Pure & AppL Chem., Vol. 47, pp. 61—73. Pergamon Press, 1976. Printed in Great Britain.

MOLECULAR DYNAMICS OF CHEMICAL REACTIONS

D. R. HERSCHBACH

Department of Chemistry, Harvard University, Cambridge, Massachusetts, USA

Abstract—Molecular beam and spectroscopic techniques allow detailed study of many dynamical properties of singlereactive collisions. The chemical scope of these methods is now very wide and includes certain unimolecular andtermolecular reactions as well as bimolecular reactions and energy transfer processes. Results for more than 50families of A + BC — AB + C atom transfer reactions reveal simple impulsive and persistent complex regimes thatcorrelate with electronic structure. Recent work has found examples of AB + CD -+AD+ BC and AB + CD + EF -AF+ BC + DE reactions that require exchange of two or three pairs of bonds in a single collision event yet proceedwith practically no activation energy. Processes akin to liquid phase reactions are also becoming accessible todynamical studies using beams of van der Waals polymers or solvation clusters.

The study of the intimate mechanics of reaction andenergy transfer in individual molecular collisions hasbecome one of the most active frontiers of chemicalphysics. The main experimental approach is molecularbeam scattering.' This involves forming the reagentmolecules into two collimated beams, each so dilute thatcollisions within them are negligible. The two beamsintersect in a vacuum and the direction and energy of theproduct molecules recoiling from the collision zone aremeasured. A host of reactions can now be studied readilyin this way, by virtue of an extremely sensitive massspectrometric detector and the use of supersonic nozzleswhich produce beams with greatly enhanced intensity andwith collision energies much higher than provided byordinary thermal sources. The coupling of spectroscopictechniques with molecular beams has provided furtheradvances in selectivity and sensitivity, particularly by useof intense laser sources.

The reaction properties now accessible include thedisposal of energy among translation, rotation, andvibration of the product molecules; angles of productrecoil; angular momentum and its orientation in space;and variation of reaction yield with impact energy,closeness of collision, rotational orientation or vibrationalexcitation of the target molecule. These experiments havestimulated many theoretical developments,5 includingvery useful diagnostic procedures based on informationtheory,6 a variety of insightful mechanical models,7 andthe burgeoning field of "computer experiments" in whichenormous numbers of collision trajectories are calculatedfor various postulated forces and compared with modelcalculations and laboratory data.8

In this paper we illustrate some of the molecular beamexperimental methods and results for a few prototypereactions. We give a heuristic discussion of two favoritereactions which exemplify the contrasting regimes ofimpulsive and persistent complex dynamics in atomtransfer processes,

A + BC -AB + C.

Single-collision data for more than fifty families of suchreactions have now been obtained and many other aspectsare treated in more comprehensive reviews.' In particu-lar, we do not discuss here the very fruitful work onion-molecule reaction dynamics.9 Our main emphasis ison recent studies dealing with remarkably facile exchange

61

reactions involving two or three pairs of bonds,

AB + CD-AD + BC

AB+CD+EF—AF+BC+DE.

These results have prompted electronic structure calcula-tions which suggest that the isotopic exchange reaction ofmolecular hydrogen and deuterium, long presumed to bebimolecular, is in fact termolecular. Finally, we brieflyconsider new work on reactions of van der Waals poly-mers and solvation clusters, which offer prospects fordynamical studies of processes akin to liquid phasereactions.

EXPERIMENTAL PERSPECTIVES

Figure 1 shows typical reactive scattering data for threesystems obtained from a crossed-beam apparatus withmass spectrometric detector.'° The detector unit ismounted on a rotatable "lid" which allows it to scan awide range of scattering angles in the plane of the reactantbeams. About 0 1% of the incoming molecules areionized. Velocity analysis is performed by installing aslotted disk "chopper" which is rotated at high speeds topulse the beam of scattered product molecules and thusallow measurement of the time of flight. A small computerconverts the flight times to a velocity distribution of theproduct, subtracts the background counts, and prints outthe data. The scattering angles indicated are measuredfrom the direction of the reactant atom beam (designatedas 0°) toward the reactant molecule beam (designated as900). At the peak of the product distribution, the signal isonly about 10—100 counts per second. The total countingtime required to collect the data shown was about fivehours for each reaction. Without going into details of theanalysis, we can see directly from Fig. 1 that the threereactions display qualitatively different dynamics. In theD + Br2 reaction, the DBr product emerges with highvelocity and the intensity increases at wide scatteringangles; this is evidence for repulsion in the exit valley ofthe potential surface. In the Cl+ Br2 reaction, the BrClemerges with rather low velocity and its intensitydistribution strongly favors low angles; this is evidencefor an attractive interaction. In the reaction of chlorineatoms with vinyl bromide, the product distribution isapproximately uniform with angle; the detailed analysisshows this system in fact displays statistical behavior

62 D. R. HERSCHBACH

Fig. 1. Velocity distributions of reactive scattering at variousangles.

which indicates a collision complex is formed that persistsfor many vibrational and at least a few rotational periods.

The methods used to generate the reactant beams varygreatly, particularly for reactive atoms or radicals. Beamsof alkali atoms (Li, Na, K, Rb, Cs) or alkaline earth atoms(Mg, Ca, Sr, Ba) are readily obtained by merely heating asuitable oven to moderate temperatures. Beams ofhydrogen or deuterium atoms (H or D) and halogen atoms(F, Cl, Br, I) are produced by thermal dissociation of theparent molecules in high temperature furnaces; forhydrogen, a tungsten furnace at about 3000°K is used, forhalogens a graphite one at about 1600°K. Beams of 0 andN atoms are obtained from a radiofrequency dischargesource. Beams of CH3 and CF3 radicals are produced bythermal decomposition of suitable parent molecules.Similar sources for CH2 radicals and numerous otherspecies are being developed in current research.

