaller than their averagedspacing

(3.6)

with p being the densityof states.This implies that in a continuousexcitationexperimentor using apulse long enough to resolve the individual levelsthe resulting fluorescencearisesfrom an individuallevel le, j) (eq. 3.3). Therefore:

(1) In a long-time pulse experimentthe fluorescencedecaysexponentiallywith lifetime = h/y1.(2) The relative intensitiesof differentabsorptionor fluorescenceexcitationlines (correspondingto

different Ie, j) levels) are given by [cf. eq. (3.4)] ~4j;na,o12

(3) The fluorescencespectrumdoesnot evolve in time.In practical experimentalscenarios,theseconclusions(which apply to an ideal case)apply with

modifications.For one thing, evenin the coldestsupersonicbeamexperiments,the initial stateis not asingle molecular eigenstate,but includes at least a few rotational states. Hence, the fluorescencespectrumcontainscontributionsfrom a numberof initial rotationallevels,and this superpositiontendsto washout someof thedescribedstructure.Note that IVR doesnot lead to broadeningof individually


resolvedmoleculartransitionsin the intermediatecase,whereasthereis broadeningin the extremestatisticallimit. As the densityof statesincreases,the energyspacingbecomessmallerthanthe widths11y

1 and so it becomesimpossible to excite individual moleculareigenstateseven with an ideallymonochromaticradiation.

In the statistical limit (see below), the manifolds le, n~,n~)become dissipative continua forintramoleculardecayof the initially excitedlevel le, na,0). The decayrateis then givenby the GoldenRule formula

= 2ir ~ ~e,nI WIe,n~)I2~ wean;’ - ~ ~e~’)], (3.7)

where n and n’ are short-handfor (na,0) and (ni, nt). The absorptionlineshapeto the resonancecentredaroundthe zero-orderstate e, n, 0) becomes,in the extremestatisticallimit, approximatelyaLorentzianwith a width f~+ ‘y~consistingof contributionscorrespondingto vibrational, radiationlesselectronic,and radiativerelaxation.The relativeyields of the direct versusrelaxedfluorescencecan becalculatedby using a “kinetic” approach[Nitzan et al. 1971; Mukamel and Nitzan 1977], which isjustified in this casebecausethe randomnatureof the couplingmatrix elementsensuresthe absenceofinterferenceeffects [Freedand Nitzan 1980].

3.4. Short-timeexperiments

Now we considerthe casewherethe excitationpulse is short enoughin time, and broadenoughinenergyso that a zero-ordernonstationarystate le, ~

2a’0) is prepared.In terms of exact moleculareigenstatesle, j) the initial state is [cf. eq. (3.3)]

= le, na, 0) = ~ A~nole, j). (3.8)

To achievethis initial excitation, the pulsetime hasto be muchshorterthanthe inverseenergyspanofthe statesfe, j) with Aj;n

0 different from zero (fig. 4). The pulsebandwidthhasto besmallerthan the

energyspacingbetween e, n5,0) levels. Assumingthat theseconditionshold, we can analyzethemolecularfluorescencein termsof the simplified model definedin section3.2 anddepictedin fig. 3. Interms of this model, the zero-orderlevel structure, the correspondingexact-statesmanifold and acharacteristicabsorptionspectrumcorrespondingto the sparseintermediate(of level spacingslargerthanlevel widths) and the denseintermediate(level spacingsmuch smallerthanlevel widths) casesareshown in fig. 5. Note that by level widths we meanthe sum of radiative and electronicnonradiativewidths ratherthanwidths associatedwith IVR.

To appreciatethe significance of the emission spectradisplayed in fig. 5, consider first the twoemissionlines which correspondto two different final (groundstate)levels Im~)and I mi). Supposethatin the zero-orderrepresentation,rn~)is exclusively coupled (via ~,the molecule—radiationfieldinteraction)to Is), while I mr) is coupled to anotherzero-orderlevel I r) which belongsto the { II) }manifold, i.e., if the moleculeis initially in Im~),a broad-bandpulsewill prepareit in stateIs). If theinitial stateis mv), similar excitationwill yield the state Ir). Figure 6 showsthis situationin the denseintermediatecasetogetherwith the schematicdescriptionof the emissionlineshapefor transitionsthatend in the rn~)and Imr) levels. Note that by definition, the Is)—* m~)transitionconstitutesa line inthe direct (unrelaxed)fluorescencespectrum,whereasthe Ir) —* Im5) emissionconstitutesone(of many


v~ ~ =w _____

~A 2 ________

=w1 ___ liii ______

Is> {IA>} {Ijz.}

Fig. 4. Schematicdiagram of the mixing of zero-ordervibrational levels (or rovibrational levels) within a single electronic state.The averagecoupling matrix elementsV~and V~are expectedto be equal. within the S1 state,the state s) has a large Franck—Condonfactor so that it isoptically accessiblefrom theS~zero-pointlevel. Thebathlevels { I I) } are“dark” on accountof small Franck—Condonfactors,The If) are theexactmoleculareigenstatesIc, j); the figure on the right shows the initial excitedstatein termsof a superpositionof moleculareigenstates.This is apictorial representationof thestate that evolves accordingto (2.32) or (3.13). (Adaptedfrom Parmenter[1983]).

overlapping) contributions to the relaxed emission (see also Mukamel [1985]). Is) and Ir) are,however, linear combinationsof the exact energy eigenstatesIi). Therefore, in the intermediateregime,eachof theselines could, in principle,be resolvedto the different jj) levels,as shownin fig. 6.

At any time following the initial preparationof the excitedmolecule,the integratedintensityof theemissionassociatedwith the transitioninto the ground level Imp) is proportionalto the population inlevel p), i.e. (pI~i(t))j

2, where ItIi(t)) is the molecular state at time t, and I~)is the excitedzero-orderstatewhich carriesall the oscillatorstrengthfor transitionsfrom lm~)in the spectralregionof interest[MukamelandJortner19771.Note that I p) canbe the initially excitedlevel Is) or any other

___ ___ ~

w ____________ w ____________ w ____________

H>} {~i>} i>}

SPARSE DENSE STa,TISTI~~LNTERMEDIATE NTERMEDI~E LI ~.1IT

Fig. 5. Zero-orderlevelstructure(upper), exactmolecularlevel structure(middle), andthecorrespondingabsorptionspectrumassociatedwith thesparseintermediatecase(left), thedenseintermediatecase(middle), and the statistical limit (right). (From Freed and Nitzan [1980]).


{Ii>}

_js>_.w, _____

ims> — —

mr>

EMISSIONINTENSITY

AIs>—..Irns> Ir>—.-Imr> w

Fig. 6. Direct (unrelaxed, Is)-Hm,)) andindirect (relaxed, r)—* rn,)) emissionsin thesimplified molecularmodel. Note that the Ir)—* Im,)emissionis only one componentcontributingto the total relaxedfluorescence.(From Freed and Nitzan [1980]).

zero-orderlevel belongingto the manifold {Il)}. In the restof this sectionwe assumethat the conditionunderwhich the populationsof the zero-orderlevelsdeterminethe instantaneousemissionspectraissatisfied.The intensity of the direct (unrelaxed)emission at time t is proportional to the population

P5(t) = I(slexp(—iHt)Is)I2 (3.9)

(henceforthwe seth = 1), and the spectrallyintegratedintensityof the relaxedemissionat a time t isproportionalto

PL(t) ~ P,(t) = ~ Ii(llexp(—iHt)Is)I2 . (3.10)

The evolution within the Is)—{Il)} manifold in the model describedin fig. 6 hasbeen intensivelystudiedin connectionwith the theoryof intramolecularelectronicrelaxation.In the intermediatecase

94 T. Uzer, Theoriesof intramolecular vibrationalenergy transfer

the populationP5(t) evolves differently at short and long times. Immediatelyfollowing the excitation,i.e., after Itfr(t = 0)) = Is) and during a time short comparedto the inverselevel spacingin the {Il)}manifold, the {Il)} manifold behavesas a continuumand the evolution of P5(t) proceedsas in thestatisticallimit (seebelow). The direct emissiondecayswith the total ratey1 + F, wherey~is thesum ofthe radiative and electronicradiationlesswidths of the levels Is) while F is given by

2 1 ~(y1+e0) 2F 2ir IW~,I — 2 t 2 ~21T(IW51IPt)’ (3.11)ir (E1 — E1) + ~(y1+

whereYt is the sumof the radiative andelectronicnonradiativewidths of the level Il), Pt 1S the densityof levelsin the {Il)} manifold and r~is the uncertaintywidth (of the order 70t) associatedwith theshort observationtime r~[Freed1970].

At long times (t greaterthan hp0), dephasingbetweenthe different levels due to their slightlydifferent energieshas takenplace (however, in principle, this time may also be long enough forrecurrencesto takeplace). As in the previoussection,the situationis bestdescribedin terms of theexactmoleculareigenstatesIi). The initial stateis

= Is) ~a51Ij), (3.12)

and at a time t it evolvesto

I qi(t)) = a~exp(-iE1t- y1t)I j). (3.13)

Equation(3.13) is basedon the assumptionthat the dampingmatrix associatedwith the exactmoleculareigenstatesIi) is diagonal. This may be justified for the radiationlesspart of ‘y1 invoking therandom-like nature of intramolecular coupling between vibrational levels. It is also true for theradiativepart of becausethe total radiativelifetime is the samefor all rovibronic levelscorrespondingto a given electronicstate.Equation(3.13) leadsto

P~(t)= I(sI~j(t))I2= ~ Ia~

1I~exp(—y1t) + 2 ~ Ia~1I2Ia

51.I2 exp[— ~(y

1+ y1)t] cos(w11.t), (3.14)

wherehw11. = E. — E1.. The secondterm in the right-handside of eq. (3.14)describesquantumbeatsinthe fluorescence.However, if more than just a few levels contribute, the sum of oscillating termsaveragesfor t> lip1 to a vanishingly small contribution. This subject is further discussedin the nextsection.When theseoscillationsmaybe disregarded,we obtain (for long enoughtime)

P1(t) = ~ Ia.,1l4 exp(—’y

1t) . (3.15)

If thereare approximatelyN> 2 strongly coupleds and 1 levelsparticipatingin this process,we mayinvoke the simpleststatisticalmodel of egalitarianmixing so that

Ia5112=N’. (3.16)

T Uzer Theories of intramolecularvibrational energytransfer 95

We then have

exp[—(y~+ F)t] (short time) , (3.17)

(1/N) exp(—y1t) (long time). (3.18)

The numberN of effectively coupledlevels maybe estimatedfrom

N~Fp=2irIWJ2p2. (3.19)

The only differencebetweenthe presentsituationand that encounteredin the theory of electronicrelaxationlies in the fact that in most electronicrelaxationproblemsthe real-life counterpartof themanifold {Il)} does not carry significant oscillator strength for radiative emission.The observedradiativelifetime (that is, the radiativepart in y

1) is then of the orderyr/N. In the current treatment,

yr, y~and are all equal.

The level widths

(3.20)

and

(3.21)

haveapproximatelythe sameorderof magnitude.If we makethe simplifying assumptionthatthey arethe same(which is rigorously true if variationsin the nonradiativewidth areneglected),wecan obtainalso the time evolution of the integratedrelaxed fluorescence.The individual P1(t) are fluctuationswithin the egalitarianmodel, havingzero average.Their sum, however,is readily evaluatedby usingthe sum rule

P~(t)+ ~ P,(t) = exp(—yt), (3.22)

wherey = y~= y1, to obtain

~ P,(t) = exp(—yt)[1 — exp(—Ft)] (short time);

~ P~(t)= NN 1 exp(—yt) (long time). (3.23)

Equations (3.17), (3.18) and(3.23) leadto the following importantconclusion.Whenquantumbeatsinthe fluorescenceareabsent,the time evolutionof the molecularfluorescencespectrum(i.e., transferofintensity from direct to relaxedemission bands) takes place only during the initial short dephasingperiod. This may also be concludedusing the exact molecularstatesrepresentation.Following theinitial dephasing,eachlevel If) evolvesin time essentiallyindependentlyof otherlevelsandcontributesto the emissionspectrumdirectand relaxedcomponentsas discussedin section3.3. Thesecomponentsdecaywith lifetime y,,. andanyevolutionin the structureof the emissionspectrumin the post-dephasingperiod is due to the accidentaldifferencesbetweendifferent lifetimes y

t of the different j levels.

96 T. Uzer, Theoriesof intramolecular vibrational energy transfer

3.4.1. The statistical limitAs seenfrom eqs. (3.17) and (3.18), as the number N of effectively coupledlevels within the

Is)—{Il)} manifold becomesvery large, the long-time componentin the evolution of P~(t)becomesnegligibly small, P~(t)decaysexponentially practically to zero with the rate y~+ F. In most casesofelectronicradiationlessrelaxation,fluorescenceis observedto decayon this timescale.In the presentcaseeq. (3.23) indicatesthat the total populationof the {Il)} states,which determinethe intensityoftherelaxedfluorescence,decayon thelong-timescalewith a rate y = KY) following an initial rapidrise.The situationis very similar to that discussedby Nitzan et al. [1972]in connectionwith consecutiveelectronicrelaxation.It is importantto notethat as long asquantumbeatsarenot observedandaslongasno attemptis madeto resolvethe intermediatelevel structurein theemission,the statisticallimit andthe intermediatecasediffer only by the magnitudeof the number N which entersinto eqs. (3.17),(3.18) and (3.23).

An interestingquestionthat hasnot been addressedbefore concernsthe populationof differentlevels 1) in the {Il)} manifold during thevibrationalrelaxation.Consideragainthe modelshownin fig.6, and focus attention on the statistical limit (or the short-time evolution in intermediatecases).Usually, the time evolutionwithin the Is) — { Il)) manifold is determinedby disregardingthe anharmoniccoupling W betweenthe Il) levels,denotedby W2 in fig. 2. The usualargumentgiven is that as long aswe areinterestedonly in P5(t) we can performa partial diagonalizationof the Hamiltonianmatrix andobtain a set of {Il)} stateswhich are not coupledto eachother. The resultsfor the populationsP~(t)and P1(t) for this caseare well-known [Goldbergerand Watson 19691,

P1(t) = exp[—(y + F)tJ, (3.24a)

P1(t) = IV~ ~t) [1+ exp(-Ft)-2 exp(-~Ft)cos(E0~t)I, (3.24b)

E15=E1—E1, F=2ir(W~0p1).

