Recent Advances in Quantum Mechanical Reactive Scattering ...3a,b) ... Caltech, Berkeley, Aarhuus...

Annu. Rev. Phys. Chern, 1990. 41." 245~31Copyright © 1990 by Annual Reviews Inc. All rights reserved

RECENT ADVANCES INQUANTUM MECHANICALREACTIVE SCATTERING THEORY,Including Comparison of RecentExperiments with Rigorous Calculationsof State-to-State Cross Sections for theH/D + H 2 ~ H 2/HD -F H Reactions

William H. Miller

Department of Chemistry, University of California, and Materials andChemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley,California 94720

KEY WORDS"chemical reactions (theory of), chemical dynamics, dynamics chemical reactions.

INTRODUCTION

It is well recognized that quantum mechanical reactive scattering theoryprovides the most complete description of an elementary bimolecularchemical reaction allowed by the basic laws of nature. Thus ever since the1960s, when crossed molecular beam experiments opened the door tostudying reactions at this most rigorous state-to-state level (la,b), therehas been intense interest and effort devoted to developing the theory tothe practical stage that reliable calculations can be carried out for realchemical reactions. The purpose of this review is to describe some ratherdramatic progress that has been made in the theoretical methodology andits application in only the last few years. These theoretical developmentshave come at a very propitious time because of a variety of new experi-mental studies of state-to-state cross sections for the fundamental reactions

2450066M26X/90/1101-0245502.00

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246 MILLER

H + H2(para) ~ H2(ortho) I a.

D+H2-~ HD+H. lb.

Comparison of the recent scattering calculations with these experimentalresults is also reviewed and provides a snapshot of how close one presentlyis to the ideal of being able to measure and to calculate from first principlesthe most fundamental attributes of a chemical reaction.

This review restricts attention to rigorous (in principle "exact") treat-ments of quantum reactive scattering, although this is not meant to implythat it is necessary (or even desirable) to approach all applications fromthis point of view. If one is interested only in thermal reaction rateconstants, for example, then it is well known that transition state theory isoften quite adequate. Furthermore, classical trajectory simulation methodsare satisfactory in many cases for describing more detailed state-to-stateproperties of reactions. There are also a variety of approxi~nate quan-tum mechanical and semMassical models that are useful in variousspecial situations. Only a rigorous quantum scattering calculation, how-ever, is guaranteed to be correct (for a given potential energy surface),so it is important to develop these capabilities to as great an extent aspossible in order to be able, in some cases at least, to provide a com-pletely reliable theoretical description. Such is the point of view of thisreview.

Earlier reviews on approximately this same subject are those by Walker& Light (2) ("Reaction M~lecular Collisions") in 1980 and by Schatz(3a,b) ("Quantum Effects in Gas Phase Bimolecular Chemical Reactions")in 1988. Comprehensive surveys of work on the H 3 family of reactions arethe early review by Truhlar & Wyatt (4a) and the more recent one Valentini & Phillips (4b). The latter paper was written just before theappearance of all the rigorous cross section calculations for these reactions,which are the primary subject of the present review.

One may ask why the last few years have seen such a revival of activityon a topic that has been of central interest for over 20 years. Certainly onereason has been the increased computer power that has been made avail-able to theorists from a variety of programs. In experimental or theoreticalresearch one gears the type of problem one addresses to the availableresources; ready access to supercomputer facilities has motivated theoristsat many locations (Texas, Paris, Northwestern, Minnesota, Los Alamos,Houston, Chicago, Florida, ,Cambridge, Caltech, Berkeley, Aarhuus .... )to start thinking again about how to carry out reactive scattering calcu-lations. Somewhat surprising, though, was the discovery of some subtletiesof basic scattering theory that have made the calculations actually much


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QUANTUM REACTIVE SCATTERING 247

more straightforward than had been believed. Thus advances both incomputational power and in theoretical methodology have been sig-nificant factors in the recent developments reviewed here.

The recent developments in scattering theory and methodology aresurveyed first--at a very qualitative level--and then the recent experi-ments and cross section calculations for reactions 1 are compared. Thetwo parts can be read essentially independently.

OVERVIEW OF RECENT THEORETICALDEVELOPMENTS

No attempt is made here to give a comprehensive review of reactivescattering theory, but rather to summarize the basic ideas behind variousapproaches, to discuss how they relate to one another, and to see why suchrapid progress has been possible recently. In order to focus on the keypoints and avoid unnecessary complexities, the collinear A + BC -~ AB + Creaction (Figure l) will be used to illustrate matters. It is important, course, that the calculations can be (and are) carried out for the physicallyrelevant three-dimensional version of the reactions.

CoordinatesThe theory of reactive scattering is more complicated than that for elasticand inelastic scattering (5a-c) because of coordinates, and different for-mulations of reactive scattering turn on how one deals with this coordinateproblem. Figure 1 depicts the situation for the collinear A + BC -, AB + Creaction. If one were treating only an inelastic scattering process (i.e.vibrational excitation),

A+ BC(n) ~ A+ BC(n’), 2.

then the standard Jacobi’s coordinates (ro, Ra) are the natural choice, andthe coupled-channel expansion of the wavefunction has the form

where {~b,} are the (known) vibrational eigenfunctions for diatom BC andn~ denotes the initial vibrational state. Substitution of this expansion intothe Schr6dinger equation leads to the standard coupled-channel equationsfor the unknown translational functions

0 = 2~ ~ En L~nl(Ra)+~n’ Vn’n’(Ra)fn’~n’(Ra)’ 4a.


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248 MILLER

(a)

(b)

(c)

Figure I Schematic depiction of a collinear A + BC ~ AB + C potential energy surface anddifferent ways of choosing coordinates. (a) Jacobi coordinates for arrangement a(A + and c(AB + C); (b) reaction path ("natural collision") coordinates; (c) hyperspherical simply pola0 coordinates.

which are a set of ordinary differential equations that are typically solvedby various finite difference :methods. [l/n.,~,(Ra) is the matrix of the inter-action potential V(R~,ra)-v(ra), where V is the total potential energyfunction and v that for the diatom BC alone,


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Vn,n’(Ra) = f dra c~n(r~) (V- 4b.

En = E- en is the translational energy for channel n, where E is the (fixed)total energy and e~ the vibrational energy for state n.]

The "coordinate problem" referred to above for reactive scattering isthat the Jacobi coordinates (ra, Ra) that are natural for describing thereactants A + BC are not appropriate for describing the products, AB + C.One way for dealing with this problem, introduced by Marcus (6) in 1966,is to use a curvilinear coordinate system ("natural collision coordinates")that goes smoothly from reactants to products. Referring to Figure lb,the reaction coordinate s is the distance along the indicated curve (thereaction path), and u the perpendicular distance away from it. Thewavefunction is then usually expanded as

n

where here {~b,} are the vibrational eigenfunctions for the u-motion atdistance s along the reaction path (a vibrationally adiabatic basis). Thisapproach has been generalized for the three-dimensional version ofA + BCreactions (7) and also even for polyatomic molecular systems (8) "reaction path Hamiltonian"). The translational functions {f,~,,} satisfycoupled differential equations similar to Eq. 4a, though the kinetic energyterm is considerably more complicated (9) because of the curvilinear andmulti-valued nature of the coordinates.

