Semiclassical IVR treatment of reactive collisionsY. Elran and K. G. Kay Citation: The Journal of Chemical Physics 116, 10577 (2002); doi: 10.1063/1.1479137 View online: http://dx.doi.org/10.1063/1.1479137 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/116/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Chaotic dynamics in multidimensional transition states J. Chem. Phys. 137, 214310 (2012); 10.1063/1.4769197 Extraction of state-to-state reactive scattering attributes from wave packet in reactant Jacobi coordinates J. Chem. Phys. 132, 084112 (2010); 10.1063/1.3328109 A simple model for the treatment of imaginary frequencies in chemical reaction rates and molecular liquids J. Chem. Phys. 131, 074113 (2009); 10.1063/1.3202438 Strong geometric-phase effects in the hydrogen-exchange reaction at high collision energies J. Chem. Phys. 128, 124322 (2008); 10.1063/1.2897920 Theories of reactive scattering J. Chem. Phys. 125, 132301 (2006); 10.1063/1.2213961

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Semiclassical IVR treatment of reactive collisionsY. Elran and K. G. Kaya)

Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel

~Received 1 February 2002; accepted 27 March 2002!

We generalize a recently-developed semiclassical uniform initial value representation~IVR!treatment of theS-matrix @Y. Elran and K. G. Kay, J. Chem. Phys.114, 4362 ~2001!# to chaoticnonreactive and reactive collinear scattering. The present modifications allow one to determine thephase of the complex IVR integrand in a unique and practical manner even when the integrand isdiscontinuous or rapidly varying. The method is applied to the collinear H1H2 exchange reactionon the Porter–Karplus surface. A strategy is introduced for adapting the integration over the chaoticchattering zones to the fractal nature of the integrand. The results indicate that the technique iscapable of good accuracy while requiring a relatively small number of trajectories perenergy. © 2002 American Institute of Physics.@DOI: 10.1063/1.1479137#

I. INTRODUCTION

Semiclassical treatments of molecular dynamics aim todescribe quantum effects accurately while maintaining theadvantage of computational simplicity, characteristic ofpurely classical methods. Of the many semiclassical ap-proaches that have been developed over the years, the popu-lar ~IVR! techniques seem especially promising.1–36 In viewof the central importance of molecular collisions to chemicaldynamics, many attempts have been made to apply thesemethods22–36 as well as other semiclassicaltechniques29,35,37–47to determine theS-matrix for molecularscattering.

The first IVR formulation of theS-matrix was developedby Miller and Marcus29–34 ~MM ! in the early 1970s. Theirexpression was a nonuniform semiclassical approximationthat was susceptible to large errors. More crucially, the ex-pression was not entirely well-defined, possessing ambigu-ities in the definition of the phase of a complex integrand.This problem, among others, surfaced clearly in attempts totreat reactive scattering, and effectively prevented applica-tions of the method to general processes of this kind.35 Fi-nally, numerical difficulties hindered applications of thistreatment even to nonreactive scattering at high-energies.36,43

To overcome these problems, we recently developed anew IVR expression for theS-matrix for the case of generalcollinear collisions.28 In contrast to the MM formula, ourexpression is a uniform semiclassical approximation and iswell-defined. In a test calculation, the new treatment wasfound to be accurate and far more numerically efficient thanrecent IVR methods of a different nature, developed fromsemiclassical expressions for the propagator.22–27 This im-proved efficiency can be traced to the differences in the di-mensionality of the integrations in the two kinds of IVRexpressions. In contrast to our expression, which involves asingle integral over classical trajectories per fragment degreeof freedom, the propagator-based expressions use two inte-grals. This greatly increases the number of classical trajecto-

ries needed to achieve convergence of the calculations.The calculations in our previous paper28 treated the rela-