In the presentcasewe are interestednot only in P~(t),but also in P1(t). The sum ~ P1(t) whichdeterminesthe integratedrelaxedfluorescenceis given by eq. (3.23). Individual populationsP1(t) maybe usefulfor sortingout differentcontributionsto the relaxedspectrum.For this purposewe would liketo keep the original characterand symmetry of the Ii) states,and in particular the selectionrulesassociatedwith their symmetry. We thereforeneed a solution to the problem where the couplingbetweenthe 1) statesis not disregarded.

Such a solutioncan beobtainedin the statisticallimit within the framework of the randomcouplingmodel [Kay 1974; Heller and Rice 1974; Gelbart et al. 1975; Druger 1977a,b; Carmeli and Nitzan1980a,b;Carmeliet al. 1980]. In this model,the couplingmatrix elementsbetweenany two statesin the

Is) — { I I) } manifolds are regardedas randomfunctionsof the state index. This picture is justified formatrix elementsinvolving excitedvibrationalwavefunctionsbecauseof the highly oscillatory natureofthesefunctions.We are interestedin P~(t)and Pr(t), given thatP~(t= 0) = 1, wherethe levels Is) andI r) areparticularmembersof adenseset of coupledstates.Thelevel Is) is uniqueonly becauseit is theonly one accessiblefrom the ground state and may thereforebe preparedinitially. The resultsare[Freedand Nitzan 19801

P1(t) = exp[—(y + F)t], (3.25)


Pr(t) = 2yt) {1 — exp(—Ft)cos(E~5t)— (FlErs) exp(—Ft)sin(Erst)}. (3.26)

As expected,the time evolutionof the initially populatedstate is the samein this modelas for thesimplified model which leads to eq. (3.24). The time dependenceof Pr(t) (for r different from s) isdifferent, but the qualitativebehavioris similar. The most significantdifferencebetweeneqs. (3.24b)and (3.26) lies in the fact that the latter predictsthatthe populationspreadover zero-orderlevelsin a(zero-order)energyrangeis twice as largeas thatobtainedin eq. (3.24b).This occursbecausein themodel which leadsto (3.26),eachlevel hasan anharmonicwidth F, while in the modelwithout couplingbetweenthe {Il)} levelsonly the initially excited level is assignedsuch a width.

3.5. Quantumbeats

In bulk room temperatureexperimentsinvolving relatively large numbersof molecules,quantumbeatsin the fluorescencearenot observed.Theinitial thermaldistributionof rotationalstatesis carriedover into the excitedstateand any oscillations in the fluorescenceoriginating in theseincoherentlyexcitedrotationallevelsareaveragedout. With low beamtemperaturesthe rotationalstructurecan beeliminatedand beatsareobserved(e.g.,Laubereauet al. [1976],Chaikenetal. [1979],Van der Meeretal. [1982a],Felkeret al. [1982],Okajimaet a!. [1982],Felkerand Zewail [1984,1985a], Roskeret al.[1986],Ha et al. [1986]).

The observabilityof quantumbeatsin molecularfluorescencedependson anotherfactor relatedtothe theoremmentionedin section 3.4. To understandthe implications of this theorem,considerthethree-levelmodel in fig. 7. Theground stateIg) is radiativelycoupledto zero-orderlevel Is) but not toIr), Is) and Ir) are coupled by the intramolecular coupling W. Partial diagonalizationof the

ZERO ORDER LEVELS EXACT MOLECULAR LEVELS”________ Ii>

Is> £

JWsr

Ir> ~ ______ Ii>

p. p.

Ig>Fig. 7. A three-levelmodel for quantumbeats. (From Freed and Nitzan [1980]).


Hamiltonianleadsto exactmoleculareigenlevelsI i) and I I) which arelinear combinationsof the states

Is) and Ir) which diagonalizethe Hamiltonian H~+ W. Both I i) and Ii) contain contributionsfromIs) andboth arethereforecoupledradiatively to ~g).We assumethat the energyspacingE1~= E, — E1 islargerelativeto the level widths so that the levels Ii) and I]) can be selectivelyexcited.A broadband,short-timepulseexcitationof the systemwhich is in Ig) initially leadsto a coherentlinear combinationof the Ii) and I I) states,

ItI’(t=O)) = Is) = a3~Ii)+ a~1Ij), (3.27)

which evolve in time accordingto

I ~i(t)) = a~1exp(—iE1t— ~y,t)~i) + a51 exp(—iE1t— ~y~t)~j) (3.28)

wherey, and y1 are the radiativewidths of thesetwo levels. For timeslonger than the inverseenergyspacinglIE11 the emissionspectrumcan be resolved; the intensity of the line at energyEjg is

71P1(t) = y1Ia~1I2exp(—y

1t). (3.29)

As long as these two emission componentsare resolved, no quantum beatsare expectedin thecorrespondingintensities.

Consider now the total integratedintensity. We have shown that this is proportional to thepopulationof the zero-orderIs) level, obtainedby taking n = 2 in (3.14),viz.,

P~(t)= IKsk’(t))I2 = Ia

51I4 exp(—’y

1t) + Ia5114 exp(—y

1t)+ 2Ia~1I2Ia

51I2exp[— ~(y

1+ y1)t] cos(E11t)

(3.30)

Quantumbeatsshould be observedin the integratedintensityfor

E1> ~(y,+ y1). (3.31)

Theobservationof quantumbeatsin fluorescencespectradependson the modeof detection.Beatsarenot expectedin the resolved componentsbut, when some other conditionsare fulfilled, should beobservedin the total intensityof theemissionwhich correspondsto a particularzero-orderlevel. Thesebeatscan be suppressedby increasingspectralresolution.

4. Vibrational energy transfer in model nonlinear oscillator chains; the Fermi—Pasta—Ulam paradox andthe Kolmogorov—Arnol’d—Moser theorem

In this andthe nextsection,we reviewtheorieswhich adopta more microscopicview of vibrationalenergytransferprocesses.Theseareclassicalmechanicaltreatmentswhich, while not strictly applicableto molecularsystemsaway from the correspondence-principlelimit, haveguidedsubsequentdevelop-mentssignificantly.


4.1. TheFermi—Pasta—Ulam(FPU) paradox

Until the relatively recentexplorationof chaoticphenomena,it was generallybelievedthat when alargenumberof linear harmonicoscillators are suitably coupledby nonlinearforces, an approachtoequilibriumwould ensue.This belief was basedon the work of Poincaré[1890,1957], furthersupportedby Whittaker [1944],and Fermi’s theorem [Fermi 1923; Ter Haar 1954], which statedthat weaklycouplednonlinearsystemswill exhibit the ergodicbehaviornecessaryfor an approachto equilibrium.For a finite numberof oscillators,deviationsfrom equilibrium areexpected[Mazurand Montroll 1960]andthe systemwill returnto its initial conditionsbecauseof Poincarérecurrences.However,whenthenumberof oscillators is large, the Poincaréperiod is presumedlarge [McLennan1959] and deviationsfrom equilibrium rare [Mazurand Montroll 1960].

Fermi, Pastaand Ulam [1955]tried to illustrate the expectedapproachto equilibrium by observingthe equipartitionof energyamongnormal modesof a systemof one-dimensionaloscillatorsobeyingequationsof the type

d2y~Idt2= (y1+1 — + y~1)+ a[(y0~1 — y~)

2— (y~— y~-t)

2], (4.1)

where i runs over the positive integersfrom unity to as high as 64. Here a is chosento be sufficientlysmall that the nonlineartermscan be treatedas a smallperturbation.Using the then-newMANIAC-Icomputerat Los Alamos, theysolved eq. (4.1) numerically andcalculatedthe energyin eachnormalmode as a function of time. Thesecalculationsrevealedno tendencytowardsequipartitionof energyamongthe normal modes.Variousinitial conditionswere usedandvarious couplingswere consideredwithout changingthe outcomesignificantly. Energy was sharedby only a few modesin a periodicfashion reminiscent of the energysharing betweentwo vertical pendulumshangingfrom the samehorizontalstring. Theseresultswere in contradictionto the previouslywidely held notionsconcerningthe approachto equilibrium.

The unexpectedresultof the FPUcalculationwas qualitatively explainedby Ford [1961]:a systemofharmonicoscillators(which, in this case,arethe normalmodesof the chainsystem)weaklycoupledbynonlinearcouplingswill not achieveequipartitionof energyas long as the frequenciesWk are linearlyindependenton the integers,i.e., as long as thereis no collection of integers{nk} for which

~~inkwk_O (4.2)

otherthan {~k} = 0. Thesefrequencymatchingsareknownas “internalresonances”in the terminologyof Kryloff andBogoliuboff [1947],andplay a crucial role in the proofof Poincaré’stheoremmentionedabove[Poincaré1890,1957], as well as in Whittaker’sconstruction[Whittaker 1944] of the constantsofmotion called for in Poincaré’stheorem.Physically, the linear independenceof uncoupledfrequenciesmeansthat noneof the interactingoscillatorsdrivesanotherat its resonancefrequency,andthislack ofinternalresonanceprecludesappreciableenergysharingin the limit as the coupling tendsto zero. Fordshowedthat for the systemselectedby FPU there were no such active resonances.Further (andquantitative)treatmentsof the FPU paradoxwere performedby Jackson[1963a,b]and ZabuskyandKruskal [1965].The first explanationin termsof “overlapping resonances”appearedin Izrailev andChinkov [1966].

100 T. Uzer, Theories ofintramolecular vibrational energytransfer

Ford’sperturbation-theoreticalreasoningwas furtherextendedby Jackson[1963a,b]whowas able toprovidea quantitativeexplanationof the FPUresults.It wasalsoJacksonwhopointedout that Fermi’stheorem [Fermi 1924; Ter Haar 19541 was valid far less often than it was presumedat the time.Concurrently,Fordand Waters [1963]provided a computerstudyof energysharingandergodicity incoupled nonlinear oscillator systems, in which they underscoredthe importance of the internalresonanceconditions, eq. (4.2), by building resonancesinto their coupledsystemandobservingtheireffect on the energysharing. The recurrencesobservedin the linear chain systemswere analyzedandexplainedby ZabuskyandKruskal [1965]with their introductionof solitons [Collins 1981]. As seenbefore [Jackson1963a,b]theseresearchersalsofound that integrableapproximationswould reproducethe FPU results, and the systemwas not chaotic.

Indeed,by this time, the earlier resultsby Kolmogorov [1954],Arnol’d [1963a,b]andMoser [19621hadbegunto be appreciated.Contraryto the expectationof the appearanceof statisticalpropertiesinweakly nonlinear systems,these results (subsequentlyknown as the KAM theorem,see below)indicated that for sufficiently small perturbations,the motion of a nonlinear system retains itsquasiperiodicnature. It alsoindicatedthat therewouldbe a critical perturbationwhenstochasticitysetsin. Suchan estimatewasprovidedby Chirikov [1960],andIzrailev andChirikov [1966]by meansof theso-called“resonanceoverlap” criterion. Theseissueswill be discussedmorefully by J. Ford in a reviewin PhysicsReports.

4.2. The Kolmogorov—Arnol’d—Moser(KAM) theoremand nonlinearresonances

In attempting to determinethe behavior of nonlinearoscillator systemsgovernedby the Hamil-toniansof the form

H=H0+V, (4.3)

whereH0 representsan integrablesystemof oscillatorsand whereV representsa weak nonlinearandnonintegrableperturbation,most investigatorshadproceededalongtwo divergentpaths.Oneapproachassumesthat the weak perturbationV changesthe unperturbedmotion only to the extentof slightlyshifting thefrequenciesof the motionandintroducingsmallnonlinearharmonics.This approachis usedmost oftenwhenthe numberof oscillatorsis smallandis exemplifiedin certainperturbationexpansionsdue to Poincaré[1957],Kryloff and Bogoliuboff [1947],and Birkhoff [1927].The second approachassumesthat V, eventhough it is weak,has a profound and pathologicaleffect on the unperturbedmotion, converting it into ergodic motion. This latter approachuses the methodsof statisticalmechanics(presumedvalid when the numberof oscillators is large) and is exemplifiedin the work ofFermi [1923]and Peierls[1929].

A brief article by Kolmogorov [1954]enunciateda theoremwhich provided the cornerstoneforlinking thesetwo divergent views on the effects of the weak perturbationV. Kolmogorov did notpresenta detailedproof of his theorem; the missing proof, which is quite long and mathematicallysophisticated,was given almost a decadelater by Arnol’d [1963a]andindependentlyby Moser [1962].The theoremwasmadeknownand accessibleto manyscientistsby, amongothers,the work of WalkerandFord [1969]on coupledtwo-degreeof freedomsystems.And it is on theselow-dimensionalsystemsthat we would like to illustrate the theorem.When action—anglevariables(J,, ç~,)areintroducedfor atwo-oscillatorsystem,the Hamiltonian (4.3) can be rewritten

H= HØ(J1, J2)+ V(J1, J2, ~ 4~). (4.4)


If we set V= 0, then Hamiltonian (4.4) generatesmotion for which the actionsJ are constant,and= w.(J1, J2)t + ~, wherethe unperturbedfrequenciesw, aregiven by

(4.5)

Following Kolmogorov, we view the unperturbedmotion in phasespaceas lying on two-dimensionaltori where (‘1~~~2) are the angle coordinateson the torus and ~ J2) are the radii of the tori. Byassumingthat V is sufficiently small andby assumingthat the Jacobianof the frequencies

o(w1,w2)I8(J~,J2)�0, (4.6)

Kolmogorov,Arnol’d and Moser (KAM) were able to showthat mostof the unperturbedtori bearingconditionally periodic motion with incommensuratefrequenciescontinueto exist, being only slightlydistorted by the perturbation.On the other hand, the tori bearingperiodic motion, or very nearlyperiodic motion, with commensuratefrequencies,or with incommensuratefrequencieswhose ratio isapproximatedvery well by (rls) wherer ands arerelativelysmall integers,aregrosslydeformedby theperturbationand remain no longer close to the unperturbedtori. Since the unperturbedtori withcommensuratefrequencieswhich are destroyed by the perturbationare dense everywhere, it isremarkablethat KAM are able to show that the majority (in the senseof measuretheory) of initialconditionsfor Hamiltonian(4.3) lie on thepreservedtori bearingconditionallyperiodic motion whenVis sufficiently small.

Thus for smallV, theKAM theoremsaysthat for most initial conditionsHamiltonian (4.3) generatesnonergodicmotion. This justifies the view that the perturbationV largely servesonly to changethefrequenciesslightly and to introduce small nonlinearharmonics into the motion. Nevertheless,therelatively small set of initial conditions leading to motion not on preservedtori are, from a physicalpoint of view, pathologicallyinterspersedbetweenthe preservedtori. Thus,if Hamiltonian(4.4) is everto provide generallyergodicmotion bestdescribedin termsof statisticalmechanics,the sourceof suchbehaviormust lie in the reasonsfor the very existenceof this relatively small set of destroyedtori.

