DOI: 10.1126/science.279.5358.1875 , 1875 (1998); 279Science

et al.Richard N. Zare,Laser Control of Chemical Reactions

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Laser Control of Chemical ReactionsRichard N. Zare

Experiments show how product pathways can be controlled by irradiation with one ormore laser beams during individual bimolecular collisions or during unimolecular de-compositions. For bimolecular collisions, control has been achieved by selective exci-tation of reagent vibrational modes, by control of reagent approach geometry, and bycontrol of orbital alignment. For unimolecular reactions, control has been achieved byquantum interference between different reaction pathways connecting the same initialand final states and by adjusting the temporal shape and spectral content of ultrashort,chirped pulses of radiation. These collision-control experiments deeply enrich the un-derstanding of how chemical reactions occur.

One of the intriguing aspects of reactiondynamics is the possibility that what welearn might allow us to control the outcomeof chemical reactions (1). The practicalreasons for seeking such control range fromsuppressing unwanted side products to syn-thesizing new structures and new materials.Control of chemical reactions is a well-established concept. Successful control iscommonly practiced by a variety of means,such as varying the external conditions ofthe reaction mixture by changing, for ex-ample, the temperature or the pressure, orfinding a suitable catalyst that selectivelylowers the activation barrier to the desiredreaction products.

Here, I explore a different approach toreaction control where we, as chemists,“guide” the reaction process during a singlereactive encounter. This guidance usuallytakes one of two forms. First, we may selectone or more internal energy states of thereagent before collision. State preparationhas a long history of successful applications.Some preparations depend simply on in-creasing the energy available to the reac-tants. Others involve more subtle effectssuch as control by promoting internal mo-tions that aid or hinder various reactionpathways and stereodynamic control inwhich the three-dimensional geometry ofthe activated complex is selected. Second,we may actively intervene during the courseof the reaction and guide the reactants bycontrolling the phase of their motions. Thisactive control may cause the reactants tofollow preferentially one of many differentreaction routes. This second procedure isless well established, but in some cases, verypromising results have been recently ob-tained. In what follows, I present a fewexamples of collision control along with anassessment of what has been accomplishedto date and what likely lies ahead.

Mode-Selective Chemistry

In 1972, Polanyi (2) proposed that vibra-tional excitation along the reaction coordi-nate would be more efficacious than trans-lational motion in promoting endoergic(uphill) reactions with a “late” reaction bar-rier, that is, reactions in which the transi-tion-state region occurs late in the passagefrom reactants to products. Since then, anumber of atom-plus-diatom reaction stud-ies confirmed this conjecture (3), but ex-tension of the idea to reactions involvingpolyatomic molecules was rather slow incoming. In 1984, Schatz and co-workers (4)carried out quasi-classical trajectory calcu-lations on the H 1 H2O 3 H2 1 OHreaction, which is estimated to have a re-action barrier of 7580 cm–1 (5). Schatz et al.(4) clearly demonstrated that the reactionrate should be enhanced by excitation ofthe H-OH stretching vibration of water,because the transition-state geometry showsa marked preference for extension of one ofthe O-H bonds in water.

An isotopic variant of water, HOD, hasproven to be an excellent candidate systemfor demonstrating the power of vibrationalcontrol of polyatomic reagents to influencethe outcome of a chemical reaction. TheH-OD and HO-D stretching frequencies areapproximately at 3800 cm–1 and 2800 cm–1,respectively. Consequently, these twostretches are quite distinct and representnearly pure stretching modes. The first ex-ample of the power of reagent vibrationalexcitation in controlling the outcome of abimolecular reaction with a polyatomic re-agent was achieved by Crim and co-workers(6), who found that reaction of thermal Hatoms with HOD prepared with four quantain the H-OD stretch produces almost exclu-sively H2 1 OD, whereas reaction of ther-mal H atoms with HOD prepared with fivequanta in the HO-D stretch produces al-most exclusively HD 1 OH. In this exper-imental study, the H atoms were generated

