energy redistribution in reacting systems

A3.13Energy redistribution in reacting systems

Roberto Marquardt and Martin Quack

A3.13.1 Introduction

Energy redistribution is the key primary process in chemical reaction systems, as well as in reaction systemsquite generally (for instance, nuclear reactions). This is because many reactions can be separated into twosteps:

(a) activation of the reacting species R, generating an energized species R:

R R (A3.13.1)

(b) reaction of the energized species to give products.

R P. (A3.13.2)

The first step (A3.13.1) is a general process of energy redistribution, whereas the second step (A3.13.2)is the genuine reaction step, occurring with a specific rate constant at energy E. This abstract reactionscheme can take a variety of forms in practice, because both steps may follow a variety of quite differentmechanisms. For instance, the reaction step could be a barrier crossing of a particle, a tunnelling process ora nonadiabatic crossing between different potential hypersurfaces to name just a few important examples inchemical reactions.

The first step, which is the topic of the present chapter, can again follow a variety of different mechanisms.For instance, the energy transfer could happen within a molecule, say from one initially excited chemicalbond to another, or it could involve radiative transfer. Finally, the energy transfer could involve a collisionaltransfer of energy between different atoms or molecules. All these processes have been recognized to beimportant for a very long time. The basic idea of collisional energization as a necessary primary step inchemical reactions can be found in the early work of vant Hoff [1] and Arrhenius [2, 3], leading to the famousArrhenius equation for thermal chemical reactions (see also chapter A3.4)

k(T ) = A(T ) exp(EA(T )

RT

). (A3.13.3)

This equation results from the assumption that the actual reaction step in thermal reaction systems can happenonly in molecules (or collision pairs) with an energy exceeding some threshold energy E0 which is close, ingeneral, to the Arrhenius activation energy defined by equation (A3.13.3). Radiative energization is at the basisof classical photochemistry (see e.g. [47] and chapter B2.5) and historically has had an interesting sideline inthe radiation theory of unimolecular reactions [8], which was later superseded by the collisional Lindemann

Copyright 2001 IOP Publishing Ltd

mechanism [9]. Recently, radiative energy redistribution has received new impetus through coherent andincoherent multiphoton excitation [10].

In this chapter we shall first outline the basic concepts of the various mechanisms for energy redistribution,followed by a very brief overview of collisional intermolecular energy transfer in chemical reaction systems.The main part of this chapter deals with true intramolecular energy transfer in polyatomic molecules, whichis a topic of particular current importance. Stress is placed on basic ideas and concepts. It is not the aim ofthis chapter to review in detail the vast literature on this topic; we refer to some of the key reviews and books[1132] and the literature cited therein. These cover a variety of aspects of the topic and further, more detailedreferences will be given throughout this review. We should mention here the energy transfer processes, whichare of fundamental importance but are beyond the scope of this review, such as electronic energy transfer bymechanisms of the Forster type [33, 34] and related processes.

A3.13.2 Basic concepts for inter- and intramolecular energy transfer

The processes summarized by equation (A3.13.1) can follow quite different mechanisms and it is useful toclassify them and introduce the appropriate nomenclature as well as the basic equations.

A3.13.2.1 Processes involving interaction with the environment (bimolecular and related)(a) The first mechanism concerns bimolecular, collisional energy transfer between two molecules or

atoms and molecules. We may describe such a mechanism by

M + R M + R (A3.13.4)

or more precisely by defining quantum energy levels for both colliding species, e.g.

{M(EMi) + R(ERi)}I {M(EMf) + R(ERf)}F. (A3.13.5)

This is clearly a process of intermolecular energy transfer, as energy is transferred between two molecularspecies. Generally one may, following chapter A.3.4.5, combine the quantum labels of M and R into onelevel index (I for initial and F for final) and define a cross section FI for this energy transfer. The specificrate constant kFI(Et,I) for the energy transfer with the collision energy Et,I is given by:

kFI(Et,I) = FI(Et,I)

2Et,I

(A3.13.6)

with the reduced mass: =

(1mM

+1mR

). (A3.13.7)

We note that, by energy conservation, the following equation must hold:

EMi + ERi + Et,I = EMf + ERf + Et,F. (A3.13.8)

Some of the internal (rovibronic) energy of the atomic and molecular collision partners is transformed intoextra translational energyEt = Et,FEt,I (or consumed, ifEt is negative). If one averages over a thermaldistribution of translational collision energies, one obtains the thermal rate constant for collisional energytransfer:

kFI(T ) =(

8kBT

)1/2 0

x exp(x)FI(kBT x) dx. (A3.13.9)


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We note here that the quantum levels denoted by the capital indices I and F may contain numerous energyeigenstates, i.e. are highly degenerate, and refer to chapter A3.4 for a more detailed discussion of theseequations. The integration variable in equation (A3.13.9) is x = Et,I/kBT .

(b) The second mechanism, which is sometimes distinguished from the first although it is similar in kind,is obtained when we assume that the colliding species M does not change its internal quantum state. Thisspecial case is frequently realized if M is an inert gas atom in its electronic ground state, as the energies neededto generate excited states of M would then greatly exceed the energies available in ordinary reaction systemsat modest temperatures. This type of mechanism is frequently called collision induced intramolecular energytransfer, as internal energy changes occur only within the molecule R. One must note that in general thereis transfer of energy between intermolecular translation and intramolecular rotation and vibration in such aprocess, and thus the nomenclature intramolecular is somewhat unfortunate. It is, however, widely used,which is the reason for mentioning it here. In the following, we shall not make use of this nomenclature andshall summarize mechanisms (a) and (b) as one class of bimolecular, intermolecular process. We may alsonote that, for mechanism (b) one can define a cross section fi and rate constant kfi between individual, non-degenerate quantum states i and f and obtain special equations analogous to equations (A3.13.5)(A3.13.3),which we shall not repeat in detail. Indeed, one may then have cross sections and rates between differentindividual quantum states i and f of the same energy and thus no transfer of energy to translation. In thisvery special case, the redistribution of energy would indeed be entirely intramolecular within R.

(c) The third mechanism would be transfer of energy between molecules and the radiation field. Theseprocesses involve absorption, emission or Raman scattering of radiation and are summarized, in the simplestcase with one or two photons, in equations (A3.13.10)-(A3.13.12):

Ri + h Rf (absorption) (A3.13.10)Ri Rf + h (emission) (A3.13.11)

Ri + hi Rf + hf (Raman scattering). (A3.13.12)In the case of polarized, but otherwise incoherent statistical radiation, one finds a rate constant for radiativeenergy transfer between initial molecular quantum states i and final states f:

kfi = 83

h2I z

(40)c|Mzfi|2 (A3.13.13)

where I z = dI z/d is the intensity per frequency bandwidth of radiation and Mzfi is the electric dipoletransition moment in the direction of polarization. For unpolarized random spatial radiation of density ()per volume and frequency, I z /c must be replaced by ()/3, because of random orientation, and the rate ofinduced transitions (absorption or emission) becomes:

kinducedfi = Bfi() (A3.13.14)

= 83

3h2(40)()|Mfi|2.

Bfi is the Einstein coefficient for induced emission or absorption, which is approximately related to the absolutevalue of the dipole transition moment |Mfi|, to the integrated cross section Gfi for the transition and to theEinstein coefficient Afi for spontaneous emission [10]:

Bfi = chGfi = c

3

8h3fiAfi (A3.13.15)

withGfi =

line

fi()1 d (A3.13.16)


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6E

-

Intramolecular (IVR)

6

\Intermolecular"

?

6

collisional

?

6

or radiative

?

- -

Reaction

Figure A3.13.1. Schematic energy level diagram and relationship between intermolecular (collisional or radiative) andintramolecular energy transfer between states of isolated molecules. The fat horizontal bars indicate thin energy shells ofnearly degenerate states.

and fi() the frequency dependent absorption cross section. In equation (A3.13.15), fi = |Ef Ei |/h.Equation (A3.13.17) is a simple, useful formula relating the integrated cross section and the electric dipoletransition moment as dimensionless quantities, in the electric dipole approximation [10, 100]:

Gfi

pm2 41.624

MfiDebye2. (A3.13.17)

From these equations one also finds the rate coefficient matrix for thermal radiative transitions includingabsorption, induced and spontaneous emission in a thermal radiation field following Plancks law [35]:

kfi = Afi sign (Ef Ei)exp((Ef Ei)/kBT )) 1 . (A3.13.18)

Finally, if one has a condition with incoherent radiation of a small band width exciting a broadabsorption band with ( ) (), one finds:

kinducedfi =()

hI (A3.13.19)

where I is the radiation intensity. For a detailed discussion refer to [10]. The problem of coherent radiativeexcitation is considered in sections A3.13.4 and A3.13.5 in relation to intramolecular vibrational energyredistribution.

(d) The fourth mechanism is purely intramolecular energy redistribution. It is addressed in the nextsection.