The reaction path formulation has been very useful for many qualitativediscussions of reactions and for various approximate treatments, and ithas also been used for rigorous scattering calculations, particularly ofcollinear A + BC systems (10, 11). For reactions in three dimensions, theiruse is complicated by the fact that there is not just one reaction path thatis appropriate for all reactant and product possibilities. However, somethree-dimensional scattering calculations (12a~) have been successfullycarried out using reaction path coordinates. My guess is that in the future,reaction path approaches will find more use in qualitative and approximatemodels for describing reactions than for rigorous scattering calculations.

Another choice of coordinates for reactive scattering that is being widelypursued by a number of groups (13-18) is various types ofhypersphericalcoordinates, which for the collinear case in Figure 1 are simply polarcoordinates. If {qS,(0;p)} are the bound state eigenfunctions for the0-motion for fixed p, then the wavefunction is typically expanded as

~O,, = ~’, ~b,(0; P)f,~n,(P), 6.


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250 MILLER

and the translational functions {fn~-nl) satisfy coupled ordinary differentialequations similar to Eq. 4a. As for reaction path coordinates (s, u), thehyperspherical coordinates (19, p) are valid for both reactants and products,but they do not have as many mathematical complexities as the former:the kinetic energy operator i,s relatively simple, and they are globally valid(i.e. no singularities).

The hyperspherical approach generalizes to three-dimensional A + BCreactions and also to more complex collisions and, as noted, it is beingpursued by a number of groups. The primary disadvantage of the approachis that hyperspherical coordinates, though well-behaved mathematically,are really not the most dynamically natural coordinates for describingA + BC or AB ÷ C motion. They are probably most useful for describing"heavy + light- heavy" reactions, e.g. I + HI -~ IH + I, for which the anglebetween the reactant and product valleys of the potential energy surface(cf. Figure 1) is very small (~7° for IHI). Indeed, Schatz (19a,b) recently carried out very interesting calculations for systems of this typein full three-dimensional space (for total angular momentum J = 0), andthis has been very helpful in understanding the photodetachment experi-ments of Neumark et al (20a-d) on IHI 5- hv ~ 15- HI + e .

Most of the recent progress in reactive scattering, however, has beenbased on the formulation (21) in which the Jacobi coordinates for thevarious "arrangements" (i.e. A + BC, AB + C, AC + B) are all used simul-taneously. For the collinear case of Figure 1, for example, the expansionfor the wavefunction in this approach is

~tZ’,n, = ~ ~gan(ra)fan~’,n,(Ra) 5-E ~gCn(rc)fc"~71nl(Re)

R

where y = a(A + BC), b(B + AC), or c(C + AB) labels the arrangement the atoms, and {q~} and {q~} are the vibrational eigenstates of diatomsBC and AB, respectively. Note that there are only two independent co-ordinates (degrees of freedom) in Eq. 7 for the collinear case shown Figure 1; i.e. r, and R~ are functions of r~ and R~, or vice-versa (specifically,they are linear combinations of each other).

The philosophy of this approach is similar to that in quantum chemistryof using multicenter (LCAO = linear combination of atomic orbitals)expansions for molecular or’bitals. For a diatomic molecule, for example,the molecular orbital 2(r) for an electron is expanded in basis functionsutilizing the coordinates of the electron with respect to both nuclear centers,

~(r) Y’ . a~b~(ro) + Z b’qb/b(rb), i


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where ra are the coordinates of the electron with respect to nucleus a and rbthose with respect to nucleus b. (Note that there are only three independentcoordinates in Eq. 8; i.e. r~ is a function of rb, or vice-versa, specifically,ra ----- rb+Rb--R~ where Ra and Rb are the coordinates of the nuclei.) Pur-suing the analogy further, the hyperspherical coordinates discussed abovecorrespond to a sinyle center expansion of the molecular orbital. Singlecenter expansions have been used in quantum chemistry, and thoughmathematically simpler, they are much less efficient (i.e. require manymore terms) than multicenter (LCAO) expansions because one set coordinates does not provide the most natural description of the orbitalin all regions of space.

One may also think of this approach to reactive scattering as a general-ization of the standard description of electron scattering (22). Thus Figure 1, consider the case that atom B = H+ (a proton), and A and C areelectrons, i.e. the collinear version of electron-hydrogen atom scattering. Inthis case mB >> mA, me, and it is clear that the two translational coordinatescoincide with the two interparticle coordinates, i.e. Rc = ra and Ra = rc,and that the two sets of terms in Eq. 7 are related to one another simplyby exchange of the electrons. (By symmetry, the two sets of terms are thesame, with a + or combination corresponding to the singlet and tripletcase, respectively.) Thus even if one did not know that the (spatial) two-electron wavefunction should be symmetric or antisymmetric upon ex-change of the two electrons, the fact that the electrons can actually inter-change by virtue of the collision (a "chemical reaction") requires that thewavefunction include both sets of terms in Eq. 7. (The chemical reactionH + IH --, HI + H is a molecular case very close to the e-- H atom limit.)The general chemical reaction is more complicated than the electron scat-tering case because the finite mass of all atoms makes the relation betweenthe various sets of Jacobi coordinates more complicated than simplyexchanging them, but the basic idea is the same.

The expansion of the wavefunction in Eq. 7 is also essentially the samebasic idea as the "resonating group model" (RGM) used in nuclear physics(23). The different sets of Jacobi coordinates define different "groups"(or groupings) of atoms, and the fact that the wavefunction is a linearcombination of these different terms allows for "resonance" (i.e. coupling,interaction) between them if there are non-zero matrix elements of theHamiltonian ("resonance integrals") connecting them.

Equation 7 is thus a natural and efficient way to represent a reactivescattering wavefunction, but it introduces the complexity that the couplingbetween terms corresponding to different arrangements are nonlocal, ex-change type interactions. The coupled-channel equations Eq. 4a are thusgeneralized as follows (21):


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252 MILLER

(h2 d2

) ~ R0 =2#a R~ E~" fan~’¢lnl(Ra)"~En, Vn’n’(u)fan’~Tlnl(Ra)

~,~:a

The exchange interaction (the last term in the above equation), whichcouples states of different arrangements, is analogous to electron exchangeinteractions in quantum chemistry that arise from matrix elements in whichthe electron coordinates haw: been permuted (i.e. exchanged). The coupledintegro-differential equations for the translation functions, Eq. 9, are thusanalogous to the Hartree-Fock equations of electronic structure theory,and as such they cannot be solved by finite difference algorithms.

Indeed, it is the presence of the exchange interaction in this formulationof reactive scattering that until recently has stymied this approach. Wolken& Karplus (24) made some: early attempts using it, but these were notcompletely successful. It has ultimately become clear that the most satis-factory way of dealing with exchange is analogous to what quantumchemists do in the Hartree-Fock problem, namely to expand the unknownwavefunctions in a basis set and determine the expansion coefficients viaa variational principle.

Variational Principles

Unlike the standard Rayleigh-Ritz variational principle for eigenvaluecalculations, though, there are several different variational principles forscattering problems and they are not all equivalent (25a,b). I do notdescribe any of these in detail here, but simply note that in all cases thecentral aspect of the calculation is evaluation of a matrix expression of thefollowing form,

AT’M-1 "B, 10.

where A and B are vector,,; (or rectangular matrices) and M is a largematrix~ the practical aspects of the various variational methods can becharacterized by the nature of the matrix M.