For permanent gases beam production is quite simple.The preferred method'1 employs a supersonic expansionof the gas from a source chamber at high pressure, oftenabove 100Ton, through a pin-hole nozzle and into thevacuum. In the nozzle and a short distance beyond(usually less than 1 cm), the collision frequency is highand thus the gas flow is hydrodynamic. Further down-stream, where the collision frequency becomes very low,the individual molecules move independently and therea true molecular beam is obtained. However, thecollisions in the initial hydrodynamic flow bring the beammolecules to a much more uniform direction and velocitythan could be obtained from a thermal effusion source.The high source pressure also gives much higher beamintensity than an effusion source. Another importantadvantage is a very convenient, nonmechanical means ofincreasing the translational energy of the beam. This isobtained by "seeding" the beam species in a large excess ofa lighter nonreactive gas such as helium. The collisions inthe nozzle bring the heavy and light molecules to the samevelocity, and thus increase the beam translational energy inthe ratio of the molecular masses; for example, ifmolecular chlorine is diluted with helium, this ratio is 17.The scope of the seeding technique has recently been muchextended by a nozzle design suitable for solid or liquidsubstances.'2

Selection or analysis of internal states can be accom-plished in several ways. For polar molecules, staticelectric fields are used to deflect or focus the reagent orproduct and thereby delimit its rotational energy'3 ororientation.'4"5 In a few favorable cases, the combinedaction of static and radiofrequency electric fields hasachieved complete resolution of the quantum states of aproduct molecule, including vibration, rotation, and spacequantization.'6 The most fruitful method for analysis ofproduct energy states is the infrared luminescencetechnique developed by Polanyi.'7 This has now providedfully resolved distributions of both vibrational androtational excitation for H + Cl2, Cl + HI, F + H2 and adozen other reactions. In an important step, McDonaldhas devised a very sensitive apparatus which extends theluminescence method to large polyatomic productmolecules.'8

Figure 2 illustrates schematically three techniquesobtained by coupling lasers with molecular beams. One isphotofragment spectroscopy, which deals with unstableelectronically excited molecular states.'9 On absorbingsuitable laser light, a molecule in the target beam isabruptly lifted to an excited state in which both its energyand orientation are well-defined. If this state is unstable,the molecule dissociates and launches the fragments onrecoil trajectories which give information about therepulsive force and about the shape of the electronicorbital occupied in the excited state.

Another laser technique enables study of the influenceof vibrational excitation on reactivity. The originalexperiment20 used the reaction of potassium atoms withhydrogen chloride,

K+HCl-KCl+H.

Like many chemical reactions, this one is energeticallyinhibited and unexcited HC1 molecules give only a very

U-

500 1000 1500 2000 500 1000 500

LABORATORY VELOCITY (meters/sec)

1000 (500

PHOTOFRAGMENT SPECTROSCOPY

_0/C(ATIONo,XCITATION

UNSTABLE

REPULSIVE

L,IGQ'STATE

UNSTABLE, REPULSIVE

FRAGMENTS

ATIONAL

\E0TETEOF

STABLE GROUND STATE

INTERATOMIC DISTANCE

VI BRATIONAL EXC ITATION

LASER MOLECULE BEAM MOLECULE

v\ STABLE GROUND STATE

INTERATOMIC DISTANCE

LASER INDUCED FLUORESCENCE

'TRONIC

''STABLE EXCITED STATE

t I I FLUORESCENCE

\hEOCITATIONOCITATION

\EMISSI'EMISSIONSTABLE GROUND STATE

INTERATOMIC DISTANCE

Fig. 2. Experiments employing laser excitation of molecularbeams.

Molecular dynamics of chemical reactions 63

small yield. The experiment exploits a laser in whichvibrationally excited HC1 is produced by a "pumping"reaction. This laser emits infrared radiation correspondingto a transition (v = 1 — 0) between the first excited stateand the unexcited ground state of HC1, in variousrotational states. The inverse transitions (v = 0 —* 1) areinduced when a beam of unexcited HC1 molecules isirradiated by a pulsed beam of the laser light. Detection ofthe reaction product KC1 is synchronized with the laserpulses. This gives quite directly the ratio of yields forexcited HC1 (produced only during the "on" cycle of thepulses) and unexcited HC1 ("off" cycle). The observedratio is about 100.

The most recent technique is laser induced fluorescence,which Zare et al.2' have shown to be an extremelypromising new state-sensitive detector for reactivescattering. In this method, a tunable laser is swept inwavelength. When the laser emission coincides with amolecular absorption line, the product molecule makes atransition to a stable excited electronic state and isdetected by observing the subsequent emission from thatexcited state. The potential sensitivity of the method isextremely high. For a sizeable class of diatomicmolecules, it may soon be feasible to measure the angulardistribution of the product in an individual rotation—vibration state. The method has been applied to thereaction of barium atoms with oxygen molecules,

Ba+02—*BaO+O

and several others. The BaO appears in a wide range ofrotation—vibration states, consistent with a statisticalcollision complex as indicated by reactive scattering data.As few as 5 x 10 BaO molecules per cc could be detectedin a particular rotation—vibration state. If the BaOmolecules were represented by ping-pong balls numberedto correspond to the quantum states, at this sensitivity theaverage distance between balls with the same numberwould be about. 2km.

Both experiment and theory have thus far dealtprimarily with energy distributions and angular distribu-tions. However, much more dynamical information can beobtained from study of directional properties related toangular momentum.22 For example, for dynamical modelswhich assume abrupt release of a repulsive force as theproducts separate,7 the scattering angle and velocity canbe calculated by specifying just the total repulsive energyreleased, whereas the rotational angular momentumrequires in addition the repulsive force as a function ofseparation distance. The spatial distribution of theproduct rotational angular momentum is just beginning tobe explored in deflection experiments15 (by varying theelectric field direction) and in trajectory calculations.23For several reactions, the product rotation has been foundto be polarized perpendicular to the reactant relativevelocity; the polarization is strong in some cases, weak inothers. The observable quantities involve averages of thedirection cosines relating the product rotational angularmomentum to the initial and final relative velocity vectors.There are several such quantities and all vary with thescattering angle and with the relative translational energyand rotation—vibration states of reagents and products.Thus each polarization parameter offers a multi-dimensional probe of the reaction dynamics analogous tothe product angular distribution but with a distinctlydifferent functional dependence.