with a microwave discharge, and the HODmolecule was prepared in a selected over-tone stretch by laser irradiation in the vis-ible wavelengths. Zare and co-workers (7)showed that the product-branching ratioselectivity was reduced but still survived forreactions of fast H atoms with HOD pre-pared in either the H-OD or HO-D stretch-ing fundamental. Here, fast H atoms weregenerated by photolysis of HI, and theHOD stetching fundamental was excited bylaser irradiation in the infrared. Broni-kowski et al. (8) went on to examine thereaction of fast H atoms with D2O, inwhich the D2O was prepared in either theasymmetric stretching fundamental or in acombination band consisting of the asym-metric stretch and one quantum of thebend. The bending motion was found to beineffective in promoting reaction, as ex-pected. Other examples of vibrational-statecontrol on polyatomic reactions are Cl 1HOD (9) and Cl 1 HCN (10). All thesestudies definitively demonstrate what iscalled mode-selective chemistry, whichmeans for the same internal energy content,the mode of internal excitation controls thereaction outcome.

Examples of mode-selective chemistryare not limited to bimolecular neutral reac-tions. For example, the NH3

1 1 ND3 re-action shows clear distinctions among thethree product branches [NH3 1 ND3

1

(charge transfer), NH2 1 ND3H1 (protontransfer), and NH3D1 1 ND2 (D-atom ab-straction)], depending on whether theNH3

1 was selectively prepared with excita-tion of its umbrella inversion mode or itsall-symmetric stretching mode (11). Chargetransfer is enhanced with umbrella inver-sion motion, and this effect has been ra-tionalized by the preference of the neutralmolecule to take on a pyramidal geometry.This argument is analogous to the Franck-Condon principle (12). The more energeticall-symmetric stretch is found to be essen-tially inactive. Theoretical calculations areable to model this behavior well (13).

Another example of how the selectionof nuclear motion of the reagent can influ-ence reaction outcome has been providedby Anderson and co-workers (14) who stud-ied the ion-molecule reaction C2H2

1 1CH4 in which the acetylene cation wasprepared with vibrational excitation in ei-ther the C-C stretch or the cis-bend. Thereaction proceeds by two mechanisms. Atlow collision energies, the dominant reac-tion is mediated by a strongly bound C3H6

1

The author is in the Department of Chemistry, StanfordUniversity, Stanford, CA 94305–5080, USA. E-mail:zare@stanford.edu

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complex that breaks apart to produce eitherC3H5

1 1 H or C3H41 1 H2. This complex

is sufficiently long-lived that the H atomsin an isotopically labeled experiment arescrambled. The complex-mediated mecha-nism decreases in importance with increas-ing collision energy and with C-C stretchvibration to a similar extent, but cis-bend-ing excitation enhances the reactivity viathis mechanism by 50 to 100%, dependingon collision energy. A direct H-atom ab-straction mechanism competes with thismechanism, yielding C2H3

1 1 CH3. Thisslightly endoergic channel is enhanced bynearly a factor of 30 by cis-bending excita-tion. At low collision energies, this bendingexcitation favors the direct over the com-plex reaction channel by a factor of 40.Collision energy and C-C stretching alsoenhance the direct mechanism at low col-lision energies, but much less markedly thanbending excitation. Recoil velocity mea-surements reveal that direct H-atom ab-straction proceeds through a weakly boundcomplex with a lifetime of about 1 ps.Anderson and co-workers proposed thattwo different bent transition-state geome-tries control the direct and complex-medi-ated reaction channels. Subsequent theo-retical studies by Klippenstein (15) are con-sistent with this model.

Just as mode-selective chemistry can beachieved by localizing vibrational energy ina bond or a motion along the reactioncoordinate, so can selectivity also beachieved by localized electronic excitation.Examples are the selective breaking of theC-I or C-Br bonds in CH2IBr and the C-Sand S-H bonds in CH3SH by Butler, Lee,and co-workers (16).