tively simple case of nonreactive collinear scattering in theSecrest–Johnson48 system. Our purpose in the present workis to apply this technique to the more demanding test case ofcollinear reactive scattering of H1H2 on the well-knownPorter–Karplus49 ~PK! surface. This system exhibits scatter-ing phenomena that do not occur for the previous system,including the existence of both reactive and nonreactive col-lisions, scattering resonances, and reactive tunneling. It is ofinterest to examine whether our method can treat these fea-tures. In addition, the semiclassical treatment of the presentsystem must contend with a number of new technical com-plications. In particular, it is well-known that classical trajec-tories of reactive systems generally undergo chaotic‘‘chattering’’ 42,46,47,50–56as small changes in initial condi-tions abruptly vary the reactive/nonreactive nature of the tra-jectories. The presence of long-lived chaotic trajectoriescomplicates semiclassical treatments in general and dramati-cally slows the convergence of all forms of IVRcalculations.6,13,16,17,19For the particular case of ourS-matrixexpression, chaos causes the IVR integrand to become dis-continuous at an infinite number of points,50,51,53 leading toobvious numerical difficulties. Such problems must be ad-dressed carefully, even when chaos affects only a small pro-portion of the phase space, since the chaotic nature of thetrajectories is often intimately involved in the mechanism forthe formation of scattering resonances.41,53However, beyondthese numerical issues, the discontinuous nature of the IVRintegrand has more fundamental consequences. When suchbehavior occurs, our simplest expression for theS-matrix nolonger contains enough information to determine the phaseof the complex IVR integrand uniquely. To treat reactive ornonreactive chaotic scattering it is, thus, necessary to over-come this problem. This requires a reformulation of our IVRS-matrix expression.

The rest of this paper is organized as follows: In Sec. IIwe describe the modifications required to treat reactive andnonreactive chaotic scattering and derive our new IVR ex-pression for theS-matrix. In Sec. III we apply this formula toa!Electronic mail: kay@mail.biu.ac.il

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 24 22 JUNE 2002

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treat the H1H2 reaction and discuss the computational re-sults. Finally, in Sec. IV, we present a summary and conclud-ing remarks.

II. IVR S-MATRIX EXPRESSION FOR CHAOTIC ANDREACTIVE COLLISIONS

We restrict our discussion here to the case of the collin-ear reactive collision of an atom with a diatomic moleculemodeled as an oscillator~not necessarily harmonic!. Gener-alization to more general collinear collisions is quite straight-forward by analogy to the treatment of Ref. 28.

Our IVR expression for theS-matrix element, describinga transition from initial vibrational staten1 to final staten2 attotal energyE, was given for this case as28

Sn2n15

i

2p E0

2p

Ca~ u !eiDa( u)/\du. ~1!

In the expression above, the integration is performed over aninitial angle-like variableu, defined by

u[u2mvR1

P1, ~2!

whereu is the angle variable for the vibrational motion of thediatomic att50. t is a time variable describing the progressof the collision, with values ranging from 0 at the ‘‘initial’’time ~which is taken to be asymptotically early so that thefragments are still infinitely far apart and do not interact! to` ~corresponding to times after the collision is completedand the fragments again do not interact!. R1 is the initialdistance between the atom and the center-of-mass of the di-atomic fragment att50 andP1 is the conjugate initial in-terfragment momentum.v is the energy-dependent fre-quency of the noninteracting oscillator andm is the reducedmass for the atom relative to the diatom.

To define the remaining quantities in Eq.~1! we mustrecall additional details describing the collision process. Ast→`, the interfragment distanceRut , evolving from R1 ,can be described as the sum of a time-independent valueRu

and a free particle distance,

Rut5Ru1Put

m, ~3!

wherePu is the time-independent value of the interfragmentmomentumPut at asymptotically late times. Similarly, ast→`, the angle coordinatequt that evolves at timet fromthe initial valueu, has the asymptotic form

qut5qu1vt, ~4!

where qu is independent oft, while the oscillator actionvariable Jut , that evolves at timet from the initial valueJ15(n111/2)\, approaches thet-independent final valueJu . Last, by analogy to Eq.~2!, we define the final angle-likevariableq u at these asymptotically late times as

q u[qu2mvRu

Pu. ~5!

With these definitions, the pre-exponential factorCa( u)in Eq. ~1! is given by

Ca~ u !5S ]qu

]u1

ia

2Ju

]Ju

]uD 1/2

, ~6!

wherea is generally a complex function ofu that may bechosen arbitrarily, except that its real part must be finite andpositive. If a is chosen to be 0, Eq.~1! becomes the originalMM-IVR expression for theS-matrix.29–34 At the other ex-treme, if a is allowed to approach infinity, Eq.~1! tends tothe MM primitive semiclassical expression~PSC! for theS-matrix.29,37–39Finally, we define

Da~u !52ia

2 S J2 lnJu

J21J22JuD2q u~J22Ju !

