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Is the simplest chemical reaction reallyso simple?Justin Jankunasa,1, Mahima Snehaa, Richard N. Zarea,2, Foudhil Bouaklineb, Stuart C. Althorpec,Diego Herráez-Aguilard, and F. Javier AoizdaDepartment of Chemistry, Stanford University, Stanford, CA 94305-5080; bMax Born Institute, 12489 Berlin, Germany; cDepartment ofChemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom; and dDepartamento de Química Física, Facultad de Química,Universidad Complutense, 28040 Madrid, Spain

Edited by Robert W. Field, Massachusetts Institute of Technology, Cambridge, MA, and approved November 18, 2013 (received for review August 19, 2013)

Modern computational methods have become so powerful for predicting the outcome for the H + H2 → H2 + H bimolecular exchange reactionthat it might seem further experiments are not needed. Nevertheless, experiments have led the way to cause theorists to look more deeply intothis simplest of all chemical reactions. The findings are less simple.

reaction dynamics | differential cross-sections

H + H2 Collision Looks Like PlayingMolecular Pool on a Warped TableThe reaction in which a hydrogen atomcollides with a hydrogen molecule to forma new hydrogen molecule plus a hydrogenatom is often referred to as the simplestneutral bimolecular reaction because thiscollision system contains only three electronsand three protons. Consequently, the forcesacting on the nuclei can be calculated usingquantum mechanics from first principles toa very high level of accuracy. It is this re-action system that we believe we can mosttrust theory to tell us what is happening.Study of this benchmark system and its re-lated isotopic cousins has led to a number offirsts in reaction dynamics: the first potentialenergy surface with chemical accuracy (lessthan 1 kcal/mol deviation over the surface),the concept of a transition state, the idea ofquantum tunneling in reaction dynamics, theuse of transition-state theory to predict re-action rates, the concept of resonances in thereactive scattering cross-section, etc. (1–3).The extension of these concepts to the F +HD and Cl + HD reactions has also beenvery fruitful (4, 5).To facilitate a discussion of this reaction,

we label the H atoms as if they were distin-guishable and examine the reaction:

Ha +HbHc →HaHb +Hc:

Many studies have taught us that reactionsare best described by considering the mini-mum energy path in going from reactants toproducts. In the case of H + H2, the config-uration with the lowest energy is for the threeH atoms to lie in a straight line (collinear

configuration) for which there exists a barrierto reaction of about 0.42 eV (about 9 kcal/mol) which must be surmounted for reactionto proceed at any significant rate (Fig. 1). Thiscan be accomplished by giving the incomingH atom sufficient kinetic energy. The top ofthe hill is the transition state. Still more in-sight is obtained by making a contour plot ofthe potential energy surface (Fig. 2) in whichthe angle between the forming Ha–Hb bondand the breaking Hb–Hc bond is held fixed at180° (collinear configuration). The internu-clear equilibrium distance of the H2 moleculeis 0.741 Å and the energy needed to pull apartthe two hydrogen atoms in H2 is ∼4.5 eV (104kcal/mol). A black dot marks the transitionstate in Fig. 2, which is located at the saddlepoint of the HaHbHc potential energy surface.Examination of Fig. 2 shows that this reactionis a concerted process in which both Ha–Hb

and Hb–Hc bonds are longer at the transitionstate than the H2 internuclear equilibriumdistance. Instead of requiring the dissociationenergy to pull HbHc apart, much less energyis needed.It is common to characterize the reaction

by measuring the translational kinetic energyof the recoiling Hc atom, the internal energyof the HaHb molecule, and the angles intowhich these products are scattered, refer-enced to the direction of the incoming Ha

atom (6). The internal energy of the HaHb

molecule is characterized by its vibrationaland rotational energy levels, which are rep-resented by the quantum numbers ν′ and j′.The distribution of products as a function ofthe scattering angle (θ) is called the differential

cross-section. Again, past experiments andtheory guide our understanding. A greatsimplification is made by separating theelectronic and nuclear motions (Born–Oppenheimer approximation) and treatingthe nuclei as classical particles. In this pic-ture, elementary chemical reactions of thistype might be regarded as molecular bil-liards (pool). Thus, the expectation is thatnearly head-on collisions cause the HaHb

product to be scattered backward (θ is closeto 180°) with little rotational excitation (lowj′), whereas more glancing collisions causethe HaHb molecule to be scattered sideways(θ < 180°) with more rotational excitation(high j′). Fig. 3 presents a simple cartoon il-lustrating this behavior.How well are these expectations fulfilled?