How effectively mode-selective chemistrycan be applied to collisions between largepolyatomic molecules remains an unsettled

issue. It seems that three conditions must bemet to achieve mode-selective chemistry:First, it must be possible to excite a state ofthe reagent (an eigenstate or a superpositionof eigenstates) that localizes energy in somepart of the molecule; second, this energymust remain relatively localized during thereaction; and third, this prepared state mustpromote or hinder reactivity of that portionof the molecule to cause a change in theproduct-branching ratio. Fast, direct reac-tions fulfill these criteria best because it isnecessary for the reaction to occur before theexcitation energy becomes statistically dis-tributed among the degrees of freedom avail-able, in which case vibrational excitation isno more effective than heat.

Stereodynamic Control

Another type of state preparation is controlof the reagent approach geometry, whichalso may be called stereodynamic control(17). The orientation of one reagent withrespect to the other is usually achieved bycolliding two beams of reagents in whichone reagent is oriented or aligned by theapplication of an external electric field,such as a strong homogeneous electric field(18) or an inhomogeneous field created bya hexapole. Alternatively, the rotationalangular momentum direction of one re-agent is controlled by the absorption oflinearly polarized light (19). For example,Brooks and co-workers (20) and Bernsteinand co-workers (21, 22) reacted alkali at-oms M with CH3I molecules oriented by ahexapole field to give MI 1 CH3 andshowed that reaction preferentially oc-curred when M collided with the I end ofthe molecule (Fig. 1). Loesch and co-work-ers (23) reacted Sr and K atoms with vibra-tionally excited HF molecules aligned withlinearly polarized light to give SrF 1 H andKF 1 H, respectively. At low collision en-ergies, reactivity of Sr with HF is favored bythe HF bond being perpendicular to the Srapproach direction, whereas reactivity of Kwith HF is favored by the HF bond beingparallel to the K approach direction. Incontrast, at higher collision energies, reac-tivity of Sr with HF reverses so that the HFbond prefers to be parallel to the Sr ap-proach direction, whereas reactivity of Kwith HF becomes insensitive to the align-ment of the HF bond. These studies showthat by controlling the plane of rotation ofHF with respect to the incoming atom, theHF reagent presents distinctly differentchemical shapes in the reaction with amonovalent K atom or with a divalent Sratom as a function of collision energy.

Not only is it possible to control thenuclear framework in a reactive encounter,but it is also possible to control the direc-

tion of the electronic charge distribution foropen-shell reagents, as was first demonstrat-ed by Rettner and Zare (24) on the reac-tions of electronically excited calcium at-oms (Ca 3s3p 1P) with various halogen-containing compounds. Since then, manyrelated studies have been carried out. Leeand co-workers (25) investigated the elec-tron transfer rates for the collisions ofaligned, electronically excited Ba atomswith NO2 to yield Ba1. The rate of electrontransfer varied by more than a factor of 2 asthe orientation of the p orbital occupied bythe electron was changed. Leone and co-workers (26) performed an extensive set ofinelastic scattering studies using orbitallyaligned, excited alkaline earth atoms. Dinget al. (27) showed that orbital alignment ofCa(1P) colliding with CH3I can influencethe CaI product vibrational distributions.Although orientation and alignment stud-ies have shown that reaction rates can onlybe changed by less than an order of magni-tude, these studies do provide an impressiveprobe of the three-dimensional geometriesof reaction pathways.

Reagent approach geometry may also becontrolled, to some extent, by starting witha loosely bound (van der Waals) complex ofthe form AB attached to CD and photolyz-ing the CD moiety to liberate C and D.Then the reaction of AB 1 C can beinvestigated, in which the approach geom-etry is set by the structure of the complex.Quite interesting experiments of this typehave been carried out by the researchgroups of Soep (28), Wittig (29), and Ze-wail (30), among others. A related ap-proach, advocated by Polanyi (31), is tophotolyze molecules adsorbed on a surface.The difficulty with interpreting this type ofexperiment is that the spatial relations be-tween AB and CD in the loose complex oron the surface may be poorly known orpoorly constrained, or both. Moreover, thespectator D may not get out of the way soonenough, thus becoming an “unwanted par-ticipant,” that interferes with the reaction.