The simplest of these is the Kohn variational principle (KVP) (26, 27),which is essentially the Rayleigh-Ritz principle modified to account forscattering boundary conditions; in this case the matrix is

(n)i,i, (z~IH-EIzr), 11.

where H is the total Hamiltonian of the system and E the total energy.Thus only matrix elements of the Hamiltonian are required, and M is given


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in terms of two energy-independent matrices (the Hamiltonian matrix andthe overlap matrix), the matrix elements of which can be computed onceand then used for calculations at various energies.

Until recently, however, the Kohn principle has not been generally usedbecause of anomalous (i.e. spurious, unphysical) singularities (28a,b) result when it is used with standing wave boundary conditions (to determinethe K-matrix). It was thus a significant step forward when it was recentlyrealized (29a,b) that application of the Kohn principle with incoming/outgoing wave boundary conditions (to determine the S-matrix) is free these "Kohn anomalies" so that it can be employed in a completelystraightforward manner. [That is, the results of the KVP are not invariantto how the boundary conditions are applied. Using the KVP with standingwave boundary conditions to compute (a variational approximationto) the K-matrix, and then using the formally exact relation S = (1 +iK)(1-iK)-1 to obtain the S-matrix, does not give the same S-matrix asusing the KVP with incoming/outgoing wave boundary conditions tocalculate (a variational approximation to) the S-matrix directly.] Thissomewhat esoteric subtlety of basic scattering theory has thus had veryimportant practical consequences, making reactive scattering calcu-lations much simpler than previously thought. McCurdy, Rescigno &Schneider (30) have also quickly realized the importance of this S-matrixversion (29a,b, 31a-d) of the Kohn variational principle and have madevery impressive advances in electron atom/molecule scattering using it.

Before the above discovery about the S-matrix Kohn approach wasmade, the Schwinger variational principle (27) had been employed (32a,b,33) to solve Eq. 9, and also has been used very widely for electron scatteringcalculations (34a,b). In this case the matrix M in Eq. 10 has the form

(M),,c = (zil A V-- VGo(E)A V[Zc) 12.

where Go(E) = (E- Ho)- isthescattering Green’s function for a reference(i.e. "zeroth order") Hamiltonian and A V is the part of the scatteringinteraction not included in the reference Hamiltonian. The matrix elementshere are much more difficult to compute (because of Go), and must recomputed anew at each different energy (because Go is energy-depen-dent), but it was thought that this was necessary in order to avoid the"anomalies" of the Kohn principle. Not only has this been seen not to betrue, but ironically the Schwinger principle itself has recently been dis-covered to have anomalous singularities (35, 36a-c)! Furthermore, these"’Schwinger anomalies" do not disappear when one changes the way theboundary conditions are applied, because the results of the Schwingervariational principle are invariant to this. (This is because the Schwinger


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254 MILLER

variational principle can be thought of as making an approximation to theinteraction potential and then solving this approximate problem exactly.)

Another variational approach has been used extensively by Truhlar,Kouri, and co-workers (33, 37a-j), namely the generalized Newton vari-ational principle applied to the amplitude density version of the coupled-channel equations, Eq. 9. The final "working formulae" of this approachcan actually be obtained from the Kohn variational principle by modifyingthe basis {Izi)} in Eq. 11 as follows

I~i) ~ G0(E)[~), 13a.

where Go(E) is again the Green’s function for a reference scatteringproblem. The matrix M occurring in the generic linear algebraic expressionEq. 10 thus has the form (since H-E = Ho-E+AV = --Go~+AV)

(M)i,c = (~il- Go(E) + Go(E)A VGo(E) 13b.

the evaluation of which has the same undesirable features as the Schwingermatrix elements above. Since: this generalized Newton approach is in factseen to be a special case of the Kohn variational principle, the K-matrixversion of it (i.e. with standing wave boundary conditions) could displayanomalous singularities, though Truhlar and Kouri have not noted anyof these in their applications. They have commented more recently (37j),though, that the S-matrix version (i.e. with incoming/outgoing waveboundary conditions) of their approach does seem better behaved thanthe earlier K-matrix version.

Though introducing a Green’s function into the basis set, Eq. 13a, leadsto more complicated and intrinsically energy-dependent matrix elements,Eq. 13b, it does have the positive feature that fewer basis functions maybe required (37c). That is, {Go(E)[~i)} is possibly a better basis set {[)~s)} because the action of G0(E) on the original basis may adapt it to the scattering problem at energy E.

Much of this advantage can be incorporated in the Kohn approachmore simply, however, through various basis set contraction schemes. Thisidea is also borrowed from quantum chemists: one begins with a large"primitive" basis ([~)), i = 1,..., N, and then forms a fewer number better basis functions I~) from them

N

I~7 = ~ Ix,)C,.~, 14.

j = 1, . . . , M < N. The Hamiltonian matrix elements are all evaluatedinitially in the primitive basis;, H~,c (a N x Nmatrix), and then transformedto the smaller contracted representation, /~.~. (an M x M matrix), via


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//-- CT" H’ C. The linear algebra calculations, Eq. 10, is then carried outin the smaller contracted space. Zhang & Miller (38) have recentlydescribed a very easy-to-use quasi-adiabatic contraction scheme, whichcontracts the basis of the internal (diatom’s rotational and vibrational)degrees of freedom for each localized (Gaussian) translational basis func-tion, and Manolopoulos et al (39) have devised another very useful strategyby diagonalizing a reference Hamiltonian (separately in each arrange-ment). Both of these approaches, as well as others, are certainly usefuldirections for making the calculations more efficient.

Finally, Wyatt & Manolopoulos (40a~i) have developed an efficientversion of the Kohn approach for the log derivative of the scatteringwavefunction. The particular way that they employ this makes it verysimilar to the R-matrix method (41), and they have carried out someexcellent calculations (see below) using it.

Summary and Proynosis

A very straightforward and general framework for carrying out chemicallyreactive scattering calculations has thus become clear, and the next fewyears should see a number of methodological developments that willfurther enhance the capabilities and usefulness of the Kohn variationalapproaches described above. Essentially all of the basis set contractionschemes and the pointwise representations that have been developed forvibrational eigenvalue calculations (42-44) in polyatomic molecules, forexample, are directly transferable to this way of doing reactive scattering,and utilizing them should be very fruitful. It may be useful to includesome explicitly energy-dependent basis functions, as in the Truhlar-Kourivariant, though how efficient this will be compared to various contractionschemes is not clear at present.

It is also not necessary in a Kohn-type calculation to adhere only to theJacobi coordinates to express the wavefunction, as in Eq. 7. Thus thegeneric form of the variational wavefunction in the S-matrix Kohnapproach is (21, 31a)

y,n i

where ~r,(Rr, rr) is an asymptotically incoming radial wave in channel vn,i.e.

lim ¢brn(Rr, rr) ~ e-~k~.~¢~(rr), 16.R~ ~

¯ ~. (the complex conjugate) is the corresponding asymptotically outgoingradial wave, and {~} is an L~ (square integrable) basis that spans the


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256 M~I~LEI~

interaction region. (The coel~icients {C~,,,.r~n,~ and {Ci,~,,n,} are the par-ameters that are determined variationally.) Though the "free" functionsqbrn need to be expressed in terms of the Jacobi coordinates of arrangement~, any convenient set of coordinates can be used to express the L2 basis{)~i}. In all applications to date these functions have also been chosen interms of the Jacobi coordinates of the various arrangements, i.e.