A3.13.2.2 Strictly monomolecular processes in isolated molecules

Purely intramolecular energy transfer occurs when energy migrates within an isolated molecule from onepart to another or from one type of motion to the other. Processes of this type include the vast field of


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molecular electronic radiationless transitions which emerged in the late 1960s [36], but more generally anytype of intramolecular motion such as intramolecular vibrational energy redistribution (IVR) or intramolecularvibrationalrotational energy redistribution (IVRR) and related processes [3739]. These processes will bediscussed in section A3.13.5 in some detail in terms of their full quantum dynamics. However, in certainsituations a statistical description with rate equations for such processes can be appropriate [38].

Figure A3.13.1 illustrates our general understanding of intramolecular energy redistribution in isolatedmolecules and shows how these processes are related to intermolecular processes, which may follow any ofthe mechanisms discussed in the previous section. The horizontal bars represent levels of nearly degeneratestates of an isolated molecule.

Having introduced the basic concepts and equations for various energy redistribution processes, we willnow discuss some of them in more detail.

A3.13.3 Collisional energy redistribution processes

A3.13.3.1 The master equation for collisional relaxation reaction processesThe fundamental kinetic master equations for collisional energy redistribution follow the rules of the ki-netic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equa-tion (A3.13.5), can be considered as a somewhat special reaction. The kinetic differential equations forthese processes have been discussed in the general context of chapter A3.4 on gas kinetics. We discuss heresome special aspects related to collisional energy transfer in reactive systems. The general master equationfor relaxation and reaction is of the type [1113, 15, 25, 40, 41]:

dcj (t)dt

= F({ck(t)}) (A3.13.20)cj (t = 0) = cj0. (A3.13.21)

The index j can label quantum states of the same or different chemical species. Equation (A3.13.20) corre-sponds to a generally stiff initial value problem [42, 43]. In matrix notation one may write:

d c(t)dt

= F [c(t)] (A3.13.22)c(t = 0) = c0. (A3.13.23)

There is no general, simple solution to this set of coupled differential equations, and thus one will usuallyhave to resort to numerical techniques [42, 43] (see also chapter A3.4).

A3.13.3.2 The master equation for collisional and radiative energy redistribution under conditions of gen-eralized first-order kineticsThere is one special class of reaction systems in which a simplification occurs. If collisional energy redistribu-tion of some reactant occurs by collisions with an excess of heat bath atoms or molecules that are consideredkinetically structureless, and if furthermore the reaction is either unimolecular or occurs again with a reactionpartner M having an excess concentration, then one will have generalized first-order kinetics for populationspj of the energy levels of the reactant, i.e. with

dpjdt

=k =j

(K jkpk K kjpj ) kjpj (A3.13.24)

pj = cj(

k

ck

)1. (A3.13.25)


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In equation (A3.13.24), kj is the specific rate constant for reaction from level j , and K jk are energy transferrate coefficients. With appropriate definition of a rate coefficient matrix K one has, in matrix notation,

d pdt

= Kp (A3.13.26)

where for j = iKji(T ) =

(8kBT

)1/2[M]

0

x exp(x)ji(kBT x) dx. (A3.13.27)

(see equation (A3.13.9)) andKjj (T ) = kj +

k =j

Kkj (T ). (A3.13.28)

The master equation (A3.13.26) applies also, under certain conditions, to radiative excitation with rate coeffi-cients for radiative energy transfer being given by equations (A3.13.13)(A3.13.19), depending on the case, orelse by more general equations [10]. Finally, the radiative and collisional rate coefficients may be consideredtogether to be important at the same time in a given reaction system, if time scales for these processes are ofthe appropriate order of magnitude.

The solution of equation (A3.13.26) is given by:

p(t) = exp(Kt)p(0). (A3.13.29)

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chap-ter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replacedby a continuous energy variable and the populations by a population density p(E), with replacement of thesum by appropriate integrals [11]. This approach can be made the starting point of useful analytical solutionsfor certain simple model systems [11, 19, 4446].

While the time dependent populationspj (t) may generally show a complicated behaviour, certain simplelimiting cases can be distinguished and characterized by appropriate parameters:

(a) The long time steady state limit (formally t ) is described by the largest eigenvalue 1 of K. Sinceall j are negative, 1 has the smallest absolute value [35, 47]. In this limit one finds [47] (with thereactant fraction FR =

j pj ):

d ln(FR(t))dt

= d ln(

j pj (t))

dt= kuni = 1. (A3.13.30)

Thus, this eigenvalue 1 determines the unimolecular steady-state reaction rate constant.(b) The second largest eigenvalue 2 determines ideally the relaxation time towards this steady state, thus:

1relax = 2. (A3.13.31)

More generally, further eigenvalues must be taken into account in the relaxation process.(c) It is sometimes useful to define an incubation time inc by the limiting equation for steady state:

ln(F stR (t)) = 1(t inc). (A3.13.32)

Figure A3.13.2 illustrates the origin of these quantities. Refer to [47] for a detailed mathematicaldiscussion as well as the treatment of radiative laser excitation, in which incubation phenomena areimportant. Also refer to [11] for some classical examples in thermal systems.


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Figure A3.13.2. Illustration of the analysis of the master equation in terms of its eigenvalues 1 and 2 for the exampleof IR-multiphoton excitation. The dashed lines give the long time straight line limiting behaviour. The full line to theright-hand side is for x = FR(t) with a straight line of slope 1 = kuni. The intercept of the corresponding dashed line(F StR ) indicates inc (see equation (A3.13.32)). The left-hand line is for x = |FR F StR | with limiting slope 2 = 1relax(see text and [47]).

As a rule, in thermal unimolecular reaction systems at modest temperatures, 1 is well separated fromthe other eigenvalues, and thus the time scales for incubation and relaxation are well separated from thesteady-state reaction time scale reaction = k1uni . On the other hand, at high temperatures, kuni, 1relax and 1incmay merge. This is illustrated in figure A3.13.3 for the classic example of thermal unimolecular dissociation[4851]:

N2O + Ar N2 + O + Ar. (A3.13.33)Note that in the low pressure limit of unimolecular reactions (chapter A3.4), the unimolecular rate constantkuni is entirely dominated by energy transfer processes, even though the relaxation and incubation rates (1relaxand 1inc ) may be much faster than kuni.

The master equation treatment of energy transfer in even fairly complex reaction systems is now wellestablished and fairly standard [52]. However, the rate coefficients Kij for the individual energy transferprocesses must be established and we shall discuss some aspects of this matter in the following section.

A3.13.3.3 Mechanisms of collisional energy transferCollisional energy transfer in molecules is a field in itself and is of relevance for kinetic theory (chapter A3.1),gas phase kinetics (chapter A3.4), RRKM theory (chapter A3.12), the theory of unimolecular reactions ingeneral, as well as the kinetics of laser systems [53]. Chapters C3.3C3.5 treat these subjects in detail. Wesummarize those aspects that are of importance for mechanistic considerations in chemically reactive systems.

We start from a model in which collision cross sections or rate constants for energy transfer are com-pared with a reference quantity such as average Lennard-Jones collision cross sections or the usually cited


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Figure A3.13.3. Dissociation (kuni = kdiss), incubation (1inc ) and relaxation (1relax) rate constants for the reactionN2O N2 + O at low pressure in argon (from [11], see discussion in the text for details and references to the ex-periments).

Lennard-Jones collision frequencies [54]

ZLJ = 2AB(

8kBTAB

)1/2&(2,2)AB (A3.13.34)

where AB is the Lennard-Jones parameter and &(2,2)

AB is the reduced collision integral [54], calculated fromthe binding energy and the reduced mass AB for the collision in the Lennard-Jones potential

V (r) = 4[(

AB

r

)12(AB

r

)6]. (A3.13.35)

Given such a reference, we can classify various mechanisms of energy transfer either by the probability that acertain energy transfer process will occur in a Lennard-Jones reference collision, or by the average energytransferred by one Lennard-Jones collision.

With this convention, we can now classify energy transfer processes either as resonant, if |Et| definedin equation (A3.13.8) is small, or non-resonant, if it is large. Quite generally the rate of resonant processescan approach or even exceed the Lennard-Jones collision frequency (the latter is possible if other long-rangepotentials are actually applicable, such as by permanent dipoledipole interaction).

Resonant processes of some importance include resonant electronic to electronic energy transfer (EE),such as the pumping process of the iodine atom laser

(EE) O2(1) + I(2P3/2) O2(3)g ) + I(2P1/2). (A3.13.36)


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Another near resonant process is important in the hydrogen fluoride laser, equation (A3.13.37), where vibra-tional to vibrational energy transfer is of interest:

(VV) HF(v) + HF(v) HF(v + v) + HF(v v), (A3.13.37)

where v is the number of vibrational quanta exchanged. If HF were a harmonic oscillator, Et would bezero (perfect resonance). In practice, because of anharmonicity, the most important process is exothermic,leading to increasing excitation v of some of the HF molecules with successive collisions [55, 56], becausethe exothermicity drives this process to high v as long as plenty of HF(v) with low v are available.