4.3. Theonsetof stochasticity

As far as intramolecularenergytransferis concerned,it would be very desirableto havepracticalguidelinesfor the onsetof instability in coupledoscillatorsystems.With this practicalaim in mind, aswell as preparation for later applications,we will follow the lead of Walker and Ford [1969]andconcentrateon analyzingthe onset of ergodicbehaviorin a two-degreeof freedomoscillatorsystem.

Let us expandthe V of Hamiltonian (4.4) in a Fourierseriesand write

H=Ho(Ji,J2)+fmn(Jt,J2)cos(m4?t+n~2)+~.., (4.7)

where we haveexplicitly written only one term in the series.The KAM formalism seeksto eliminatethe angle-dependenttermsusing a convergentsequenceof canonicaltransformations,eachof which isclose to the identity transformation, thus obtaining a Hamiltonian which is a function of thetransformedaction variablesaloneandis closeto the initial Hamiltonian(for explicit implementationsin the chemical physics literature, see Chapmanet al. [1976],Swimm and Delos [1979],Jaffé andReinhardt[1982]).If this can be accomplishedin somegeneralsense,thenoneimmediatelyfinds thatthe perturbedmotions lie, for the most part, on tori close to the unperturbedtori.


As an example, let us eliminate the explicit angle-dependentterm in the Hamiltonian (4.7) byintroducingthe canonicaltransformationgeneratedby [Goldstein1980]

F2 = J~çb1+ J~42+ Bmn(J~J~)sin(m41+ n~5), (4.8)

where(J~,J~)aretransformedactionvariablesto be determined.We note that if Bmn = 0, we havetheidentity transformation.Introducingthe canonicaltransformationgeneratedby (4.8) into Hamiltonian(4.7), we obtain

H= H0(J;, J~)+ {[mw~(J,J) + nw2(J,J~)]Bmn(J~,~) ~fmn(~’ J~)}cos(mq~1+ nq~2)+...,

(4.9)

w~(J~,J~)= 0H0(J,J~)10J,

We may eliminatethe given angle-dependentterm by setting

B ‘J’ J’\__ fmn t’ 2 410mn~0’ 2) mw1(J,J~)+nw2(J~,J)

andprovidethat the denominatorin eq. (4.10) is not zeroor very small comparedwith fmn If it is verysmall, the coefficient B is large, the transformationgeneratedin eq. (4.7) is not closeto the identitytransformation,and the transformedcoordinatemotion is not close to the unperturbedmotion. As aconsequence,if thereexists a band of frequenciesw1 for the Hamiltonian (4.4) satisfying

Irnwt(J~, J2) + nw2(J1, J2)I ~ Ifmn(Jt’ J2)I, (4.11)

thenthe angle-dependentterm cos(m4.~+ nçb2) grosslydistorts an associatedzoneof unperturbedtoribearingthe frequenciessatisfying the inequality (4.11).

Moreover,when a zoneof unperturbedtori is grosslydistortedby a specifiedangle-dependenttermcos(rn41+ n42), one must in generalanticipatethat there will be a host of angle-dependenttermscos(m’~t+ n’42) in Hamiltonian (4.4) whosem’/n’ ratios are sufficiently close to the specifiedratiorn/n that the analogof inequality (4.11) is satisfiedfor them also. Hence the zoneof unperturbedtoridistorted by cos(rn41+ ncb2) will simultaneouslybe affected by a large number of other angle-dependentterms. Physicallyspeaking,inequality (4.11) is a resonancerelationshipwhich, if satisfied,assertsthat cos(rn4i1 + n~2)resonantlycouples the oscillators when their frequencieslie in thedesignatedband. If a numberof angle-dependentresonantterms couple the oscillators in this band,thenonehasthe situationenvisionedin the quantum-mechanicalGoldenRule in which aninitial stateis resonantlycoupledto a densityof final statesleadingto statisticallyirreversiblebehavior.In analogy,onewould anticipatethat the motiongeneratedby Hamiltonian (4.4) in theoverlappingresonantzonesof destroyedtori is highly complicatedandperhapsergodic.

When V is very small and henceall fmn are small, the inequality (4.11) showsthat the resonancezonesarevery narrow. Moreover,KAM showthat the totality of all resonantdestroyedzonesis smallrelativeto the measureof the allowed phasespace[Rosenbluthet a!. 1966]. However, as the V andthef,,~increase,or equivalently as the total energyincreases,one anticipatesfrom inequality (4.11) thatthe measureof the overlappingresonantzonesmayincreaseuntil most of phasespaceis filled withhighly complicatedtrajectoriesmoving under the influenceof many resonances.In short, the KAM


theory indicates the existenceof an amplitude instability for conservativenonlinearsystemswhichpermits a transition from motion which is predominantlyconditionally periodic to motion which ispredominantlyergodic.The Walker andFordpaper[1969]establishes,in pictorial form, this transitionand makes the influence of resonancezone overlap evident [Ford 1978; Berry 1978]; for theterminology, see Scheweand Gollub [1985].For the transition from classicalmechanicsto statisticalmechanics,seeFord [1975];alsoTabor [1981,1989].

For systemswith threeor moredegreesof freedom,the approachto stochasticitydiffers fundamen-tally from the precedingexample [Lichtenbergand Lieberman 1983]. Briefly, resonancesbetweenoscillatorsin two degreesof freedomleadto theformation of resonancelayerseachof whichplays hostto stochasticmotion. However,energyconservationpreventslarge excursionsof the motion alongthelayer— only motion acrossthelayer is important.For near-integrablesystemswith a weakperturbation,the stochasticlayers are isolatedby KAM surfaces.However, this is not true for resonancelayers inthreeor more degreesof freedom.Instead,they intersect,forming a connectedweb dense in actionspace(for an illustration, seeTennysonet al. [1980]).Also, conservationof energyno longer preventslarge stochasticmotionsof the actionsalonglayersover long times. The systemcan thereforeexploreall regionsof phasespaceconsistentwith energyconservationby steppingfrom layer to layer in thisinterconnecteddenseset of layers. The basic mechanismof diffusion alongresonancelayers is calledArnold diffusion [Arnold 1963b]. While there is no critical perturbationstrength for its presence,Arnold diffusion is knownto vanish with vanishingperturbationstrength.

Oneof the first numericaldemonstrationsof an amplitudeinstability wasmadein an investigationofunimoleculardissociationby the chemicalphysicistsThiele and Wilson [1961]andBunker [1962](seethe review by Hase[1976]).Theseinvestigatorsnotedthat for small-amplitudemotion, the harmonicmodesof model triatomicmoleculesexhibitedvery little mode—modeenergytransfer.As the energyofthe moleculewas increased,an amplitudestability set in which allowedfree and rapid interchangeofenergybetweenharmonicmodes.Consequently,as the energywasfurther increasedto slightly abovethat needed to dissociate one atom from the molecule, almost all initial configurations led todissociation.Thiele and Wilson [1961],who were using coupledMorse oscillators, thenused theseresults to arguethat nonlinearity must be given a centralrole in developingastatistical theory ofunimolecular reactions.While these molecularsystemswere not analyzed in terms of the KAMtheorem,Walker andFordhaveshown[1969]that the instability observedby Thiele and Wilson [1961]occursconcurrentlywith large-scaledisappearanceof preservedtori.

The secondmajor numerical demonstrationof amplitude instability occurredin an astronomicalsystem,when Hénon and Heiles [1964]studiedthe Hamiltonian

H ~(p~+ q~+p~+ q~)+ q~q2— ~ (4.12)

which now bears their name. This Hamiltonian has made frequentappearancesin the theory ofmolecularenergy transferwhere it describesa molecularpotential with a C3~,symmetry. Anotherpopularsystemfor demonstratingKAM instability is the Hamiltonian

H ~ (P~+ w~q~)+ a ~ A~J~q0q,q~+ A ~ ~ (4.13)2 k i,j,k h,i,j,k

The studiesby FPU [1955],FordandWaters[1963],Jackson[1963a,b]andZabuskyandKruskal [1965]were doneon Hamiltoniansof this form. But Izrailev andChirikov [1969]were the first to suggestthatHamiltonian (4.13) shouldexhibit a KAM instability leadingto statistical behavior.


Many criteria have been proposedfor the determinationof the transition to global stochasticity[Chirikov 1960, 1979; Izrailev and Chirikov 1969; Jaegerand Lichtenberg1972; Greene1968, 1979;Escandeand Doveil 1981; Percival 1974]. For a summary,see Lichtenbergand Lieberman[1983];thesubjecthasrecentlybeenreviewedby Escande[1985].The earliestcriterion, dueto Chirikov [19601,isknown as the “resonanceoverlap” criterion, andconstitutesan approximateyet elegantand powerfulintuitive devicefor understandingthe physicsof this transition. In its simplestform, it postulatesthatthe last KAM surface between two lowest-order resonancesis destroyedwhen the sum of thehalf-widths of their island separatrices(seesection5.1) formedby them(but calculatedindependentlyof each other) equals the distance betweenthe resonances.The distancesare measuredeither infrequencyor action space,whicheveris moreconvenient.Becauseof its highly practicalprescription,this criterion hasbeenwidely used.

The effect of resonanceoverlapwas exhibitedby Ford andLunsford [19791on a multiple-oscillatorsystemwith very small nonlinear couplings. Among the numerousrecent implementationsof the“overlap of resonances”criterion are those of Rechesterand Stix [1979], and Petrosky [1984].Resonanceoverlap can occur for thesesystemsin the limit of arbitrarily small couplingswhen thefrequenciesof the systemfulfill commensurabilityconditionswhich allow the nonlinearcouplings tocouple all internal degreesof freedom. The stability of the periodic orbits for the Henon—Heilesoscillator was studiedby Lunsford and Ford [19721,who were able to demonstratethe existenceof adenseor nearly denseset of unstableperiodic orbits throughoutits stochastic(unstable)regions ofphase space. They also showedthe intimate relation betweenthe Henon—Heilessystem, the FPUsystemas well as the so-calledToda lattice [Lichtenbergand Lieberman1983].

Energytransferprocessesin nonlinearlycoupledchainsand,in particular,the approachto statisticalequilibrium in such systemsis still a subjectof interestand research.For example,see Henry andOitmaa [1983,1985a,b],Florencio and Lee [1985],Henry and Grindlay [1986,1987,1988].

5. Classical microscopic theory of energy sharing; energy transfer through nonlinear resonances

In the previoussectionwe haveseenthecrucial role that nonlinear(or “internal”) resonancesplay incausinginstabilitiesin dynamicalsystems.The effect of resonanceshadbeenappreciatedby molecularphysicists before the study of intramolecular energy transfer since resonancesshowed up in thedistortion of vibrationalspectra.In fact, the splitting of the almostdegeneratevibrational energylevelsin the carbondioxide moleculewas explainedby Fermi [1931]as being due to an almost exact 2: 1frequencyresonancebetweenthe bendingandthe stretchingmodesof this molecule(for more recenttreatmentsof the quantizationanddynamics,seeHeller et al. [1979,1980], Noid et al. [1979],Siberteta!. [19831,Uzer [1984],Brabhamand Perry [1984],Voth et al. [1984],Peyerimhoffet al. [1984],VothandMarcus[1985],Kato [1985b],Baggottetal. [1985,1986], Farrelly [1986a],FarrellyandUzer [1986],Kellman andLynch [l986a,b,1988a,b],Martensand Ezra [1987a,b],Greenet al. [1987],Wong et al.[1987],QuappandHeidrich [1988],Quapp [1989],Colbert and Sibert [19891).

Since these frequency resonancesare central to both the dynamical and static propertiesofmolecules,wewill presenttheclassicalandquantummechanicsof energytransferthrough a resonance.In reviewing this field, our task is considerablyeasedby the existenceof several informative andcomprehensivereviewsandmonographs:Ford [1973],Rice [1981],as well as Brumer[1981]andTabor[1981,19891, where the emphasisis on the onsetof statisticalbehavior. Ratherthan recountingthematerial andthe numerousreferencescontainedin thesereviews,we will focuson developmentsof theinterveningdecade.


As an aside, much effort in chemical physics has gone into the relatedbut distinct subject ofsemiclassicalquantizationof nonlineardynamical systems(e.g., Eastesand Marcus [1974],Noid andMarcus [1975],Sorbie andHandy [1976],Chapmanet al. [1976],Swimm andDelos [1979],Jaffé andReinhardt [1982], Farrelly and Smith [1986],Farrelly [1986b], Ratner et a!. [1987],Taylor andGrozdanov[1987],GerberandRatner[1988],De Leon andMehta [1988],Kummer andGompa[1988],Sibert [1988]).Comprehensivereviewson bothgeneralandspecialproceduresareavailable(Noid et al.[1981a,b],Reinhardt [1982],Delos [1986],Bowman [1986],Ezra et a!. [1987],Reinhardtand Dana[1987],Skodjeand Cary [1988],Reinhardt [1989]).

5.1. Nonlinear resonancesand the classical mechanicsof the pendulum

Like many other applicationsof nonlinear dynamics in molecularphysics, the pendulum or thehinderedrotor pictureoriginatedin plasmaphysicsresearch.This paradigmaticpicturewas recognizedanddeveloped[Chirikov 1959; Zaslavskii andFilonenko1968] to understandthe trappingof chargedparticles.Soonit was realizedthat thispicturecould be appliedgenerallyto a numberof othertrappingphenomena,including trappingandleakagein phasespace.For an extensivereviewof the derivationofthependulumpicturethe readeris referredto Chirikov [1979],as well as to standardtexts[LichtenbergandLieberman1983, Zaslavsky1985]. For our purposes,the following brief derivation(seesection4.3)will suffice [Brumer 1981].