)~, = 4)~(r~,)ut( R~,), 17.

where {u~(Rr)} is a translational basis [typically of distributed Gaussians(45)] and i is thus the collective index = (V, n, t), but thisis not necessary.The {~} basis could be taken as a pointwise basis (31c,d), for example, it could be a basis in hyperspherical coordinates. This latter choice wouldobviously be good for "heavy+light-heavy" reactions. Thus once onehas decided to use a variational method to solve the coupled-channelequations, the S-matrix Kohn approach provides the framework for incor-porating a variety of seemingly different methodologies under the simplerubric of how to choose basis functions in the most useful fashion.

COMPARISON OF THEORETICAL CALCULATIONSWITH RECENT EXPERIMENTS

All rigorous quantum scattering methods use conservation of total angularmomentum to simplify the calculations (46). The S-matrix--the primaryresult of the calculation--is thus determined separately for each value ofJ, the total angular momenturn quantum number, and physically relevantcross sections are then assembled from the separate calculations. Theintegral cross section, for example, summed and averaged over therotational projection quantum, numbers of the diatomic molecules, is givenby,

The vibrational quantum number n of the previous section’s collinearexample has been expanded as n ~ vjl for the three-dimensional case,where (v,j) are the vibration-rotation quantum numbers of the diatom andl that for the orbital angular momentum of relative atom-diatom motion.For a given value of J, the sums over l and l’ in Eq. 18 are limited by theusual angular momentum triangle relations, i.e. l = IJ-Jl, - ¯ ¯, J+J andl’ = [J-J’l ..... J+f .

The matrices involved in calculating the S-matrix increase in sizeapproximately linearly with or, so the J = 0 (S-wave) calculation is the


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easiest. There have thus been quite a number of recent calculations ofreaction probabilities,

s (2j+ 1)-~P~’v’s’~r~j(E) =- El JS~.vv,t,,rv~.~(E)] 19.

for a variety of atom-diatom systems for the special case J = 0; in addi-tion to many for various isotopic forms of the H3 system (12c, 13, 37a,37g, 47-49), these include F+H2--,HF+H (50-52a,b), F+HD~HF+D,DF+H (53a), F+D2 ~ DF+D (53b), O+H~ ~ OH+H (37b),I+H2 ~ HI+H (37d), O+HC1 ~ OH+C1 (19b). These are not directlycomparable to experiment, however, except in the case of a Franck-Condonprocess, e.g. photodissociation,

ABC + hv ~ A + BC, 20a.

or photodetachment

ABC- + hv ~ A + BC + e-, 20b.

where J = 0 of the initial bound state of the triatomic molecule ABCmeans that (approximately) only J = 0 is needed for the scattering systemA+ BC in order to calculate the experimental observable (i.e. the bound-free Franck-Condon factor). Schatz (19a) has carried out such calculationsfor X + HX ~ XH + X (X = CI, I) systems, and they have been very helpfulin interpreting the photodetachment experiments by Neumark’s group(20a,b,c),

XHX- +hv ~ X+HX+e-. 21a.

Zhang & Miller (54) have similarly used their J = 0 F+H~ ~ HF+Hscattering calculation to obtain the photodetachment cross section for

H2F- +hv ~ H2+ F, HF+H+e-, 2lb.

for which Ncumark’s group has obtained preliminary results (D. M.Neumark, private communication).

Scattering cross sections, however, require the S-matrix for all values ofJ that contribute significantly to the partial wave sum, Eq. 18, and only forthe two reactions noted in the Introduction have such rigorous calculationsbeen recently carried out in the high energy region (> 1 eV translationalenergy) relevant to the recent experiments. Values of J up to Jm~x ~ 2~30 are required for convergence of the partial wave sum at these energies.The remainder of this section is devoted to comparing results of thesecalculations to various recent experiments.


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258 MILLER

H4. H2(para) ~ H2(ortho) 4- HTheoretical calculations long ago predicted the existence of scatteringresonances, i.e. short-lived collision complexes, in the H+H2 reaction,first in the collinear model (56) of the reaction and then later in the three-dimensional version (57) of the reaction for J - 0. Figure 2, for example,shows the reaction probability, Eq. 19, for J = 1 (which is essentially thesame as that for J = 0) from the ground state v = j = 0 to various finalvibrational states, summed over final rotational states. The width of theresonance feature at E -~ 1.0 eV is ,-~ 0.05 eV, corresponding to a Lifetimeof ,-~ 10-15 femtosec for the collision complex. (The resonances in the three-dimensional calculation are about twice as broad as the corresponding onesin the collinear calculation on the same potential energy surface.) Theprimary theoretical question more recently has been whether the sumover total J in Eq. 18 would average out this resonance structure (cf.inhomogenous broadening) in physically observable integral crosssections. The early work by Schatz & Kuppermann (57) suggested thatresonance structure would survive the partial wave sum, but more recentwork by Schatz (3a,b, 58) suggested that it might not.

The paper by Nieh & Valentini (59a,b) reporting observation of resonancestructure in state-to-state integral cross sections for the H+H2 reaction

p J= IV~O0

0.6

0.5

0.4

0.5

0.2

0.1

00.5

~ I ~ I [ I ’ I ’

V=O

0.7 0.9 I.I 1.3

E (eV)

.5

Figure 2 Reaction probability for 1:-I + H z(I~ = j = 0) --, H2(v’, all j’)+ H, for total angularmomentum J = 1, as a function of total energy E. The results for v’ = 1 and v’ = 2 havebeen multiplied by factors of 2 and 10, respectively, for convenience in displaying them.


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has thus received considerable attention. Figures 3 and 4 show their resultsfor the specific transitions

H+H2(v = 0,j = 0and2) ~ H2(v’ 1, f = 1, 3)+H; 22.

the reactant is a Boltzmann average over para states (52% j = 0 and 48%j = 2), while individual final (v’,f) states are detected. Since the v’ cross section peaks at the resonance energy and the v’= 0 one has aminimum (cf. Figure 2), the ratio of v’ = 1 to v" = 0 cross sections shownin Figure 4, each summed over oddf (ortho) values, enhances the resonancefeatures.

Also shown in Figure 3 are the results of Zhang & Miller’s (60) cal-culations obtained using the S-matrix Kohn method discussed in theprevious section. Though the absolute magnitude of the experimental andtheoretical cross sections are in good agreement--which is nontrivial,since the experiments are absolute cross sections--the calculations showno hint of the resonance structure that is present in the experimentalresults. That is, though resonance structure exists in the energy dependenceof the reaction probability for individual values of J (cf. Figure 2), thesum over J washes it out, as Schatz had anticipated (3a,b). (Note thatthese theoretical calculations are for initialj = 0, and not an average overj = 0 and 2. Calculations by Mladenovic et al (37e) of reaction probabilities

0.08

0.06

0 0.040

b 0.02-

).95 I .05 I.I 5 I. 25 1.35ToIol Energy (eV)

Figure 3 Integral cross section for the para ~ ortho reaction H+H2(v = 0,j)~ H2(v’ l,f) + H, as a function of energy. The solid and dashed lines are the theoretical resultsof Ref. (60) for j’ = 1 and 3, respectively; they both are for the rotational ground state(j = 0). The open and closed circles are the corresponding experimental results of Ref. (59a,b),for which the initial rotational state is a Boltzmann distribution of para states (52% j O, 48%j- 2).