Stereodynamic control is also possiblefor gas-surface scattering reactions. For ex-ample, Jacobs and co-workers (32) studiedcollisions of fast NO1 with the (111) faceof single-crystal silver. In this gas-surfacescattering system, both NO– and O– aredetected, although the yield of negativeions is less than 10–4. The NO1 is preparedfrom the NO neutral precursor via reso-nance enhanced multiphoton ionization,and the polarization of the radiation is usedto select whether the ion collides end-on orside-on with the surface. Jacobs and co-workers found that end-on collisions en-hance O– production by a factor of twocompared to side-on collisions, but hardlyaffect NO– production. They interpret their

I

Fig. 1. Observed steric effect for the Rb 1 CH3I3RbI 1 CH3 reaction. An oriented beam of CH3I iscrossed with an atomic beam of Rb; the RbI prod-uct is measured as a function of CH3I orientation.Reactivity to produce back-scattered RbI is max-imal for direct collisions of Rb with the I end of theC-I bond axis; reactivity decreases when the Rbimpacts the side of the methyl iodide moleculeand becomes vanishingly small in a region labeledas the “cone of nonreactivity.” The Rb atom strikesthe CH3I molecule from any direction; the arrowshown is just an example. This figure is adaptedfrom Parker and Bernstein (22).

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data as involving fast neutralization of theion on the surface followed by collision-induced dissociation and electron capture,the order of these two latter events beingundetermined. This mechanism suggeststhat the end-on approach geometry is fa-vored for O– production because it leads tothe lengthening of the N-O bond uponsurface impact (Fig. 2). Classical trajectorycalculations (32) support this explanationand suggest that vibrational excitation ofthe NO1 would enhance O– production.Indeed, in subsequent studies of NO1 onthe (110) face of gallium arsenide, it wasshown that that NO1 (v 5 6) is 2.4 timesmore effective than NO1 (v 5 0) in pro-duction of O–, whereas if the same energyof vibrational excitation is placed intoNO1 (v 5 0) translation, the O– yieldincreases only by a factor of 1.14 (33).Thus, both stereodynamic and vibrationalcontrol should be expected to be applicablefor such processes.

Quantum Control

I turn next to phase control, which is mosteasily understood by reference to Young’scelebrated two-slit experiment in optics.Here, a monochromatic plane wave is inci-dent on two small circular apertures (slits).At each aperture, a new spherical waveemerges and travels to an observationscreen at a distance large compared to thewavelength of the light. Each spherical wavehas a fixed phase relation to the incidentlight wave and, hence, to each other. Theobservation screen shows an alternating pat-tern of bright and dark fringes. This interfer-ence pattern depends on whether the twowaves are in or out of phase at any point onthe observation screen, that is, whetherthey interfere constructively or destructively.Mathematically, the intensity of the screen

spot is proportional to the square of the totalelectric field E at that spot, that is,E1 1 E22 5 E12 1 E22 1 2 E1E2cosf, where f is the phase difference. Thegeneralization to more than two slits isstraightforward. Control of the phase differ-ence f governs the intensity at any point.

The molecular scattering equivalent toYoung’s two-slit experiment arises whenan initial molecular state can follow twodifferent paths, characterized by wavefunctions C1 and C2, that lead to thesame final scattering state. The probabilityof being in that final state is then propor-tional to the square of the total wavefunction. Once again, the two wave func-tion amplitudes must be summed togetherbefore squaring, and the probability of theprocess contains a cross term involvingthe phase of the two wave functions. Itfollows that by altering the relative phasesof a set of competing paths, it is possible tocontrol the transition probability. Thisidea for phase control of a molecular scat-tering event was first put forward byBrumer and Shapiro (34) and was success-fully demonstrated by Chen et al. (35) foratoms and by Park et al. (36) for mole-cules. The basic idea they use is to excitean atom or a molecule by two differentoptical pathways that lead to the sameupper state from which various processessuch as ionization or dissociation can oc-cur by the absorption of a subsequent pho-ton (37). In this manner, the upper state iscreated in a coherent superposition of (atleast) two different product channels. Theparticular superposition depends on theoptical pathway, that is, on how manyphotons are used to produce the excita-tion. It is to be expected that differentoptical pathways have different relativephases.