Supposeone is consideringthe dynamicsof acoupledoscillatorsystem,thezero-orderfrequenciesofwhichhavea resonance,i.e. a commensurabilityrelation,like eq. (4.2). It is convenientandcustomaryto analyzethe nonlinearcoupling in terms of the action—anglevariablesof the unperturbedsystem,

(J,~),

H(J, f4=H0(J)+ ~ V{rn)(J)exp(im~41), (5.1){rn

wherem is a multidimensionalvectorof integers.In particular,for two coupledoscillators,

H(J, 4) = I-10(J) + ~ Vmjm2(J)exp[i(rn1çl)1 + rn2~2)], (5.2)m1 rn2

or concentratingon a single resonance,

H(J, ~)— H0(J) + Vmm(J)exp[i(m1~1+ rn242)]. (5.3)

If the systemcontainsa resonance,

rn~w~(J)+ m2w2(J) 0, (5.4)

for a particularset of actionf, thentermslike exp[i(mi~t+ rn2cti2)] areslowly varying in time (cf. the

small-denominatorproblemin the precedingsection).Introducingthe new independentanglevariables,

~k~p.ki~i (k1,2), (5.5)

where ~kj are constantssuch that p.~= rn1, p.12 = m2, p.21 = rn1, p.2~= rn2. The new actions I are


obtainedby the generatingfunction

= ~ (i~+ ~ Ikp.kI)~I, (5.6)

giving

(5.7)

wherep.~are matrix elementsof the matrix ~

The new momenta1 measuredeviations from J[• Since qi, does not occur in the resonanceHamiltonian (5.3), 17 is a constantof the motion, and (5.3) can be rewritten

Hr(I, ~)= ~ I~p.~w1+ + V~1m,(fl cos ~. (5.8)

Here, Mkl = ~J p.~f(aWf/8I1)p.11 is a generalizedmass.At the resonancecenter, (5.4) can be applied, and (5.8) becomes

Hr(J, ~/J)= (1/2M11)I~+ Vrnm(I)cos t~i~ . (5.9)

The approximationto the dynamicshasbeenobtainedby averagingoverthe fastmotion (seealsoBorn[1960],Kryloff andBogoliuboff [1947]),leaving only the slowly varying terms.Therefore, this processhasresultedin a lower-dimensional“resonanceHamiltonian” Hr, eq. (5.9),which can be examinedforsomeof the salientfeaturesof the original Hamiltonian.

Special casesarise when the anharmonicity disappears,i.e., the system becomes“intrinsicallydegenerate”[Izrailev 1980; Martensand Ezra 1987b] as opposedto accidentally degenerate,which isthe casewe treat. Often one can approximatethe Fourier coefficientsat the centerof the resonance,i.e., when condition (5.4) is fulfilled.

While it is not necessaryto make this approximation for subsequentanalysis, it simplifies themathematicsand sometimesis accurate(for semiclassicalevalutionof matrix elements,see Noid et a!.[1977],Koszykowskiet a!. [1982a],Wardlaw et a!. [1984],Ozoriode Almeida[1984a],Voth andMarcus[1985],Voth [1986],Noid andGray[1988]).Examplesof allowing the Fouriercoefficientsto vary aretobe found in Gray and Child [1984],Shirts [1987a],Sahm and Uzer [1989a],Sahm et al. [1989].

The resulting approximateHamiltonian (the subscriptsare not needed),

Hr(I, ~)=(1I2M)I2+V(I~)cos~j=E~, (5.10)

is that of a pendulum,or a hinderedrotor (for an applicationof this procedureto two coupledMorseoscillators,see Jaffé and Brumer [1980]).It is clearly an integrablesystem,andit has threetypes ofmotion. When the energyis less than the potentialbarrier,

Er<V(Ir), (5.11)

the motion is that of “libration”. When the energyis abovethe barrier, it is “rotation” [Noid and


Marcus1977]. Thesetermsoriginate from the motion of a physicalpendulum,but arecommonlyusedwhen the pendulumvariablesareabstractquantities,as well. The third typeof motion is the motion onthe separatrix,i.e., when the energyis preciselyequalto the barrierheight.The phasespaceprofile ofthesethreemotionsis shownin fig. 8. The pendulumis sucha usefulsystemthatwe quote theresultsofits analysisin full. Its action—anglevariablesaregiven by Zaslavskii and Filonenko[1968],Rechesterand Stix [1979],LiebermanandLichtenberg [1984],

E(K)—(1—K2)K(K), K<1, (5.12a)J=(8ITr)(VM)112x

K>1, (5.12b)

K<l, (5.12c)

2[K(K’)]~F(~/2,Kt), K >1, (5.12d)

2k2=1+ErIV, (5.12e)

whereK, E are completeelliptic integralsof the first and secondkinds, F is an incompleteellipticintegralof the first kind, and

K sin i~= sin(~i/2). (5.12f)

E

E

~iIIl

(a)

rotationE separatrix

E b E ~

libration(b)

Fig. 8. Correspondencebetweena, an energydiagram, and b, a phase spacediagram for the pendulumHamiltonian. (From LichtenbergandLieberman[1983]).


The approximatewidth of the resonancein action andangle, (~J)t and (~W)r,respectively,is given byZaslavskiiandChirikov [1972],Shuryak[1976],Brumer[1981][in termsof the original variablesof eq.(5.9)],

(~J)r =2m(MuVmm)’~2, (~W)r= (Vmm/M~)’2, (5.13)

where’m= (m,, rn2). The explicit time dependenceof the variables,on the otherhand is [Zaslavskii

and Filonenko1968]

1(t) =

2W

0K cn(w0t) , K < 1; 1(t) =

2W0K dn(w0t) , K > 1 , (5.14a,b)

where w0 = (VIM)’2 and the frequencyof the pendulumis

w(K)Iw() = ~1~-[K(K)]~’ , K <1; w(K)/w0 = ITK/K(K’), K >1. (5.15a,b)

Near the separatrix

~ir/ln[4(1 — K2)O21, K <1,

lim(w1w00)=

K1 ~/ln[4(K2 — 1)_1/21, K > 1.

The behaviorof w/w0 with varying K is plotted in Lichtenbergand Lieberman[1963],fig. 2.2.

5.2. Vibrational energytransferand overlappingresonances

In section4.3, we haveseena classical-mechanicaldemonstrationof the appearanceof a dynamicalinstability due to overlapof nonlinearresonancezones(the quantal interactionof nonlinearresonancescan be found, amongothers, in Shuryak [1976],and Bermanet al. [19811).The appreciationof theeffect of nonlinear resonancesin intramolecularenergytransferprocessescan be said to havebeenphrasedin thesetermsfirst by Oxtoby andRice [1976],who computedregionsof resonanceoverlapinphasespacefor severalcoupledoscillator systemsin order to correlatethe extentof intramolecularenergytransfer with the fraction of phasespacedisplaying overlappingresonances.Nonlinearreso-nances,the pendulumpicture, andthe overlapof resonancescriterion was alsoused in the earlyworkof Kay [1980].Later applicationswerevery muchacceleratedby the studyof local modes(for a review,see Child and Halonen [1984]), where it could be assumedthat the initial excitation was indeedlocalizedin oneidentifiable oscillatorin the molecule,usuallyconsistingof a hydrogenatom boundto acarbonor oxygen atom, which is too heavy to participateextensivelyin the (highly anharmonic,i.e.,nonlinear)vibrations of its light partner. It is well known that the vibrations of local modescan beapproximatedby Morse oscillators (Child and Halonen [19841),and therefore,most of the earlytheoreticalapplicationshavebeen done on coupledMorse oscillators [Kay 1980; Jaffé and Brumer1980], the simplestsuch systembeingthe nonrotatingwatermoleculeconsistingof two OH stretchingbendingmodesand a frozen bending angle [Sibert et al. 1982a,b]. For an overview of calculationsconcerninglocalizationof energy,see Reinhardtand Duneczky[1988].

Jaffé and Brumer [1980],and Sibert et al. [1982a,b]studiedthe classicaland quantaldynamicsoflocal modes in the water molecule by meansof the pendulum picture, and provided a pictorialexplanation of local-mode dynamics. They realized that the energy transfer dynamics could be


approximatedvery compactlyby the time dynamicsof the pendulumvariables;indeed, the canonicaltransformation(5 6) in this casereducesthe original Hamiltonian to one for the variable action

~ ‘2’ (5.16)

which, when multiplied by a representative,averagefrequency,approximatesthe energydifferencebetween the two stretching motions. In this picture, the action of the nonlinear resonanceas aself-regulatingenergytransfermechanismis particularlyobvious.A frequencymatchingis requiredtodrive the mode—modeenergytransfer;on the otherhand,whenenergyis transferred,the frequencieschangeand detune from each other, causing the energy flow to stop. The pendulum picture alsoprovidesa dynamicalview of the local versusnormal mode classificationin molecules.Local modescorrespondto the rotating solution (5.12b),and normal modesto the librating solution (5.12a).Theoriginal analysis was performed in local-mode actions, and the alternative, normal-mode baseddescriptionwas providedby Kellman[1986].

Sibertet al. [1982b]useda one-dimensionalintegrableapproximationto the original Hamiltoniantofind the quantum-mechanicallocal-modeenergy levels. When the action variablesare replacedby1—* (h/i) d/d4, in accordancewith the traditional recipeof obtainingquantum-mechanicaloperatorsfrom classical-mechanicalHamiltonians(seee.g., Augustin and Rabitz [1979]),a Mathieu differentialequation[McLachlan1964]results,the eigenvaluesof which approximatethe locationsof the quantizedenergylevelsof thewatermoleculeratherwell. Moreover,theperiodsfound by Sibertet al. [1982b]forthe energytransferfunctionsturnedout to berelated(throughtheuncertaintyrelation) to thesplittingsbetweenadjacentquantizedlocal modes,as would be expected[Lawton and Child 1979, 1981].

The quantizationof a nonlinearresonancein nonautonomoussystemshadalso led to the Mathieuequation[BermanandZaslavskii 1977], anduniform quantizationschemes[Uzer et at. 1983; Ozorio deAlmeida 1984b; Uzer and Marcus 1984; Voth and Marcus1985; Farrelly and Uzer 1986; Voth 1986;Graffi et al. 1987, 1988] havealso beenused to understandavoidedcrossingsin nonlinearsystemsinterms of signaturesof resonances[Noid et al. 1980, 1983; Ramaswamyand Marcus 1981]. Forexperimentalobservationof quantalstatestructureneara resonance,see Holtzclaw and Pratt [1986]and Fredericket a!. [1988].

This microscopicformulationwas soonextendedto a morecomplicatedsystem,namelythe transferof energyout of local CH oscillatorsinto the vibrationsof frameof the benzenemolecule[Sibertet al.1983, 1984a,b].In this case,a simple analysisshowsthat the relaxationshould be most effectivewhenthere is a Fermi (i.e., 2:1) resonancebetween the CH stretching and HCC bending (so-called“wagging”) motions [Sibert et al. 1983]. After reaching the wagging motion, vibrational energydispersesinto the numerousvibrations of the frame principally through 1: 1 resonances.Theseconclusionswere first reachedby meansof classical trajectory calculations[Sibert et al. 1984a1byanalyzingthe widths of resonancesat eachstepof the relaxationprocess.For an experimentalanalysisof CH fundamentalrelaxation,seeStewartandMcDonald [1983].

Classical (or quasiclassical)trajectory calculations,which are reviewed, amongothers,by Hase[1976],and Raff and Thompson[1985],are the mostwidely usedprocedurefor studyingthe classicalmechanicalenergyflow in molecularsystems.Their applicationsto realisticpotentialenergysurfaces,andthe degreeto which theycan simulatethe true dynamicsfaithfully havebeendiscussed[Hase1982;Haseand Buckowski 1982; Hase1986]. The basis for the validity of classicaltrajectory calculationsinthe chaoticregime is relatedto the so-called$-shadowtheorem(e.g., Hammelet a!. [1987]).One ofthe persistentpractical difficulties in simulatingvibrations has been the contribution of zero-point


energyto the energyflow andreactivity. The amountof zero-pointenergycontainedin the numerousmodesof a polyatomicmoleculecan be considerable.Recentresearch[Bowmanet al. 1989; Miller etal. 1989] addressesthis problemwith schemesto preventclassicaloscillatorenergiesfrom falling belowtheir zero-point energy.

In their quantum-mechanicalwork on benzene,Sibertet al. [1984b1used“tiers” of coupledstatesinvery muchthe sameway that individual quantumstateswere usedin the fluorescentcalculationin thefirst sectionof this review(IVR processesin benzenehavebeenreviewedby Riedle et al. [1984]).Thetime-dependentprobability flow betweenthesestateswas computedand the timesfor coarse-grainedprobability transferwere thenconvertedto lifetimes and linewidths for given initial excitations.Themost striking conclusion of these calculations was an explanation of the unexpecteddecreaseinlinewidth of benzeneovertonevibrations as the energyis increased(recentexperimentsarePerryandZewail [1984],Pageet at. [1987,19881;recenttheory by Lu andHase[19881).Sincethe densityof statesincreaseswith increasingenergy,onewould expectthe widths to increase.On the otherhand, by thetime one reachesthose high overtones,the systemhasmoved out of the widths of theseessentialresonances,therebyslowing down energytransferconsiderably.Here is a casewherethe density ofstates,which is commonly such a paramountfactor in driving energyrelaxation,is secondaryto whatcould be called the microscopicdetail, namely the couplingand the resonancesbetweenmodes(foralternative treatments,see Stone et al. [1981], Prasad[1981], Buch et al. [1984], Mukamel andIslampour[1984],Shi and Miller [19851,Dietz and Fischer[1987],and Voth [1988]).

The work of Sibert et al. [1984a,bl on benzene and its derivatives has been followed by aconsiderablenumber of other classical and semiclassicalcalculations employing increasingly refinedmodels— sometimesin the spirit of “reduceddimensionality” models (reviewedby Bowman [1985]).Refinementsintroducedby Lu et al. [1986],and Garcia-Ayllonet al. [1988],include, for instance,theattenuationof the bendingforce constantwith the stretchingvibrations[SwamyandHase1986; Wolf etal. [1986]and quasiclassicalprescriptionsfor calculatingthe lineshapeandwidth [Lu andHase1988].The calculationsof Bintz et a!. [1986a,b;1987] as well as Lu and Hase [1988],Clarke and Collins[1987a,b]and Garcia-Ayllon et al. [19881havetendedto affirm the mechanisminitially proposedbySibert et al. [1984a,b].

Parallel calculationson vibrational energyredistribution in other hydrocarbons[Hutchinsonet al.1983, 1984, 1986] (experimentdoneby Nesbittand Leone[1982]),concludedthat the energytransferwas driven mainly by 1: 1 resonances,or combinationresonances[Hofmannet al. 1988], whereassomeresearchershavereportedthat a similar resonancepicture is only loosely applicable [SumpterandThompson 1987b] or inapplicablein their modeling of energy transfer in alkanes[DuneczkyandReinhardt1990]. The tier-by-tier energy flow picture has been applied to IVR in highly excitedacetyleneand its isotopic variants,and the connectionbetweenenergyflow pathwaysand resulting“clump” spectrahas beenexploredby Holme and Levine [1988a,b;1989]. For recentexperimentsonenergyflow in alkanesfollowing CH overtoneexcitation, see Mcllroy and Nesbitt [19891.For otherchains,see,e.g., Shin [1989].