Resonant rotational to rotational (RR) energy transfer may have rates exceeding the Lennard-Jonescollision frequency because of long-range dipoledipole interactions in some cases. Quasiresonant vibrationto rotation transfer (VR) has recently been discussed in the framework of a simple model [57].

Non-resonant processes include vibrationtranslation (VT) processes with transfer probabilities de-creasing to very small values for diatomic molecules with very high vibrational frequencies, of the order of10 4 and less for the probability of transferring a quantum in a collision. Also, vibration to rotation (VR)processes frequently have low probabilities, of the order of 10 2, if Et is relatively large. Rotation to trans-lation (RT) processes are generally fast, with probabilities near 1. Also, the RVT processes in collisions oflarge polyatomic molecules have high probabilities, with average energies transferred in one Lennard-Jonescollision being of the order of a few kJ mol1 [11, 25], or less in collisions with rare gas atoms. As a generalrule one may assume collision cross sections to be small, if Et is large [11, 58, 59].

In the experimental and theoretical study of energy transfer processes which involve some of theabove mechanisms, one should distinguish processes in atoms and small molecules and in large polyatomicmolecules. For small molecules a full theoretical quantum treatment is possible and even computer programpackages are available [6063], with full state to state characterization. A good example are rotational energytransfer theory and experiments on He + CO [64]:

He + CO(j ) He + CO(j ). (A3.13.38)

On the experimental side, small molecule energy transfer experiments may use molecular beam techniques[6567] (see also chapter C3.3 for laser studies).

In the case of large molecules, instead of the detailed quantum state characterization implied in thecross sections fi and rate coefficients Kfi of the master equation (A3.13.24), one derives more coarse grainedinformation on levels covering a small energy bandwidth around E and E (with an optional notationKFI(E

, E)) or finally energy transfer probabilities P(E, E) for a transition from energy E to energy Ein a highly excited large polyatomic molecule where the density of states (E) is very large, for examplein a collision with a heat bath inert gas atom [11]. Such processes can currently be modelled by classicaltrajectories [6870].

Experimental access to the probabilities P(E, E) for energy transfer in large molecules usually involvestechniques providing just the first moment of this distribution, i.e. the average energy E transferred in acollision. Such methods include UV absorption, infrared fluorescence and related spectroscopic techniques[11, 28, 7174]. More advanced techniques, such as kinetically controlled selective ionization (KCSI [74])have also provided information on higher moments of P(E, E), such as (E)2.

The standard mechanisms of collisional energy transfer for both small and large molecules have beentreated extensively and a variety of scaling laws have been proposed to simplify the complicated body of data[58, 59, 75]. To conclude, one of the most efficient special mechanisms for energy transfer is the quasi-reactiveprocess involving chemically bound intermediates, as in the example of the reaction:

O2(v, j ) + O O3 O2(v, j ) + O. (A3.13.39)


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Such processes transfer very large amounts of energy in one collision and have been treated efficiently by thestatistical adiabatic channel model [11, 19, 30, 7679]. They are quite similar mechanistically to chemicalactivation systems. One might say that in such a mechanism one may distinguish three phases:

(a) Formation of a bound collision complex AB:

A(v, j ) + B AB. (A3.13.40)

(b) IVRR in this complex:AB AB. (A3.13.41)

(c) Finally, dissociation of the internally, statistically equilibrated complex:

AB A(v, j ) + B. (A3.13.42)

That is, rapid IVR in the long lived intermediate is an essential step. We shall treat this important processin the next section, but mention here in passing the observation of so-called supercollisions transferringlarge average amounts of energy E in one collision [80], even if intermediate complex formation may notbe important.

A3.13.4 Intramolecular energy transfer studies in polyatomic molecules

In this section we review our understanding of IVR as a special case of intramolecular energy transfer.The studies are based on calculations of the time evolution of vibrational wave packets corresponding tomiddle size and large amplitude vibrational motion in polyatomic molecules. An early example for theinvestigation of wave packet motion as a key to understanding IVR and its implication on reaction kineticsusing experimental data is given in [81]. Since then, many other contributions have helped to increase ourknowledge using realistic potential energy surfaces, mainly for two- and three-dimensional systems, and wegive a brief summary of these results below.

A3.13.4.1 IVR and classical mechanics

Before undergoing a substantial and, in many cases, practically irreversible, change of geometrical structurewithin a chemical reaction, a molecule may often perform a series of vibrations in the multidimensionalspace around its equilibrium structure. This applies in general to reactions that take place entirely in thebound electronic ground state and in many cases to reactions that start in the electronic ground state near theequilibrium structure, but evolve into highly excited states above the reaction threshold energy. In the lattercase, within the general scheme of equation (A3.13.1) a reaction is thought to be induced by a sufficientlyenergetic pulse of electromagnetic radiation or by collisions with adequate high-energy collision partners. Inthe first case, a reaction is thought to be the last step after a chain of excitation steps has transferred enoughenergy into the molecule to react either thermally, by collisions, or coherently, for instance by irradiation withinfrared laser pulses. These pulses can be tuned to adequately excite vibrations along the reaction coordinate,the amplitudes of which become gradually larger until the molecule undergoes a sufficiently large structuralchange leading to the chemical reaction.

Vibrational motion is thus an important primary step in a general reaction mechanism and detailedinvestigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions.In classical mechanics, vibrational motion is described by the time evolution q(t) and p(t) of general internalposition and momentum coordinates. These time dependent functions are solutions of the classical equationsof motion, e.g. Newtons equations for given initial conditions q(t0) = q0 and p(t0) = p0. The definition ofinitial conditions is generally limited in precision to within experimental uncertainties q0 and p0, more


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fundamentally related by the Heisenberg principle q0p0 = h/4 . Therefore, we need to consider aninitial distributionF0(qq0, pp0), with widthsq0 andp0 and the time evolutionFt(q q(t), pp(t)),which may be quite different from the initial distribution F0, depending on the integrability of the dynamicalsystem. Ideally, for classical, integrable systems, vibrational motion may be understood as the motion ofnarrow, well localized distributions F(q q(t), p p(t)) (ideally -functions in a strict mathematical sense),centred around the solutions of the classical equations of motion. In this picture we wish to consider initialconditions that correspond to localized vibrational motion along specific manifolds, for instance a vibrationthat is induced by elongation of a single chemical bond (local mode vibrations) as a result of the interactionwith some external force, but it is also conceivable that a large displacement from equilibrium might beinduced along a single normal coordinate. Independent of the detailed mechanism for the generation oflocalized vibrations, harmonic transfer of excitation may occur when such a vibration starts to extend intoother manifolds of the multidimensional space, resulting in trajectories that draw Lissajous figures in phasespace, and also in configuration space [82] (see also [83]). Furthermore, if there is anharmonic interaction,IVR may occur. In [84, 85] this type of IVR was called classical intramolecular vibrational redistribution(CIVR).

A3.13.4.2 IVR and quantum mechanics

In time-dependent quantum mechanics, vibrational motion may be described as the motion of the wave packet|(q, t)|2 in configuration space, e.g. as defined by the possible values of the position coordinates q. Thismotion is given by the time evolution of the wave function (q, t), defined as the projection q|(t) ofthe time-dependent quantum state |(t) on configuration space. Since the quantum state is a completedescription of the system, the wave packet defining the probability density can be viewed as the quantummechanical counterpart of the classical distribution F(q q(t), p p(t)). The time dependence is obtainedby solution of the time-dependent Schrodinger equation

ih

2d|(t)

dt= H |(t) (A3.13.43)

where h is the Planck constant and H is the Hamiltonian of the system under consideration. Solutions dependon initial conditions |(t0) and may be formulated using the time evolution operator U (t, t0):

|(t) = U (t, t0)|(t0). (A3.13.44)Alternatively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P (t) =|(t)(t)| at time t can be obtained as the solution of the Liouvillevon Neumann equation:

P (t) = U (t, t0)P (t0)U (t, t0) (A3.13.45)where U (t, t0) is the adjoint of the time evolution operator (in strictly conservative systems, the time evolutionoperator is unitary and U (t, t0) = U1(t, t0) = U (t0, t)).