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260 MILLER

0.3

0.2

>~o.,

01.05 1.15 I.;~5 1.35

ToIol Energy (eV)

Figure 4 Ratio of v’ = 1 to v’ = 0 experimental (59a,b) para-ortho cross section, eachsummed over final odd (ortho) rotational states; i.e.

/~ a(v" = l,j" .,,--- v = O,j = O, 2)/~-~, a(v’ = O,f ’-- v = O,j = O, 2).

J’odd t t J’~,~

for total J = 0, 1, and 2, however, suggest that this will have little effecton the cross sections; if anything, further averaging out any structure thatmight exist.)

Moreover, Manolopoulos & Wyatt (40b) shortly afterward reportedcalculations using their log derivative version of the Kohn method, andeven more recently Launay & LeDourneuf (17c) have obtained resultsusing a hyperspherical coordinate approach, both of which are essentiallyidentical to the theoretical results in Figure 3. There thus seems little doubtthat these arc the correct results for this [LSTH (61a,b)] potential energysurface.

Calculations have also been carried out for the integral cross sectionsby Manolopoulos & Wyatt (40d), and for a few values of total J Auerbach et al (62), using the DMBE (63) potential energy surface, very minor changes in the results are observed. Figure 5 shows the con-tribution to the integral cross section from total J = 10, for example, of(a) the energy dependence of the (0, 0) -~ (1,1) and (0, 0) transitions, and (b) the complete product state distribution (0, 0) (v’ ,f)at one value of energy E-- 11.2 eV. This is roughly the same degree ofdifference that persists in the converged integral and differential cross


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I -- 0MBE j’=3m- LSTH

8

"~ 0.007

0.80 0.92 1.04 1.16 1.28 1.40(a) E (eV)

0.148

~o 0.111--/11’~---- LSTPI _

8 0.074 --

0.037

0 ~ I I(01) (03) (05) (07) (09) (11) (13)

(b) (v’j ")

Figure 5 Comparison of reaction probabilities (with total J = 10) for H + H2(v = j = 0) H2(v’,j’)+H for LSTH (6la,b) and DMBE (63) potential energy surfaces: (a) as a of total energy for final states (v’,j’) = (1, 1) and (1, 3); (b) as a function of the final 0Y,f) for total energy E = 1.2 eV.

sections. Thus the theoretical results appear not to be extremely sensitiveto modest changes in the potential energy surface.

Thus unless there are some rather major deficiencies in the H3 poten-tial energy surface--which seems unlikely, though not impossible--the


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262 MILLER

unavoidable conclusion of l:he theoretical calculations is that the Nieh &Valentini experiments are actually not observing the integral cross sectionfor the processes in Eq. 22. What might be the resolution of this lack ofagreement? From their earlier calculations for total J--0, l, and 2,Mladenovic et al (37e) noted that if the partial wave sum for the crosssection, Eq. 18, were artificially truncated at Jmax, then l’or Jmax = 0, l, or2 the cross section ratio actually does show structure somewhat reminiscentof the experimental ratio in Figure 4. Figure 6 shows this ratio for valuesof Jmax ---- 0, l .... ,24, the final values giving the ratio of the fully convergedcross sections. Thus even though the cross section ratio has by no meansreached its converged value for Jmax as small as 2, this does suggest that ifthe experiments were somehow observing only products from small Jcollisions--and not the true integral cross section--perhaps this wouldexplain the resonance structure that was reported.

Zhang & Miller (60) thus :~uggested that there might be some unknownkinematic aspect of the experiments that would result in only back scatteredproducts being observed, which would have their dominant contributionfrom small J. To test this, Manolopoulos & Wyatt (40b) and Zhang Miller (64) independently calculated the energy-dependence of the differ-

0.4

~" 0.3

"~.. o.~

rr0.1 Jmax~24

0.9 1.0 1.1 1.2 1.3 1.4 1.5E (eV)

Fiyure 6 Theoretical results of Re[. (60) for the ratio in Figure 4 (with initialj = 0), wherethe upper limit in the sum over total J values in Eq. 18 is truncated at Jm,x- The uppermostcurve is the result for Jma~ -- 0, and those with Jm~ = l, 2 .... are the successively lowercurves. The lowermoa’t curve (Jm,~ = 24) is thus the physically relevant ratio of convergedcross sections.


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ential scattering cross section for back-scattering (0 = 180°) and also forother smaller angles. Figure 7 shows the results of these calculations (whichagain are in essentially complete agreement with each other) for the (0, 0) (1, 1) and (0, 0) -~ (1, 3) transitions. There is indeed more structure than in the integral cross section, but it is still considerably broader thanthe features reported by Nieh & Valentini. Thus this "explanation" mayor may not be relevant to unraveling the discrepancy.

Krause & Shapiro (65) have suggested that nonlinear aspects of theCARS detection scheme used by Nieh & Valentini may enhance the detect-ability of the collision complex and thus give increased signals at theresonance energies. Muga & Levine (66) have suggested a classical mech-anism whereby high impact parameter (and thus large J) collisions leadto complex formation. The region of Jvalues and coordinate space relevantfor this, however, are in principle contained in the basis sets of the quantumcalculations so that whatever effect this contributes should be containedin the quantum calculations.

The matter must thus be left unreconciled at present. Other experimentswould obviously be welcome, as would even more accurate calculationsof the potential energy surface. If the Nieh & Valentini experiments areindeed measuring something other than the integral cross section, and ifone can understand what this quantity is, then perhaps these experimentswill lead to a much more sensitive way to study scattering resonances, atrue "spectroscopy of the transition state." When better potential surfacesbecome available, it is clear that there are now several groups that canreadily carry out the scattering calculations. They are actually relativelyeasy at this point: with the latest basis set contraction scheme of Zhang &Miller (38), for example (which is probably still far from optimum), S-matrix Kohn calculations require no more than 2-3 minutes (on CRAY) per energy to obtain the fully converged integral and differentialcross sections (including all necessary values of J).

Note added in proof Dramatic new experimental results for this reaction(Eq. 22) were reported at the recent Boston meeting of the AmericanChemical Society (April 22-27, 1990) by the Stanford group (D. A. Kliner, D. Adelman, and R. N. Zare). The experiment is essentially thesame as that of Nieh & Valentini (59a,b) except that the product moleculeHz(v’,f) is detected by a multiphoton ionization scheme rather than CARS. Relative integral cross sections for the two final states in Eq. 22were reported for the range of total energy E = 1.15-1.27 eV, whichincludes the important region in which Nieh & Valentini see resonance-like structure (cf. Figure 3). Zare et al see no such structure, and theirresults, when normalized to the theoretical curves in Figure 3, are in


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264 ~R

(a)

00.9

I I I

~ (deg)

1,0 1.1 1.2E (eV)

1180

(b)

00.9

I I I I

e (deg) :

///\\

1.0 1.1 1.2 1.3E (eV)

Figure 7 Energy dependence of the differential cross section a~7.oo(0) for H+H2(v =j = O) ~ H2(tY,j’)+ H at various fixed center-of-mass scattering angles. Final state (v’,./’) (1, I) in (a) and (l, 3) in


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excellent agreement with them. The energy dependence of the ratio of thesetwo cross sections, a quantity independent of any normalization, is inquantitative agreement with the ratio of the theoretical cross sections overthis energy range.