1F~Ju ,J1!, ~7!

whereJ25(n211/2)\ is the action variable associated withthe final quantum numbern2 . F(Ju ,J1) is the classical gen-erating function for the canonical transformation from(J1 ,u,E,t50) to (Ju ,q u ,E,t→`) that appears in the clas-sicalS-matrix theory of Miller and Marcus.31,37,38,40It can beexpressed as

F~Ju ,J1![ limt→`

FF2~r u0 ,J1!2F2~r ut ,Jut!

1P1R12PutRut1E0

tS put8

]r ut8]t8

1Put8

]Rut8]t8 Ddt8G , ~8!

wherer ut and put , respectively, denote the diatomic vibra-tional coordinate and momentum at timet evolving from theinitial valuesr u0 andpu0 , and

F2~r ,J!5E0

r

p~r 8,J!dr8 ~9!

is theF2-type generating function for the canonical transfor-mation between the Cartesian variables (r ,p) and the angle-action variables (q,J) of the noninteracting diatomic frag-ment.

As with any IVR expression, care has to be taken toensure that the phase of the integrand of Eq.~1! is deter-mined correctly. In particular, the branch of the square rootappearing in Eq.~6! for the pre-exponential factor must bechosen appropriately. As long as the variablesJu andq u arecontinuous functions ofu ~as was the case in our treatment28

of the Secrest–Johnson system!, the phase of the prefactorcan be determined by requiring that it be a continuous func-tion of u. However, in reactive systems, these variables arediscontinuous at values ofu that lie at the boundaries, sepa-rating zones of reactive and nonreactive trajectories.35,42,46,52

More generally, even in nonreactive systems, these variablesare discontinuous at an infinite number ofu when the dy-namics is chaotic.47,50,51,53–55Clearly, in such cases, thephase of the prefactor cannot be determined simply by con-

10578 J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Y. Elran and K. G. Kay

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tinuity with respect tou. This problem can be overcome byreplacing Ju with Jut and likewise using Eqs.~3!–~5! toexpressq u in terms of t-dependent quantities, so that thepre-exponential factor is obtained as a function of timetalong the trajectories. Since this function must then be con-tinuous with respect to time, the asymptotict→` phase ofCa( u) can be determined by requiring that~a! it be continu-ous with respect tou at the initial timet50, and~b! it varycontinuously with respect tot during the course of each tra-jectory.

However, even if the discontinuity in the pre-exponentialfactor is resolved in this manner, there remains a relatedproblem concerning the factor exp@iDa(u)/\# in Eq. ~1! forthe S-matrix. Since this factor is not a 2p-periodic functionof q u @see Eq.~7!#, we need to know the full value ofq u ,not just its value mod(2p), to determine the correct overallphase of the IVR integrand. Thus, it is not sufficient to obtainq u from the values ofput andr ut at the end of the collision.In addition, when the dynamics is chaotic, continuity withrespect tou cannot be used to supply the missing informa-tion aboutq u . Finally, this difficulty cannot be convenientlyovercome by imposing continuity with respect tot duringthe course of each trajectory since, ifJut50 at a certain timealong the trajectory, the value ofqut becomes undefined andone loses the ability to determine its full value at subsequenttimes. The vanishing ofJut also complicates the direct nu-merical solution of Hamilton’s equations in terms of action-angle variables (Jut ,qut).

We solve these problems in this work by expressing theintegrand of Eq.~1!, to the extent practicable, in terms of thetime-dependent Cartesian variables (put ,r ut ,Put ,Rut). Wethen rewrite the integrand as a product of a new set of com-plex factors, each having a phase that can be determinedunambiguously, either by following it as a function of timeor by simply substituting the values of the Cartesian vari-ables. Since Eq.~1! was derived28 from an expression for thetime-independent wave function in terms of thesevariables,57,58 our present reformulation involves, to someextent, reversing the derivation presented in Ref. 28.

We proceed by first relatingCa( u) to the quantity

But5U ]Put

]u22ig1

]Rut

]u

]put

]u22ig2

]r ut

]u

]Put

]t22ig1

]Rut

]t

]put

]t22ig2

]r ut

]t

U 1/2

, ~10!

that serves as a pre-exponential factor in the time-independent scattering wave function of Ref. 28. The quan-tities g1 andg2 appearing above are ‘‘arbitrary parameters.’’The general theory57,58actually permits these quantities to bearbitrary complex functions oft, u, R, andr , provided that`.Reg1, Reg2.0. In the present context, these quantitiesare related to the parametera of Eq. ~1! in a manner to beestablished below.

