Intramolecular vibrational redistribution and energy relaxation

in solution: A molecular dynamics approachy

Gunter Kab,* Christian Schroder and Dirk Schwarzer

Max-Planck-Institut fur Biophysikalische Chemie, Am Faßberg 11, D-37077 Gottingen, Germany

Receivved 14th Auggust 2001, Accepted 30th October 2001First published as an Advvance Article on the web 7th January 2002

Recent IR-pump=UV-probe investigations on the intramolecular and intermolecular vibrational energy flow forCH2I2 dissolved in CDCl3 are complemented by classical molecular dynamics simulations. Directnonequilibrium molecular dynamics simulation is able to reproduce significant features of the energy gain and

loss in Franck–Condon-active CI2 stretch modes as observed via the UV-probe pulse. A key finding of oursimulations is the fact that most of the excess energy deposited by the IR-pump pulse is dissipated into thesolvent via the CH-mode of lowest frequency (CH2 rocking vibration) while the mode of lowest frequency (CI2bending) is merely a spectator in the overall process of energy redistribution and relaxation.

1 Introduction

Intramolecular vibrational redistribution (IVR) and inter-molecular vibrational energy transfer (VET) are undoubtedlyof fundamental importance in chemical reaction dynamics.The validity of statistical reaction rate theories, such as themicrocanonical RRKM or canonical transition state theories,depends crucially on how rapidly an equilibrium is establishedbetween the nonreactive modes or solvent and the chemicalreaction coordinate. Therefore, IVR has been investigatedunder isolated molecule conditions and in low pressure gasesfor quite some time1 and can be considered a relatively maturefield of research in chemical physics, both with respect toexperiment and theory.2–4 Similarly, intermolecular vibrationalenergy transfer of diatomic molecules in gases and liquids hasbeen studied by experimentalists and theorists alike using avariety of different techniques. For reviews see ref. 5–13. Keyissues are the notion of frequency dependent friction or theapplicability of the isolated binary collision (IBC) model inliquid state vibrational energy relaxation.In studies of large polyatomics (e.g. ref. 14), owing to the

large density of states, it is almost always assumed that thevibrational excess energy is nearly statistically distributedamong all modes on the timescale of VET (rapid IVR).Molecular dynamics simulations seem to justify this assump-tion.15,16 Small polyatomic molecules, on the other hand, dueto a substantially smaller density of states, at least for low tomoderate excitations, offer the possibility to selectively exciteand interrogate certain vibrational motions and therefore togain mode specific information on vibrational energy flow inmolecules and solutions.17–19 Moreover, due to slower IVR,one can hope to dissentangle intramolecular and inter-molecular energy relaxation processes. Undoubtedly, two- andthree-dimensional vibrational spectroscopies20 as well asIR=UV-pump–probe schemes21,22 will continue to contributeto a more detailed molecular understanding of energy path-ways in polyatomic gases and liquids.Before we go on to describe our model system and simula-

tions, it may be helpful to consider briefly some general issuesand models of IVR and VET pertaining both to the gaseous

and condensed phases. Firstly, even if a timescale separation isassumed for intramolecular and intermolecular vibrationalenergy transfer, these processes are by no means dynamicallydecoupled. A high frequency quantum of energy, �hO, may betransformed into two (or more) vibrational quanta of lowfrequency, �ho1 and �ho2 , according to

�hO ¼ �ho1 þ �ho2 þ �hophonon ; ð1Þ

where some portion of energy ends up in the solvent(�hophonon). Here, �hophonon represents the bath (gas or liquid)spectral density at a certain frequency.23,24 Secondly, iterationof eqn. (1) leads to a cascaded energy relaxation (‘ ladderrelaxation ’) via Fermi resonances (couplings) of low order(cubic or quartic resonances).20 Thirdly, it is often assumedthat most of the vibrational excess energy initially deposited inthe dissolved molecule is transfered to the solvent, after lowfrequency modes (�hO) have been significantly populated:

�hO ¼ �hophonon: ð2Þ

The basic argument is that, due to frequency dependent fric-tion, low frequency modes are very efficient in intermolecularenergy dissipation, while the most efficient way for dissipationof high frequency quanta affords participation of a smallnumber of intramolecular excitations as expressed by eqn. (1).Owing to their assumed importance in the intermolecularenergy relaxation, low frequency modes are sometimes called‘doorway vibrations ’20 or ‘gateway modes ’.25 The concept of‘doorway vibrations ’ implies that, at least for initial excitationof high frequency molecular modes, IVR is essentially com-plete before the largest part of the excess energy is transfered tothe solvent bath. Although, as stated above, IVR and VET arenot independent of one another, in the following we use theseterms when focusing on intramolecular and intermolecularenergy flow, respectively. The term VER (vibrational energyrelaxation), although often used interchangeably with VET, isused when referring to the overall energy relaxation processcomprising all molecular vibrational modes.Recently, fs-IR-pump=UV-probe investigations were

undertaken by Crim and coworkers18 as well as Abel andSchwarzer and coworkers19,26 on diiodomethane and similarmolecules in a variety of organic solvents. In these experi-ments, CH-overtones or combination bands were excited usingfs IR pulses, and the dynamic energy population and

y Presented at the annual meeting of the Deutsche Bunsen-Gesellschaft fur Physikalische Chemie, Stuttgart, May 24–26, 2001.