The calculation of the time evolution operator in multidimensional systems is a formidable task and someresults will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics asdemonstrated, among others, by Heller [8688], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers(see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). Thismethod basically consists of replacing the -function distribution in the true classical calculation by a Gaussiandistribution in coordinate space. It allows for a simulation of the vibrational quantum dynamics to the extentthat interference effects in the evolving wave packet can be neglected. While the application of semi-classicalmethods might still be of some interest for the simulation of quantum dynamics in large polyatomic moleculesin the near future, as a natural extension of classical molecular dynamics calculations [68, 96], full quantum


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mechanical calculations of the wave packet evolution in smaller polyatomic molecules are possible with thecurrently available computational resources. Following earlier spectroscopic work and three-dimensionalquantum dynamics results [81, 97100], Wyatt and coworkers have recently demonstrated applications offull quantum calculations to the study of IVR in fluoroform, with nine degrees of freedom [101, 102] and inbenzene [103], considering all 30 degrees of freedom [104]. Such calculations show clearly the possibilitiesin the computational treatment of quantum dynamics and IVR. However, remaining computational limitationsrestrict the study to the lower energy regime of molecular vibrations, when all degrees of freedom of systemswith more than three dimensions are treated. Large amplitude motion, which shows the inherently quantummechanical nature of wave packet motion and is highly sensitive to IVR, cannot yet be discussed for suchmolecules, but new results are expected in the near future, as indicated in recent work on ammonia [105, 106],formaldehyde and hydrogen peroxide [106108], and hydrogen fluoride dimer [109111] including all sixinternal degrees of freedom.

A key feature in quantum mechanics is the dispersion of the wave packet, i.e. the loss of its Gaussianshape. This feature corresponds to a delocalization of probability density and is largely a consequence ofanharmonicities of the potential energy surface, both the diagonal anharmonicity, along the manifold in whichthe motion started, and off diagonal, induced by anharmonic coupling terms between different manifolds inthe Hamiltonian. Spreading of the wave packet into different manifolds is thus a further important feature ofIVR. In [84, 85] this type of IVR was called delocalization quantum intramolecular vibrational redistribution(DIVR). DIVR plays a central role for the understanding of statistical theories for unimolecular reactions inpolyatomic molecules [84, 97], as will be discussed below.

A3.13.4.3 IVR within the general scheme of energy redistribution in reactive systemsAs in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrenceof IVR in polyatomic molecules, depends both on the Hamiltonian and the initial conditions, i.e. the initialquantum mechanical state |(t0). We focus here on the time-dependent aspects of IVR, and in this casesuch initial conditions always correspond to the preparation, at a time t0, of superposition states of molecular(spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occursif these levels involve at least two distinct vibrational manifolds in terms of which the total (vibrational)Hamiltonian is not separable [84]. In a time-independent view, this requirement states that the wave functionsbelonging to the two spectroscopic states are spread in a non-separable way over the configuration spacespanned by at least two different vibrational modes. The conceptual framework for the investigation of IVRmay be sketched within the following scheme, which also mirrors the way we might investigate IVR in thetime-dependent approach, both theoretically and experimentally:

|(t1)Uprep(t0,t1) |(t0) Ufree(t,t0) |(t). (A3.13.46)

In a first time interval [t1, t0] of the scheme (A3.13.46), a superposition state is prepared. This step cor-responds to the step in equation (A3.13.1). One might think of a time evolution |(t1) |(t0) =Uprep(t0, t1)|(t1), where |(t1) may be a molecular eigenstate and Uprep is the time evolution operatorobtained from the interaction with an external system, to be specified below. The probability distribution|(q, t0)|2 is expected to be approximatively localized in configuration space, such that |(q, t0)|2 > 0 forposition coordinates q M belonging to some specific manifoldM and |(q, t0)|2 0 for coordinatesq M belonging to the complementary manifoldM =M. In a second time interval [t0, t1], the superpo-sition state |(t0) has a free evolution into states |(t) = Ufree(t, t0)|(t0). This step corresponds to theintermediate step equation (A3.13.47), occurring between the steps described before by equations (A3.13.1)and (A3.13.2) (see also equation (A3.13.41)):

R R . (A3.13.47)


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IVR is present if |(q, t)|2 > 0 is observed for t > t0 also for q M. IVR may of course also occur duringthe excitation process, if its time scale is comparable to that of the excitation.

In the present section, we concentrate on coherent preparation by irradiation with a properly chosenlaser pulse during a given time interval. The quantum state at time t1 may be chosen to be the vibrationalground state |(g)0 in the electronic ground state. In principle, other possibilities may also be conceived for thepreparation step, as discussed in sections A3.13.1A3.13.3. In order to determine superposition coefficientswithin a realistic experimental set-up using irradiation, the following questions need to be answered: (1) Whatare the eigenstates? (2) What are the electric dipole transition matrix elements? (3) What is the orientationof the molecule with respect to the laboratory fixed (linearly or circularly) polarized electric field vector ofthe radiation? The first question requires knowledge of the potential energy surface, or the HamiltonianH0(p, q) of the isolated molecule, the second that of the vector valued surface (q) of the electric dipolemoment. This surface yields the operator, which couples spectroscopic states by the impact of an externalirradiation field and thus directly affects the superposition procedure. The third question is indeed of greatimportance for comparison with experiments aiming at the measurement of internal wave packet motion inpolyatomic molecules and has recently received much attention in the treatment of molecular alignment andorientation [112, 113], including non-polar molecules [114, 115]. To the best of our knowledge, up to nowexplicit calculations of multidimensional wave packet evolution in polyatomic molecules have been performedupon neglect of rotational degrees of freedom, i.e. only internal coordinates have been considered, althoughcalculations on coherent excitation in ozone level structures with rotation exist [116, 117], which could beinterpreted in terms of wave packet evolution. A more detailed discussion of this point will be given belowfor a specific example.

A3.13.4.4 Concepts of computational methodsThere are numerous methods for solving the time dependent Schrodinger equation (A3.13.43), and someof them were reviewed by Kosloff [118] (see also [119, 120]). Whenever projections of the evolving wavefunction on the spectroscopic states are useful for the detailed analysis of the quantum dynamics (and thisis certainly the case for the detailed analysis of IVR), it is convenient to express the Hamiltonian based onspectroscopic states |n:

H0 =n

h

2n|nn| (A3.13.48)

where n are the eigenfrequencies. For an isolated molecule H = H0 in equation (A3.13.43) and the timeevolution operator is of the form

U (t, t0) =n

exp(in(t t0))|nn|. (A3.13.49)

The time-dependent wave function is then given by the expression:

(q, t) =n

c0n exp(int)n(q). (A3.13.50)

Here, n(q) = q|n are the wave functions of the spectroscopic states and the coefficients c0n are determinedfrom the initial conditions

(q, t0) =n

c0nn(q), c0n = n|(t0). (A3.13.51)

Equation (A3.13.49) describes the spectroscopic access to quantum dynamics. Clearly, when the spectralstructure becomes too congested, i.e. when there are many close lying frequencies n, calculation of all


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spectroscopic states becomes difficult. However, often it is not necessary to calculate all states when certainmodel assumptions can be made. One assumption concerns the separation of time scales. When thereis evidence for a clear separation of time scales for IVR, only part of the spectroscopic states need to beconsidered for fast evolution. Typically, these states have large frequency separations, and considering onlysuch states means neglecting the fine-grained spectral structure as a first approximation. An example forseparation of time scales is given by the dynamics of the alkyl CH chromophore in CHXYZ compounds,which will be discussed below. This group span a three-dimensional linear space of stretching and bendingvibrations. These vibrations are generally quite strongly coupled, which is manifested by the occurrence of aFermi resonance in the spectral structure throughout the entire vibrational energy space. As we will see, thecorresponding time evolution and IVR between these modes takes place in less than 1 ps, while other modesbecome involved in the dynamics on much longer time scales (10 ps to ns, typically). The assumption fortime scale separation and IVR on the subpicosecond time scale for the alkyl CH chromophore was foundedon the basis of spectroscopic data nearly 20 years ago [98, 121]. The first results on the nature of IVR inthe CH chromophore system and its role in IR photochemistry were also reported by that time [122, 123],including results for the acetylenic CH chromophore [124] and results obtained from first calculations ofthe wave packet motion [81]. The validity of this assumption has recently been confirmed in the case ofCHF3 both experimentally, from the highly resolved spectral structure of highly excited vibrational overtones[125, 126], and theoretically, including all nine degrees of freedom for modestly excited vibrational overtonesup to 6000 cm1 [102].

A3.13.4.5 IVR during and after coherent excitation: general aspectsModern photochemistry (IR, UV or VIS) is induced by coherent or incoherent radiative excitation processes[47]. The first step within a photochemical process is of course a preparation step within our conceptualframework, in which time-dependent states are generated that possibly show IVR. In an ideal scenario, energyfrom a laser would be deposited in a spatially localized, large amplitude vibrational motion of the reactingmolecular system, which would then possibly lead to the cleavage of selected chemical bonds. This isbasically the central idea behind the concepts for a mode selective chemistry, introduced in the late 1970s[127], and has continuously received much attention [10, 117, 122, 128135]. In a recent review [136], IVRwas interpreted as a molecular enemy of possible schemes for mode selective chemistry. This interpretationis somewhat limited, since IVR represents more complex features of molecular dynamics [37, 84, 134], andeven the opposite situation is possible. IVR can indeed be selective with respect to certain structural features[85, 97] that may help mode selective reactive processes after tailored laser excitation [137].