Figure 3 compares polarization data from a deflectionexperiment with a model calculation for the Cs + CH3I

Fig. 3. Moments of direction cosines relating rotational angularmomentum J' of CsI from Cs + CH3I reaction to initial and finalrelative velocity vectors, V and V. Curves are calculated fromimpulsive model, points indicate experimental results. Coordinatesystem has z axis along V vector, y axis perpendicular to thescattering plane and x axis in the plane. For an isotropicdistribution, the x2, y2, z2 moments would be 1/3 and the xz and ymoments zero; for a distribution with azimuthal symmetry aboutV the x2 and y2 moments would be equal and the xz and y

moments again zero.

reaction.'5 At present, the experimental method providesonly two parameters (a sum rule requires x2 + y2 + z2 = 1),does not determine the angular dependence, and averagesover all energy states. The results are instructive,however. The mean square projection of the CsIrotational momentum perpendicular to the scatteringplane is very large. This shows that, as the freshly formedCsI recoils from the collision, its internuclear axis tends tospin in or near the plane of the initial and final relativevelocity vectors. Model calculations'5 and trajectoryresults23 find such alignment is characteristic for A +BC -* AB + C reactions when strong repulsion occursbetween the B and C atoms. Laser-induced fluorescencecan be expected to give much more detailed informationof this kind since polarization of the product rotation willresult in polarization of the fluorescence spectrum. Again,the great sensitivity of the method may allow thepolarization of the individual rotation—vibration states tobe measured as a function of the scattering angle. Suchexperiments promise new insight into the "stereochemis-try" of reaction dynamics and the underlying potentialenergy surfaces.

IMPULSIVE REGIME

For atom exchange reactions a wide domain belongs todirect or impulsive mechanisms. The simplest evidencefor this is forward—backward asymmetry of the productangular distribution with respect to the initial relativevelocity vector. Such asymmetry implies that the collisioncomplex usually breaks up before it can rotate through ahalf-turn. Here we consider a favorite example7"°

H+C12-+HC1+Cl.

Figure 4 gives a contour map of the distribution in angle

(I-)

zw

-Jz0Id

0

0.5

0

SCATTERING ANGLE, 6

80'

64 D. R. HERSCHBACH

and velocity of the HC1 product molecules. Successivecontours correspond to 10% increments in intensity. Themap is constructed directly from the experimental data(such as Fig. 1), by a geometrical transformation. Thecoordinate system used has its origin at the center-of-mass of the three atoms, which in this case nearlycoincides with the midpoint of the Cl2 bond. In thissystem, the reagents approach with equal and oppositetranslational momenta; the direction of the incident Hatoms is designated as 00 and that of the Cl2 molecule as180°. The map is symmetric about the 0—180° line, since thescattering must have azimuthal symmetry about the initialrelative velocity. Likewise, in this system, the productsrecoil with equal and opposite momenta, so the contourmap for the Cl atom (not shown) is the "mirror image" ofthat for the HC1 molecule.

The product angular distribution is seen to be broad butvery anisotropic, with the HC1 recoiling backwards withrespect to the incoming H atom and the Cl recoilingforwards. The product velocity is very high, about1600 m/sec at the peak of the distribution. About half ofthe chemical energy released in the bond transformation(—45 kcal/mol) appears in the translational recoil of HC1and Cl. The rest appears as vibrational and rotationalexcitation of the HCI molecule, which is observed asinfrared luminescence.'7 The form of the angular distribu-tion implies that the preferred configuration of the atomsat the onset of reaction is essentially collinear, H—Cl—Cl.The high recoil energy shows that strong repulsive forcesare abruptly released as the Cl—Cl bond breaks.

This repulsion shows a remarkable resemblance tophotodissociation. Figure 4 includes a contour map for theCl atoms recoiling from the repulsive electronic state ofthe Cl2 molecule excited by light absorption (as in Fig. 2,top panel). The direction of the incident photons is 0°. As

required by the selection rules for absorption, thetransition probability is largest for configurations with theCl—Cl axis along the photon direction. The recoil velocityis seen to be nearly the same as in the H + Cl2 reaction.The large repulsive energy release in photodissociationreflects the strongly antibonding character of the lowest-lying unoccupied orbital in the halogen molecule, aorbital involving out-of-phase axial overlap of the valencep orbitals of the halogen atoms. A qualitative analysis ofthe molecular orbitals3 for a collinear H—Cl—Cl reactioncomplex suggests that this antibonding o' orbital alsogoverns the product repulsion.

Other reactions of hydrogen atoms with halogensprovide instructive contrasts. As Cl2 -+Br2—12, therepulsive energy release becomes a smaller fraction ofthat in photodissociation and the hydrogen halide angulardistribution shifts from backwards to sideways withrespect to the H atom direction. The molecular orbitaltreatment3 relates both trends to the variation in halogenelectronegativity, which produces a systematic change incharacter of the highest occupied orbital in the reactioncomplex. The H + ICl reaction is an especially interestingcase. On an energetic purely statistical basis, reaction atthe "Cl-end" would be more favorable than at the"I-end", since the H—Cl bond strength is much larger thanH—I (by 32 kcal/mol). The molecular orbitals suggest theH atom should prefer to attack the I-end, however. As aconsequence of the electronegativity difference, in IC!both the highest occupied orbital (IT*) and the lowestunoccupied orbital (o.*) are predominantly I atomorbitals. Beam studies'°'24 show the HI yield is at leastcomparable to HC1; actually much more HI is detectedbut a dispute about experimental factors which discrimi-nate against HC1 remains unresolved. The angulardistribution of HI peaks sideways, much like that from the12 reaction, whereas the HC1 distribution peaks back-wards, like that from the Cl2 reaction. An infraredluminescence study25 of H + IC! observes only HC1 andnot HI, but this is compatible with the beam resultsbecause the dipole derivative of HI is exceptionally smalland hence the infrared emission is very weak. The HC1energy distribution has a very unusual bimodal form.Comparison with trajectory calculations25 suggests that18% of the HC1 results from direct attack at the Cl-endand 82% from indirect reaction at the I-end. These tworeaction modes produce HC1 in low and high rotation—vibration states, respectively. At present it is uncertainwhether this difference comes from dynamical effects ona single potential surface25 or to reaction on twosurfaces.26 In any case, the preference for attack at theI-end is consistent with the molecular orbital asymmetry.

Analogous correlations of reaction dynamics withqualitative electronic structure have .been pursued formany other atom transfer processes.3'27 For example,strong repulsive energy release is often found in reactionsof alkali atoms with halomethane molecules. The lowalkali ionization potential allows the valence electron totransfer to the target molecule. In many cases the electronenters a strongly antibonding orbital with a radial nodebetween the central carbon atom and an adjacent halogenatom, which is then ejected as a halide anion andcombines with the alkali cation to form the product saltmolecule. The dissociative electron attachment sufferedby the reagent molecule resembles photodissociation.Examples involving orbital asymmetry are numerous.Thus, reactions of Br, 0, and CH3 with ICI all formpredominantly iodides. Analysis of triatomic molecular

20 90' 60

Fig. 4. Comparison of contour maps for H + Cl2 reactivescattering and for phçtodissociation of the Cl2 molecule. Tic

marks along radial lines indicate velocity intervals of 200 m/sec.