DOI: 10.1039/b107256k Phys. Chem. Chem. Phys., 2002, 4, 271–278 271

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depopulation of lower frequency Franck–Condon (FC) activemodes (CI2 a stretch and bend of methylene iodide) probedby UV excitation of a dissociative electronic transition. Theresultant pump–probe signals display simple rise and fallbehaviour, where trise and tfall are interpreted in terms of modespecific IVR and VET, respectively.Although the analysis of pump–probe signals by Bingemann

et al.18 and Charvat et al.19 is different, the models used to fitthe data are closely related. Bingemann et al. use a time-dependent temperature model, where the temperature-depen-dent UV absorption cross section of CH2I2 between 370 and430 nm obtained for temperatures from 275 to 320 K is linearlyextrapolated to higher temperatures relevant for the inter-pretation of transient signals. Charvat et al.19 extend the rangeof temperatures from 285 K to around 1440 K by measuringhigh temperature shock tube spectra. These data were used tocalibrate optical signals to mean internal energies, using thefact that for a polyatomic molecule microcanonical andcanonical spectra are identical provided the mean energies arethe same.Rise and fall times obtained from these investigations on

diiodomethane, which is the model system investigated here,are in general agreement for solvents CDCl3 and CCl4 underthe same conditions, where IVR and VET times are trise’10 psand tfall’ 60 ps, respectively.The results on intra- and intermolecular vibrational energy

transfer for methylene iodide in solution raise a couple ofquestions, which cannot be answered from inspection ofexperimental data alone. Although IVR and VET rates shouldgenerally depend on which modes are excited and interrogated,and thus the observed rise and fall times of the pump–probesignal reflect the specific choice of modes, the observations areinterpreted in terms of ‘global ’ quantities, i.e. internal tem-perature or mean energy. To some extent, an excitation modedependence of IVR times is ruled out by the experiments ofCharvat et al.,19 where different excitation energies correspondto different CH-overtones or combination bands. Therefrom itcan be concluded that equilibration among dominantly CH-vibrations is more rapid than energy transfer to the lowerfrequency CI-modes and especially the FC-active CI2 anti-symmetric stretch vibration. But how is the energy transmittedfrom the initially excited vibrational states to the Franck–Condon active modes interrogated by the probe pulse? Is aglobal picture of IVR appropriate, i.e. is IVR complete on thetimescale of the observed rise times, or do the IVR times notreflect the longest timescale of intramolecular energy redis-tribution? Can certain IVR bottlenecks, e.g. restricted IVRamong CH-excitations, be identified by simulation? Are IVRand global energy relaxation (VER) separable with respect totimescales?In the following, we will address some of these questions

by classical molecular dynamics simulation of diiodomethanein CDCl3 solution. Our paper is organized as follows. Insection 2 we outline the basic theory for performing andanalyzing molecular dynamics (MD) simulations of liquidstate vibrational energy relaxation of polyatomics, where weput special emphasis on analytical methods which allowdetailed investigation of intra- and intermolecular energyflow. Section 3 contains the model potential energy surface(PES) used for methylene iodide in CDCl3 solvent as well asinformation on the simulation procedure. In section 4 wepresent and discuss our results obtained by direct none-quilibrium MD simulation. Section 5 concludes by summar-izing our main findings.

2 Basic theory

In the following we briefly introduce the theoretical conceptsand methods which will be used to analyze and interpret the

molecular dynamics (MD) simulations of vibrational energyrelaxation. The basic theoretical framework has been describedelsewhere.27

The total energy of an isolated polyatomic (solute) moleculein the ground state of the electronic manifold can be written as

Hsolute ¼XNa¼1

1

2majvaj2 þ VðfragÞ ; ð3Þ

where the kinetic energy is separable according to

XNa¼1

1

2majvaj2 ¼

1

2mcjvcj2 þ

XNa¼1

1

2majvacj2; ð4Þ

yielding the center of mass (c.o.m.) translational energy andthe kinetic energy with respect to the c.o.m. (rotation, vibra-tion). vac¼ va vc defines the velocity of the ath atom relativeto the center of mass and is equal to

vac ¼ or � rac þ ua; ð5Þ

where or� rac and ua are the rotational and vibrational velo-cities of atom a, respectively. or denotes the rotational angularvelocity. Insertion of eqn. (5) into the last term of eqn. (4)yields the relative kinetic energy separated into rotational andvibrational components,

XNa¼1

1

2majvacj2 ¼

XNa¼1

1

2majðor � racÞj2 þ

XNa¼1

1

2majuaj2

þXNa¼1

maðor � racÞ � ua; ð6Þ

where the last term is called the Coriolis energy, reflecting thecoupling of vibration and rotation.In order to define the vibrational (and rotational) motions of

atoms in a polyatomic molecule, a prescription is necessary forseparating rotation and vibration. While for a diatomicrotational and vibrational atomic velocities are always per-pendicular to each other and thus the Coriolis energy is zero,for a polyatomic molecule there are different ways to define themolecular reference frame. The Eckart frame28,29 provides amolecule-fixed cartesian coordinate system, which is designedsuch as to minimize the vibration–rotation interaction andthus the Coriolis energy. The Eckart frame is defined by theequations (the Eckart–Sayvetz conditions28,30,31)