To be more specific, we assume that for a possible preparation step the Hamiltonian might be givenduring the preparation time interval [t1, t0] by the expression:

H = H0 + Hi(t) (A3.13.52)where H0 is the Hamiltonian of the isolated molecule and Hi is the interaction Hamiltonian between themolecule and an external system. In this section, we limit the discussion to the case where the external systemis the electromagnetic radiation field. For the interaction with a classical electromagnetic field with electricfield vector E(t), the interaction Hamiltonian is given by the expression:

Hi(t) = E(t). (A3.13.53)

where is the operator of the electric dipole moment. When we treat the interaction with a classical field inthis way, we implicitly assume that the field will remain unaffected by the changes in the molecular systemunder consideration. More specifically, its energy content is assumed to be constant. The energy of theradiation field is thus not explicitly considered in the expression for the total Hamiltonian and all operators


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acting on states of the field are replaced by their time-dependent expectation values. These assumptions arewidely accepted, whenever the number of photons in each field mode is sufficiently large. For a coherent,monochromatic, polarized field with intensity I = 0/0| E|2 1 MW cm2 in vacuo, which is a typicalvalue used in laser chemical experiments in the gas phase at low pressures, the number N of mid infraredphotons existing in a cavity of volume V = 1 m3 is [138, p 498]:

N = IVc0h

1021. (A3.13.54)

Equation (A3.13.54) legitimates the use of this semi-classical approximation of the moleculefield interactionin the low-pressure regime.

Since Hi(t) is explicitly time dependent, the time evolution operator is more complicated than in equa-tion A3.13.49. However, the time-dependent wave function can still be written in the form

(q, t) =n

cn(t)n(q) (A3.13.55)

with time-dependent coefficients that are obtained by solving the set of coupled differential equations

idcn(t)

dt=n{Wnn + Vnn(t)}cn(t) (A3.13.56)

where Wnn = nnn (nn is the Kronecker symbol, n were defined in equation (A3.13.48)) and

Vnn(t) = 2hn|Hi(t)|n

= 2hn| |n E(t). (A3.13.57)

The matrix elements n| |n are multidimensional integralsn(q) (q)n(q) d of the vector valued

dipole moment surface. The time-independent part of the coupling matrix elements in equation (A3.13.57)can also be cast into the practical formula

V 0nn/(2c0 cm1) = 0.46093n|/Debye|n

I0/MW cm2, (A3.13.58)

where is the direction of the electric field vector of the linearly polarized radiation field with maximal intensityI0. The solution of equation (A3.13.56) may still be quite demanding, depending on the size of the systemunder consideration. However, it has become a practical routine procedure to use suitable approximationssuch as the QRA (quasiresonant approximation) or Floquet treatment [35, 122, 129] and programmes for thenumerical solution are available [139, 140].

A3.13.4.6 Electronic excitation in the FranckCondon limit and IVR

At this stage we may distinguish between excitation involving different electronic states and excitation occur-ring within the same electronic (ground) state. When the spectroscopic states are located in different electronicstates, say the ground (g) and excited (e) states, one frequently assumes the FranckCondon approximationto be applicable:

(g)n | |(e)n ge(g)n |(e)n . (A3.13.59)Such electronic excitation processes can be made very fast with sufficiently intense laser fields. For example,if one considers monochromatic excitation with a wavenumber in the UV region (60 000 cm1) and a coupling


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strength ( ge E)/hc 4000 cm1 (e.g.ge 1 Debye in equation (A3.13.59), I 50 TW cm2), excitationoccurs within 1 fs [141]. During such a short excitation time interval the relative positions of the nuclei remainunchanged (Franck approximation). Within these approximations, if one starts the preparation step in thevibrational ground state |(g)0 , the resulting state |(t0) at time t0 has the same probability distribution as thevibrational ground state. However, it is now transferred into the excited electronic state where it is no longerstationary, since it is a superposition state of vibrational eigenstates in the excited electronic state:

|(t0) =n

(e)n |(g)0 |(e)n . (A3.13.60)

Often the potential energy surfaces for the ground and excited states are fairly different, i.e. with significantlydifferent equilibrium positions. The state |(t0) will then correspond to a wave packet, which has nearlya Gaussian shape with a centre position that is largely displaced from the minimal energy configuration onthe excited surface and, since the Franck approximation can be applied, the expectation value of the nuclearlinear momentum vanishes. In a complementary view, the superposition state of equation (A3.13.60) definesthe manifold M in configuration space. It is often referred to as the bright state, since its probabilitydensity defines a region in configuration space, the FranckCondon region, which has been reached by theirradiation field through mediation by the electric dipole operator. After the preparation step, the wave packetmost likely starts to move along the steepest descent path from the FranckCondon region. One possibility isthat it proceeds to occupy other manifolds, which were not directly excited. The occupation of the remaining,dark manifolds (e.g.M) by the time-dependent wave packet is a characteristic feature of IVR.

Studies of wave packet motion in excited electronic states of molecules with three and four atoms wereconducted by Schinke, Engel and collaborators, among others, mainly in the context of photodissociationdynamics from the excited state [142144] (for an introduction to photodissociation dynamics, see [7], andalso more recent work [145149] with references cited therein). In these studies, the dissociation dynamicsis often described by a time-dependent displacement of the Gaussian wave packet in the multidimensionalconfiguration space. As time goes on, this wave packet will occupy different manifolds (from where themolecule possibly dissociates) and this is identified with IVR. The dynamics may be described within theGaussian wave packet method [150], and the vibrational dynamics is then of the classical IVR type (CIVR[84]). The validity of this approach depends on the dissociation rate on the one hand, and the rate ofdelocalization of the wave packet on the other hand. The occurrence of DIVR often receives less attention inthe discussions of photodissociation dynamics mentioned above. In [148], for instance, details of the wavepacket motion by means of snapshots of the probability density are missing, but a delocalization of the wavepacket probably takes place, as may be concluded from inspection of figure 5 therein.

A3.13.5 IVR in the electronic ground state: the example of the CH chromophore

A3.13.5.1 Redistribution during and after coherent excitationA system that shows IVR with very fast spreading of the wave packet, i.e. DIVR in the subpicosecondtime range, is that of the infrared alkyl CH chromophore, which will be used in the remaining part of thischapter to discuss IVR as a result of a mode specific excitation within the electronic ground state. The CHstretching and bending modes of the alkyl CH chromophore in CHXYZ compounds are coupled by a generallystrong Fermi resonance [100, 151]. Figure A3.13.4 shows the shape of the potential energy surface for thesymmetrical compound CHD3 as contour line representations of selected one- and two-dimensional sections(see figure caption for a detailed description). The important feature is the curved shape of the V (Qs,Qb1)potential section (V (Qs,Qb2) being similarly curved), which indicates a rather strong anharmonic coupling.This feature is characteristic for compounds of the type CHXYZ [84, 100, 151153]. Qs, Qb1 and Qb2 are(mass weighted) normal coordinates of the CH stretching and bending motion, with symmetry A1 and E,


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Figure A3.13.4. Potential energy cuts along the normal coordinate subspace pertaining to the CH chromophore in CHD3.Qb1 is the A coordinate in Cs symmetry, essentially changing structure along the x-axis (see also figure A3.13.5), andQb2 is theA coordinate, essentially changing structure along the y-axis. Contour lines show equidistant energies at wavenumber differences of 3000 cm1 up to 30 000 cm1. The upper curves are one-dimensional cuts along Qb2 (left) and Qs(right). The dashed curves in the two upper figures show harmonic potential curves (from [154]).

respectively, in the C3v point group of symmetrical CHD3. A change of Qs is a concerted motion of all atomsalong the z-axis, defined in figure A3.13.5. However, displacements along Qs are small for the carbon anddeuterium atoms, and large for the hydrogen atom. Thus, this coordinate essentially describes a stretchingmotion of the CH bond (along the z-axis). In the same way, Qb1 and Qb2 describe bending motions of theCH bond along the x- and y-axis, respectively (see figure A3.13.5). In the one-dimensional sections thepositions of the corresponding spectroscopic states are drawn as horizontal lines. On the left-hand side, inthe potential section V (Qb2), a total of 800 states up to an energy equivalent wave number of 25 000 cm 1has been considered. These energy levels may be grouped into semi-isoenergetic shells defined by multipletsof states with a constant chromophore quantum number N = vs + 12vb = 0, 12 , 1, 32 , 2, 52 , . . ., where vs andvb are quantum numbers of effective basis states (Fermi modes [97, 152, 154]) that are strongly coupled bya 2:1 Fermi resonance. These multiplets give rise to spectroscopic polyads and can be well distinguished inthe lower energy region, where the density of states is low.