Molecular dynamics of chemical reactions 65

orbitals28 has led to a general criterion which we refer toas "the electronegativity ordering rule." This predicts thatH—X— V will be a more favorable conformation than H—Y—X, if X is less electronegative than Y (where these areany atoms with p electrons.) Likewise, in X—Y—Zsystems, the preferred conformation has the leastelectronegative atom in the middle. If the electronegativ-ity difference becomes so large that electron transferoccurs, other considerations enter.29 However, otherwisethis rule holds for almost all known stable triatomicmolecules, linear or bent, and it now appears to apply alsoto preferred reactive conformations.

PERSISTENT COMPLEX REGIME

When the duration of a reactive collision is longcompared to vibrational periods of the transient collisioncomplex, product formation is indirect since the interact-ing atoms or bonds undergo multiple encounters duringthe collision. The distributions in energy and angle of thescattered products give information about the unimolecu-lar decomposition of the lingering or persistent complex.The beam technique thus offers a single-collision versionof the chemical activation method of unimolecularkinetics.30 This is illustrated here for the reaction ofchlorine atoms with vinyl bromide,3'

Cl + CH2 = CHBr-CH2 = CHC1 + Br

Figure 5 gives the contour map. The symmetricalforward—backward peaking with respect to the initialrelative velocity shows that the intermediate collisioncomplex persists long enough to rotate several times andthereby "forget" the approach direction before decom-posing to form the products. This implies that the meanlifetime of the complex exceeds about 5 x 10-12 sec. Thatis much longer than the periods of typical vibrationalmotions within the complex, which are about 10's sec,and much longer than the collision duration for impulsivereactions, which is also typically of the order of l0' orshorter. The relatively long lifetime of the complex

lOOm/sec Ia'- 90'

50

l8 i

20'

10

6

50

0'

0'Cl RBr

Cl RBr

requires that attractive rather than repulsive forcesgovern the reaction. In this example, the complex indeedcorresponds to a known, stable species, the chloro-bromoethyl radical,

H'I{BrThe intermediate radical is vibrationally excited (by

-'30 kcal/mol) since energy is generated by addition ofthe chlorine atom, which converts the cargon—carbondouble bond to a single bond and forms a newcarbon—chlorine bond. In an ordinary chemical environ-ment, the vibrational excitation would be quickly de-graded by subsequent collisions but this does not occur ina beam experiment. The vibrational excitation thusremains "trapped" in the complex, shuffling among thevarious bonds until eventually one bond ruptures and thecomplex dissociates. For this system, the only dissocia-tion paths energetically accessible involve reforming thedouble bond and releasing either the Cl atom or the Bratom. The latter path is much more favorable since theC—Br bond is weaker than the C—Cl bond. The longlifetime of the complex allows the vibrational excitationenergy to become random before decomposition. Thus,the product energy distribution can be predicted from asimple statistical model, akin to the Rice—Ramsberger—Kassel—Marcus theory for unimolecular reactions.32

The main qualitative features of the statistical theory33are illustrated in Fig. 6. Consider first a complex which isnot rotating. In partitioning the available energy amongrelative translation, vibration, and rotation of the pro-ducts, the statistically favored situation puts only a smallpart of the energy into translation, since the vibrationaland rotational modes are more numerous. Thus, theprobability distribution declines rather rapidly withincrease in product translational energy. This declinebecomes more rapid as the number of atoms in thecomplex increases and hence the number of vibrationalmodes increases. Another important effect enters whenthe complex is rotating, however. The energy incentrifugal motion of the rotation complex is not availablefor statistical distribution among the other modes. Ondecomposition of the complex this centrifugal energy isconverted into relative motion of the emerging productmolecules. This changes the shape of the producttranslational energy distribution. The region of lowtranslational energy is determined by the centrifugalcontribution, the region of high translational energy by thestatistical contribution.

\,,,_NONROTATINC COMPLEX

- STATISTICAL REGION

Fig. 5. Contour maps for bromo-olefin products from reactions ofchlorine atoms with (Ia) vinyl bromide and with (ha) allylbromide. Tic marks along radial lines indicate velocity intervals of Fig. 6. Distribution of product relative translational energy

100 m/sec. predicted by statistical theory.

ENERGY OF PRODUCTS

66 D. R. HERSCHBACH

The shape of the product angular distribution is alsorelated to rotational properties of the complex.34 How thiscomes about may be illustrated by two limiting casesshown in Fig. 7. Consider first a complex having all atomscollinear in the critical configuration or transition-state fordecomposition. The rotation of such a complex occursabout an axis perpendicular to the line of atoms. If thecomplex rotates a few times before dissociating, theproducts are spread uniformly over all azimuthal anglesabout the rotation axis, like water from a lawn sprinkler.The total distribution is obtained by averaging uniformlyover all orientations of the sprinkler with respect to thedirection of approach of the reactants. The resultcorresponds to the intersections of circles of latitude andlongitude on a globe. The product intensity becomes veryhigh in the "polar" regions, which represent scatteringdirections parallel or nearly parallel to the initial reactantapproach, but the intensity is low in the "equatorial"regions.

Consider next a complex having all atoms situated intwo coaxial hoops which maintain a face-to-face configu-ration in the transition-state. Suppose also that therotational motion of the complex consists entirely of thespinning of these hoops about their axis of separation asthe complex dissociates. Since the axis of rotation of thecomplex is again perpendicular to the direction of reactantapproach, in this case the product separation axis must beperpendicular to it also. The product intensity therefore ishigh in the equatorial regions and low in the polar regions.

The arrangement of atoms and the rotational motions inthe transition-state thus can leave its imprint in the productangular distribution, even though the complex oftendissociates after a few rotations. A complex such as thechlorobromoethyl radical has some atoms situated close toor on the axis of product separation and others located wellaway from that axis. The rotation of the complex hencecorresponds to a superposition of the limiting cases of Fig.7. There is both centrifugal rotation of the separation axis

COLLINEAR COMPLEX

and spinning of the product molecules about that axis. Theshape of the angular distribution reveals the relativecontribution of the centrifugal and spinning motions. Thiscan be regarded as roughly analogous to the rotationalspectrum of a stable molecule, for which the moments ofinertia about the principal axes indicate the mass-weightedmean square average distances of atoms from those axes.In the chlorobromoethyl case, the pronounced forward—backward peaking indicates that centrifugal momentum isdominant in the dissociation of the collision complex.