An example of the power of this type of

phase control is the study by Gordon andco-workers (38) who excited a beam of DImolecules with one photon at 118 nm andthree photons at 354 nm. The experimentwas begun with some phase relation be-tween the two laser beams, and this phaserelation was varied by passing both beamsthrough a transparent gas medium. Becausethe refractive indices at the two differentwavelengths vary, the phase can be con-trolled by changing the pressure inside thegas cell. The excited DI decays either byautoionization to produce DI1 1 e– or bypredissociation to produce a ground-state Datom and an excited I(2P1/2) atom, whichabsorbs an additional photon to yield I1. Asthe phase between the two laser beams isvaried, the I1 and DI1 ion signals varysinusoidally in and out of phase, and thedepth of modulation typically reaches 15%.The relative yields are predicted to be asinusoidal function of this phase differenceand a function of the relative intensities ofthe fundamental and third harmonic of thelight. Recently, Bersohn and co-workers(39, 40) studied the interference betweenthe 19 1 2 and 3 1 2 multiphoton ioniza-tion of methyl iodide (Fig. 3), ammonia,

A B C

––

Fig. 2. Schematic drawing illustrating the effect of stereodynamic control in the production of O– incollisions of aligned NO1 with a single-crystal silver surface: (A) NO1 approaches the Ag(111) surfaceeither end-on or side-on, and in either case, the ion is rapidly neutralized; (B) the end-on collision causesa compression of the N-O bond, whereas the side-on collision does not compress the bond; (C) thecompressed N-O bond is more likely to break, causing the formation and escape of O– whereas theuncompressed molecule is less likely to rupture. The time ordering between N-O bond rupture andelectron transfer from the surface is not known. This figure is based on the work of Jacobs andco-workers (32).

A

B

Fig. 3. Illustration of quantum control in the mul-tiphoton ionization of CH3I. (A) Three-photon andone-photon transitions from the ground state ofmethyl iodide to the same intermediate Rydbergstate interfere in the (3 1 2) and (19 1 2) photo-ionization of methyl iodide. As the argon pressureis increased in a gas cell through which both the(one-photon) red and (three-photon) ultravioletbeams pass (red and blue arrows, respectively),the phase varies between them, which causes theobserved modulation in the CH3I ionization signal(B). This figure is adapted from Bersohn and co-workers (39).

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trimethylamine, triethylamine, cycloocta-tetraene, and 1,1-dimethylhydrazine. [Here,the notation (n 1 m) indicates that nphotons are absorbed in the transitionfrom the ground state to the intermediatestate and m photons are absorbed in thetransition from the intermediate state tothe ionization continuum. A prime is add-ed to a number to indicate a differentfrequency from the unprimed number.] Inall experiments, interference was ob-served, which suggests that molecular sizemay be little obstacle to this type of phasecontrol, provided that the molecule hassufficiently strong absorptions at the fun-damental and at the frequency-tripled out-put of the laser beam to a bound interme-diate state. One intriguing possibility forfuture work is the proposal (41) of con-trolling bimolecular reactions by preparingone of the reagents in a coherent superpo-sition of its internal states.

The most general scheme for control-ling the evolution of a molecule undergo-ing some dynamical process is to controlthe interaction of the molecule and anelectromagnetic field whose spectral andtemporal characteristics are continuouslytuned (chirped) throughout the evolutionof the molecular system in such a manneras to favor a particular outcome of thetime evolution of the molecular state. Oneimplementation of this general schemehas been put forward by Tannor and Rice(42), who proposed to vary the intervalbetween an initial pump pulse of radiationthat transfers amplitude from the groundstate of the molecule to the excited stateand a dump pulse of radiation that trans-fers amplitude in the opposite direction.As has been pointed out by Rabitz andco-workers (43) and by Kosloff, Rice, andco-workers (44), optimal control theorymay be used to find the pulse shape andspectral content of the pulse that maxi-mizes the yield of the specified product.