To answer this question, we conducted ex-periments on the related reaction H + D2 →HD + D, in which the deuterium atom D isa heavy isotope of the hydrogen atom hav-ing one additional neutron in its nucleus (3).The use of isotopes allows us to distinguishreadily the products from the reactants. Briefly,a mixture of HBr and D2 is coexpanded as

Author contributions: J.J., R.N.Z., S.C.A., and F.J.A. designed re-

search; J.J., M.S., F.B., and D.H.-A. performed research; J.J., M.S.,

F.B., S.C.A., D.H.-A., and F.J.A. analyzed data; and J.J., R.N.Z., F.B.,

S.C.A., and F.J.A. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

1Present address: Ecole Polytechnique Federale de Lausanne, In-

stitute of Chemical Sciences and Engineering, CH-1015 Lausanne,

Switzerland.

2To whom correspondence should be addressed. E-mail: zare@stanford.edu.

This article contains supporting information online at www.pnas.org/

lookup/suppl/doi:10.1073/pnas.1315725111/-/DCSupplemental.

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a molecular beam into an evacuated chamberand fast H atoms with a controlled trans-lational energy are generated by photolyz-ing HBr at a variable wavelength (3). Someof these fast H atoms collide with the su-personically cooled D2 which is in theground vibrational state (v= 0) and the firstfew rotational levels (j = 0, 1, and 2). TheHD(v′, j′) product is detected state-specifi-cally. In terms of theory, the nuclei move onthe same potential energy surface, althoughwe must now introduce an energy-levelcorrection in that D2 and HD have differ-ent vibrational frequencies and differentzero-point energies, which is indicatedby the purple curve in Fig. 1. In theseexperiments, we measured the differentialcross-sections for HD products in differ-ent vibrational and rotational states. Theexperimental results for four different com-binations of ν′ and j′ are compared in Fig. 4against the best theoretical calculations(which are fully quantum mechanical) (7).Although theory and experiment do notperfectly agree, and some puzzling dis-crepancies do exist (8), the match is closeenough that we have great confidencein both. Likewise, the quasiclassical cal-culations agree fairly well with the accu-rate quantum mechanical results (Figs.S1 and S2), lending credence to the clas-sical description of the reaction. More-over, they both show that our billiard-ball picture for this reaction describeswell what is found, although not all re-active collisions proceed only throughdirect recoil (9). The HD molecules inhigher rotational states were found atlower values of θ, meaning they werescattered more sideways. It might seem

then that the H + H2 reaction system isreally simple to understand. Indeed, wefirst thought so (10).

Warped Billiard Table Changes as thePool Balls Move and Is Weirder thanFirst ImaginedHowever, then we did one experiment toomany; we examined the differential cross-sections of the HD products that were pro-duced with more vibrational excitation(ν′ = 4) (11). Fig. 5 presents these results.To our initial surprise, we find that withincreasing rotational excitation, the HDproduct is scattered more backward (thedistribution moves closer to 180°). Theoryagrees with the observations, but what isthe explanation? Do we need to abandonour model of billiard-ball collision dy-namics? If the last is true, it is a huge set-back. What we hope to extract from thestudy of H + H2 is not only how this simplereaction system behaves but hopefully gen-eralizations that can be applied to othermore complicated chemical reactions thatcannot be modeled to the theoretical accu-racy achievable for H + H2.

Fig. 1. Plot of the potential energy versus the reaction coordinate for H + D2 →HD + D.The symmetric black curve isthe classical minimum energy path and the purple curve is the vibrationally adiabatic minimum energy path for H +D2(v = 0) going to HD(v = 0) + H.

Fig. 2. Plot of the PES for the collinear approach of a hydrogen atom Ha to the hydrogen molecule HbHc tobreak the old Hb–Hc bond and to form the new Ha–Hb bond in a bimolecular exchange reaction. In this diagram,the contours are equally spaced by 0.2 eV. The saddle point region, marked by a black dot, lies in the center about0.42 eV above the asymptotes. The entrance valley starts at the lower right and the exit valley ends at the upperleft of the diagram in the process Ha + HbHc → HaHb + Hc. This figure is adapted from Bin Jiang and Hua Guo(2013) J Chem Phys 138:204104.