While in the precedingcalculationsthe energyexchangeamongmodesproceedsthroughpotentialand/orkinetic couplings,somerecentwork hasexaminedthe interactionsof vibrationswith rotations[FrederickandMcClelland 1986a,b;Ezra 1986; Shirts 1986,1987b; Lawranceand Knight 1988; Knight1988a,b] (see also the monograph by Ezra [1982]) and internal rotations [Sumpteret a!. 1988],vibration—rotation energy flow [Frederick et al. 1985; Ezra 1986; McClelland et al. 1988] andintramolecularvibration mixing induced by rotational (Coriolis) couplings[Clodiusand Shirts 1984;Uzer et al. 1985a;Burleigh et al. 1988; Lin 1988; Gray and Davis 1989; Sibert 1989; Sahmand Uzer

T. Uzer, Theories of intramolecularvibrational energy transfer ill

1989a,b;Sahmet al. 1989; Parson1989a,b].For recentexperimentalstudies,seeGarlandet al. [1983],Watanabeet al. [1983],Nathansonand McClelland [1984],Dai et al. [1985a,b],Felker and Zewail[1985c],Amirav andJortner[1986],Apel and Lee [1986],Garlandand Lee [1986],Bain et a!. [1986],Ohtaet al. [1987],De Souzaet al. [1988],Neusseret al. [1988],Lawranceand Knight [1988],Frye etal. [1988].

Both rotation-inducedvibration mixing and vibration-inducedrotation mixing havebeenexaminedin a two-modemodel of the formaldehydemolecule[Burleigh et al. 1988; Sibert 1989; GrayandDavis1989] andits deuteratedanalog[Parson1989, 1990]. Theseinvestigationsshowthe consequencesof thecollective operationof theseeffects. They also suggestthat in regionsof experimentalinterest, theintramoleculardynamicsis not significantly influencedby resonances.An examinationof vibrationalenergyflow under the influenceof both Coriolis and potential couplings [Fischerand Dietz 1987;Hiroike and Fujimura 1988; Burleigh et a!. 1988; SahmandUzer 1989a,b]hasled to the developmentof a pictureof the joint action of rotationalandpotential couplings[Sahmand Uzer 1989a,b;Sahmetal. 1989], seealso nextsection.Vibrationsand rotationsinteractthroughboth Corio!is and centrifugalcouplings.The relativerolesof thesemechanismshavebeendiscussedin Lawranceand Knight [1988],andreviewedin Knight [1988a,b].For a recentrovibronic IVR modelcalculation,seeDietz andFischer[1987].

Classicaland/orquantalpropagationmethodshavebeenused to trackenergyflow throughpotentialenergyblockers [Uzer and Hynes 1986], for example,a heavymetal atom [Lopezand Marcus1982;SwamyandHase1985;Ledermanet al. 1986;Lopezet al. 1986; UzerandHynes1987; Ledermanet a!.1989; Uzer and Hynes 1989]. Experimentallythe possibility seemsto exist [Rogerset al. 1982, 1983]that energymaybe localized in onereactiveportionof an energizedmoleculeby a heavymetalatom;RRKM behaviorinvolving rapid redistributionand reactionremote from the bond excitationsite isthwarted,thoughstatisticalbehaviorhasbeenreportedby Wrigley et al. [1983,1984] in somemoleculeswith heavymetal“blockers”. The possibilityof theseblockersinducingnonstatisticalreactivebehavioris discussedby Uzer and Hynes [1986,1989].

The precedingcalculations are primarly (but not exclusively) concernedwith vibrational energyredistribution leading to relaxation.Thereare othercalculationsin which the redistributionof energycan lead to unimolecular reaction(e.g. Rai and Kay [1984],Uzer [1988]).While such calculations,startingwith Thiele andWilson [1961]andthe extensivework of Chapman,Bunker,Hase,Thompsonand colleagues(for a review of their work, see Hase[1976,1981], Raff and Thompson[1985],Hase[1986]), have commonly simulatedreagentsthat develop from collisions (e.g., Viswanathanet al.[1984,1985], NoorBatchaet a!. [1986],Rice et a!. [1986]) or chemical activation (Wolf and Hase[1980a],Haseet al. [1983,1984]), more recentexampleshave probed the consequencesof mode-specific excitation [Sloane and Hase 1977, Marcus et a!. 1984, Hase 1985]. Among these areinvestigationsof overtone-inducedunimolecular dissociation(for an overview, see Uzer and Hynes[1987]),and isomerizationreactions[Hutchinson1989]; i.e., processesin which the reactionenergyissupplied by exciting the overtones of a OH or CH local mode (for overviews of the experimentalsituationin isomerization,seeJasinskiet al. [1983,1984], Segall and Zare [1988];andin bondfission,Crim [1987])or NH modes [Foy et a!. 1989] (for recent theoretical work, see Hase [1985],Holme andHutchinson[1986],Hutchinson [1989],Marshall and Hutchinson [1987],Hutchinsonand Marshall[1988];for generaltreatmentsof the dynamicsof localizedstatessee,e.g.,Moiseyev [1983],Davis andHeller [1984]).

Sincethe reactionenergyreachesthe reactioncoordinateby vibrationalenergyredistribution,theseprocessesform an interfacebetweenIVR and reactiondynamics.The role of nonlinearresonancesin


model isomerization reactionswere investigated by Uzer and Hynes [1985], and researchon theisomerizationof hydrogencyanide[Smithet at. 1987] hasassigneda role to combinationresonancesinchangingthe rate of reaction[Holme andHutchinson1985]. The energytransferprocessesleadingtoisomerizationin HONO have been examinedby Guanet a!. [1987].Other studiesof the interfacebetweenisomerizationand IVR areSumpterandThompson[1987c],Hutchinson[19871,Spearset al.[1988a,b],R.P. Muller et a!. [1988],Raff [1989].

The overtone-induceddissociationof hydrogenperoxide has becomea prototypical unimolecularfission reactionwhere many of the ideasconcerningintramolecularvibrational relaxation, reactivity,andthe interpretationof linewidths havetendedto cometogether.Calculationsof the energytransferleading to reaction [Sumpterand Thompson1985, Uzer et al. 1985a,1986, Sumpterand Thompson1987a, 1988, Uzer et al. 1988, Getino et al. 1989], have, using different potential energysurfaces,tendedto confirm aslow, sequential,two-stepenergytransfermechanism[Uzeret al. 1985a,1986]. Theenergymigratesfrom the OH stretchingmode to the OOH waggingmode, which, in turn, channelsitinto the restof the molecule,including the reactive00 stretchingmotion. This transfertakesplaceona timescaleslow comparedwith molecularvibrations,andleavesthe initially unexcitedOHstretchingmodeas an inactivespectatorbecauseits high frequencyprecludeseffectiveresonanceswith the restofthemolecule.While the detailedresultsare sensitiveto thechoiceof force field [Getino et at. 1989], therobustnessof the energytransfermechanismsuggeststhat hydrogenperoxidehassomepotentialformode-specificdissociation[Schereret a!. 1986, Schererand Zewail 1987, Uzer et al. 1988]. Indeed,Chuang et al. [1983] and Gutow et al. [1988] report mode-specificity in the overtone-induceddissociationof the relatedsystem, t-butyl hydroperoxide.Dynamical experimentswhich, insteadofdeducinglifetimesindirectly from linewidths [Rizzoet a!. 1984,Ticich et al. 1986, Butler et al. 1986b,Luo et al. 1988], usereal-timeinvestigationsof the dissociationprocess[Schereret al. 1986, SchererandZewail 1987] areexpectedto helpresolvesuchissues.Amongreactions,predissociation,especiallyin Van der Waals molecules [Miller 1986], is a processwhere the role of IVR hasbeenprobedextensively(e.g., Grayet a!. [1986a],Ewing [1987],Ewing et a!. [19871,JucksandMiller [1987],Evardet a!. [1988],Wozny and Gray [1988]).For the intricaciesinvolved in connectingunimoleculardecayratesfrom linewidths, the readeris referredto the appendix.

5.3. Generalizingthe pendulum

It was mentioned earlier that during the reduction of the full nonlinear Hamiltonian into a pendulum[eqs. (5.9), (5.10)], it was necessaryto approximatethe Fourier coefficients by constants— by, forexample,evaluatingthem atthe resonancecenter.This approximationis only good whenthe couplingbetweenthe degreesof freedomis weak.For strongcouplings,the variation of the Fouriercoefficientswith the momentumneedsto be takeninto account.An exampleof this procedureis providedby theHamiltonian

~ (5.17)

A single-resonanceapproximation to (5.17), when expressedin the action—anglevariablesof theunperturbedHamiltonian, reads[Sahmand Uzer 1989a;Sahm et al. 1989]

H= ~(3k—k12)~2+ ~k

12(I2—~2)cos4~. (5.18)

This Hamiltonian is similar to that of an asymmetricrotor in torque-freespace.The relationbetweenthis and the pendulum Hamiltonian can be made more transparentby transforming the rotor


Hamiltonianinto conjugatevariables(K, x)~whereK is the z-componentof the angularmomentumonthebody-fixed axes. This transformation[Deprit 1967; Augustin andMiller 1974; Augustin and Rabitz1979; Ezra 1986],

Jx=_(J2_K2)H2sinX, J~=_(J2_K2)v2cosx, J~K, (5.19)

convertsthe asymmetrictop Hamiltonian

HAR=CJ~+BJ~+AJ~ (5.20)

into

HAR = ~(B + C)J2 + [A — (B + C)I2]K2 + ~(B — C)(J2 — K2) cos2x, (5.21)

which approximatesthe pendulumHamiltonian (5.10) for small variationsof K.The replacementof the pendulum by an asymmetrictop has been applied to vibrational energy

transferdue to rotations, or Coriolis-inducedvibrational mixing [Clodiusand Shirts 1984; Uzer et al.1985a; SahmandUzer 1989a; Sahmet al. 1989]. The transformations,when applied to the modelvibration—rotationHamiltonian

H=Hv+Brr~—2BLir7, ir7=~(p2q1—p1q2), (5.22)

with B, ~,L constants,showthat the energy-transferpicturebetweenthetwo modescan bedeterminedby consideringthe intersectionof two solid figures

H CJ~+ BJ~+ AJ~+ cJ~, J2 = J~+ J~+ J~, (5.23a,b)

in the spacespannedby the vibrationalvariables~ J~,,J~)[Sahmet a!. 1989].The semiaxesof the ellipsoid (5.23a),which is a “constant energysurface” [Harter and Patterson

1984; Harter 1984, 1988], aredeterminedby the vibrationalconstants,andthe radiusof the spherebythe initial excitations.The angularmomentumof the moleculedeterminesthe distancebetweenthecentersof the two solids. The samepicturehasalso beenappliedto vibration-inducedrotation mixing(“K-mixing”) by SahmandUzer [1989b].

The basis for thesedevelopments is the homomorphism between the SU(2) and R(3) symmetrygroups[Harter 1986]; thesehavesomegeneralityfor resonancesotherthan 1: 1 becausean arbitrarym: n resonancecan be transformedinto a 1: 1 resonance,i.e., can be madeto showSU(2) symmetry[Farrelly 1986a]. Group-theoreticalideas have been very fruitful in understandingand classifyinginteractionsbetweenvibrations[Kellman 1982,1985, 1986; Lehmann1983,BenjaminandLevine 1983;Van Roosmalenet al. 1983, 1984; Levine and Kinsey 1986; Kellman and Lynch 1986a,b; 1988a,b; Jaffé1988; Xiao and Kellman 1989].

5.4. Nonstatisticaleffectsandphasespacestructures

In the introduction,wementionedthe two keyassumptionsof RRKM theory: Intramolecularenergyrelaxation is complete and fast compared with the time scale of unimolecular dissociation (statisticaltheorieshavebeen reviewedby Truhlar et a!. [1983],Hase[1983],as well as Wardlawand Marcus


[1987]). Notwithstandingthe widespreadsuccessesof the RRKM theory,even early on a numberofexperimentalhypothesesdid not seemto apply; amongthese“nonrandomdecay” caseswere someunimolecularreactionsinvolving chemicalactivation [Forst 19731. When statisticaltheory is appliedtosuch reactionswithout anymodifications,the resultsare at variancewith the experiment.A particularseverecaseis the predissociationof Van der Waals moleculeslike He12,where a moleculewith morethan enoughenergyto dissociatelives for a very long time becausethe dissociationenergyis in the“wrong” mode and tendsto stay there(seeWozny and Gray [1988]for references).

Togetherwith the realizationthat energyrelaxationmight sometimesbeincompletecameempiricalways of accounting for such nonstatistical effects in reaction rate calculations. Corrections fornonstatisticalrelaxationincludethe exclusionof certainmodesfrom ratecalculations,therebypractical-ly treatingthe moleculeas a smallermolecule.For example,a vibrational modemayhavemuchhigherfrequencythan othersand cannotcouple with them strongly, andis thereforeinactive in the energytransfer process[Davis 1985]. A recent example involves reaction rate calculations for hydrogenperoxide [Brouweret al. 1987]. For otherempirical proceduresfor correctingfor slow relaxation,seefor example,Marcus et al. [1984].

With the applicationof classicalmechanicsto the study of molecularsystems(explained in theclassicaltrajectoriessection) came the opportunity to observe,in microscopicdetail, the dynamicalprocessesthat areassumedin the statisticaltheories.An immediateoutcomeof thesestudieshasbeenthe recognitionthat phasespaceis very complex;for realisticsystems,it is not homogeneous,but full ofstructures.Moreover, all of thesestructuresinfluencethe motion of phasepointsin differentways. It isworth rememberingat this point that the inclusion of quantummechanicsalters thesephasespacestructuresandtheir effecton dynamics.For moreon the quantum-mechanicalstructureof phasespace,see,for instance,Davis andHeller [1981],Heller andDavis [1981],Heller [1983],DeLeonet al. [1984],Davis andHeller [1984],Davis [1988],andSkodjeetal. [1989].Ratherthandelving into the impactofmodernnonlineardynamicson the studyof molecularsystems,let usstartwith an observation:whenthe timedependenceof modeenergieswas studied(say in a modelof OCS— moreof that later),it wasfound that somesystemswere riddled with long-time correlations.Mode energiesdid not decay (orincrease)monotonicallyto their statisticallimit, but instead,phasepoints sometimesappearedto betrappedin certainregionsof phasespacefor many vibrationalperiods. In otherwords,thesesystemsfailed to relaxstatistically.Clearly, onecan expectthe dissociationbehaviorof such molecularsystemsto be nonstatisticaland nonrandom,also.

It hadbeenknown that at high energies,phasespacemay be divided such that quasiperiodicandchaoticregionscoexist [Chirikov 1983]. While quasiperiodiccomponentsin the availablephasespaceimply incompleterelaxation, long-time correlationslead to slow relaxation. For applying statisticaltheories, such “nonrandom” componentsmust be discounted. Thus, DeLeon and Berne [19811correctedtheir RRKM calculationfor quasiperiodicmotion by excludingit from their measureof phasespace,much in the spirit of empirical correctionsto statisticaltheories.