These new experimental results thus appear to resolve the situationregarding this reaction. A remaining question is whether the CARS detec-tion method of Nieh & Valentini is measuring something interesting aboutthe reaction dynamics or is an uninteresting artifact.

D+ H2 --~ HD+ H

Many aspects of this reaction are obviously very similar to H+H2, butthere have been a greater number and variety of experiments regarding itbecause the isotopic difference makes for simpler detection. Zhang &Miller (67) have carried out a comprehensive set of calculations for thereaction from ground state reactants

D+H2(v =j = 0) ~ HD(v’,j’) 23.

over a wide range of total energy (E = 0.9-1.4 eV), and have reportedboth differential and integral cross sections, so that many experimental/theoretical comparisons are possible.

As this review was being completed, two preprints were received fromZhao, Truhlar, Schwenke & Kouri (68, 69) reporting calculations that theyhave carried out for the D + H2 reaction using the DMBE potential energysurface (whereas Zhang & Miller used the LSTH potential). Comparisonswith this work have thus been added in appropriate places in the discussionbelow. In general, these calculations (68) for Reaction 23 show relativelyminor differences from those of Zhang & Miller (67), essentially the samedegree of differences as those seen for the H+H2 calculations (40d, 62)with the LSTH and DMBE potentials as discussed above. Zhao et al (68),however, have carried out the calculations also for rotationally excitedreactants, j = 1. This is very useful because it allows one to see to whatdegree the earlier comparisons made by Zhang & Miller are affected bytheir restriction to the rotational ground state, j = 0.

ENERGY-DEPENDENCE OF INTEGRAL CROSS SECTIONS Figure 8 first showsthe energy dependence of the reaction probability for J = 0, and one againsees resonance features, though not as pronounced as for H+H2 (cf.Figure 2). After summing over all values of total J to obtain the integralcross section shown in Figure 9, however, one sees that the resonancestructures at E ~ 0.95 eV survive only as weak shoulders in the crosssections.

Also shown in Figure 9 are experimental values reported by Phillips,


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266 MILLER

0.6

0.5

0.4

0.3

0.2

0.1

I ’ I ’ I ’ I ’ I

0.6 0.8 I .0 I .2 I .4

E (v}Figure 8 Reaction probability for D+H2(v =j = 0) ~ HD(v’, allj’)+H, for total J as a function of total energy.

7l(~ I I ’ I = I ’ I ’ /6 .~

1

2100

0

1.0

0.4 0.6 0.8 i.o 1.2 i .4E (eV)

~#,r~ ~ Im¢~r~l cross section ~or D+H~(~ =j = 0) ~ HD(~’, ~llj’)+H, as a Junction total Cn~@y. The points connected by the so/M c~r~ arC the t~¢orCtical results o~ RcC (67)(usin@ th~ ESTH potential surF~cc), and th¢ solM(6~) FoF i~itial # = 0 a~d I. ~sp~cIiv~ly (usin@ t~¢ D~BE potential). (The two th~or~tic~Iresults arc ~ss~tially indi~tin~,i~h~l~ For ~’ = I.) Tb~~xpcrim~t~I results o[ R~[. (?0).

Levene & Valentini (70) for these cross sections. The energy dependencefor the v" = 0 product is in good agreement, but the absolute values differby about a factor of 2. For the v’ = 1 product, the absolute values agreequite well, but not the energy dependence. In particular, Phillips et al


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interpret the decrease in the cross section with energy for v’ = 1 as evidenceof resonance structure in the cross section, whereas the theoretical resultsindicate that this should not be the case.

The results obtained by Zhao et al (68) using the DMBE potentialsurface are also shown in Figure 9. As with earlier LSTH-DMBE com-parisons (40d, 62) for H + H2, the DMBE cross sections are slightly larger(presumably because its barrier is slightly lower). It is interesting that theserotationally summed integral cross sections are relatively insensitive to theinitial rotational state (at least forj = 0 or 1).

ROTATIONAL STATE DISTRIBUTION There have been several determinationsof the integral cross section to specific final rotational states, ~rvv,~j(E).These experiments do not yield absolute cross sections and they have beencarried out at only a few energies, but they probe more detailed aspects ofthe reaction than the rotationally summed cross sections discussed above.They are presumably sensitive to the angular dependence of the potentialenergy surface.

Figure 10 shows the theoretical results (67) (solid points connected curves) at total energies 0.93 eV (Figure 10a) and 1.086 eV (Figure for ground state reactants (v =j = 0). The open points are the experi-mental results of Phillips et al (70), and one sees that the agreement perhaps reasonable but certainly not excellent. One point to keep in mindis that the experiments involve a Boltzmann average over the initialrotational states (14% j = 0, 67% j = 1, 11% j = 2, 8% .j = 3) whereasthe calculations are for the ground state (100% j = 0). This may partiallyexplain the differences in Figure 10.

The dashed lines in Figure 10 show the theoretical results of Zhao et al(68) using the DMBE potential surface; these values involve an appro-priately weighted average over initial rotational states j = 0 and 1. (Thetwo sets of theoretical results are absolute values with respect to oneanother.) The modest differences in the two theoretical results are com-parable to those seen in the earlier LSTH-DMBE comparisons for H + H2,and are much less than their difference with the experimental values. Thefact that the theoretical integral cross sections are not sensitive to the initialj state--at least for j = 0 and l--suggests that the Boltzmann averageover initial j state may not be the source of the disagreement of theexperimental and theoretical results in Figure 10.

More recently, Kliner, Rinnen & Zare (71) have determined therotational state distribution for vibrationally excited product, HD(v’ = 1),at a total energy for ~ 1.32 eV, and these results are shown in Figure 1 la.Also shown here are the results of Zhang & Miller’s (67) calculations andthose of Zhao et al (68). The two quantum calculations are in quite good


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268 ~ER

(a)

(b)

12.0

lO.O

8.0

6.0

4.0

2.0

o6

12.0

’- 10.0

~ ~.o._~

¯ 6.0

£ 4.0

2 4 6 8 10j,

o v’=l \

2 4 6 8 10 12j,

Figure 10 (a) Integral cross sections (arbitrary units) to specific final rotational states D + H 2(v = j -- 0) --, I tD(t/,f)+ H at total energy E = 0.93 eV. soli d curves connectingthe solid points are the theoretical results of Ref. (67), the dashed line the theoretical resultof Ref. (68) (using the DMBE potential surface), and openpoints the experimental resultsof Ref. (70). (The two theoretical results are indistinguishable lbr v’ = I); (b) E = 1.086 eV.

agreement--the differences are thc same order of magnitude the LSTH-DMBE comparisons (40d, 62) for H ÷ H2--and also in excellent agreementwith the experimental results. Here, too, the reactant rotational states inthe experiment are a Boltzmann distribution of initial j states, but thisseems not to matter. Zhao et al included an average overj = 0 and 1, butthis has little effect on the rotational state distribution.


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(a)

(b)

0.3

0.21

0.1

v=j=l I T

0 2 4 6 8 10 12 14j,

Figure 11 (a) Same as Figure 10, except for v’ = 1 and total energy E = 1.32 eV. The solidcurve connects the experimental values of Ref. (71), the dotted curve the theoretical valuesof Ref. (67) (using the LSTH potential), and dash-dot curve the theoretical values of Ref.(68) (using the DMBE potential). (b) Same as (a) except for initial state v =.j = 1 and energy E = 1.92 eV. The solid curve connects the theoretical values of Ref. (69), and the opensquares (with error bars) are the experimental values of Ref. (72).