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Happily, we can still keep our billiard-ballmodel, but we must recognize that the pic-ture we have painted of the H3 potentialenergy surface as everywhere purely repulsiveneeds drastic revision for HD productsformed in higher vibrational levels. Asshown in Fig. 2, the only minimum in thepotential energy surface (PES) occurs whenthe H atom is infinitely far away from theH2 molecule. We are only interested in“chemical” wells, and ignore the meager(∼10−3 eV) van der Waals wells. Withinthis framework, the H + H2 collision is anarchetype of a direct chemical reaction: inthe absence of any wells, the reactants ap-proach one another, and promptly recoilaway. Such a simple picture of the H + H2

dynamics is incapable of explaining thepuzzling scattering behavior exemplifiedby the differential cross-sections for HD(v′ = 4, j′) products.Fig. 6 shows the minimum energy path of

the H + D2 reaction for different vibration-ally excited HD(v′) products. These so-called“vibrationally adiabatic paths” are created byassuming that the D2 reagent and the HDproduct have the same vibrational quantumnumber (v = v′). We find for high vibrationalexcitation that the HD2 PES develops at-tractive wells! To illustrate this behavior moredramatically, we present in Fig. 7 a 3D PESfor a frozen reactant bond length appropriatefor the outer turning point of HD(v = 4).Wells are readily apparent. The origin of

these attractive lakes in the PES is the in-cipient pairwise bonding among all of theatoms when the D2 molecule becomes suffi-ciently stretched in the reaction producinghighly vibrationally excited HD. The sametype of behavior was observed before inthe inelastic scattering process H + D2(v =

0) → H + D2(v′ = 3) + H, in which it wasfound that vibrational excitation of D2 isnot achieved by compressing the D–Dbond but rather by stretching it as the Datom closest to the H atom is attracted toit (12). In that sense, we understand theinelastic scattering process leading tohighly vibrationally excited D2 to be anexample of a frustrated chemical reactionin which vibrational excitation arises fromthe D atom snapping back to its originalD-atom partner when it is no longer ableto catch the H atom to form HD.Actually, the reaction path for H + D2(v =

0) → HD(v′ = 4) + D collision will be evenmore complicated than the adiabatic poten-tials shown in Fig. 6, owing to obvious non-adiabatic coupling between the v = 0 and v =4 potentials, presumably in the region ofclose interaction between the three atoms, i.e.,in the vicinity of the transition-state config-uration. We present in Movies S1–S7 somerepresentative reactive collisions forming HD(v′ = 4, j′ = 0–6), where the impact param-eters and the collision times are indicated ineach movie file. In some reactive collisionsthe incoming H atom ends up bound not tothe first D atom it encounters but rather itspartner via an insertion mechanism [see thesecond collision trajectory in Movie S6 show-ing the formation of the HD(v′ = 4, j′ = 5)product]. Clearly, the dynamics for the pro-duction of HD(v′ = 4) molecules is verydifferent from, say, HD(v′ = 0 or v′ = 1)

Fig. 3. H + D2 → HD(v′, j′) + D reactive collision showing typical trajectories in the center-of-mass frame.The symbolu denotes the velocity, j′ the rotational quantum number of the HD product, and θ the scattering angle of the HDproduct measured with respect to the incoming H atom. The b is the impact parameter, defined by the distance ofclosest approach if the H atom travels in a straight line. The quantity L is the orbital angular momentum of the collision,and is the quantum mechanical analog of the classical quantity b. (Upper) Nearly head-on collisions (b ∼ 0, L ∼ 0);(Lower) More glancing collisions (b > 0, L > 0).

Fig. 4. Comparison of the experimental (red dots) and the blurred time-dependent quantum mechanical (blackcurve) differential cross-sections for the production of (A) HD(v′ = 1, j′ = 2) and (B) HD(v′ = 1, j′ = 5) both at a collisionenergy Ecoll of 1.97 eV (46.5 kcal/mol), and (C ) HD(v′ = 2, j′ = 3) and (D) HD(v′ = 2, j′ = 6) both at a collision energyEcoll of 1.61 eV (40.0 kcal/mol). With increasing HD product rotational excitation, the HD is scattered more sideways.To aid visualization of this effect arrows have been added to A and B because the differential cross-section containstwo peaks.