The adventof nonlineardynamicshasprovidedthe neededsupportfor a microscopicexaminationofphasespacein general,and the problemof long-time correlationsin particular [Contopoulos1971,Chirikov 1979, DeLeon and Berne 1982a,b;Vivaldi et a!. 1983; Meiss et al. 1983; Karney 1983;Casartelli 1983, Kato 1985a]. Some long-time correlationshave been interpretedas “vague tori”[Reinhardt and Jaffé 1981; Jaffé and Reinhardt1982; Shirts and Reinhardt1982; Reinhardt1982;Kuzmin et a!. 1986a]. In the backgroundto the theory, we discover the commonthreadthat runsthrough this review, namely the influence of nonlinear resonances.For our purposes,findings ontwo-degreeof freedomsystemswill be sufficientfor makingour point (for a discussionof systemswithmoredegreesof freedom,see Lichtenbergand Lieberman[1983],Martenset al. [1987]).


In the discussionof an isolated resonance,we haveseenhow resonancezones are boundedbyseparatrices.Irregularmotion appearsaroundseparatricesat low energies.At higher energies,evenasseparatricesfragment, irregular motion may still fail to spreadwidely becauseit may be restrictedtotori andKAM curves.The last KAM curve to disappearin the StandardMap is a torusthat is formedwhen the ratio of (the two) local frequencies,a, is relatedto the GoldenMean y, viz.

a=l+y 7(s1/2_1)/2. (5.24)

The break-upof this “Golden Torus” (which is a “cantorus” in the terminologyof Percival [1979])hasattractedthe attentionof researcherssearchingfor a transitionto a global stochasticity[Greene1968;Berry 1978; Greene1979; EscandeandDoveil 1981; Shenkerand Kadanoff 1982; MacKay 1983].Crudely speaking,the Golden Mean is the irrational numberwhich is hardestto approximatebyrationals[Berry 1978]. Intuitively, the “Golden Torus” as the last obstacleto unrestrictedtransportinphasespaceis appealing,becauseit is found in phasespaceregionswherethe coupledoscillatorshavethe greatestdifficulty of driving eachotherresonantly[Davis 1985].

Basedon the cited earlier work on transition to global stochasticity,MacKay et al. [1984]andBensimonandKadanoff[1984]developeda quantitativeapproachto transportin area-preservingmapsthat gives ratesof relaxationfor systemsof two degreesof freedom,evenwhenlong-timecorrelations,which are such stumbling blocks to traditional statisticalrelaxationapproaches,exist [Meissand Ott1986]. While researchinto the fundamentalsof diffusion is outside the scope of this review, theinterestedreadercan consultLichtenbergandLieberman[1988],LichtenbergandWood [1989a,b]forcoverageof somerecentresearch.The applicationof thesemethodsto intramolecularenergytransferwas undertakenby Davis [1984]who usedthe term“intramolecularbottleneck”to describephasespacestructuressuch as canton since eventhe brokenremnantsof aKAM torus actasbottlenecksto the flowof phasepoints. This terminologyemphasizesthe connectionof this recentwork to earlier statisticaltheories;indeed,the work of MacKayet a!. [1984]andBensimonandKadanoff[1984],possessesmanyof the featuresof statisticalreactionrate theories.It is also reminiscentof periodic-orbit approachtochemicalreactions[Pollaket a!. 1980; Pollak 1985].

Using this work on phase-spacetransport,Davis [1985]modeledenergyrelaxationin highly excitedOCS restrictedto collinear geometry[Carter and Brumer 1982, 1983; Hamilton et al. 1982], wherelong-timecorrelationswerepreviouslyobserved[DavisandWagner1984;Davis 1984]. Figures9a andbillustrate a Poincarésurfaceof sectionfor a single trajectory.The surfaceof section is for the normalmodeof OCSwhich is principallyCS stretchin character,about80% of CS characterin the low-energylimit. In addition to the trajectory, which appearsas dots, thesefigures show three curves. Theoutermostcurve illustratesthe energyboundary.The middle curveshowsa dividing surfacewhich is anapproximation to the last KAM curve separatingthe two primary resonancezones.The innermostcurve showsthe boundaryof a quasiperiodicregion. Davis [1984]was able to calculatethe flux throughthe dividing surfaceand calculatethe relaxationrate for systemswith suchlong-time correlationsandslow randomization. A comparisonof the quantum-mechanicalintramolecular dynamics with theclassical one for OCS has beenperformedby Gibson et al. [1987].These calculationshave beenextendedand the quantum-mechanicalnatureof statesaroundthesephase-spacestructures,especiallyin the chaotic regime, have been discussedby Davis [1988]. Although outside the scope of thisoverview, we mention that the role of phase-spacebottlenecksin unimolecularreactionshasbeenexaminedby Davis and Gray[1986],Grayet a!. [1986b]as well as Grayand Rice [1987]andTersigniand Rice [1988].Therearesimilar studieson bimolecularreactionsby Davis [1987],SkodjeandDavis[1988],andSkodje[1989].The quantum-mechanicalmanifestationsof thesebarriershaverecentlybeen


~~

~ ~

:: ~ ~ ‘~ ~

-100 0 100 200 -100 0 100 200

0 0Fig. 9. Theeffect of astrong bottleneckis shownfor collinear OCSat E = 20000cm ‘. a. For thefirst 24.6 ps of propagationthetrajectory movesoutsideof thedividing surface.b. For thenext 4.5 ps thetrajectory is inside the dividing surface,but cannotenterthe inner quasiperiodicregion.(From Davis [1985]).

uncoveredby Brown and Wyatt [1986a,b],Geisel et at. [1986],and this work has beensurveyedbyRadonset al. [19891.

6. “Mesoscopic” descriptionof intramolecularvibrationalenergytransfer

In the precedingfour sections,we have seenthe various determinantsof intramolecularenergytransfer.We startedwith a quantum-mechanicaldescription,and studiedenergytransferin a state-by-statefashion with few assumptionsabout the detailednatureof the couplings. Later, we went on tononlinearoscillatorchainsand saw the importanceof nonlinearresonancesas well as the relevanceofthe KAM theoremfor thesescenarios.Finally, we describedenergytransferthrougha single as well asoverlappingresonancesin greatdetail by meansof classical mechanics.In this sectionwe presenta“mesoscopic”combinationof the quantumstatistical-mechanicalview with the detail of the nonlinear-dynamicalpoint of view [Kuzminand Stuchebryukhov1985; Kuzminet a!. 1986a;Stuchebrukhovet at.1986; Stuchebryukhov1986], which usesquantalGreenfunctionmethods(seee.g.,Couley [1963])tocalculatequantitiesandtrendsof interestfor IVR whenthe coupling is explicitly known. The approachreportedhereis in contrastwith thework of Hansonet a!. [1985]andMeiss andOtt [1986],on Markovchainsand trees in area-preservingmapswherea (discrete-time)masterequationis deriveddynami-cally.

It will emerge,againexplicitly, that undersuitableconditions,therecan be energyregionswhereagiven excitedmodefails to relax(for generalreferences,seeKuzmin et al. [1987a,b]).Obviously,whenmodesin a highly excitedmoleculeare decoupledin this mannerabovethe dissociationenergy, thedissociationof the moleculewill not be governedby the ordinarystatisticalreactionratetheories.Suchhighly excitedsystemshave been discussedby Noid and Koszykowski [1980],Wolf and Hase [1980b],Waite and Miller [1980,1981,1982], Hedgesand Reinhardt[19831,Bai et a!. [1983],HoseandTaylor[1984],Christoffel and Bowman [1982,1983], Chuljian et al. [1984],Moiseyev and Bar-Adon [1984],Skodjeet al. [1984a,b],Swamyet a!. [19861,Changet al. [1986],andKuzmin et al. [1986b].Theworkin this sectionhas recently been reviewed, and its implication to molecular lineshapesand widthsemphasized[Stuchebrukhovet a!. 1989].


6.1. Self-consistentdescription of quasiharmonicmoderelaxation

In a molecule,the frequenciesof variousmodesareenergydependent,as arethe relaxationratesofthe excited modes. At high excitation levels, the molecule can be describedin the self-consistentquasiharmonicapproximation [Kuzmin and Stuchebrukhov1985; Stuchebrukhov1988] as an en-semble of vibrational quasimodes{k} with energy-dependent frequencies wk(E). While the modefrequenciesvary, each vibrational mode is modeled by a series of equally spacedlevels (a briefjustification will be given in the next section).Moreover,eachmodehasa corresponding(exponential)relaxationrate yk(E). At low energies,the frequenciesturn into normal-modefrequencies,and the

yk(E) = 0. As the energyof the moleculeis raisedabovea certaincritical valueof E~,eachmodebeginsto undergodampingat its own rate Yk’ which describesthe energyexchangeprocessamongthe modes.This thresholdbehaviorof transition to a quasidissipativeregimehasbeen observedexperimentally(e.g., Bagratashviliet al. [1981],Felkerand Zewail [1984])andin numericalmodeling[TennysonandFarantos1984; Livi etal. 1985]. In this model, the appearanceof nonzeroycorrespondsto a transitionto chaotic intramoleculardynamicswith exponentialdampingof the correlations[Koszykowskiet al.1981; Bermanand Kolovsky 1983; Zaslavsky1985].

As an example,considerthe Hamiltonian

H= ~ ww(p~+ q~)+ ~ X1J~q~q1q~, (6.1)i= I jj/~~=1

where q, are the normal coordinatesand p, are the correspondingmomenta. Assume that thenonlinearities{X} aremuchsmallerthanthe unperturbedfrequencies,i.e., that this is a “degenerate”systemvery muchlike the one discussedby Fordand Lunsford [1970];if the unperturbedfrequencies

~ satisfycertainresonancerelations,randomizationcan occurevenat very smallmolecularenergies.But for large and intermediatesize molecules, a more typical situation is one where the variousresonancesdoes not satisfy exact resonancerelations, but approximateones with characteristicdetuningsof the order 10—100wavenumbers,and the randomizationbeginsat finite energiesE~.Theappearanceof relaxationin this model hasa thresholdcharacterat finite energies,andmoreoveris inharmony with experimentaland simulationstudies,as discussedbelow (the qualitativechangein themotion at a critical energyhas been analyzedusing critical point analysis in Cerjan andReinhardt[1979]).

The existenceof close resonancesfor severaldegreesof freedomleadsto a situationwhere thecritical energyE~is much lower than the dissociation energyD, in contrastto muchsmallersystemslikethe Hénon—HeilesHamiltonian (Oxtobyand Rice [1976]),wherethe resonancesoverlapwith respectto onecoordinate,andE~is closeto D (for the inconclusiveevidenceconcerningthe existenceof E~formore than two degreesof freedom, seeLichtenbergand Lieberman[1983],chapter6).

Information relevant to IVR, such as correlation functions, the absorption spectrum, and therelaxationcharacteristics,are contained in the system’s linear responsefunction, i.e., the retardedGreenfunction [Couley 1963] which is used to computethe vibrational relaxationrate.

The calculationproceedsby the computationof the responseof an ensembleof systems(6.1) to aninfinitely weakexternalmonochromaticfield andfollowing how this responsechangesas the energyofthe moleculeis increased.Molecularcollisions areignoredand the only role thatthe ensembleplays isto ensurethe appropriateaveraging.The linear responseof the systemis determinedby the correlationpropertiesof the unperturbedmotion; therefore,we are actually studyingthe motion of the isolatedsystem(6.1) with averagedinitial data. To simulate experimental conditions the calculation wasperformedfor a thermaldistribution.

118 T. Uzer, Theories of intramolecularvibrational energy transfer

The responseof the ensemblewith the Hamiltonian (6.1) to anexternalperturbationis given by the

retardedGreenfunctionG1~= iO(t)([q,(t), q~(0)~, (6.2)

wherethe q,(t) arethe operatorsin the Heisenbergrepresentation,0(t) is the Heavisidestepfunction,and the averagingis over the thermaldistribution with the exact Hamiltonian (6.1)

..) =Tr(e H .)/Tre~ (6.3)

The imaginary part of the Fourier transformGR(w) is proportionalto the absorptionspectrumof thesystem. Consider the form of GR(W) for a few simple cases.For a molecule in the harmonicapproximation,i.e., when XlJk = 0, the frequency-dependentGreenfunction is

G~°~(w)= ~6lk[1/(w + Wok) — 1 /(w — wok)] . (6.4)

The absorptionspectrumconsistsof a single line at the frequencyWok, (the samefrequencywith theoppositesign gives the stimulatedemissionspectrum).If eachof the harmonicmodeswere exponential-ly attenuatedat the rate Yk’ as a resultof, for example,contactwith an infinite energyreservoir,thenwe have

G~°~(w)= ~k( 1 , — 1 (6.5)I ‘\w+Wk+lyk/

2 WWk+1YkI2I

The relaxationof energyin eachof the modesto the equilibrium stateoccursat a rate Yk~In the case(6.5), the contributionof the kth modeto the total absorptionis given by the Lorentzian

1 Yk’2

(6.6)

The details of the Greenfunction calculationaregiven in the articlesof Kuzmin et a!. [1986a]andStuchebryukhov[19861.The exact expressionfor the G~containsa very largenumberof closelineslying in the absorptionspectrumabout(hp)~apart,but thelow-lying lines in the absorptionspectrumof the system,or the envelopeof the spectrumcan beobtainedreadily. The result is a setof nonlinearequationsfor the quasimodefrequenciesWk and the relaxationratesYk’ viz.