So far, experimental examples of thistype of control are rather sparse, but quiterecently, Wilson and co-workers (45) in-vestigated the effect of chirped femtosec-ond pulses on the three-photon absorptionof I2. They demonstrated a significant en-hancement compared to transform-limitedzero-chirped pulses. For a positivelychirped pulse, the low-frequency compo-nents arrive before the high-frequencycomponents, whereas the order is reversedfor a negatively chirped pulse. An ultra-short pulse with its concomitantly broadspectral character prepares a coherent su-perposition of excited molecular states,and the corresponding wave packetevolves in time. In this experiment, theabsorption process is resonantly enhancedand corresponds to the sequence X3 B3

C 3 D, where the capital letters denotedifferent electronic states of the I2. mole-cule. A fluorescence signal at 340 nm fromcollisionally relaxed D 3 A emission isdetected. Wilson and co-workers foundthat at 570 nm, a positively chirped pulseyields an increase by a factor of 3 in theD 3 A fluorescence signal compared tothe shortest, transform-limited pulse.They suggest that the three-photon ab-sorption enhancement involves severalfactors. These include a time-delay reso-nance mechanism, in which the high-fre-quency components of the positivelychirped pulse arrive as the moving wavepacket comes into resonance with the B3 C transition; a wave packet–followingmechanism, in which the change in pulsefrequency follows in time the change inthe wave packet’s electronic transition en-ergy; and a sequential resonance mecha-nism, in which the chirp provides frequen-cies that are first resonant with X 3 B,then with B 3 C, and finally with the C3 D transitions.

The extension of this type of quantumcontrol to large molecules in solution, whichis the medium in which most chemistry isdone, is a challenging task both because ofthe complexity and because of the lack ofdetailed understanding about such a system.It should be possible, however, to use thebehavior of the molecular system in an iter-ative manner (feedback control) to “teach”the experimenter how to tailor the radia-tion guidance (46). The first preliminarybut promising report of such a success hasjust been made by Wilson, Warren, andco-workers (47) who tailored femtosecondpulses by a computer-controlled acousto-optic pulse shaper to excite fluorescencefrom an infrared laser dye in solution.

Concluding Remarks

Collision control is presently a topic ofintense research activity. The increasingversatility of laser sources and laser tech-niques underpins almost all advances. Untilthe cost of performing photon-inducedchemical reactions is greatly reduced, how-ever, no practical applications can be antic-ipated for this type of chemical reactioncontrol. Moreover, in many cases the effectsare too small to be useful. Nevertheless, thereal gain in pursuing laser-based collisioncontrol is likely to be the increased under-standing of how chemical reactions occur,which in time will no doubt lead to impor-tant applications.

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39. G. Xing, X. Wang, X. Huang, R. Bersohn, B. Katz,J. Chem. Phys. 104, 826 (1996).

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48. Special thanks go to M. C. Everest, S. A. Kandel, C.Matthews, J. C. Poutsma, and T. P. Rakitzis forassistance in preparing this manuscript and to R.Bersohn, P. Brumer, R. J. Gordon, S. L. Anderson,and K. R. Wilson for correspondence and for sharingunpublished data. Supported by NSF under grantNSF CHE-93-22690.

Quantum Theory of ChemicalReaction Dynamics

David C. Clary

It is now possible to use rigorous quantum scattering theory to perform accurate cal-culations on the detailed state-to-state dynamics of chemical reactions in the gas phase.Calculations on simple reactions, such as H 1 D2 3 HD 1 D and F 1 H2 3 HF 1 H,compete with experiment in their accuracy. Recent advances in theory promise to extendsuch accurate predictions to more complicated reactions, such as OH 1 H23 H2O 1H, and even to reactions of molecules on solid surfaces. New experimental techniquesfor probing reaction transition states, such as negative-ion photodetachment spectros-copy and pump-probe femtosecond spectroscopy, are stimulating the development ofnew theories.