Figure 8 compares the observed and calculated producttranslational energy distributions for three reactions:oxygen atoms and bromine,35 chlorine atoms with vinylbromide,3' and cesium with sulfur hexafluoride.36 Thecentrifugal contribution and the number of contributingvibrational modes vary considerably among these reac-tions but the statistical theory gives good agreement ineach case. Long-range attraction of the form —C/r6 wasassumed for both the reactants and products in evaluatingthe centrifugal factor; the force constants C were

ostimated from customary approximations involvingmolecular polarizabilities. All vibrational modes wereassumed to be active in intramolecular energy transfer.For the vinyl bromide reaction, however, the contribu-tions from hydrogen atom motions are almost negligiblebecause of the large mean vibrational spacings for thesemodes.3' This simple approximate theory has given goodagreement with translational energy data for many otherreactions.3 Similar results have been obtained with thecorresponding angular distribution theory. Thus, in thepersistent complex regime the transition-state approxima-tion provides a useful interpretative tool.

Figure 5 includes a contour map for the reaction ofchlorine atoms with ally! bromide,

Cl + CH2 = CHCH2Br-*C1CH2CH = CH2+ Br.

The forward peak of the angular distribution is more

COAXIAL HOOPS COMPLEX

ProductsEmitted

Uniformly

ReactantDirection

of Approach

ReactantDirection

Average RequiredOver All OrientationsOf Rotation Axis

ROTATION AXISOF COMPLEX

ROTATION AXIS

Of Rotation Axis

PRODUCT DISTRIBUTIONFAVORS POLAR REGIONS

PRODUCTS CONFINEDTO EQUATORIAL REGION

Fig. 7. Angular distribution of products predicted by statistical theory for two special cases.

Molecular dynamics of chemical reactions 67

z

4

4

0x-JU-

Fig. 8. Comparison of product relative translational energydistributions derived from experiment (full curves) and from

statistical theory (dashed curves).

pronounced than the backward peak, by a factor of aboutthree. This indicates that the complex is dissociating as itrotates. For such cases an "osculating complex" model37has been formulated in which the dissociation occurs withan exponential distribution of lifetimes. Comparison withthe data shows the mean lifetime of the complex is aboutone-half of a rotational period, or about 1 x 10-12 sec. Inaccord with this shorter lifetime, the product translationalenergy distribution for this reaction is markedly nonstatis-tical. It is displaced upwards, indicating the only abouttwo-thirds of the vibrational modes associated with thefive heavy atoms (carbon or halogen atoms) are active inthe intramolecular energy transfer before dissociation ofthe complex.

In the context of other work, the contrast between thereactive scattering of chlorine atoms with vinyl bromideand allyl bromide suggests a qualitative difference inmechanism. The vibrational excitation of the intermediatechlorobromoalkyl radical is practically the same(—30 kcal/mol) and the dissociation energy for bromineatom emission from these radicals derived from ther-mochemical data is also about the same (—18—20 kcal/mol). The customary theory of unimolecularprocesses predicts that the allylic reaction should proceedmore slowly and hence show more nearly statisticalbehavior, since it involves one more heavy atom andtherefore three more accessible vibrational modes thanthe vinylic reaction. The lifetime for bromine atomemission estimated from the theory is less than about2 X 1012 sec for both the vinylic and allylic systems. Somespecial feature apparently intervenes to make the vinylicreaction more statistical than the allylic reaction. Theinitial stage in both reactions almost certainly involvesattack on the double bond rather than the C—Br bond38 andthere is extensive evidence for anti-Markovinkov additionto olefins.39 Thus, a likely hypothesis3' is that the vinylicreaction proceeds via a 1,2-chlorine atom migration,

whereas the allylic reaction goes via a 1,3-bond migration,

Cl + —Br Cl——-----BrCI—" + Br

Since the heavy atom migration should be much slowerthan the bond migration, this might provide a rate-limitingprocess which makes the vinylic reaction more statisticalthan the allylic one.

These mechanisms require the product chlorolefin tohave Cl on the carbon to which Br was originally bondedin the vinylic case, but on the "most distant" carbon in theallylic case. This was verified by analysis of the fragmention mass spectra of chlorolefins formed in correspondingreactions with a methyl group added to "label" one oranother carbon atom.3' The vinylic reactants,

— Me/\ 1/ \ anuiiC Dr Br

were indeed found to yield only the chlorolefin with Cl atthe original Br position in each case. For the allylicreactants,

yields only

and

yields only

The methyl substitutions also produce interestingchanges in the reactive scattering. For the vinylic cases,the angular distributions show variations which reflectchanges in the rotational motions caused by the methylgroup. For the allylic cases, the product translationalenergy distributions become statistical, as if the methylgroup diverts the intramolecular energy flow and therebymakes it more nearly random.

Similar studies have now been conducted for aboutthirty other families of reactions governed by persistentcollision complexes.3'4 In almost every case the inter-mediate complex corresponds to a known chemicalspecies or to an electronic structure with favorablebonding overlaps. As with the chlorobromolefins consi-dered here, however, many such systems show nonstatis-tical features. We cite three examples. (1) Analogousreactions of fluorine atoms with olefins have beenextensively studied by Lee and coworkers in crossed-beams4° and by McDonald in i.r. luminescence.'8 Thevibrational excitation of the intermediate radicals is quitehigh (—60 kcal/mol) by virtue of the large C—F bondstrength, and several competing dissociation channelsinvolving scission of C—C bonds and migration of CHgroups or H atoms become accessible. The translationaland vibrational energy was found to show systematicdeviations from statistical behavior; often the populationof high modes is low and that of low modes is high. (2)Reactions of alkali atoms with alkali halide molecules

0.5PRODUCT TRANSLATIONAL ENERGY, f E'/E,,

Cl +Br Cl\_%Br Br C'

==— + Br

68 D. R. HERSCHBACH

Na + K+Cr±(NaK)FC1 NaC1 + K

have also been much studied.34'4' The reactant andproduct salt molecules are essentially ion-pairs andelectronic structure calculations42 show the ionic dialkalihalide (NaK)Cl is likewise a stable species. Statisticalmodels give good agreement with the product transla-tional and angular distributions but for more than 20 casesoverestimate the ratio of reactive to nonreactive decay ofthe collision complex, often by a factor of 3—5 or more.This has been attributed to preferred reaction geometry.34The potential surfaces predict the most stable configura-tion of the dialkali halide complex is triangular but themost favorable direction of approach is collinear, with Naattacking the K end of the salt molecule. Since thecentrifugal momentum in the collision often restrains theroughly collinear (NaK)Cl configurations from bendinginto the triangular configurations required for reaction,this "wrong-end" approach strongly favors nonreactivedecay of the complex. (3) Reactions of oxygen atomswith halogen molecules35 also conform to the statisticalcomplex model yet show clear evidence for preferredreaction geometry. In particular, the 0 + ICl reactionyields only JO + Cl and not ClO + I, again the resultconsistent with the "electronegativity ordering rule".