XNa¼1

marac ¼ 0 ð7Þ

XNa¼1

mar0ac � Drac ¼ 0; Drac ¼ rac r0ac: ð8Þ

Here, the first condition defines the center of mass and {r0ac} isthe dynamic, i.e. rotated, equilibrium geometry of the mole-cule. The three Eckart basis vectors fi (i¼ 1, 2, 3), groupedcolumn-wise, define an orthogonal transform f, which rotatesthe predefined static molecular equilibrium geometry {d 0ac}(laboratory frame) into its new orientation,

r0ac ¼ f d 0ac; ð9Þ

the time dependence of which is determined (in the lab frame)by

_r0ac ¼ or � r0ac: ð10Þ

For details of how the Eckart frame vectors are generated thereader is referred to ref. 29.Using the solute-fixed Eckart frame calculated at every time

step, the rotational angular velocity is computed. The rota-

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tional and hence vibrational atomic velocities are then deter-mined through eqn. (5), and the rotational kinetic energy is

XNa¼1

1

2majðor � racÞj2 ¼

1

2or Ior ; ð11Þ

where I is the symmetric but non-diagonal inertia tensor.Certainly, the main MD observables in simulations of

vibrational energy transfer are the time-dependent energies ofvibrational modes, where we use cartesian normal modesderived from the isolated solute molecule potential energyfunction. The normal mode vectors, i.e. linear combinations ofatomic displacements with respect to the c.o.m., grouped col-umn-wise into a matrix Tq , define the transformation betweennormal mode displacements (or velocities) and mass-weightedatomic displacements (velocities) according to ref. 31 and 32

Dq M1=2 Dr ¼ Tq Q; ð12Þ

where Dr and Dq contain the cartesian and mass-weightedcartesian displacements of all solute atoms in one vector. M1=2

is the diagonal square-root mass matrix and the vector Qcontains the scalar normal mode displacements. For CH2I2there are a total of 3N¼ 15 normal modes. Because the c.o.m.kinetic energy as well as the rotational energy are calculatedfrom atomic velocities, only the remaining 3N 6¼ 9 normalmode vectors are explicitly needed to calculate the vibrationalnormal mode energies.The vibrational energy of the solute molecule up to quad-

ratic order in the displacements is then

X3N6

n¼1En ¼

X3N6

n¼1

1

2_Q2n þ

1

2o2nQ

2n

� �; ð13Þ

where on is the vibrational angular velocity of the nth normalmode and o2n its force constant. Anharmonicities given byhigher polynomial orders of the potential energy in terms ofnormal mode displacements (cubic, quartic terms, etc.) are ne-glected in a first approximation. However, these terms, apartfrom contributing to the diagonal normal mode energies, leadto energy exchange between normal modes and thus drive in-tramolecular vibrational redistribution. Additional anharmo-nicities, including bilinear terms, are to be expected due tothe solute–solvent interaction energy.Besides the time-dependent energies of translational, rota-

tional and vibrational degrees of freedom (DoF) of the dis-solved molecule, the concept of time-dependent capacity

(power) associated with each DoF is of central importance toour approach. The capacity of external forces is defined through

NsoluteðtÞ dEsolutedt

¼XNa¼1

F eaðtÞ � vaðtÞ ; ð14Þ

where F ea(t) is the external force at time t exerted by the solvent

environment upon the ath solute nucleus and va(t) is the velo-city of that atom. The capacity is a measure of the solute energychange, as expressed by eqn. (14). From the time-dependent capacity (power), the external work done by theenvironmental degrees of freedom can simply be obtained byintegration:

W esoluteðtÞ ¼

Z t

0

NsoluteðtÞdt: ð15Þ

Separation of atomic velocities into translational, rotationaland vibrational components leads to

NsoluteðtÞ ¼ F ec � vc þ T e � or þ

XNa¼1

F ea � ua ; ð16Þ

where T e is the external torque exerted upon the rotationalangular velocity vector. As can easily be recognized by

differentiation of eqn. (6) with respect to time, completeseparability of external force capacities into rotational andvibrational components on the level of each atom can only beachieved if part of the time-derivative of the Coriolis energy isassigned to the rotational and vibrational terms, respectively.In addition, kinematic forces (i.e. centrifugal and Coriolis for-ces) arise from the dynamic coupling of rotation and vibration,which leads to energy exchange between these DoF as well as tomixing of vibrational modes. While for a diatomic the corre-sponding capacity terms are straightforwardly evaluated,33

they are rather complicated for a general polyatomic mole-cule,27 and therefore are skipped here for simplicity. Rotation–vibration work terms are not reported in this paper.For the purposes of our study, the vibrational capacity (last

term in eqn. (16)) is conveniently expressed in terms of normalmode quantities as