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270 MILLER

Finally, Kliner & Zare (72) have very recently reported (relative) integralcross sections to specific finalf states from vibrationally (and rotationally)excited reactants,

D+H2(v =j = 1) -+ HD(v’ = 1,j’)+H. 24.

Figure 1 lb shows these results, compared to calculations by Blais et al(69) using the DMBE potential surface. The agreement is reasonable,though certainly not as quantitative as for the vibrational ground state ofreactants, Figure 1 la.

DIFFERENTIAL CROSS SECTIONS AT 0 = 180° Buntin, Giese & Gentry (73a) havereported relative cross sections to specific final reactant states (v’,j’) of HDat "scattering angles near 180°" for total energy E -~ 1.22 eV (with 20%resolution). For the final vibrational state distribution (summed over finalrotational states) there is essentially complete agreement between theirresults (84% v’ = 0 and 16°,/o v’ = 1) and the theoretical calculations Zhang & Miller (67).

The cross section at 0 = 180° to specific final rotational states is shownin Figure 12. First comparing the two theoretical results (67, 68), one seesthat averaging over initial j := 0 and j = 1 values (dotted curve) has moreof an effect here than for the integral cross sections discussed above; i.e.the solid (j = 0 only) and dotted (average overj = 0 and 1) curves showmore differences, at least for v’ = 0. In particular, for the state-selected

0.15 , ,

~ 0.05

"o 2 4 6 8 10 12j,

Figure 12 Differential cross section at 0 = 180° to specific final rotational vibrational statesfor D + H z(v = 0,j) ~ HD0~’,j’) + H. Solid lines a re the theoretical results of Ref. (67) (j = LSTH potential surface), the dotted line those of Ref. (68) (averaged over./" = 0 and 1, DMBEpotential surface), and the dashed lines the experimental results of Ref. (73b) (Boltzmannaverage over j). The two theoretical results are essentially indistinguishable for v’ = 1.


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QUANTUM REACTIVE SCATTERING " 271

case (solid curve, j = 0) there is an interference-looking structure in thefdistribution that is partially quenched by averaging overj = 0 and 1 (dottedcurve); this is a typical occurrence that is well known from semiclassicalscattering theory (74). Other than this, the two theoretical results are quitesimilar, and for v’ = 1 they are hardly distinguishable on the scale of thefigure. The experimental results of Buntin et al (dashed curve), whichwere obtained from their time-of-flight measurements by a deconvolutionprocedure, show a somewhat flatter and broader j" distribution for bothv’ = 0 and 1.

Since the average over j = 0 and 1 has a significant effect on smearingout thej’ distribution at 0 = 180°, it is possible that a complete Boltzmannaverage overj (14%j = 0, 67%j = 1, ll%j = 2, 8%j= 3) might givebetter agreement with these experiments. Averaging over the experimentalangular (~ 15°) and energy (~ 20%) resolution would also tend to quenchthe interference structure and broaden the f distributions. Conversely, ifthe experiments can be carried out with a high degree of angular andenergy resolution and with mostly j = 0, there is the chance of seeingsignificant new structure in thef distribution at specific scattering angles.

Note added inproof Buntin et al (73b) have recently carried out a detailedanalysis of their experiment results using the theoretical differential crosssections calculated at various energies by Zhang & Miller (67) and Zhaoet al (68). Similar to Continetti et al (75a,b) (see below), they average theoretical cross sections via a sophisticated Monte Carlo procedure overall experimental parameters in order to simulate the actual time-of-flightspectra that they measure.

The authors find negligible difference in their calculations between thetwo sets of theoretical cross sections for initial rotational state j -- 0, andusing the j = 1 results of Zhao et al (68) to average over initial j = 0 and1 had only a small effect on the simulated TOF spectra. The calculationsshow good agreement with the experimental results for the vibrationalstate distribution (v’ = 0 and 1) of back-scattered (0 ~ °) products, buta rather large discrepancy between the relative intensity of back-scatteredproducts and forward-scattered (0 ~ 70°) products (which were notresolved into v’= 0 and 1). The calculations also show less rotationalexcitation for the back-scattered products than seen experimentally (as shift and broadening of the TOF peaks).

Since the independent scattering calculations of two different groupsagree well with each other, Buntin et al conclude that these discrepanciesare due to shortcomings of the potential energy surface.

COMPLETE ANGULAR DISTRIBUTIONS Continetti, Balko & Lee (75a) havevery recently completed an extraordinary series of crossed molecular beam


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272 MILLER

experiments on the D ÷ H2 reaction that provide essentially the completeangular distribution, with time-of-flight analysis of the reaction products.The reactant molecule H2(v,j) is in the ground vibrational state v = 0 andmostly the ground rotational state (70-80% j = 0, the remainder j = 2).The experiments were carried out at two values of total energy E = 0.78and 1.25 eV (with FWHM resolution of 0.08 eV).

Though these experiments do not explicitly resolve individual (v’,f)states of the product HD, the shape of the time-of-flight distribution hasthis information contained in it. Figure 13 shows such experimental resultsat one laboratory scattering angle. Individual final vibrational states v’ = 0and 1 are clearly resolved, and the shift and shape of these peaks is relatedto the j’ distribution that accompanies each value of v’. The solid curve inFigure 13 is the predicted time-of-flight distribution obtained with Zhang& Miller’s (67) state-to-state differential cross sections o-vy~vj(0), averagedvia a sophisticated Monte Carlo procedure (75b) over all experimentalconditions (angular resolution, time-of-flight resolution, energy resolution,etc.). The agreement is excellent in many regards, but there is a noticeabledifference in the depth of the local minimum of the distribution betweenthe very large (unresolved forward scattered) peak and the v’ = 1 peak.

If one integrates over the time-of-flight distribution at each scatteringangle, then the total (non-state-resolved) angular distribution of reactionproducts is obtained. Figure 14 shows this LAB angular distribution atthe two energies, compared to that obtained by averaging the theoreticalresults appropriately. At this level of comparison one sees almost completeagreement between theory and experiment.

In their deconvolution wocedure, Continetti et al (75a) start with thetheoretical cross sections and then alter them as needed in order to fit the

27.5°

50 1 O0 150 200 250Flight Time,

Fi#ure 13 Time-of-flight spcctrura for D+H2(v =j = 0)-~ HD(v’,j’)+H at LAB angle27.5°, from Ref. (75a,b), for total energy E = 1.25 eV.


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1.5

1.0N(O)

0.5

1.5

1£N(o)

00

1 o 20 30 40 50Laboratory Angle (deg)

10 20 30 40 50Laboratory Angle (deg)

Figure 14 Total angular distribution for D+H2(v =j = 0) -~ HD(v’,j’)+H, summed all final (v’,f) states. Points with error bars are the experimental results of Ref. (75a,b), andthe solid curve the result obtained by suitably averaging the theoretical cross sections of Ref,(67). Total energy E = 1.25 eV (a) and 0.Tg eV

time-of-flight data at all laboratory scattering angles as best as possible.In so doing they obtain state-to-state (center-of-mass) differential crosssections that can be compared directly with the theoretical ones to seemore precisely where differences exist. Figure l 5 shows this comparisonfor a number of final (v’,f) states, and it is seen that the major discrepanciesoccur at large scattering angles for high values off. Though it is hard toassign error bars to these deconvoluted experimental cross sections--orto know how unique the deconvolution process is--these differences arecertainly real and are necessary to fit the time-of-flight data at all scatteringangles.