FACILE MOLECULAR REACTIONS

One of the chief criteria invoked in postulatingelementary steps jn chemical reaction mechanisms is thatonly processes in'olving attack by an open-shell atom orfree radjcal have very low activation energies, less than 5or 10 kcal/mol. Reactions of molecules with molecules,which inyolve the concerted making and breaking of twoor three pairs of bonds, typically have activation energiesabove 2 kcal/mol when fully "allowed" in theWoodward—Hoffmann sense and about 40 kcal/mol when"forbidden". It is thus instructive that recent molecularbeam experiments find certain classes of bimolecular andtermolecular reactions of diatomic molecules proceedwith remarkably low activation energies, zero or at most afew kcal/mol.

Before describing these results, we briefly considerrelated work on the isotopic exchange of hydrogen anddeuterium,

H2+D2-÷2HD.

This is a prototypical example (although numericallyatypical) of a "forbidden" four-center reaction. It 'hasaroused much interest because a severe conflict betweentheory and experiment has persisted for several years.Figure 9 illustrates the correlation of reactant and productmolecular orbitals for the simplest case of a coplanar pathproceeding through a square transition state.43 The nodalproperties of the composite orbitals formed by in-phaseand out-of-phase (dark shading) overlap of the diatomicbonding o and antibonding O!* orbitals require thefour-center reaction to raise two of the four electronsfrom o to cr* orbitals. This suggests the potential Ønergybarrier should be comparable to or larger than the H2dissocation energy (about 110 kcal/mol). Electronicstructure calculations indeed find the square planar H4transition state lies more than 140 kcal/mol above theH2 + H2 ground state. A variety of other possible reactionpaths have also been examined, including tetrahedral,

H\

H/ \H H

'p

S$c__ —$S

cPsZ 8s..c9s8

Fig. 9. Orbital correlation diagrams for H4 and H6 systems.

trapezoidal, rhomboid, kite, and Y- or T-shaped transitionstates; allappeartolieabovetheH2dissociationenergy.

Experiments on hydrogen—deuterium exchange inshock tubes45 find an activation energy of only about40 kcal/mol. Elegant evidence that vibrationally excitedH2 or D2 are involved has been obtained by use of astimulated Raman laser. The observed rate law iscomplex, as a consequence of multistage vibrationalenergy transfer, and the empirical activation energy is notsimply related to the potential energy barrier. However,the HD yields are roughly 6—9 orders of magnitude toolarge to reconcile with the H4 potential surface calcula-tions.

Figure 9 includes a correlation diagram for thetermolecular, six-center reaction proceeding through anH6 hexagon. This is a fully "allowed" process. Forsimplicity, the path shown preserves a symmetry planebisecting one of the initial and final H2 bonds and theorbitals are classified as symmetric (S) or antisymmetric(A) with respect to that plane, but the nodal propertieswhich govern the qualitative correlations are similar forother paths. Wright47 proposed that the hydrogen—deuterium exchange might go by this six-center route andshowed the energy barrier for an H6 hexagon is at leastlower than for an H4 square. Following the recent beamresults on termolecular reactions, Dixon48 undertook

such as H H H—H

H H H—H

$cc

0.. •cc0 cc

Molecular dynamics of chemical reactions 69

more accurate calculations for H6 and found the potentialbarrier in fact lies far below the H2 dissociation energy.Figure 10 shows some of his results obtained withsuccessively better wavefunctions. By extrapolation, thepotential barrier for the H6 hexagon is estimated to be nohigher than 60 kcal/mol and perhaps substantially less.Kinetic calculations show that, despite the reducedprobability of termolecular collisions as compared withbimolecular ones, the six-center processes

H2 + H2 + D2 - 2HD+ H2

H2 + D2 + D2 -2HD+ D2

with vibrationally excited reactants are consistent withthe experimental data. A special form of "vibrationalcatalysis" becomes possible here. If a substantial part ofthe energy needed to surmount the potential barrier goesinto vibration of the H2 or D2 regenerated by thesix-center reaction, rapid vibration—vibration energytransfer in bimolecular collisions of these molecules withthe ambient gas can raise the vibrational temperaturesignificantly and hence accelerate the overall exchangerate. It may be possible to test the termolecularmechanism by looking for this effect in future experi-ments.

The crossed beam studies of molecular reactions wereprompted by the classic work of Sullivan49 on the favorite"textbook" case,

H2 + I2—2HI.

He showed this does not occur as a four-center reactionbut involves dissociation or near-dissociation of '2followed by I + H2 + I. Noyes5° likewise cast doubt onBr2 +12 and other reactions long presumed to bebimolecular. The subsequent orbital correlation argu-ments43 indicated that most bimolecular exchange reac-tions of covalently-bonded diatomic molecules are forbid-den. In this context, several beam experiments were triedin pursuit of four center reactions. For example, theHI + DI reaction was examined51 using the seededsupersonic beam technique and no detectable HD yieldfound at collision energies far above the empiricalactivation energy. This is certainly an allowed reaction, at

I I I I

H4N

ISON

__I00 i;Lt±i_—--- H

Ui — —

zLii

50

I I I H—DZ DZ+CI DZP+CI EXACT 2

Basis Sets

Fig. 10. Energy of square planar H4 relative to 2H2 and hexagonalplanar H6 relative to 3H2, for various approximate electronicstructure calculations: DZ denotes double zeta; CI configurationinteraction with all single and double excitations; P polarization.Corresponding results for the dissociation energy of H2 are also

shown.

least as the reverse of I + HD + I, but apparently vibra-tional rather than translational activation is required.However, a sizeable class of four-center reactions wasfound52 which proceeds readily at thermal collisionenergies of a few kcal/mol. These all involve ionic bonds.