NvibðtÞ ¼XNa¼1

F ea � ua ¼

X3N6

n¼1F e

n_Qn ; ð17Þ

where the vector F eQ of external normal mode forces F e

n iscomputed from atomic forces F e

a (contained in the vector F e)through

F eQ ¼ T T

q M1=2 F e: ð18Þ

Finally, for the analysis of microcanonical vibrationalenergy equilibration we define a coarse-grained informationentropy in terms of the partitioning of vibrational excessenergy among normal modes. Therefore we use the ‘ energyoccupation probabilities ’

pn ¼En

Evib

� �’ hEni

hEvibi; ð19Þ

where brackets denote ensemble averaging and Evib is the sumof normal mode energies En . If microcanonical equipartitionis established at a given vibrational energy, the occupationprobabilities assume the value pn¼ 1=Nvib for all n. The coarse-grained entropy

SvibðtÞ ¼ XNvibn¼1

pn ln pn; Smaxvib ¼ lnNvib ð20Þ

is a global measure of microcanonical vibrational energypartitioning. Using the ansatz Svib(t)¼ lnNeff , an effectivenumber of participating (energy sharing) vibrational modes,

NeffðtÞ ¼ expðSvibÞ; ð21Þ

can be obtained, which provides a physically intuitive inter-pretation of the nonequilibrium entropy Svib(t).

3 Model and simulation procedure

In the present work we report on direct nonequilibriummolecular dynamics simulations of VER for diiodomethane inCDCl3 solution at 300 K after excitation of the solute n1þ n6combination mode (CH2 s=a stretch). We used a simple modelPES designed to accurately reproduce the spectrum of funda-mental frequencies for CH2I2 in the gas phase. The isolatedmolecule potential energy functions for methylene iodide andCDCl3 essentially comprise quadratic stretch and bendingangle potentials. Only the C–I bonds are described by Morsefunctions. Table 1 collects the resultant harmonic normalmode frequencies for diiodomethane compared to theirexperimentally observed values, together with mode assign-ments19,34 and symmetries. Intermolecular interactions areparametrized through Lennard–Jones (LJ) 12-6 pair-wisefunctions (short range) and coulombic forces between atomicpartial charges (long range).

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The minimum energy geometry as well as the partial chargesof CH2I2 are derived from DFT quantum chemical calcula-tions (B3LYP/DZVP(I),35 Sadlej-pVTZ(C,H);36 Gaussian9837), where atom charges have been scaled to reproduce thedipole moment of the solute.38 Intramolecular interactionparameters as well as atomic partial charges for the CDCl3solvent are used as derived by Fox and Kollman.39 As for thediiodomethane solute, these partial charges are scaled to theexperimental dipole moment. Lennard-Jones parameters forsolute–solvent and solvent–solvent intermolecular forces aretaken from the AutoDock 3.040 force field. Tables 2 and 3report the force constants and equilibrium bond lengths andangles for solute and solvent, respectively.MD simulations have been performed for one solute mole-

cule embedded in a cubic box (periodic boundary conditions)containing 245 solvent molecules employing the DLPOLY2.1141 (and DLPROTEIN 1.242) molecular dynamics package.The dimension of the simulation box is chosen as equal totwice the short range cut-off defined as four times the largestLennard-Jones diameter. In order to preclude artificial highfrequency components of solvent forces arising from atomsmoving across the cut-off region, the truncated LJ potentialsare shifted such that ULJ(rcut)¼ 0, and smoothed to make ULJdifferentiable at r¼ rcut . Long range coulombic forces beyondthe LJ cut-off are computed by Ewald summation.43

Nonequilibrium MD simulations proceed as follows. Initi-ally, excess vibrational energy is favourably stored as potentialenergy by appropriately displacing the respective normalmodes, while thermalized modes may be initiated with randomvibrational phase at a given canonically sampled energy. Sincevibrational phases are rapidly randomized within a few pico-seconds, the energies of all normal modes may also be initiallystored in the potential. In addition, we have chosen to includethe zero point energy (ZPE) for all vibrational modes, in orderto start nonequilibrium simulations in the energy region whereinitial states are prepared by the experimental pump scheme.In our first attempt to simulate the pump–probe scenario, thepumped CH modes (n1 and n6) thus initially each contain oneexcess quantum of energy, while the energies of thermalized

modes are sampled from a quantum statistical distribution atT ¼ 300 K. In a classical simulation however, account must betaken of unphysical effects due to zero point energy flow.Despite the neglect of intrinsic quantum properties of highfrequency vibrations in classical dynamical simulations, webelieve that essential dynamical features of IVR and VET arecaptured by classical MD.Typically, we start from a solute embedded in a fcc lattice of

solvent pseudo-particles. The first initial state is drawn after aperiod of �100 ps, where the mean solvent temperature (300K) is fixed by velocity scaling. Subsequent starting configura-tions are obtained after additional 10 ps periods of time. Afteran ensemble of independent initial states of the total many-particle system has been prepared subject to the above-men-tioned constraints, each member of the ensemble is propagatedin time within the NVE ensemble (250 ps). The latter ensembleensures strictly Hamiltonian, time-reversible dynamics, wheretotal energy and momentum are conserved on the average.Conservation of total energy is absolutely essential in none-quilibrium studies of IVR and VET in solution, when mean-ingful results shall be obtained. The heat capacity of the system(system size) can be chosen to keep unavoidable rises of tem-perature within an acceptable range (4 10 K). Ensembleaveraging over 30 to 50 statistically independent none-quilibrium trajectories is typically sufficient.