In the potential section V (Qs), shown on the right hand side of Figure A3.13.4, the subset of A1 energystates is drawn. This subset contains only multiplets with integer values of the chromophore quantum numberN = 0, 1, 2, . . .. This reduction allows for an easier visualization of the multiplet structure and also represents


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Figure A3.13.5. Coordinates and axes used to describe the wave packet dynamics of the CH chromophore in CHX3 orCHXYZ compounds.

the subset of states that are strongly coupled by the parallel component of the electric dipole moment (seediscussion in the following paragraph). The excitation dynamics of the CH chromophore along the stretchingmanifold can indeed be well described by restriction to this subset of states [97, 154].

Excitation specificity is a consequence of the shape of the electric dipole moment surface. For the alkylCH chromophore in CHX3 compounds, the parallel component of the dipole moment, i.e. the componentparallel to the symmetry axis, is a strongly varying function of the CH stretching coordinate, whereas itchanges little along the bending manifolds [155, 156]. Excitation along this component will thus inducepreparation of superposition states lying along the stretching manifold, preferentially. These states thusconstitute the bright manifold in this example. The remaining states define the dark manifolds and anysubstantial population of these states during or after such an excitation process can thus be directly linkedto the existence of IVR. On the other hand, the perpendicular components of the dipole moment vector arestrongly varying functions of the bending coordinates. For direct excitation along one of these components,states belonging to the bending manifolds become the bright states and any appearance of a subsequentstretching motion can be interpreted as arising from IVR.

The following discussion shall illustrate our understanding of structural changes along dark manifoldsin terms of wave packet motion as a consequence of IVR. Figure A3.13.6 shows the evolution of the wavepacket for the CH chromophore in CHF3 during the excitation step along the parallel (stretching) coordinate[97]. The potential surface in the CH chromophore subspace is similar to that for CHD3 (figure A3.13.4above), with a slightly more curved form in the stretchingbending representation (figures are shown in[97, 151]). The laser is switched on at a given time t1, running thereafter as a continuous, monochromaticirradiation up to time t0, when it is switched off. Thus, the electric field vector is given as

E(t) = h(t t1)h(t0 t) E0 cos(Lt). (A3.13.61)

where h(t) is the Heaviside unit step function, E0 is the amplitude of the electric field vector and L = 2cLits angular frequency. Excitation parameters are the irradiation intensity I0 = 30 TW cm2, which correspondsto a maximal electric field strength E0 3.4 1010 V m1, and wave number L = 2832.42 cm1, whichlies in the region of the fundamental for the CH stretching vibration (see arrows in the potential cut V (Qs)of figure (A3.13.4)). The figure shows snapshots of the time evolution of the wave packet between 50 and70 fs after the beginning of the irradiation (t1 = 0 here). On the left-hand side, contour maps of the


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Figure A3.13.6. Time evolution of the probability density of the CH chromophore in CHF3 after 50 fs of irradiation withan excitation wave number L = 2832.42 cm1 at an intensity I0 = 30 TW cm2. The contour lines of equiprobabilitydensity in configuration space have values 2 105 u1 pm2 for the lowest line shown and distances between thelines of 24, 15, 29 and 20 105 u1 pm2 in the order of the four images shown. The averaged energy of the wavepacket corresponds to 6000 cm1 (roughly 3100 cm1 absorbed) with a quantum mechanical uncertainty of 3000 cm1(from [97]).


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time-dependent, integrated probability density

|(Qs,Qb, t)|2 =b

|(Qs,Qb, b, t)|2 db (A3.13.62)

are shown, whereQs is the coordinate for the stretching motion andQb =Q2b1 + Q

2b2 , b = arctan(Qb2/Qb1)

are polar representations of the bending coordinatesQb1 andQb2 . Additionally, contour curves of the potentialenergy surface are drawn at the momentary energy of the wave packet. This energy is defined as:

E(t) =n

Enpn(t) (A3.13.63)

wherepn(t) = cn(t)cn(t) (A3.13.64)

are the time-dependent populations of the spectroscopic states during the preparation step (the complexcoefficients cn(t) in equation (A3.13.64) are calculated according to equation (A3.13.55), the spectroscopicenergies En = h2 n are defined in equation (A3.13.48); the dashed curves indicate the quantum mechanicaluncertainty which arises from the superposition of molecular eigenstates). The same evolution is repeated onthe right-hand side of the figure as a three-dimensional representation.

In the treatment adopted in [97], the motion of the CF3 frame is implicitly considered in the dynamicsof the normal modes. Indeed, the integrand |(Qs,Qb, b, t)|2 in equation (A3.13.62) is to be interpretedas probability density for the change of the CHF3 structure in the subspace of the CH chromophore, asdefined by the normal coordinates Qs, Qb1 and Qb2 , irrespective of the molecular structure and its changein the remaining space. This interpretation is also valid beyond the harmonic approximation, as long as thestructural change in the CH chromophore space can be dynamically separated from that of the rest of themolecule. The assumption of dynamical separation is well confirmed, both from experiment and theory, atleast during the first 1000 fs of motion of the CH chromophore.

When looking at the snapshots in figure A3.13.6, we see that the position of maximal probability oscillatesback and forth along the stretching coordinate between the walls at Qs = 20 and +25

u pm, with an

approximate period of 12 fs, which corresponds to the classical oscillation period = 1/ of a pendulumwith a frequency = c0 8.5 1013 s1 and wave number = 2850 cm1. Indeed, the motion of thewhole wave packet approximately follows this oscillation and, when it does so, the wave packet motion issemiclassical. In harmonic potential wells the motion of the wave packet is always semiclassical [157159].However, since the potential surface of the CH chromophore is anharmonic, some gathering and spreadingout of the wave packet is observable on top of the semiclassical motion. It is interesting to note that, at thisinitial stage of the excitation step, the motion of the wave packet is nearly semiclassical, though with modestamplitudes of the oscillations, despite the anharmonicity of the stretching potential.

The later time evolution is shown in figure A3.13.7, between 90 and 100 fs, and in figure A3.13.8, between390 and 400 fs, after the beginning of the excitation (time step t1). Three observations are readily made:first, the amount of energy absorbed by the chromophore has increased, from 3000 cm1 in figure A3.13.6, to6000 cm1 in figure A3.13.7 and 12 000 cm1 in figure A3.13.8. Second, the initially semiclassical motionhas been replaced by a more irregular motion of probability density, in which the original periodicity is hardlyvisible. Third, the wave packet starts to occupy nearly all of the energetically available region in configurationspace, thus escaping from the initial, bright manifolds into the dark manifolds. From these observations,the following conclusions may be directly drawn: IVR of the CH chromophore in fluoroform is fast (in thesubpicosecond time scale); IVR sets in already during the excitation process, i.e. when an external force fieldis driving the molecular system along a well prescribed path in configuration space (the bright manifold);IVR is of the delocalization type (DIVR). Understanding these observations is central for the understandingof IVR and they are discussed as follows:


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Figure A3.13.7. Continuation of the time evolution for the CH chromophore in CHF3 after 90 fs of irradiation (see alsofigure A3.13.6). Distances between the contour lines are 10, 29, 16 and 9 10 5 u1 pm2 in the order of the four imagesshown. The averaged energy of the wave packet corresponds to 9200 cm1 (roughly 6300 cm1 absorbed) with a quantummechanical uncertainty of 5700 cm1 (from [97]).


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Figure A3.13.8. Continuation of the time evolution for the CH chromophore in CHF3 after 392 fs of irradiation (see alsofigures A3.13.6 and A3.13.7). Distances between the contour lines are 14, 12, 13 and 14 10 5 u1 pm2 in the orderof the four images shown. The averaged energy of the wave packet corresponds to 15 000 cm1 (roughly 12 100 cm1absorbed) with a quantum mechanical uncertainty of 5800 cm1 (from [97]).


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(a) A more detailed analysis of quantum dynamics shows that the molecular system, represented by thegroup of vibrations pertaining to the CH chromophore in this example, absorbs continuously more energyas time goes on. Let the absorbed energy be Eabs = N,abs(h/2)L, where N,abs is the mean number ofabsorbed photons. Since the carrier frequency of the radiation field is kept constant at a value close to thefundamental of the stretching oscillation, L N=1 N=0 (N being the chromophore quantum numberhere), this means that the increase in absorbed energy is a consequence of the stepwise multiphoton excitationprocess, in which each vibrational level serves as a new starting level for further absorption of light after it hasitself been significantly populated. This process is schematically represented, within the example for CHD3,by the sequence of upright arrows shown on the right-hand side of figure A3.13.4. N,abs is thus a smoothlyincreasing function of time.