1 V 2/ (n~+nJ+l)(wI+wJ—wk) (n~+nI+1)(w~+WJ+wk)Wk = Wok — ~ 0=’ XIJk~( + (V

1 — Wk)2 + (y~+ + Yk)’4 + (w

1 + (01 + Wk)2 + (y~+ + Yk)’4

+ (nJ—n~)(W~—wJ—wk)2 + (nI—nJ)(W1—wJ—Wk) 2 ) (6.7)

(w~—w~—wk)-l-(y~+yJ+yk)I4 (W/—&~—Wk) +(y~+yJ+yk)/4

— 1 ±x2 ~ (n~+n1+l)(y,+yJ—yk) + (n~+nJ+l)(y~+yJ+y~)

Yk~•, IJk~(W+W_W)2+(y+y+y)2/4 (w1+wJ+Wk)2+(y~+yj+yk)2I4

+ (nJ—n~XyI—yJ—yk)2 + (~I—~JX~—YJ—Yk) 2 )~ (6.8)(wI—~—wk)+(y~+y]+yk)/4 (w~—wj—wk) +(y~+yJ+yk)I4


which mustbe solvedself-consistentlyat everyinternalenergy.Here,n, arethe Boltzmannpopulationsof the ith mode at a temperatureT (for experimentaldeterminationof vibrational temperature,seeWeitz andFlynn [1981]).In computing these,the unrenormalizedfrequenciescan beusedbecausethefrequencyshifts arevery small comparedto the frequenciesthemselves.The dependenceof Wk and Yk

arisesfrom their dependenceon ~ Clearly, when the systemis harmonic,thenthe only solutionstothe systemare the normal modeswhich do not decay.These solutions survive at high energiesbutthere,thereexists anothersolutionwith y different from 0, which becomesthe principal solution. Forhigher temperatures,a number of useful expressionscan be derived from (6.8) indicating how theFermi Golden Rule is reachedwith the densityof three-frequencyresonances[Bagratashviliet al.1985]. The model becomesless valid near the transition energyE~,as is characteristicof othermean-fieldtheories.

When theseequationsareapplied [Kuzminet a!. 1986a]to smallpolyatomicmoleculeslike CFCI2Br,the resultsarefound to be in agreementwith experiment[Bagratashviliet al. 1981; Felkerand Zewail1984]. In general, the ‘ show a nonmonotonicincreasewith total moleculeenergybecauseof theintricate way manynonlinearresonancesact in concertto bring about relaxation.Sincethe frequenciesof the modesare energydependent,the correspondingresonancedetuningscan either increaseordecreaseas the moleculeenergyincreases.Of course,whendetuningsincrease,-r will tendto decrease.In particular, occasionallyy can vanish at E> E~(seefig. 10). This implies that thereareno resonancesfavorablein thismoleculeto drive relaxation.In the samefigure, the detuningof the nearestresonancesis shown. In order for gammato be nonzero,at least two resonancesmust overlap. The intimateconnectionbetweenrelaxationand the overlapof resonancesis shown in fig. 11, which shows thedependenceof “1 for this moleculeon thenormal-modefrequencywe,. We can clearlyseetheresonancestructureof the dependenceof y,(Wo,) for small anharmonicityconstants.For large anharmonicity

Yi,~1 / 20 ~16.8

&cmi / 16 ~ -i-U) (05+0)7— -5.3 0 8.2 12.3 — 6~ 8 ,e, ~

8 1 8

6.72 13.44 20.16 X,cm1 955 970 985 1000 1015

X~1 Xc2 X~~3 (010,cm

1

Fig. 10. Dependenceof the intramolecularrelaxationratefor mode 1 Fig. 11. Dependenceof the intramolecularrelaxationratefor mode 1on the anharmonicityconstantX = X

1~.The system is one of nine on thenormal-modefrequencyw0~.The systemis thesameasfor fig.oscillatorswith frequencies1012, 945, 844, 777, 558, 527, 461, 446and 10; ihe energyE= 4677cm~.(From Kuzmin et al. [1986a]).380cm’. The energy of the system is constant, and equal to4677cm’.The dependenceof y(E) has a similar form: y=O in theregionsE< E~and E~2<E< E~1.(From Kuzmin et al. [1986a]).


6 E~X2=const

rII~3.4 6.7 10.0 X,cm1

Fig. 12. Dependenceof the critical energyE~for the CFCI2Brmolecule on themagnitudeof the anharmonicityconstantX. (From Kuzmin et al.

[1986a]).

constantsX, the effective interaction leads to the immediate overlappingof several resonances.Thereforethe vanishingof y whenE> E~(fig. 10) can ariseonly in the caseof moderateanharmonicityconstants.

As to the dependenceof the limit E~on the systemparameters,fig. 12 showsthis dependenceon theanharmonicityconstantX = X.J~.As expected,for very large anharmonicities,evenzero-pointvibra-tions are not independentof eachother. The approximatedependenceof E~on S, the number ofcoupleddegreesof freedom is, for three-frequencyinteractions,E~— (X

2S3)~,and this relation isaccurateto 25% [Kuzmin et al. 1986a].

6.2. Relaxationof individual quasiharmonicmodesand its spectralsignature

We haveseenin the precedingsectionthat the relaxationof modesin amoleculecan be describedbya generalset of nonlinearcoupledalgebraicequations,one setfor the quasimodefrequenciesWk~andanotherfor the exponentialrelaxationrates Yk.

In this section,we will focus on the relaxationcharacteristicsof a single excitedmode [Makri andMiller 1987], sincemany modernexperiments(e.g., thoseinvolving overtoneexcitation, to nameoneclass) approximatethis scenario,and will examinethe effect IVR hason the IR spectrum[Hetler andMukamel 1979; Kuzmin et at. 1987a,b;StuchebrukhovandKhromov 1987; Ionov et a!. 1988]. For theeffect of collisional relaxationon spectralfunctions,see Koszykowskiet at. [1982b1.Characteristicofthe subsystem

Y1=’~({~k},{Yk}) (6.9)

is the fact that if the energyexceedsthe thresholdenergyE~greatly, the dependenceof the right-handsides 1 on the variables{~}is very weak, and the correlation between the various solutions y,disappears,and each of them can be found independentlyof the othersby meansof “Golden-Rule”type of recipes

2 res= 2~r~Vç(E)~p (Wi), (6.10)


where (V1(E)~is the effectiveinteraction,whichdependson the energyof the molecule,andp’~is thedensity of the Fermi resonances.This implies that for such energies,we can in the treatmentofmolecular mode relaxation, treat the remaining modes as an energy reservoir, the correlationcharacteristicsof which can be foundindependentlyof thestateof modei. In this sectionwe will, alongwith Stuchebryukhov[1986]considersuch a situation,and see that such an assumptionenablesus toinvestigateconditionsin which purely phaserelaxationcan be separatedfrom energytransferprocesses.

The validity of this approachat high energiescan be demonstrated[e.g., Zaslavsky 1985] bycomparingrelaxationtimes ~ of the modeof interest,andthe dampingtime r~of the correlationsofthe correspondingvariablesof the reservoir.For the separationof a selectedmodeand the dissipativereservoir, it is necessarythat the former be much larger than the latter. (For high energies,thiscondition is valid for molecules,as shown in Bagratashviliet at. [1985],Stuchebryukhov[1986]).

The stateof the moleculeconsisting of an excitedmode (one which hasbeen excited by a laser, say)anda reservoirof modesinactive in suchabsorptionis specifiedby the densitymatrix r of the excitedmodeand the vibrational temperatureof the reservoir.This is also known as the “pumped mode”approachin multiphoton excitation (e.g., Mukamel [1978],Stephensonet a!. [1979],Letokhov andMakarov [1981],Kay [1981],Stone et at. [1981],Lupo and Quack [1987]). For various theoreticaldescriptionsof the multiphotonexcitationprocesses,seeBagratashviliet at. [1985],Von Puttkamereta!. [1983],Lupo and Quack[1987],Grayet al. [1985],Rashev[1985],RashevandKancheva[1987],aswell as referencestherein.The PM approachis not as practicalas its alternative,manytimesreferredtoas the NME (nuclear molecular eigenstate)approach [Scheckand Jortner 1979] since it cannotdeterminethe populationsof excited levelsdirectly. But within its frameworkit is possibleto describein detail the intramolecularrelaxationprocesses(of the T, and T2 kinds, e.g.,Kay [1981],Stoneet al.[1981],Budimir andSkinner[1987]).The procedureadvocatedby Bagratashviliet at. [1985]is differentfrom a numberof thesemultiphoton excitation (MPE) theories,becausein addition to treating thevibrationaltemperature,the population in variousvibrationallevelsandthespectroscopicanharmonici-ty constants,it also takesinto accountthe relationsbetweenthe vibrational frequencies,which turnsout to be vital since it is the densityof intermoderesonancesratherthan the full level density thatdeterminesthe efficiency of MPE. The multiphotonexcitationdynamicsin thelaserfield is describedbya systemof closedequationsfor if and T (thesevariablesare coupled as a result of the possibleexchangeof energybetweenthe excitedmode and the reservoir).

On the otherhand, the absorptionspectrum,as well as the relaxationof the nearequilibriumstatescan be studiedat a fixed reservoirtemperature.The interactionof the excitedmodewith the reservoiras a resultof the anharmoniccoupling terms gives rise to the processesof dephasing(e.g., Mukamel[1978])and relaxationof the energyfrom the excitedmode,andthis is describedby the correspondingrelaxationoperatorI’~in theequationfor o. Furthermore,the anharmonicinteractionleadsto a changein the dynamicalpart of the equationfor if, a situation which can, in certaincases,be describedas aredefinitionof the Hamittonianof the excitedmode.

The simplestderivationof the kinetic equationis obtainedwhen: (1) the Markovianapproximationis valid, i.e., whentherelaxationtime rof the systemis muchlongerthanthe correlationtime r~for thereservoir;and (2) thereis no frequencydegeneracyin the system,i.e., if the pairsof levelsm, n andm’,n’, do not coincide,then

k0mn — (OmflI ~ T1, (6.11)

whereT is the relaxationtime of the system (this is often called the “secular approximation” in spin

122 T. Uzer, Theoriesof intramolecular vibrational energy transfer

relaxation). The first condition is necessaryfor the kinetic equationto be differential ratherthananintegrodifferentialequation,as in the caseof a systemwith memory [Mukamel 1981]. The secondconditionleadsto asituationin which theoff-diagonalelementsof the system’sdensitymatrix attenuateindependentlyof eachother, (the random-phaseapproximation)andthe off-diagonalpart of the kineticequationhas the form

dcrmn/dt ~(1IT2)mnUmn , (6.12)

whereT2 is the phaserelaxationtime, therebeingsucha time for eachpair of levels m, n. In the casewhen condition 2 is fulfilled, the off-diagonal elements relax like a set of uncoupledtwo-levelsubsystems.In the casethat the systemhasfrequencydegeneracy,as in the caseof the harmonicoscillator [Sazonovand Stuchebryukhov1981, 1983; Parris and Silbey 1987], to take an extremeexample,the relaxationof the off-diagonal elementsoccursmuch more slowly. The various matrixelementsare coupledand in the generalcase,the equationhas the form

= ~ ~ (6.13)

where the Rmnkl are coefficients. Thereby there is a distinctive coherencein the damping of theoff-diagonal elements,and the systemrelaxesas a whole.

Since the relaxationof the off-diagonalelementsdeterminesthe absorptionspectrumof the system,the questionof how this relaxationoccursis relevantfor the study of the linear absorptionspectrum(for samplecomputationsof the spectrum,seeAkulin and Kartov [1980],MakarovandTyakht [1982]).It hasbeenpointedout by Kay [1981]that allowancefor the coherenceeffectsdiscussedabovecan leadto a significant narrowingof theabsorptionspectrum.For furtherdiscussionof theeffect of IVR on thespectrum,seeTrue [1983].

Assumeat first thatthe relaxationtime of the system,T, is muchlargerthanthe correlationtime ‘re.It is easilyarguedthat condition2 doesnot hold for an excitedmolecule.The quantity T

1 determinesthe characteristicwidth of the absorptionspectrum.This quantity is much largerthanthe anharmoniclevel shifts6, andit is indeedthis conditionthat makesmultiphotonexcitationof polyatomicmoleculespossible in the quasicontinuum.The quantity 6 also characterizesthe remotenessof the levels of avibrational mode, and lies in the range of 1 to 4 wavenumbers.The spectral width of the excitedmoleculesrangesfrom valueslarger than 10—30 wavenumbersand the condition

6 (6.14)

is fulfilled. But 6 is alsocharacteristicof the frequencydifferencek°mn— ~~mn’ andthereforethe secondconditionaboveis not fulfilled. Indeed,the oppositeis true,whichindicatesthat the nonequidistanceoflevels is not importantfor the relaxation,andthe vibrationalmode can be modeledby a setof equallyspacedlevels. This is the quasiharmonicmodel [Bagratashviliet al. 1985] wherethe modefrequenciesdependon the modeenergy,but the levels remainequidistant.

In most past work it is implicitly assumedthat condition 2 is fulfilled. This may lead to anoverestimateof the width of the IR spectrum,which could be as large as an order of magnitude.Incontrastto a numberof articleswherethe above-discussedkinetic equationfor the densitymatrix isused,the work of Kay [1981]is carefulto takeinto accountthe secondcondition.It is alsoworth notingthat condition 2 becomesevenharderto satisfy in the limit h—~0 [Parson1988].


Based on the preceding argument, many authors [Sazonovand Stuchebryukhov1981, 1983;Bagratashvitiet a!. 1985; Kuzmin et al. 1986a;Stuchebrukhov1988] haveopted to usethe quasihar-monic approximationratherthanthe random-phaseapproximation.The simplestvariantof the kineticequationwith quasiharmonicmodeswas consideredin Bagratashviliet a!. [1985],wherea specialformfor the nonlinearcoupling was assumed,and purely phaserelaxationwas not considered.

Using the generalmethodof Lax [1964],Stuchebryukhov[1986]derivedkineticequationsdescribingthe relaxationof selectedvibrational modesof a moleculeto the equilibrium state,and the role ofvarious anharmonicterms causingrelaxationare analyzed.They discussthe types of interactionsthatgive rise to purely phaserelaxation(by a similar, thoughnot phenomenologicalprocessas Narduccietal. [1977]),andwhentheir contributionto the shapeof the absorptionspectrumcan beseparatedfromthe contributionsof the energyrelaxationprocess.Purelyphaserelaxationandits effecton the spectraof highly excitedmoleculeshasbeenconsideredrecentlyby MakarovandTyakht[1987],who, by usinga formalism similar to that of Nitzan and Persson [1985] for adsorbateson surfaces (for othertreatments,seePerssonand Persson[1980],Van Smaalenand George[1987],Volotikin et al. [1986],Lin et al. [1988]), were able to suggestprescriptionsfor extractingdynamicalinformation from thespectraof highly excitedmolecules,and to comparethis relaxationwith energyrelaxation.

In the work of Stuchebryukhov[1986], it is shown that when allowance is made for high-orderanharmonicinteractions,therelaxationoccursnonexponentiatty(for otherexamplesof nonexponentiatrelaxation,see Dumont and Brumer [1987,1988]) and the absorptionspectrumof a highly excitedmolecule is in general not Lorentzian (see, for example Bagratashvili et at. [1987]). Then thedependenceof the longitudinaland transverserelaxationrates(in the casethat they areseparable)onthe anharmonicity, the molecular frequencies, and the number of degreesof freedom can beinvestigated.They show that the effective molecularnonlinearity,which governsthe relaxationprocess,dependson the energyof the molecule. Whereasat low energiesthe relaxation is governedby athree-frequencyinteraction, i.e., by nonlinearitiesof third order, at high energiesthe higher-orderanharmonicitiesbecomeactivatedand the relaxationproceedsby many-phononprocesses.It is worthnoting that at high energies,we can identify the nonlinearity order m* that makes the greatestcontributionto the relaxationprocess.This orderdependson the energythe molecule.At low energiesit is three,and at energiesclose to the dissociationthreshold,can attain the value of ten.