4 Results and discussion

In this section we report the results of nonequilibrium MDsimulations for our model system CH2I2 in CDCl3 solution at300 K and a liquid density of r¼ 12.3 mol L1, initially pre-pared by excitation of the CH2 symmetric(s)=antisymmetric(a)stretch (n1þ n6) combination mode. The latter combinationband is prepared, together with the dominant 2n1 overtone,when exciting the CH2I2 solute at wavelength 1700 nm.

Fig. 1 Exponential fit to the ensemble averaged total vibrationalkinetic energy, yielding a VER time of 23 ps. Additional averaging ofthe energy decay along the time axis has been done over periods of100 fs.

Table 1 Vibrational normal mode frequencies of CH2I2 obtainedfrom our model force field compared to experimental values34

Normal mode Symmetry (C2v) n~exp=cm1 n~fit=cm

1

Cl2 bend n4 A1 121 122Cl2 s stretch n3 A1 486 493Cl2 a stretch n9 B2 570 567CH2 rock n7 B1 716 715CH2 twist n5 A2 1028 1199CH2 wag n8 B2 1105 1356CH2 bend n2 A1 1351 1361CH2 s stretch n1 A1 2968 2966CH2 a stretch n6 B1 3049 3051

Table 2 The CH2I2 force field used in this work

Bond kbond=eV+A2 DMorse=eV bMorse=

+A1 r0=

+A

C–I 2.293720 1.594221 2.164C–H 30.895555 1.095

Angle kangle=eV rad2 y0=degrees

I–C–I 7.365001 116.409I–C–H 4.025785 107.169H–C–H 2.917914 111.853

Table 3 The CDCl3 force field39

Bond kbond=eV+A2 r0=

+A

C–D 29.487692 1.100C–Cl 20.155705 1.758

Angle kangle=eV rad2 y0=degrees

D–C–Cl 3.304356 107.700Cl–C–Cl 6.738805 111.300

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Fig. 1 shows the ensemble-averaged decay of total vibra-tional kinetic energy, Kvib¼

PnKn , where additional aver-

aging along the time axis has been performed over periods of100 fs to smooth residual oscillations. As clearly visible, VERis exponential to a very good approximation, the associatedrelaxation time being tVER¼ 23 ps. Therefrom it can alreadybe concluded, that a timescale separation between IVR andVET must exist at least for the subspace of those vibrationalmodes, which significantly contribute to VER. Otherwise,nonexponential VER kinetics would be obtained.Turning to normal mode vibrational energies (Fig. 2–5), we

observe that energy transfer from the initially excited n1 and n6modes to lower frequency dominantly CH-vibrations is veryrapid (with times of the order of 1 ps or below), to the extentthat rise and fall behaviour is observable, with a time resolu-tion of 100 fs, only for the CH modes of lowest frequency (n5and n7 , see Table 4). As an example, we consider in Fig. 2 thedecay of the n6 vibration (3051 cm

1, CH2 a stretch) togetherwith the vibrational kinetic energy of the n7 mode (715 cm

1,CH2 rocking). The former decay is biexponential, with timeconstants t1¼ 2.3 ps (63%) and t2¼ 23 ps (37%). Time evo-lution of the n7 kinetic energy is characterized by rise and falltimes, trise¼ 0.4 ps and tfall¼ 23 ps, respectively. The n2 and n8normal vibrations are instantaneously populated (Fig. 3 and4), on a timescale of 100 fs, and only their decays are resol-vable. The associated energy decay times are equal (see

Table 4), which can be traced back to the fact that in ourparametrization the frequencies of these normal vibrations are,by accident, very close to each other (see Table 1).For the lower frequency vibrations mostly localized on the

CI2 moiety, including the Franck–Condon active n9 mode,energy rise times are longer by factors of 2.4 to 21 as comparedto energy population times observed for the higher frequencyCH modes (Table 4). Fig. 5 displays the rise and fall behaviourexhibited by the FC active CI2 antisymmetric stretchvibration (n9¼ 567 cm1). The associated rise and fall times are

Fig. 3 The same as Fig. 2 for vibrational modes n1 , n2 and n5 .

Fig. 2 Time evolution of the vibrational kinetic energy of CH2I2normal modes n6 and n7 (ensemble averaging plus time averaging, seeFig. 1). The total vibrational excess energy divided by the number ofstrongly coupled modes (eight) is drawn for comparison, illustratingthe extent of microcanonical equilibration.

Fig. 4 The same as Fig. 2 for vibrational modes n3 and n8 .

Fig. 5 The same as Fig. 2 for vibrational modes n4 and n9 .