Integration of these deconvoluted cross sections over scattering angleproduces integral cross sections to specific final (v’,j’) states, so it is alsopossible to compare theory and experiment at this level. Figure 16 showsthis comparison for the two experimental energies, and the agreement is


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274 MILLER

~ ~ ,d *d o

II ~

÷.2

I:=

,2.


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0.8

0.6

0.4

0.2

(a)

QUANTUM REACTIVE SCATTERING

--

0 2 4 6 8 10 12

0 2 4 6 8 10j,

275

(b)

~ \\\v’ = 0

I l I i I i I I I i0 2 4 6 8

j,

Figure 16 Integral cross section to specific final rotational states for D+Hz(v =j = 0) HD(v’,f)+H. The solid curves are the theoretical results of Ref. (67), and the dashed curvethe experimental results obtained in Ref. (75a,b). (a) Total energy = 1.25 eV, and upperand lower figures are for v’ = 0 and 1, respectively. (b) Total energy = 0.78 eV, v’ =

seen to be rcasonably good, particularly at the higher energy (Figure 16a).The primary differences are that the experimental distribution off showssomewhat more rotational excitation than the theoretical result.

The experimental-theoretical comparisons for these experiments are


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276 MILLER

thus reasonably good for the total angular distribution (i.e. summed overall product v’,j’ states) anti for the integral cross section (i.e. integratedover scattering angle) to specific final (d,j’) states. At the most detailedlevel, though, namely the angular distribution to specific final states (essen-tially a doubly differential cross section), there are nonnegligible differ-ences. This is a good illustration of why such detailed experimentalmeasurements are necessary to provide a rigorous test of theoreticalmodels.

What might be the source of these remaining discrepancies? For oncein this field it is probably saf~ to rule out any shortcomings in the scatteringcalculations: Zhao et al’s (68) calculations confirm those ofZhang & Miller(67) in all aspects, essentially all minor differences being attributable the slightly different potential energy surfaces that were used. Zhao et al’scalculations do show that the initial rotational statej can have noticeableeffects on the state-specific differential cross section, but these experimentsare thought to have the reactants mostly (70-80%) in the ground rotationalstate. If there were more rotationally excited reactants in these experimentsthan is suspected, then averaging over initialj might bring the theoreticalresults into closer agreement with experiment. There is also the ever-present question of the potential energy surface. The LSTH-DMBE com-parisons show that modest changes in the surface do not drastically affectthe results, but one must remember that these potentials (61a,b, 63) aresimply different fits to the same ab initio calculations. If more accuratequantum chemistry calculations were to give significantly different results,this could certainly change matters. In light of the theoretical-experimentaldiscrepancies seen here, the most critical regions of the surface to exploremore accurately are bent geometries high up (> 1 eV) in the interactionregion.

H+D2 ~ HD+D

It should be noted here that there have also been a number of recentexperiments (76a,b, 77a--e) that report integral cross sections to specificfinal quantum states for this isotopic version of the H3 system. So far,though, rigorous quantum scattering calculations have not been carriedout to obtain these cross sections. This will surely happen before long,however, and will provide even more possibilities for experimental-theo-retical comparisons.

FINAL REMARKS

What can one conclude from this recent flurry of experimental and theor-etical work on these most fundamental reactions? First, with regard to


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theoretical methodology for carrying out reactive scattering calculations,there is good news. A general, straightforward approach is evolving thatreduces reactive scattering to a very standard quantum mechanical cal-culation, i.e. choosing basis functions, computing matrix elements of theHamiltonian, and performing a linear algebra calculation. The overallframework is now reasonably clear, and one can see many avenues forfurther enhancing the capabilities of the general approach. One should seean increasing number of rigorous quantum scattering calculations forsimple systems in the next few years. There are, of course, still limitations:as the energy increases, the mass of the atoms increases, or as one increasesthe number of atoms (i.e. the number of degrees of freedom) in the dynami-cal system, the number of basis functions needed to represent the wavefunc-tion--and thus the size of the Hamiltonian matrix--will increase. Appli-cations to more complex systems will thus be a challenging calculation,but such is the nature of a rigorous quantum mechanical description.

With regard to the theoretical calculations for the H/D+H2 reactions,matters are in very good shape. Zhang & Miller’s calculations (60, 64,67) of the state-specific integral and differential cross sections have beenconfirmed by several groups (17c, 40b, 68) in essentially all aspects. Thisis true for a given potential energy surface, though, so it is now of interestto have an even more accurate ab initio determination of it in order tozero in on remaining theoretical-experimental discrepancies.

The most recent experiments discussed above fall into two categories:spectroscopic-based methods that determine integral cross sections tospecific final quantum states, and crossed molecular beam methods thatdetermine the angular distribution (differential cross section) and thoughquite a bit of information is obtained about the product state distribution(via time-of-flight analysis), specific final quantum states are not detectedexplicitly. One would like, of course, to couple spectroscopic detection ofproduct states with a crossed molecular beam experiment in order tomeasure the state-specific differential cross sections directly, but this capa-bility is not yet available.

From a global perspective, the agreement of the theoretical calculationswith these recent experiments is quite good; one quickly accepts this,however, and then proceeds to focus on the discrepancies. These areperhaps most dramatic for the H + H 2 reaction because the differences arequalitative: the theoretical calculations show no resonance-like structure,whereas the experiments do. Because the differences are qualitative, itseems unlikely that reasonable changes in the potential energy surface willalter the situation, though it is of course not impossible. Independentconfirmation of the experimental results is thus very desirable. (See thenote added in proof on page 271.)


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278 M~,LER

For the D+H2 reaction the situation is also not completely clear. Theintegral cross sections to specific finalj’ states, for example, agree extremelywell with the experiments of Kliner et al (71), but not so well with theexperiments of Phillips et al (70). The experiments of Continetti et al (75a)surely provide the most detailed test of the theory. Though the totaldifferential cross section (without product state resolution) and the integralcross section to specific final states are in reasonable agreement withtheory, the more detailed state-specific differential cross sections inferredfrom these experiments do show non-negligible differences with the theo-retical calculations. Continetti et al suggest that the potential surface needsto be modified so that the bending potential in the interaction region issofter. It will be very interesting to see whether indeed this will providethe resolution of these remaining differences between theory and experi-ment for this reaction.

ACKNOWLEDGMENTS

I would like to thank Dr. R. E. Continetti for many discussions about hiscrossed molecular beam experiments with Balko and Lee and also, ofcourse, Dr. J. Z. H. Zhang for discussions about the reactive scatteringcalculations. I also appreciate the pre-prints sent by Profs. G. C. Schatz,R. E. Wyatt, W. R. Gentry, R. N. Zare, and D. G. Truhlar. Researchsupport by the Director, Office of Energy Research, Office of Basic EnergySciences, Chemical Science,,; Division of the US Department of Energyunder Contract No. DE-AC03-76SF00098 and by the National ScienceFoundation, Grant No. CHE84-16345, is also gratefully acknowledged.

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