Acknowledgement

This article in its current form would havebeenimpossiblewithout the help and advice of manycolleagues.I am grateful to J.M. Bowman,F. Budenholzer,M.J. Davis, D. Farrelty, M.R. Flannery,R.F. Fox, B.R. Foy, K.F. Freed, W.L. Hase,K.G. Kay, A.E.W. Knight, V. Lopez, R.A. Marcus, A.Nitzan, D.W. Noid, C.S. Parmenter, M. Quack, W.P. Reinhardt, D.W. Pratt, E.L. Sibert III, D.L.Thompson, G.A. Voth, and A. Zangwi!l for their contributions. Special thanks go to W.H. Miller forcontributingthe appendix,to R. ParsonandA.A. Stuchebrukhovfor manyclarifications,as welt as toE.W. McDaniel and Audrey D. Ralston,who were instrumentalin the writing of this article. I alsothank the United States National Science Foundation which supportedme (through grant CHE86-19298to the GeorgiaInstitute of Technology)duringmostof the writing of this article. I am gratefulto the Atomic Collision InformationCenterat the Joint Institutefor LaboratoryAstrophysics,Boulder,Coloradofor performinga literaturesearch.

I dedicatethis article to my colleagueJoeFord on the occasionof his 65th birthday.


Appendix. On the relation between absorption linewidths, intramolecular vibrational energyredistribution (IVR), and unimolecular decayrates

W.H. Miller

Here, a simple model (the standardone from the theory of radiationlesstransitions) is used toillustrate the relation (or lack thereof) between line widths in photoabsorptionspectra and theunimolecular decay rate of the excited molecule. Among the relevant experimentsare the recentphotoabsorptionspectraof ethylenedimersin a molecularbeam.The point madeis that a Lorentzian-like absorptionline may indeed be unresolvable,even in principle, and still not be related to thelifetime of the molecule(i.e., the unimoleculardecayrate).

With the current intense level of activity in high resolution vibrational spectroscopyand photo-dissociationof polyatomicmolecules,thereis considerablediscussionas to the interpretationof linewidthsin the absorptionspectrumwith regardto ratesof “intramolecularvibrational energyredistribu-tion” (IVR) and, in the caseof photodissociation,to unimoleculardecayrates.The typical situationisthat one observesa Lorentzian-likeabsorptionline andwishesto interpret the width of the line as arate;the question,which is addressedin this paper,is therate of what? (Thediscussionthroughoutthispaper relates to isolated molecule behavior; i.e., collisional and other environmentaleffects areassumedto be unimportant.)

The experimentsthat one has in mind are the excitation of overtones of CH stretch modes inpotyatomicmolecules[Henry 1977; Bray andBerry 1979; Reddyet al. 1982] and,morespecifically, thephotodissociationof Van der Waals clustersin supersonicmolecularbeams,[Smalleyetal. 1976; Levy1980; Cassassaet al. 1981; Hoffbaueret at. 1983; Fischeret a!. 1983] thoughthe sameconsiderationsarise in many other contexts. In the former case there is no dissociationinvolved, so there is noquestion of unimoleculardecay; the essentiallyuniversal point of view [Heller and Mukamel 1979;FreedandNitzan 1980; StannardandGelbart1981; Sibertet al. 1982a,b]is thatthe —~100cm~width ofthe Lorentzian-tikeline gives the IVR rate of the initially excited local modestate.Underidealized,infinite resolution,of course,this Lorentzian-like “line” is actually a densebundle of Dirac deltafunctions,the coefficientsof which havea Lorentzian-likedistribution, but this doesnot alter the IVRinterpretation.

In the latter case,however— e.g., the photodissociationof ethylene dimersin a molecularbeam[Cassassaet al. 1981; Hoffbaueret al. 1983, Fischeret a!. 1983]— it is not immediatelyobviouswhatthe‘—10 cm~width of the IR absorptiontine means.The Lorentzian-like line appearsto be “homoge-neous”by themost rigorousstandards;i.e., unlike thecaseof CH overtoneexcitation,it appearsthat itwould remain a single, smooth Lorentzianline evenin the limit of infinite resolution.Assumingthatthis is indeed true, does the width of the line give the unimolecutardecay rate (via the standarduncertaintyrelation)?

The purposeof this appendix is to illustrate the answer to this questionby consideringa simplemodelsystem.The model is borrowedfrom the standardtheoryof radiationlesstransitions,andindeedthe absorptionspectrumresulting from it is quite standard.The relation of this to the unimolecu!ardecayrate is the somewhatnonstandardpoint that is the purposeof the appendix.

The model is the standardone in the theory of radiationlesstransitions[Avouris et al. 1977]: azeroth-orderstate a~ is embeddedin, andcoupledto, abath of zeroth-orderstates n~.It is state athat carriesthe oscillatorstrengthfrom the initial, e.g., groundstateof the molecule; i.e., thereis nodipole matrix element from the ground state g) to any bath state n~.The bath statesIn) are


characterizedby complexenergies,E~— iI,12, E~beingtheir energyand f~their width for unimolecu-lar decay (rate= I,Ih). The optically allowed state a) is assumedto be nondissociative, i.e.,characterizedby the real energyEa.

The molecularsystemis thuscharacterizedby the complexHamiltonian matrix [Heller et al. 1978]

Ea ‘ Han

Hna, E~—iI~I2, 0 (A.1)‘1. 0

whereHan is the matrixelementcouplingstate a) to variousbath statesIn). The complexeigenvaluesof this matrix give the energiesand lifetimes of the eigen-metastablestatesof the isolatedmolecule.

The absorptionspectrumof the system,initially in its ground state, is given by

1(E)=—~Im(g~~G(E)~g), (A.2)

G(E)=1irn(E+ir—H)~, E=hW+Eg. (A.3)

Becauseof the assumptionthat pjg) only hascomponentsalongstate a), this becomes

1(E) = (al ~.~~g)J2(ir’) Im(alG(E)Ja) . (A.4)

Also of interestis the survival probability of the initial state a)

Pa,_a(t) (ale~hlt~la)l2, (A.5)

and the survival probability P(t) of the moleculeitself,

P(t) Pa_a(t)+ ~ P,,_~(t), (A.6)

Pn_a(t) = I (n I eh/~t~a) 2 (A.7)

The survival probability of state a), Pa_a(t), is the probability that at time t the systemis still in theinitially excitedstate a), white the survival probability of the molecule,P(t) is the probability that bythe time t the moleculehasnot decomposed.If the Hamiltonian matrix in eq. (A.1) were Hermitian(i.e., I~= 0), then the propagatore~hJt~would be unitary and the total probability P(t) conserved.This is the situationusuallyconsideredfor radiationlesstransitions,andfor exampleappliesto the CHovertoneexcitation discussedin the introduction: P(t) = I for all t, but is still the casethat Pa=a(t)

decays.Matrix elements of the propagatorin eqs. (A.5), (A.7) are most easily obtained by Fourier

transformingthe Green’sfunction,

etHt~= —(2~i)’f dEe~t~G(E), (A.8)


so that

Pa_a(t) = —(2~i)~J dE e~t~(alG(E)Ia) 2 (A.9a)

Pn_a(t) = —(2~i)~f dEe’~(nIG(E)la) 2 (A.9b)

Therefore, to determinethe absorptionspectrum, eq. (A.4), and the survival probabilities [Eqs.(A.5)—(A.7)], it is necessaryonly to havethe a—a anda—n matrixelementsof the Green’sfunction. Forthe model Hamiltonian of eq. (A.1) one readily obtains

HI2 -1(alG(E)la) = (E — Ea — E — E~+i~!2) (A.lOa)

(nIG(E)Ia) = Hna(E~E~+if~I2)~(alG(E)Ia). (A.lOb)

To proceedfurther it is necessaryto havethe energiesEa~{E~},the lifetimes {I~},and matrixelements Hna~which would require a very detailed molecular model (at best ab initio quantumchemistrycalculations).For illustrative processeswe considerherethe simplified versionof the modelfirst usedby Bixon andJortner[1968]:the bathenergies{E~}are assumedto be equallyspacedandthecoupling elementsHna and decaywidths 1,, constant,

E~=ne, n=0,±1,±2,...; 1~=1D; Hna=V. (A.lla,b,c)

The sumover bath statesin eq. (A.10) can then be evaluated,and one obtains

(aIG(E)la) = [E — Ea— ~T cot(ITE/r + iITTD/2r)]’ , (A.12a)

(nIG(E)Ia) = (rF/2ir)”2(E — ne + ~~~DY’(alG(E)la), (A.12b)

= 2irV2Ie. (A.13)

The model is thuscharacterizedby the unimoleculardecaywidth TD andthe width (i.e., rate) for IVR,[, which is seen to be the Golden Rule width for the decayof state a). For the remainderof theappendixenergyis measuredin units of the bathlevel spacinge — i.e., onesetse= 1 — andalso Ea is setto zero. (Taking Ea different from zero has only minor effectsthat are of no concernfor the presentconsiderations.)

Utilizing eq. (A.12) (with e= 1, Ea = 0), eq. (A.4) for the absorptionspectrumbecomes

1(E) = ~ — ~)2 + (1/2)21, (A.14)

T(1 + t2) F t(1 — T2)— ‘ T2 + t2 ‘ — 2 T2 + t2 . a,

T. Uzer, Theoriesof intramolecular vibrationalenergytransfer 127

t = tan(irE), T= tanh(~1rFD). (A.16a,b)

Examplesof the spectrumwill beshown in the nextsection,but hereit is usefulto notethat for FD ~- 1(which in practice meansthat FD~2)one has T—+1, so that the quantitiesin eq. (A.15) become

F—*F1, ~—*0, (A.17a,b)

i.e., when the unimoleculardecaywidth1’D is larger than the level spacing,1(E) becomesa simple

Lorentzianwhosewidth is the IVR width 1.It is somewhatmore difficult to determinethe survival probabilities Pa,...a(t) and P(t) in simple

analytic form for arbitraryvalues of FD. For FD ~- 1 (i.e., ~2), however, it is easyto do so, and theresultsare

Pa_a(t) = ~ P(t) (F1 e’~’~— FD el~t~)I(Fi— FD). (A.18a,b)

First, considerthe absorptionspectrum;this is very standardfare. Figures13, 14 and 15 show1(E),from eqs. (A.14), (A.15) for FD = 0.5, 1 and2, respectively.F~= 10 for alt cases.Recall that the energyspacingof the bath,e, is unity.

Figure 13 shows that onehasa resolvedspectrumwhich “sees” individual moleculareigenstateswhenthe unimoleculardecaywidth

T’D is lessthan the energylevel spacingof the bath. Figure 14 showsthe intermediatecase = 1, and fig. 15 the casethat T’D = 2 (twice the energylevel spacing).If FD isincreasedto anyvalue>2, (evenFD-.-*co), the spectrum1(E) is imperceptiblychangedfrom thatin fig.15.

Next considerthe survival probabilities.From eq. (A.18a) one seesthat in all casesthe survivalprobability of state a), Pa...a(t) decayswith the IVR rate F

11/l. The unimoleculardecayrate that oneinfers from eq. (A.18b) is

k~~1= min(FD, F1)//i. (A.19)1(E)

I I I I II I I I I I II I it

-10 -5 0 5 10

EFig. 13. Absorption spectrum(in arbitrary units) given by eqs. (A.14)—(A.16), for the caserD = 0.5. (I = 10 for figs. 13—15, and thebathlevelspacinge= 1.)


JJ~/JAJ~E\~I LI I I I I I I I I I I it

-10 -5 0 5 0

E

Fig. 14. Absorption spectrum(in arbitrary units) given by eqs. (A. 14)—(A. 16), for the case = 1. ([ 10 for figs. 13—15, andthe bath levelspacinge= 1.)

For all the casesin figs. 13—15, therefore,the unimoleculardecay rate is FDIIi., but if FD > ~ (=10here), then it is given by F1/h. This hasthe obvious interpretationthat for T’D < 1, decayof state a)into the bath is fast (rate= F11h) and decayout of the bath slow (rate = FDIh), while the converseistrue for FD > I~.In either case the unimolecular decay rate is the rate limiting step, the smallerof [/h and I’D/h. [For ~‘D = ~ F eq. (A.18b) gives P(t) = (1 + Ft/h) e_ut~,but still one has—lim~~dln P(t)/dt = F/h.]

The important observationsfrom theseresultsare:(1) if

T’D is lessthanthe molecularenergylevel spacing,thenthe width of the individual lines doesindeedgive the unimoleculardecayrate I’D/h;

1(E)

I I I I I I I I I I I I I_i~_.l-10 -5 0 5 10

EFig. 15. Absorption spectrum(in arbitrary units) given by eqs.(A.14)-(A.16),for the caseID = 2. (II = 10 for figs. 13-15, and thebath levelspacingv = 1.)


(2) however,if I’D is greaterthan the level spacing, thereis no way to identify the unimolecutardecayrate from the absorptionspectrum;and

(3) in anyevent,I’D <1 or I’D > 1 (i.e., figs. 13, 14 or 15), the width of the completeband,resolvedor unresolved,may be interpretedas the rate of IVR, I, which is the rate of decayof the initiallyexcitedoptically active state a).

What, then,can oneconcludein the casethata smoothLorentzian-!ikeline is observed,as for thephotodissociationof ethylene dimer discussedin the introduction, which is clearly broaderthan themolecularenergylevel spacing (which can be estimatedby statistical approximations)?First, onecanclearly identify the observedwidth as I, the width for IVR, which gives the rate for decay of theoptically active zerothorderstate initially excited. Alt that one can concludeaboutthe unimoleculardecayrate,however,is that it lies betweenc/h (e molecularenergylevel spacing)andF1/h. Thereisno way in principle, evenwith arbitrarily high resolution, to determinethe unimoleculardecayratefrom theabsorptionspectrum.(It shouldbe noted, though,that otherexcitationmechanismsmayexist,involving operatorsotherthan the dipole operator,which excite individual eigenmetastablestates,andin this casethe linewidth would indeedgive the unimoleculardecayrate of the individual eigenmeta-stablestates.)

Acknowledgement

This work has beensupportedby the National ScienceFoundationGrant CHE-79-20181.

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