Table 4 Parameters obtained from a multiexponential fit of time-dependent normal mode vibrational kinetic energies. In addition to en-semble averaging, data have been averaged over periods of 100 fs alongthe time axis. Decay parameters for vibrations n1 , n3 , n5 , n6 , n7 and n9have been determined with reasonable accuracy by fixing the second(longer) relaxation time to tVER¼ 23 ps

ni t1=ps A1=eV t2=ps A2=eV hti=ps

n4 — — — — —n3 5.8±0.2 0.102 23±4 0.100 —n9 8.5±0.5 0.106 23±5 0.107 —n7 0.4±0.1 0.054 23±2 0.084 —n5 2.4±0.1 0.087 23±3 0.101 —n8 19±2 0.105 — — 19n2 19±2 0.104 — — 19n1 3.7±0.1 (56%) 23±3 (44%) 12n6 2.3±0.1 (63%) 23±5 (37%) 10

Evib 23±1 0.723 — — 23

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trise¼ 8.5 ps and tfall¼ 23 ps, respectively, where the energyrise time is in remarkable agreement with experiment,18,19

while the decay time is too low by a factor of �3 (which isnevertheless satisfactory).Interestingly, the vibrational energy of the lowest frequency

n4 vibration (122 cm1, CI2 bending) stays close to equilibrium

for all times (Fig. 5). Seemingly, this mode is merely a spec-tator in the overall process of vibrational redistribution andrelaxation, thereby questioning the general concept of lowestfrequency ‘doorway vibrations ’. This fact will become moreevident below, when we analyze the mode specific inter-molecular energy transfer.In Fig. 2–5, the total excess vibrational energy from Fig. 1

divided by 8 (the number of strongly coupled vibrationalmodes, neglecting the n4 spectator mode) is drawn as a func-tion of time for comparison. This helps to illustrate thatessentially all modes, except n4 , after an initial period definedby the longest IVR timescale, decay with similar time constantsin the range 19 to 23 ps, which is approximately equal to theglobal VER time (see Table 4). Thus, the picture of IVR andVET which emerges from our analysis so far is, that theinitially excited CH stretch modes (or the ‘bright state ’) decaymainly due to IVR to lower frequency vibrations at earlytimes. IVR is essentially complete after 8.5 ps. As a result, allmodes which are strongly coupled to each other, and thereforeshare the remaining excess vibrational energy, relax on thesame (VER) timescale.The mode selective energy transfer data obtained from our

simulations exhibit some interesting patterns worth mention-ing. Firstly, rise times, when observed, are shorter for highfrequency CH modes than for low frequency CI modes,implying that excitation transfer from the initially excited n1and n6 vibrations to the CI2 symmetric and antisymmetricstretches (n3 and n9) is not facilitated by direct low order (cubicand quartic) coupling terms in the potential. Rather, energytransfer between vibrational modes far apart on the frequencyscale seems to be dominated by sequential low order processesvia intermediate modes.In addition, symmetry restrictions seem to be operative

(compare Table 1). We note that n1 and n6 vibrations (CH2 s=astretches) are of symmetry A1 and B1 , respectively, under theC2v point group. As an example, the faster population of n7(CH2 rocking, B1 symmetry) as compared to n5 (CH2 twist, A2)may be explained by a symmetry derived propensity rule. Thesame applies to the more rapid energy gain of the n3 CI2symmetric stretch mode (A1) as compared to the CI2 anti-symmetric stretch vibration (n9 , B2 symmetry).We now turn to an analysis of microcanonical equilibration.

Table 5 reports the decay parameters obtained for therelaxation of occupation probabilities pn , defined in eqn. (19),towards their microcanonical average 1=Nvib¼ 1=9. For theinitially excited modes (n1 and n6), microcanonical equiparti-tion is largely established within a few picoseconds. For the n2 ,n7 and n8 vibrations microcanonical equilibration cannot even

be resolved and is assumed to occur within the first few 100 fs.The same trends as already observed for the rise times (Table4), with respect to frequency and symmetry of the respectivevibrations, are also reflected in the mode specific micro-canonical equilibration times.We emphasize that the apparent equilibration time of 73 ps

for the lowest frequency n4 (CI2 bending) vibration is an arti-fact. Since the energy of that mode stays at equilibrium for alltimes, the relaxation of pn towards microcanonical equiparti-tion reflects merely the decay of total vibrational energy, whichoccurs in the denominator of pn . With the decay parametersfor hKvibi in hand, it can indeed be shown that the (non-exponential) relaxation of 1=hKvibi towards equilibrium leadsto such a long apparent decay time for pn . This makes clearthat a meaningful interpretation of the pn decay in terms ofmode specific IVR times can only be obtained if the timescalesobserved are significantly below the VER time, i.e. therelaxation time for the total vibrational energy (see also thesecond decay component for the n6 vibration in Table 5).Nevertheless, observation of such long (apparent) pn relaxationtimes implies the presence of slow IVR processes, with timescales on the order of or longer than the VER time.A global picture of the IVR process can be obtained by