Experiments and theory on chemical reac-tion dynamics aim to study the details ofreactions beyond just simple rate constantsat room temperature (1). Reactant andproduct molecules can have different trans-lational energies, varying angles of orienta-tion and different internal quantum states.Developing experiments and theory tostudy such microscopic aspects of chemicalreactions, and to probe directly the struc-ture and lifetimes of reaction transitionstates (2, 3), has become a major field. Theresults provide the most detailed insightinto the mechanisms of chemical reactionsand are useful also in a variety of othercontexts. For example, an understanding ofboth atmospheric (4) and combustion pro-cesses (5) can require data on the chemicalreactions of molecules in selected vibration-al quantum states, and modeling of thenonequilibrium environments of interstel-lar clouds (6) needs information on thereaction rates of molecules in selected rota-tional states for low collision energies.

Theory has a special role to play inchemical reaction dynamics as it is oftenessential for extracting useful informationfrom experimental results. Furthermore, re-action dynamics theory can now providedetailed predictions with an accuracy thatcan rival experiment on simple reactions inthe gas phase (7), and the theory has beenextended, albeit more approximately, to

more complex problems such as reactions ofpolyatomic molecules (8) and reactions onsolid surfaces (9) and in solution (10).

Quantum Reaction Dynamics

The best theory of chemical reactions usesquantum mechanics. The Born-Oppenhei-mer approximation is usually invoked, al-lowing electronic and nuclear motion to beseparated, so that a calculation reduces totwo separate steps: solution of the Schrod-inger equation for the electrons with fixedpositions of the nuclei to obtain a potentialenergy surface, followed by solution of the“quantum scattering” Schrodinger equationfor the nuclei moving on this potentialenergy surface.

Significant advances in ab initio quan-tum chemistry techniques are enabling po-tential energy surfaces to be calculated to ahigh accuracy for simple reactions (11).However, one problem is that reaction rateconstants k(T) often depend exponentiallyon the height Ea of the barrier in a potentialenergy surface according to the Arrheniusexpression k(T) 5 Aexp(2Ea/RT), whereA is a constant, R is the gas constant, and Tis temperature. This exponential depen-dence shows that Ea needs to be calculatedextremely accurately. This constraint pre-sents a major problem for quantum chem-istry calculations on reactions involvingmolecules with several electrons.

In a quantum scattering calculation on achemical reaction A 1 BC(v, j) 3 AB(v9,

j9) 1 C in the gas phase, it is necessary touse coordinates that go smoothly from re-actants to products and can describe theinitial (v, j) and final (v9, j9) vibration-rotation molecular states accurately. Devis-ing the most appropriate coordinates hasnot been straightforward. A useful “hyper-spherical coordinate” approach is to takethe bonds being broken and formed in thereaction and transform them into polar co-ordinates (r, u) (12). The “hyper-radius” rhas the advantage that it can be used as acommon scattering coordinate for both theentrance and exit channels of the reaction.The time-independent Schrodinger equa-tion HC 5 EC can then be solved, andscattering boundary conditions can be ap-plied to calculate reaction probabilities se-lected in all of the vibration-rotation statesof reactants and products and with well-defined collision energy. This method hasbeen used quite widely on reactions involv-ing three atoms (13) and has also recentlybeen applied to four-atom reactions (8).

A different approach is to solve thetime-dependent Schrodinger equation forthe nuclear motion. This method has thecomputational advantage that reaction outof individual reactant quantum states canbe computed efficiently (14). However, ithas been difficult until recently to extendthis “wave packet” technique to calculatestate-to-state reaction probabilities. There-fore, total reaction probabilities summedover all product states are often computed.

The large number of coupled vibrationaland rotational states of molecules that can beinvolved in chemical reactions has made thesolution of the nuclear Schrodinger equationcomputationally expensive. Therefore, thefield of theoretical reaction dynamics hashad to wait for the modern generation of fastcomputers with large memory capabilitiesbefore reliable calculations have been possi-ble. This is why the field of quantum reac-tion dynamics is now suddenly blossoming.

Simple Reactions

How well does the best theory do whencompared with the best experiments? Oneof the simplest chemical reactions is H1 D2(v 5 0, j 5 0) 3 HD(v9, j9) 1 D.This reaction has been studied in one of themost detailed experiments that can be per-formed, crossed–molecular beam experi-

The author is in the Department of Chemistry, UniversityCollege London, London WC1H OAJ, United Kingdom.

ARTICLES

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