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products! We are only beginning to under-stand in detail this reaction system.Each collision of H with D2 is character-

ized by an impact parameter b and an an-gular momentum L, whose magnitude isclassically proportional to b and to the ve-locity of the H atom with respect to the D2

molecule (in the center-of-mass frame). Thepotential experienced by H contains an extrarepulsive term, the centrifugal barrier, whosemagnitude grows with increasing L (actually,proportional to L2). We suggest (11) that thecentrifugal barrier in the exit channel ofthe H + D2 → HD + D reaction is re-sponsible for “choking off” higher values of Lwith increasing product rotational quantumnumber j′. A qualitative (pictorial) illustra-tion of the choking off idea is given in Fig. 8,which shows how the effective potentialchanges with increasing L. Loosely speaking,certain highly internally excited HD productsdo not have enough recoil kinetic energy toovercome the centrifugal barrier in the exitchannel of the reaction, when the H and D2

reactants collide with an L = 26 amount oforbital angular momentum (Fig. 8). Suchtrajectories are nonreactive. Thus, such ahighly internally excited HD product is a re-sult of a collision between a hydrogen atomand a deuterium molecule with a smaller

orbital angular momentum, such as L = 20:the centrifugal barrier is lower for such acollision (Fig. 8), and the HD and D productsin question have sufficient energy to pass

over this obstacle. This explains why, forexample, HD(v′ = 4, j′ = 6) product is morebackscattered than HD(v′ = 4, j′ = 3): theformer has even less kinetic energy, and thusthe maximum centrifugal barrier that canbe overcome is even lower. Correspondingly,lower values of L will contribute to the pro-duction of HD(v′ = 4, j′ = 6), compared withfaster HD(v′ = 4, j′ = 3) products; the j′ = 6product will therefore be more back-scattered than j′ = 3. Stated another way,among the high-energy products, theproducts with high j′ must come fromcollisions that are closer to collinear (bcloser to 0) because otherwise there wouldnot be enough kinetic energy for the HDproduct to escape and the reaction to oc-cur. These arguments are general, that is,not restricted to HD(v′ = 4). They areexpected to hold for all other final vibra-tional states at different collision energiesprovided the product kinetic energy is notenough to overcome the centrifugal barrier.Another factor also contributes to favor-

ing increased backward scattering with in-creasing j′. The differential cross-section forH + D2 → HD(v′ = 4, j′ = 0,1,2) actuallyhas two peaks, one of which arises fromdirect scattering and the other is a time-delayed reaction path whose origin hasbeen well-documented as a quantum bot-tleneck state (13, 14). The time-delayedpathway enhances forward scattered featuresin the differential cross-section. For j′ = 2product state, the magnitude of the twopeaks is comparable, but with increasing

Fig. 5. Comparison of the experimental (red dots) and the blurred time-dependent quantum mechanical (blackcurve) differential cross-sections at Ecoll = 1.97 eV (46.5 kcal/mol) for the production of (A) HD(v′ = 4, j′ = 1), (B) HD(v′ = 4,j′ = 2), (C) HD(v′ = 4, j′ = 3), (D) HD(v′ = 4, j′ = 5), and (E) HD(v′ = 4, j′ = 6). With increasing HD product rotationalexcitation, the HD is scattered more backward.

Fig. 6. Vibrationally adiabatic potentials for the H+D2(v)→HD(v´)+D reaction showing the formation ofwells for v= v′= 3and 4. (Left) Orbital angular momentum L = 0; (Right) L = 25.The dotted line is when one quantum of bending motion in theHD2 triatomic is also included. The horizontal dashed line at the top of the figure represents the total energy of the collision.

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product rotational excitation the contribu-tion from the time-delayed pathway dimin-ishes so that for j′ = 5 it is negligible (Fig. 9).Both the choking-off argument and thetime-delayed mechanism are subtly inter-twined; only through theory and experimentworking together is it possible to explain the

measured and calculated angular distributionsshown in Fig. 5.