(b) The disappearance of the semiclassical type of motion and, thus, the delocalization of the wavepacket, is understood to follow the onset of dephasing. With increasing energy, both the effective anharmoniccouplings between the bright stretching mode and the dark bending modes, as well as the diagonal anhar-monicity of the bright mode increase. The larger the anharmonicity, the larger the deviation from a purelyharmonic behaviour, in which the wave packet keeps on moving in a semiclassical way. In quantum mechan-ics, the increase in anharmonicity of an oscillator leads to an effective broadeningeff > 0 in the distributionof frequencies of high-probability transitionsfor transitions induced by the electric dipole operator usuallythose with a difference of 1 in the oscillator quantum number (for the harmonic oscillator eff = 0).On the other hand, these are the transitions which play a major role in the stepwise multiphoton excitationof molecular vibrations. A broadening of the frequency distribution invariably leads to a broadening of thedistribution of relative phases of the time-dependent coefficients cn(t) in equation (A3.13.55). Although thesum in equation (A3.13.55) is entirely coherent, one might introduce an effective coherence time defined by:

c,eff = 1/eff . (A3.13.65)For the stretching oscillations of the CH chromophore in CHF3 c,eff 100 fs. Clearly, typical coherence timeranges depend on both the molecular parameters and the effectively absorbed amount of energy during theexcitation step, which in turn depends on the coupling strength of the moleculeradiation interaction. A moredetailed study of the dispersion of the wave packet and its relationship with decoherence effects was carriedout in [106]. In [97] an excitation process has been studied for the model of two anharmonically coupled,resonant harmonic oscillators (i.e. with at least one cubic coupling term) but under similar conditions as forthe CH chromophore in fluoroform discussed here. When the cubic coupling parameter is chosen to be verysmall compared with the diagonal parameters of the Hamilton matrix, the motion of the wave packet is indeedsemiclassical for very long times (up to 600 ps) and, moreover, the wave packet does probe the bendingmanifold without significantly changing its initial shape. This means that, under appropriate conditions, IVRcan also be of the classical type within a quantum mechanical treatment of the dynamics. Such conditionsrequire, for instance, that the band width heff be smaller than the resonance width (power broadening) ofthe excitation process.

(c) The third observation, that the wave packet occupies nearly all of the energetically accessible regionin configuration space, has a direct impact on the understanding of IVR as a rapid promotor of microcanonicalequilibrium conditions. Energy equipartition preceding a possible chemical reaction is the main assumptionin quasiequilibrium statistical theories of chemical reaction dynamics (RRKM theory [161163], transitionstate theory [164, 165] but also within the statistical adiabatic channel model [76, 77]; see also chapter A3.12and further recent reviews on varied and extended forms of statistical theories in [25, 166172]). In the caseof CHF3 one might conclude from inspection of the snapshots at the later stage of the excitation dynamics(see figure A3.13.8) that after 400 fs the wave packet delocalization is nearly complete. Moreover, thisdelocalization arises here from a fully coherent, isolated evolution of a system consisting of one moleculeand a coherent radiation field (laser). Of course, within the common interpretation of the wave packetas a probability distribution in configuration space, this result means that, for an ensemble of identically


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prepared molecules, vibrational motion is essentially delocalized at this stage and vibrational energy is nearlyequipartitioned.

However, the wave packet does not occupy all of the energetically accessible region. A more detailedanalysis of populations [97, table IV] reveals that, during the excitation process, the absorbed energy isinhomogeneously distributed among the set of molecular eigenstates of a given energy shell (such a shell isrepresented by all nearly iso-energetic states belonging to one of the multiplets shown on the right-hand sideof figure A3.13.4). Clearly, equipartition of energy is attained, if all states of an energy shell are equallypopulated. The microcanonical probability distribution in configuration space may then be represented by atypical member of the microcanonical ensemble, defined e.g. by the wave function

micro 1Nshell

nshell

exp(irandomn )n (A3.13.66)

whereNshell denotes the number of nearly iso-energetic statesn of a shell andrandomn is a random phase. Such astate is shown in figure A3.13.9. When comparing this state with the state generated by multiphoton excitation,the two different kinds of superposition that lead to these wave packets must, of course, be distinguished. Inthe stepwise multiphoton excitation, the time evolved wave packet arises from a superposition of many statesin several multiplets (with roughly constant averaged energy after some excitation time and a large energyuncertainty). The microcanonical distribution is given by the superposition of states in a single multiplet (ofthe same averaged energy but much smaller energy uncertainty). In the case of the CH chromophore in CHF3studied in this example, the distribution of populations within a molecular energy shell is not homogeneousduring the excitation process because the multiplets are not ideally centred at the multiphoton resonance levelsand their energy range is effectively too large when compared to the resonance width of the excitation process(power broadening). If molecular energy shells fall entirely within the resonance width of the excitation, suchas in the model systems of two harmonic oscillators studied in [97], population distribution within a shellbecomes more homogeneous [97, table V]. However, as discussed in that work, equidistribution of populationsdoes not imply that the wave packet is delocalized. Indeed, the contrary was shown to occur. If the probabilitydistribution in configuration space is to delocalize, the relative phases between the superposition states mustfollow an irregular evolution, such as in a random phase ensemble, in addition to equidistribution of population.Thus, one statement would therefore be that IVR is not complete, although very fast, during the multiphotonexcitation of CHF3. Excitation and redistribution are indeed two concurring processes. In the limit of weakfield excitation, in the spectroscopic regime, the result is a superposition of essentially two eigenstates (theground and an excited state, for instance). Within the bright state concept, strong IVR will be revealed by aninstantaneous delocalization of probability density, both in the bright and the dark manifolds, as soon asthe excited state is populated, because the excited state is, of course, a superposition state of states from bothmanifolds. On the other hand, strong field stepwise IR multiphoton excitation promotes, in a first step, thedeposition of energy in a spatially localized, time-dependent molecular structure. Simultaneously, IVR startsto induce redistribution of this energy among other modes. The redistribution becomes apparent after sometime has passed and is expected to be of the DIVR type, at least on longer time scales. DIVR may lead to acomplete redistribution in configuration space, if the separation between nearly iso-energetic states is smallcompared to the power broadening of the excitation field. However, under such conditions, at least during aninitial stage of the dynamics, CIVR will dominate.

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packetfor the isolated molecule, what are typical time scales and, if initial conditions are such that an entire energyshell participates, does the wave packet resulting from the coherent dynamics look like a microcanonicaldistribution? Such studies were performed for the case of an initially pure stretching Fermi mode (vs, vb =0), with a high stretching quantum number, e.g. vs = 6. It was assumed that such a state might be preparedby irradiation with some hypothetical laser pulse, without specifying details of the pulse. The energy of that


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Figure A3.13.9. Probability density of a microcanonical distribution of the CH chromophore in CHF3 within the multipletwith chromophore quantum number N = 6 (Nshell = N + 1 = 7). Representations in configuration space of stretching(Qs) and bending (Qb) coordinates (see text following equation (A3.13.62) and figure A3.13.10). Left-hand side: typicalmember of the microcanonical ensemble of the multiplet with N = 6 (random phases, equation (A3.13.66)). Right-handside: microcanonical density Pmicro = 1Nshell

nshell |n|2 for the multiplet with N = 6 (Nshell = 7). Adapted from [81].

state is located at the upper end of the energy range of the corresponding multiplet [81, 152, 154], which hasa total of Nshell = 7 states. Such a state couples essentially to all remaining states of that multiplet. Thecorresponding evolution of the isolated system is shown as snapshots after the preparation step (t0 = 0) infigure A3.13.10. The wave packet starts to spread out from the initially occupied stretching manifold (alongthe coordinate axis denoted by Qs) into the bending manifold (Qb) within the first 3045 fs of evolution(left-hand side). Later on, it remains delocalized most of the time (as shown at the time steps 80, 220 and380 fs, on the right-hand side) with exceptional partial recovery of the initial conditions at some isolated times(such as at 125 fs). The shape of the distribution at 220 fs is very similar to that of a typical member of themicrocanonical ensemble in figure A3.13.9 above. However, in Figure A3.13.9, the relative phases betweenthe seven superposition states were drawn from a random number generator, whereas in figure A3.13.10 theyresult from a fully coherent and deterministic propagation of a wave function.

IVR in the example of the CH chromophore in CHF3 is thus at the origin of a redistribution processwhich is, despite its coherent nature, of a statistical character. In CHD3, the dynamics after excitation of thestretching manifold reveals a less complete redistribution process in the same time interval [97]. The reasonfor this is a smaller effective coupling constant ksbb between the Fermi modes of CHD3 (by a factor of four)when compared to that of CHF3. In [97] it was shown that redistribution in CHD3 becomes significant in thepicosecond time scale. However, on that time scale, the dynamical separation of time scales is probably nolonger valid and couplings to modes pertaining to the space of CD3 vibrations may become important andhave additional influence on the redistribution process.