The simplest case is the exchange reaction of twoalkali halides such as

CsCl + K4T -CsI + KCl.

This might be termed a "no-electron" reaction, since thesalt molecules are essentially closed-shell ion-pairs.Accordingly, no restraints are imposed by orbital correla-tions. Since alkali halides form rhombic quadrupolardimers with dissociation energy —30—50 kcal/mol, thepotential surface for the exchange reaction has a deepbasin. The beam data are indeed consistent with this,except that the ratio of nonreactive to reactive decay ofthe complex was 2 or 3 times larger than statistical.34 As inthe analogous atom + salt case, this can be attributed togeometrical isomerism. Ionic model calculations predictless stable, linear chain isomers exist in addition to therhombic dimer. These linear chain isomers may oftendissociate nonreactively rather than rearrange to thecyclic form required for the exchange process; again, thisis especially likely when the centrifugal angular momen-tum in the collision keeps the chain "ends" apart.Trajectory calculations53 for this reaction have confirmedthe role of geometrical isomerism.

Four-center reactions involving one ionic and onecovalent bond52 also proved to be facile at thermalcollision energies; for example,

CsBr + ICI-* CsCl + IBr.

The collision complex very likely corresponds to atrihalide salt, Cs(BrICl). Although apparently unknownin the gas phase, trihalides have been much studied insolution and in the solid state. In agreement withmolecular orbital theory,28 the trihalide anions are linearor nearly linear, the middle atom is always the leastelectronegative (I, in this case) and it has a small positivecharge whereas the end atoms share the negative charge.Striking evidence for this structure appears in somefeatures of the reactive scattering. There is no observableyield of CsI + BrC1, even at collision energies more than20 kcal/mol above the energetic threshold for thischannel. Furthermore, as seen in Fig. 11, the productangle-velocity contour map has a very unusual skewedshape. The lefthand product peak has distinctly higherintensity and velocity than the righthand peak. This shows

IBr from CsBr

100 rn/sec

Fig. 11. Contour nfap of IBr product distribution from CsBr+ IClreaction atcollisionenergy of 39 kcal/mol.

Ic: CsBr

70 D. R. HERSCHBACH

that collisions from which IBr and CsC1 reboundbackwards with respect to the incident ICl and CsBr,respectively, are more probable and involve morerepulsive energy release than collisions from which IBrand CsC1 emerge in the same direction as the incident ICland CsBr, respectively. These properties are consistentwith the expectation that in CsBf + IC! reactiveconfigurations, Bf tends to be collinear with ICl whileCs in "reaching" for Cl must come near the central Iatom. The positive charge acquired by I as the trihalideforms then repels Cs, which picks up the emerging Cland departs quickly in the direction opposite to theincident salt.

Similar four-center ionic + covalent reactions such asCsF + HC1 were likewise found to go with near zeroactivation energy.52 It will be interesting to see whathappens for systems with only partial ionic bonds. A largeclass of four-center 1,2 elimination reactions ofhaloolefins which have been described in terms of asemi-ion pair transition state54 have activation energiesnear 20—30 kcal/mol, again much lower than comparablecovalent + covalent cases.

Six-center reactions involving three covalent diatomicmolecules were found in beam experiments55 undertakenin response to tantalizing evidence obtained by Noyes5°for a third order reaction in solution,

Br2 + 212 —* 2IBr + 12.

For convenience we examined the analogous systeminvolving a bromine molecule and two chlorine molecules.In preliminary work, King and Dixon established thatunder single collision conditions the four-center process

Br2+ Cl2-2BrCl

yields no product attributable to bimolecular reaction, attranslational energies up to -25 kcal/mol. The prospectfor observing the termolecular Br2 + 2C12 process ap-peared dubious. Analogy to other "allowed" six-centercases (such as Diels—Alder reactions) suggested that thepotential barrier might well be —20 kcal/mol, and it mightalso require chiefly vibrational rather than translationalexcitation. Thus, there was consternation when King andDixon found large yields of BrCl which appeared to comefrom the termolecular process even at thermal collisionenergies of only —3 kcal/mol. Their results have nowwithstood numerous refinements in technique and consis-tency tests, however, and similar results have beenobtained for several other systems involving concertedtransformation of three pairs of bonds with practically noactivation energy.48

The beam technique for study of termolecular reactionsagain exploits supersonic nozzles. Expansion of chlorinegas through a pin-hole generates dimeric Cl2 . Cl2molecules, held together by a weak van der Waals bond56with dissociation energy of 1—2 kcal/mol and equilibriumbond distance roughly 43 A (between centroids of themonomers). Source conditions were eventually found forwhich the dimer fraction is a few percent but trimers andhigher polymers are almost undetectable. This hasallowed study of three termolecular reaction paths,

Br2 +Cl2 - 2BrCl+ Cl2

-ErCl"Cl2+BrCl-+Br2Cl2 + Cl2.

As usual, the reactant beams were monitored and theproducts detected with mass and time-of-flight velocityanalysis. The BrCl, BrCl3, and Br2Cl2 signals all correlatedwith the (Cl2)2 signal. Marked differences were found inthe dependence on collision energy and the productangle-velocity distributions which show the signals comefrom three distinct reaction modes; in particular, verylittle of the BrCl signal can be attributed to fragmentationof Br2C12 in the detector. Inelastic scattering of (Cl2)2 withvelocity changes due to translation-to-vibration energytransfer is also very prominent. At higher energiescollision induced dissociation to form Br2 + Cl2 + Cl2becomes the major process.

Figure 12 compares the reaction yields and thedependence on collision energy.55 Since (Ri) and (R2)involve making and breaking three pairs of chemicalbonds, these paths presumably require the incidentbromine molecule to interact with both chlorine moleculesin a cyclic configuration. However, (R3) merely ex-changes van der Waals bonds among the three moleculeswithout disrupting the chemical bonds. Path (R3) thus isaccessible for many noncyclic collisional orientations forwhich (Ri) and (R2) cannot occur. The energy depen-dence apparently reflects this steric distinction. Thepreponderance of noncyclic configurations probablyaccounts for the predominance of (R3) seen at lowcollision energies. At higher collision energies (R3)declines rapidly and probably goes over to collision-induced dissociation to form Br2 + Cl2 + Cl2, whereas (Ri)increases and (R2) remains about constant. The corres-ponding reaction paths have been studied in the same wayfor the HI + Cl2 Cl2 system.