inspection of the time-dependent coarse-grained informationentropy Svib(t), defined through eqn. (20), and the effectivenumber of participating vibrational modes, Neff(t) (eqn. (21)),derived therefrom. Fig. 6 depicts the time evolution of Neff in asemilogarithmic plot, where separate timescales are immedi-ately visible through the occurrence of turning points. In fact,the decay of Neff towards the value Nvib¼ 9 at microcanonicalequilibrium is triexponential, with decay times t1¼ 0.6 ps(21%), t2¼ 3.0 ps (65%) and t3¼ 68 ps (14%), respectively(Table 6). The first two relaxation times are too close to beseparable in the semilogarithmic plot of Fig. 6. AlthoughSvib(t), and therefore also Neff(t), is a highly nonlinear functionof the mode specific occupation probabilities pn , inspection ofthe ranges of Neff , to which the respective decay times corre-spond (Table 6), points to a simple and straightforwardinterpretation of timescales. t1 (Neff ¼ 5.3 to 6.0) is associatedwith microcanonical equilibration among the 6 CH2 vibra-tions, t2 (Neff ¼ 6.0 to 8.3) can be interpreted as the equili-bration time of CH modes with the two CI2 stretches (n3 andn9), and t3 (Neff ¼ 8.3 to 8.8), which is actually very close to theapparent IVR time of the CI2 bending vibration (n4), reflectsthe IVR bottleneck separating the modes of higher frequencyfrom the latter. Note that for a quantum statistical distribution

Table 5 Parameters obtained from a multiexponential fit of time-de-pendent occupation probabilities pn of vibrational modes

ni t1=ps t2=ps

n4 73±3 —n3 3.7±0.4 —n9 6.0±0.3 —n7 — —n5 1.0±0.2 —n8 — —n2 — —n1 0.5±0.2 (43%) 5.7±0.6 (57%)n6 2.1±0.2 (85%) 92±17 (15%)

Fig. 6 The effective number of vibrational modes participating in themicrocanonical distribution of energy. Relaxation times obtained froma triexponential fit may be used to characterize the global IVR process.

276 Phys. Chem. Chem. Phys., 2002, 4, 271–278

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of mode energies, with the two CH2 stretches each carrying anexcess quantum of energy, the theoretical value of Neff at timezero is Neff ¼ 4.8. Thus, during the first 100 fs (our timeresolution) the effective number of participating vibrations hasincreased by 0.5 units.Finally, we come to an analysis of the mode specific inter-

molecular work terms (see eqn. (14)–(17)). Fig. 7 contains theexternal work (in percent) done by the environmental DoFupon the vibrational normal modes, as computed from eqn.(15) and (17), after the total simulated time interval of 250 ps.These work terms correspond to the direct solute–solventenergy flow via the respective vibrations. While the CH2 stretchvibrations (n1 and n6 , pumped by the IR pulse) each contributeto the overall VER process to an amount of 9 to 10%, thelargest part of the vibrational excess energy is transferred tothe solvent via the n7 (CH2 rocking) mode. Intermediatevibrations (on the frequency scale), i.e. n5 , n8 and n2 , make lesscontributions. Approximately 7.5% of the excess energy isdissipated via each of the the CI2 stretches (n3 and n9), whereasthe lowest frequency mode (n4 , CI2 bending) is of minorimportance.At first sight, these findings are in qualitative agreement with

the rapid population of the n7 vibration, and the longer risetimes observed for the CI2 stretch modes, especially theFranck–Condon active n9 vibration (Table 4). However, thedifferent contributions of normal mode vibrations to inter-molecular energy transfer, except for the n4 vibration, do notreflect nonequilibrium effects due to mode specific IVR times.Rather, the excess vibrational energy transiently accumulatedby those modes showing rise and fall behaviour already cor-responds to a statistical distribution of the excess energy(restricted microcanonical equilibrium) as clearly visible fromFig. 2–5. At times beyond approximately 6–8.5 ps, which is thelongest mode specific IVR time observed (Tables 4 and 5), allmodes except n4 are at microcanonical equilibrium. Thus, thedifferent mode contributions to VET largely reflect the VETefficiencies of vibrations while the excess energy is sharedamong strongly coupled modes. As expected from the so-called

Landau–Teller expression6,12,44–46 for the classical VET rateconstant of a harmonic oscillator,

kVET ¼ 1

tVET¼ 1

mnkBT

Z 1

0

hdF enðtÞdF e

nð0Þi cosðontÞdt; ð22Þ

the VET efficiency of the nth normal mode vibration, i.e. therelative rate, is governed by the vibrational frequency on aswell as the spectral density of the fluctuating mode specificexternal solvent force dF e

n . The latter depends upon theproperties of the mode specific solute–solvent interaction, therepulsive part of which is steeper for C–H vibrations than forC–I modes. As a result, the decay of the mode specific externalforce spectral densities with increasing frequency is expected tobe less steep for C–H modes than for C–I vibrations.The work terms of Fig. 7 thus reflect effects of restricted

IVR, due to slow population of the lowest frequency n4vibration (IVR bottleneck), as well as the different efficienciesof normal mode vibrations in intermolecular vibrationalrelaxation at (restricted) microcanonical equilibrium. Theoverwhelmingly large contribution of the CH2 rocking vibra-tion n7 to VET, which is explained by the stiff hydrogen–sol-vent potential as compared to the iodine–solvent interaction,renders the general applicability of the concept of ‘doorwayvibrations ’ questionable under realistic nonequilibrium con-ditions, where it is assumed that in a polyatomic molecule themodes of lowest frequencies do most of the work in dissipativeintermolecular energy transfer. In addition to IVR bottlenecks,a competition between the efficient damping of low frequencyvibrations and the effect of mode specific frictional spectraldensities has to be taken into account.