Future DirectionsClearly, the H + H2 reaction system is notquite as simple as first imagined. And, wehave only begun to familiarize ourselves withits complex nature. Moreover, the situation isabout to become markedly more complicatedwith increasing translational energy of theincoming H atom or with extra energy in thereactant’s internal degrees of freedom. So far,we have been fortunate to consider only onePES as governing the motion of the nuclei.However, the H3 system is a Jahn–Tellersystem having a second PES connected to thefirst at a conical intersection seam whoseminimum is located about 2.7 eV (64 kcal/mol) above the asymptotes of the groundstate (6). Once this is energetically accessed,we can expect electronic nonadiabatic tran-sitions to occur in which we must considermotion of the nuclei on both PESs at thesame time (15, 16). This hurts the classicalmind. We will need to consider playingmolecular pool on interconnected warpedbilliard tables whose shapes change with themotion of the pool balls. Well, no one everpromised us that chemical reactions would

be simple, even for what is called the simplestchemical reaction of them all.Clearly, one inviting direction for future

experiments is to increase the translationalenergy of the two collision partners to accessthe region of the conical intersection of the H3

PES and beyond. It is believed that the the-oretical apparatus for calculating the behaviorof such high-energy collisions is sufficientlywell developed, but experimental tests remainto be made. Past history teaches us that somesurprises might be expected, even though wethink we understand how to treat fully non-adiabatic couplings for this simple system.Another inviting direction is to remove the

averaging over the 2j + 1 different magneticsublevels m of the rotational state j all of thesame energy, which presently must be in-cluded to describe the molecular reactantsand products. Classically, a molecule in thestate characterized by the quantum numbersj andm has its rotational angular momentumvector j making a projection m on the axis ofquantization and might be regarded as the jvector spinning at a uniform rate about thequantization axis. By achieving m-state res-olution either of reagents or products we willbe able to explore more deeply the quantumnature of the scattering process. However,even more information might be achieved bypreparing molecular reagents in a coherentsuperposition of m states. This feat involvingthe control of phase of how j precesses about

Fig. 7. PES plot for the H + D2 system and its associated 2D projection, with an internuclear HD separation fixed atr = 1.26 Å, corresponding to HD(v′ = 4) vibrational quantum state. It is evident that for compact collinear HD2 ge-ometries, a deep well is present as H approaches D2 along the Rx axis for Ry = 0. Rx and Ry stand for the Cartesiancoordinates of the H atom with respect to the D2 center of mass.

Fig. 8. Cartoon illustration of the “centrifugal barrier”mechanism responsible for the choking off reaction fromhigher orbital angular momentum L. The production ofcertain highly internally excited HD(v′, j′) product states isimpossible when a hydrogen atom and a deuterium mole-cule collide with an L = 26 amount of orbital angular mo-mentum (red traces). Such trajectories are nonreactive. TheHD(v′, j′) productswith low kinetic energymay be producedwhen the centrifugal barrier in the exit channel is lower(green trace), corresponding to L = 20. Consequently, suchHD(v′, j′) products are more backscattered compared withHD(v′, j′) products with more recoil kinetic energy.

Fig. 9. Ecoll – θ plot for (A) HD(v′= 4, j′= 2) and (B) HD(v′=4, j′ = 5) product states, showing that the HD(v′ = 4, j′ = 2)product has a substantial contribution from the time-delayedmechanism, which enhances the forward features in a dif-ferential cross section. In contrast, the HD(v′ = 4, j′ = 5)product proceeds only via a direct mechanism. The blackcurve in A shows a clear separation between the directand time-delayed reaction mechanisms in the H + D2 →HD(v′ = 4, j′ = 2) + D reaction.

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the quantization axis might allow us toidentify reaction paths looping in oppositesenses around the conical intersection. Ofcourse, we should not overlook some ulti-mate experiments that detect in coincidence

all of the characteristics of the scatteringprocess and thereby provide many differentvector correlations (6), although this dreamseems presently beyond what we know howto realize experimentally.

ACKNOWLEDGMENTS. J.J., M.S., and R.N.Z. thankthe US National Science Foundation for support; S.C.A.acknowledges support from the UK Engineering andPhysical Sciences Research Council; and F.J.A. andD.H.-A. thank the Spanish Ministry of Science and In-novation (Grants CTQ2008-02578, CTQ2012-37404-C02,and CSD2009-0038).

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