A3.13.5.2 IVR and time-dependent chirality

IVR in the CH chromophore system may also arise from excitation along the bending manifolds. Bendingmotions in polyatomic molecules are of great importance as primary steps for reactive processes involvingisomerization and similar, large amplitude changes of internal molecular structure. At first sight, the one-dimensional section of the potential surface along the out-of-plane CH bending normal coordinate in CHD3,shown in figure A3.13.4, is clearly less anharmonic than its one-dimensional stretching counterpart, also shownin that figure, even up to energies in the wave number region of 30 000 cm 1. This suggests that coherentsequential multiphoton excitation of a CH bending motion, for instance along the x-axis in Figure A3.13.5,


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Figure A3.13.10. Time-dependent probability density of the isolated CH chromophore in CHF3. Initially, the system isin a Fermi mode with six quanta of stretching and zero of bending motion. The evolution occurs within the multipletwith chromophore quantum number N = 6 (Nshell = N + 1 = 7). Representations are given in the configurationspace of stretching (Qs) and bending (Qb) coordinates (see text following equation (A3.13.62): Qb is strictly a positivequantity, and there is always a node at Qb = 0; the mirrored representation at Qb = 0 is artificial and serves to improvevisualization). Adapted from [81].


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may induce a quasiclassical motion of the wave packet along that manifold [159, 160], which is significantlylonger lived than the motion induced along the stretching manifold under similar conditions (see discussionabove). Furthermore, the two-dimensional section in the CH bending subspace, spanned by the normalcoordinates in the lower part of figure A3.13.4, is approximately isotropic. This corresponds to an almostperfectCv symmetry with respect to the azimuthal angle (in the xy plane of figure A3.13.5), and is related tothe approximate conservation of the bending vibration angular momentum ?b [152, 173]. This implies that thedirect anharmonic coupling between the degenerate bending manifolds is weak. However, IVR between thesemodes might be mediated by the couplings to the stretching mode. An interesting question is then to what extentsuch a coupling scheme might lead to a motion of the wave packet with quasiclassical exchange of vibrationalenergy between the two bending manifolds, following paths which could be described by classical vibrationalmechanics, corresponding to CIVR. Understanding quasiclassical exchange mechanisms of large amplitudevibrational motion opens one desirable route of exerting control over molecular vibrational motion and reactiondynamics. In [154] these questions were investigated by considering the CH bending motion in CHD3 and theasymmetric isotopomers CHD2T and CHDT2. The isotopic substitution was investigated with the special goalof a theoretical study of the coherent generation of dynamically chiral, bent molecular structures [174] and ofthe following time evolution. It was shown that IVR is at the origin of a coherent racemization dynamics whichis superposed to a very fast, periodic exchange of left- and right-handed chiral structures (stereomutationreaction, period of roughly 20 fs, comparable to the period of the bending motion) and sets in after typically300500 fs. The main results are reviewed in the discussion of figures A3.13.11A3.13.13.

The wave packet motion of the CH chromophore is represented by simultaneous snapshots of two-dimensional representations of the time-dependent probability density distribution

|sb(t,Qs,Qbi )|2 =Qbj

dQbj |(t,Qs,Qbi , Qbj )|2 (i = j) (A3.13.67)

and|bb(t,Qb1 ,Qb2)|2 =

Qs

dQs|(t,Qs,Qb1 ,Qb2)|2. (A3.13.68)

Such a sequence of snapshots, calculated in intervals of 4 fs, is shown as a series of double contour line plots onthe left-hand side of figure A3.13.11 (the outermost row shows the evolution of |bb|2, equation (A3.13.68),the innermost row is |sb|2, equation (A3.13.67), at the same time steps). This is the wave packet motionin CHD3 for excitation with a linearly polarized field along the the x-axis at 1300 cm1 and 10 TW cm2after 50 fs of excitation. At this point a more detailed discussion regarding the orientational dynamics ofthe molecule is necessary. Clearly, the polarization axis is defined in a laboratory fixed coordinate system,while the bending axes are fixed to the molecular frame. Thus, exciting internal degrees of freedom alongspecific axes in the internal coordinate system requires two assumptions: the molecule must be oriented oraligned with respect to the external polarization axis, and this state should be stationary, at least during therelevant time scale for the excitation process. It is possible to prepare oriented states [112, 114, 115] in thegas phase, and such a state can generally be represented as a superposition of a large number of rotationaleigenstates. Two questions become important then: How fast does such a rotational superposition stateevolve? How well does a purely vibrational wave packet calculation simulate a more realistic calculationwhich includes rotational degrees of freedom, i.e. with an initially oriented rotational wave packet? Thesecond question was studied recently by full dimensional quantum dynamical calculations of the wave packetmotion of a diatomic molecule during excitation in an intense infrared field [175], and it was verified thatrotational degrees of freedom may be neglected whenever vibrationalrotational couplings are not importantfor intramolecular rotationalvibrational redistribution (IVRR) [84]. Regarding the first question, because ofthe large rotational constant of methane, the time scales on which an initially oriented state of the free moleculeis maintained are likely to be comparatively short and it would also be desirable to carry out calculations that


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CHD

3

~

E

0

k

~

x

~

E

0

k

~

y

50 fs

54 fs

58 fs

62 fs

66 fs

6

Q

b

1

p

upm

50

50

70 fs

6

Q

b

2

p

upm

50

50

-

Q

b

2

=

p

upm

50 50

-

Q

s

=

p

upm

20 50

-

Q

b

1

=

p

upm

50 50

-

Q

s

=

p

upm

20 50

Figure A3.13.11. Illustration of the time evolution of reduced two-dimensional probability densities |bb |2 and |sb |2,for the excitation of CHD3 between 50 and 70 fs (see [154] for further details). The full curve is a cut of the potentialenergy surface at the momentary absorbed energy corresponding to 3000 cm1 during the entire time interval shown here(6000 cm1, if zero point energy is included). The dashed curves show the energy uncertainty of the time-dependentwave packet, approximately 500 cm1. Left-hand side: excitation along the x-axis (see figure A3.13.5). The verticalaxis in the two-dimensional contour line representations is the Qb1 -axis, the horizontal axes are Qb2 and Qs, for |bb|2and |sb|2, respectively. Right-hand side: excitation along the y-axis, but with the field vector pointing into the negativey-axis. In the two-dimensional contour line representations, the vertical axis is the Qb2 -axis, the horizontal axes are Qb1and Qs, for |bb|2 and |sb|2, respectively. The lowest contour line has the value 44 105 u1 pm2, the distancebetween them is 7 105 u1 pm2. Maximal values are nearly constant for all the images in this figure and correspondto 140 105 u1 pm2 for |bb|2 and 180 105 u1 pm2 for |sb|2.


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Figure A3.13.12. Evolution of the probability for a right-handed chiral structurePR(t) (full curve, see equation (A3.13.69))of the CH chromophore in CHD2T (a) and CHDT2 (b) after preparation of chiral structures with multiphoton laser exci-tation, as discussed in the text (see also [154]). For comparison, the time evolution of PR according to a one-dimensionalmodel including only the Qb2 bending mode (dashed curve) is also shown. The left-hand side insert shows the timeevolution of PR within the one-dimensional calculations for a longer time interval; the right-hand insert shows the PRtime evolution within the three-dimensional calculation for the same time interval (see text).

include rotational states explicitly. Such calculations were done, for instance, for ozone at modest excitations[116, 117], but they would be quite difficult for the methane isotopomers at the high excitations consideredin the present example.

The multiphoton excitation scheme corresponding to excitation along the x-axis is shown by the uprightarrows on the left-hand side of figure A3.13.4. In the convention adopted in [154], nuclear displacements


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800 fs

804 fs

808 fs

812 fs

816 fs

6

Q

b

2

p

upm

100

100

820 fs

6

Q

b

2

p

upm

100

100

-

Q

b

1

=

p

upm

100 100

-

Q

s

=

p

upm

40 70

-

Q

b

1

=

p

upm

100 100

-

Q

s

=

p

upm

40 70

Figure A3.13.13. Illustration of the time evolution of reduced two-dimensional probability densities |bb |2 and |sb |2,for the isolated CHD2T (left-hand side) and CHDT2 (right-hand side) after 800 fs of free evolution. At time 0 fs the wavepackets corresponded to a localized, chiral molecular structure (from [154]). See also text and figure A3.13.11.


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along Qb1 occur along the x-axis, displacements along Qb2 are directed along the y-axis. One observesa semiclassical, nearly periodic motion of the wave packet along the excited manifold with a period ofapproximately 24 fs, corresponding to the frequency of the bending vibrations in the wave number regionaround 1500 cm1. At this stage of the excitation process, the motion of the wave packet is essentiallyone-dimensional, as seen from the trajectory followed by the maximum of the probability distribution andits practically unchanged shape during the oscillations back and forth between the turning points. The latterlie on the potential energy section defined by the momentary energy E(t) of the wave packet, as de

energy redistribution in reacting systems

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