5 Conclusion

In the present work we have complemented recent IR-pump=UV-probe investigations on IVR and VET for diiodo-methane in CDCl3 solution by classical molecular dynamicssimulations. The vibrational dynamics following ultrafastexcitation of CH combination bands has been studied usingdirect nonequilibrium molecular dynamics. Despite the inher-ent inconsistencies arising from a classical dynamical treat-ment of high frequency modes due to zero point energy effects,classical MD simulations are able to reproduce significantfeatures of the energy gain and loss for the Franck–Condonactive CI2 stretch vibration as observed via the UV-probepulse.On the theoretical side, we have discussed the powerful

concept of capacity (power) recently developed and applied inour group, which allows for a detailed investigation of modespecific energy flow in IVR and VET processes. An intuitiveglobal picture of the IVR process may be obtained in terms ofthe effective number of energy sharing vibrations, derived froma coarse-grained nonequilibrium vibrational entropy.The energy rise time of the FC-active n9 antisymmetric

stretch obtained from simulations is in remarkable agreementwith experiment, although our simple model PES was fitted tofundamental solute frequencies only, and intermolecularinteraction parameters were drawn from a general force field.The energy decay time for the CI2 antisymmetric stretch is toolow by a factor of �3, which can be attributed to properties ofthe intermolecular potential. However, our observation that atlong times essentially all vibrational modes decay with thesame rate, determined by the global VER time, is in accordwith the interpretation of experiments.While microcanonical equilibration of vibrational energy

among the high frequency CH-modes is found to be rapid andcomplete on a sub-ps to ps timescale, the energy flow to thelower frequency CI2 stretches takes longer (6 to 9 ps). Thelowest frequency CI2 bending vibration is only weakly coupledto all the other modes. The overall 3-phase (restricted) IVR

Table 6 Parameters obtained from a multiexponential fit of time-de-pendent effective number of participating modes, Neff

t1=ps A1 (%) t2=ps A2 (%) t3=ps A3 (%) hti=ps

0.6±0.2 21 3.0±0.2 65 68±3 14 12Neff 5.3–6.0 6.0–8.3 8.3–8.8

Fig. 7 Mode specific intermolecular work terms (in percent) for vi-brational normal modes after the total simulated time interval of 250ps. Vibrations appear in order of increasing frequency (see Table 1).

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kinetics is also nicely reproduced in the global picture in termsof the effective number of energy sharing vibrations.A key finding of our simulations is the fact that most of the

excess energy deposited by the IR pump pulse is transferred tothe environment via the CH2 mode of lowest frequency,namely the CH2 rocking vibration, while the mode of lowestfrequency (CI2 bending) is merely a spectator in the overallprocess of energy redistribution and relaxation. Nevertheless,IVR and VET appear to be globally separable with respect totimescales.While we argue that IVR is driven largely by low order

processes, i.e. cubic and quartic terms in the intra- and inter-molecular potential, this issue deserves further investigation.Forces and thus work terms responsible for IVR can, inprinciple, be calculated using the concepts outlined in thetheory section. A comparison of IVR in the isolated CH2I2molecule to results obtained in solution may help to clarify therole of solvent in intramolecular vibrational redistribution.Although we believe that essential dynamical features of

IVR and VET are captured by classical MD, we have to admitthat classical simulations suffer from inescapable incon-sistencies arising from the intrinsic quantum properties of highfrequency vibrations, most notably zero point energy effectsand dynamical tunneling. Methodologies are available todescribe solute dynamics in terms of a quantum masterequation for the reduced system density matrix, employing(force–force) correlation functions from classical simulations.In addition, we have begun to investigate the applicability ofhybrid quantum/classical methods to the direct non-equilibrium simulation of vibrational energy flow in solution.

Acknowledgements

This work was made possible by a generous funding of com-puting facilities in our department by the Max-Planck-Gesellschaft. The authors are grateful to Prof. Jurgen Troe forhis continuous support, as well as to Dr. Bernd Abel andcoworkers for many helpful discussions. We thank Prof.Vyacheslav Vikhrenko, who has contributed much to ourexperience in statistical mechanics and molecular dynamics inrecent years. G.K. thanks Prof. Martin Quack for a discussionof the coarse-grained entropy used in this paper, pointing tothe necessity of relating this quantity to the most fine-grainedquantum version of entropy. Finally, we thank Prof. F.Fleming Crim for providing data on liquid phase CH2I2vibrational relaxation in the region of fundamentals prior topublication.25

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