>-

Co sion Energy, E (kcal /mo)

Fig. 12. Variation of yields with initial relative translationalenergy for five Br2 + (Cl2)2 collision processes: reactions produc-ing (1) BrCl, (2) BrCl3, and (3) Br2C12; inelastic scatteringproducing vibrationally excited (Cl2)2; and collision induceddissociation (CID) producing Br2 + Cl2 + Cl2. Experimental pointsare included for reaction (1) to illustrate quality of the data.Absolute cross-section for reaction (3) estimated to be of the order

5—50A2.

Figure 13 shows contour maps of the reactive scatteringfrom (Ri) for the Br2 and HI cases. The BrC1 productpeaks sharply forwards and backwards along the initialrelative velocity vector whereas the ICI peaks backwardswith respect to the incident HI direction. The Cl2 product

(Ri) distribution is broad for both reactions but in each case it

'R2'peaks close in angle and velocity to the interhalogenproduct. The forward—backward symmetry of the reactive

(R3) scattering for Br2 might be attributed either to statistical

0 5 10 15 20 25

Molecular dynamics of chemical reactions 71

Br2 - (Cl2)2KOm/sec

Fig. 13. Contour maps of product distributions from reactions ofBr2 and HI with (Cl2)2 at collision energies near 3 kcal/mol. Fullcontours pertain to BrC1 or IC1 and dashed contours to Cl2

product.

dissociation of the collision complex or to recoil of thetwo BrCl molecules in opposite directions. The latterappears more likely, in view of the asymmetry found forthe HI case. The experiments at higher collision energygive further evidence for the nonstatistical character ofthese reactions, both in the angular distributions and inthe product translation.

Figure 14 outlines the reaction sequence inferred fromthe scattering data.55 The sharp peaking of the BrC1 andIC1 angular distributions shows these molecules areemitted with very high centrifugal momentum; thebroader distribution of Cl2 shows it emerges with lowercentrifugal momentum. These properties indicate a chainstructure (perhaps quite nonlinear) for dissociativeconfigurations of the reaction complex,

X-Cl

This presumably results from quick scission of the Ybond as the complex traverses the cyclic configuration.The nonstatistical character of the scattering indicatesthat at least one of the two weak Cl Cl bonds also breaksquickly (within 1O sec). The comparable velocitiesfound for the Cl2 and the X—Cl products suggest that thesecond Cl Cl bond persists longer, at least until X—ClCl—Cl and Cl—Y separate sufficiently to approachtheir asymptotic exit translational momenta (equal andopposite). For the Br2 reaction, the first ClCl bond tobreak might be either one. For the HI case, the asymmetryof the Cl2 distribution indicates it is usually the bondwhich releases HC1 and ICl Cl2. There is some Cl2corresponding to the opposte case, however, and thisportion (at right in Fig. 13) shows higher velocitiesquantitatively consistent with the different mass distribu-tion of the initial fratments (IC1 and HCl"Cl2). The (R2)path is a corollary of this sequence; it occurs instead of(Ri) when the second ClCl bond does not break.

5 ANGSTROMS

Fig. 14. Schematic reaction sequence for Br2 + (Cl2)2, illustratingthe formation of van der Waals bonds (shown dashed), six center

exchange, and three successive bond scissions.

Comparison of the angle-velocity distributions for (R2)and (Ri) is quantitatively consistent with this interpreta-tion.48 Several aspects of the data thus offer evidence thatthree sequential bond scissions can be resolved in thesesingle collision experiments.

Reactive scattering for a host of other van der Waalsmolecules can be studied in the same way. Some reactionsof dimeric hydrogen iodide and various organic moleculeshave been examined already.48 The van der Waals domainlikewise poses new questions in collision dynamics andelectronic theory. For instance, the drastic collisionalenergy transfer observed is probably due to the very weakand floppy bonding; the data now invite model calcula-tions. It is interesting to find that the asymmetric scissionof the two ClCl bonds of the six-center reactioncomplex resembles that seen in photodissociation of CH3—Cd—CH3 and other symmetric molecules.57 The electronicchanges involved in switching between the extreme bondtypes will surely prove instructive. Merely adding an extrareagent molecule, linked by the very weak van der Waalsbond, catalyzes what would otherwise be a forbiddenchemical reaction.

Scattering and spectroscopic studies using much largervan der Waals polymers or molecular clusters arefeasible, as such clusters are readily generated by in-creasing the source pressure behind a supersonic nozzle.For example, recent crossed-beam experiments5°find evidence that an incident atom or molecule will"stick" or "condense" on a sufficiently large polymer(typically with more than 20 atoms) to form an adductwhich persists long enought to travel to the detector(i04 sec). Figure 15 illustrates another theme.6° In thescattering of HI from ammonia polymers (NH3), newmass peaks appear that arise from proton transfer with

(Cl2)2 + HI

9O

+1-Il

(CJ2)2 2

72 D. R. HERSCHBACH

CCI2F2 A

I lIlt IlINH3 POLYMER NUMBER (n)

Fig. 15. Mass spectra (1% resolution) of ammonia polymers(NH3)0 with n = 6 to 21, scattered at 100 from the ammonia beamby collisions with Kr, Cl2, CC12F2, and HI. For the HI caseadditional peaks are seen which correspond to NH4I(NH3)m

withm =2to13.

solvation of the resulting salt molecule,

HI + (NH3)0 (NH4T)(NH3)m + (NH3)n_m_i.

In such experiments the polymer numbers n and m can bescanned by adjusting the beam source conditions. Aninteresting possibility now being explored is the use ofsuch beams with laser-induced fluoresence to studysolvation. The emission from the solvated solutemolecule would show how its vibrational force constantand bond distance changes with the number of solventmolecules. This would almost amount to watching thesolute dissolve.

The vigor and spirit with which experiments and theoryin quest of "single collision chemistry" are now beingpursued in many laboratories is reminiscent of theopening up of molecular spectroscopy and structure fortyyears ago. The results already obtained and the oppor-tunities defined suggest a comparable development ofbasic facts and insight into reaction dynamics can beexpected. This will apply to chemical phenomena farbeyond the realm of collision chambers and computers.

Acknowledgements—It is a pleasure to thank my students andcolleagues devoted to the study of reaction dynamics and toacknowledge gratefully support of the work at Harvard by theNational Science Foundation.

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