Regardingr ut and put as functions of the action-anglevariablesJu andqut for t→`, the chain rule gives

]r ut

]u5

]r ut

]Ju

]Ju

]u1

]r ut

]qut

]qut

]u~11!

]put

]u5

]put

]Ju

]Ju

]u1

]put

]qut

]qut

]u~12!

]r ut

]t5

]r ut

]qut

]qut

]t5v

]r ut

]qut~13!

]put

]t5

]put

]qut

]qut

]t5v

]put

]qut, ~14!

for the oscillator variables at asymptotically late times,where the frequencyv can be identified as

v5v~Ju !5]eu

]J, ~15!

in terms of the asymptotic vibrational energy of the oscillatoreu . Thus, applying Eq.~3!, we may write

But

25vS ]put

]qut

22ig2

]r ut

]qutD F S 122ig1

t

mD ]Pu

]u

22ig1

]Ru

]uG12ig1

Pu

mS ]put

]qut

22ig2

]r ut

]qutD

3S 2a0

2iJ u

]Ju

]u1

]qut

]uD , ~16!

where

a0522iJ uS ]put

]Ju22ig2

]r ut

]JuD Y S ]put

]qut

22ig2

]r ut

]qutD . ~17!

We now apply energy conservation in the form

Pu5A2m~E2eu!, ~18!

to write

]Pu

]u52

m

Pu

]eu

]u52

mv

Pu

]Ju

]u, ~19!

and reexpressBut

2as

But

25xutFmv2

PuS 122ig1

t

mD ]Ju

]u22ig1v

]Ru

]u

22ig1

Pu

m

a0

2iJ u

]Ju

]u12ig1

Pu

m

]qut

]uG , ~20!

where

xut5]put

]qut22ig2

]r ut

]qut. ~21!

Since Eqs.~4!–~5! imply

10579J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Semiclassical reactive collisions

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qut5q u1mvRut

Pu5q u1mv

Ru

Pu1vt, ~22!

it follows that

]qut

]u5

]qu

]u1

mv

Pu

]Ru

]u2

mvRu

Pu

2

]Pu

]u1

mRut

Pu

]v

]Ju

]Ju

]u.

~23!

Substituting this into Eq.~20! and again using Eq.~19! there-fore gives

But

252ig1xut

Pu

mF S 2

m2v2

2ig1Pu

2 1mv2t

Pu

2 1m2v2Ru

Pu

3

2a0

2iJ u

1Rutm

Pu

]v

]JuD ]Ju

]u1

]qu

]uG . ~24!

Applying Eq. ~3! in the form

t5~Rut2Ru !m

Pu, ~25!

and definingg1 as

1

g15

1

g31

2iRut

Pu1

2iRutPu

mv2

]v

]Ju, ~26!

in terms of a new arbitrary parameterg3 with a positive realpart, Eq.~24! becomes

But

252ig1xut

Pu

mF S 2

m2v2

2ig3Pu

2 2a0

2iJ uD ]Ju

]u1

]qu

]uG .

~27!

All we need to do now is to identifya as

a5m2v2Ju

Pu

2g3

1a0 , ~28!

to obtain the desired result which expressesBut in terms ofCa( u),

But5F 2ig1xut

Pu

m S ia

2Ju

]Ju

]u1

]qu

]uD G 1/2

5F2ig1xut

Pu

mG 1/2

Ca~ u !. ~29!

The parametersg1 andg2 appearing inBut are determinedfrom a by choosinga0 ~nonuniquely! according to the con-dition that Rea0,Rea and then applying the relations

g252i

2S i

]put

]Ju1

a0

2Ju

]put

]quD Y S i

]rut

]Ju1

a0

2Ju

]rut

]quD~30!

and

1

g15

~a2a0!Pu

2

m2v2Ju12i

Rut

Pu12i

RutPu

mv2

]v

]Ju, ~31!

which can be derived from Eqs.~17!, ~26!, and~28!.

We remark that Eq.~26! is a slight generalization of Eq.~63! of Ref. 28 which treated the special case of a harmonicfragment so that the last term involving]v/]Ju was absent.Similarly, Eq. ~28! is a generalization to Eq.~65! of Ref. 28in which a0 was taken to have the special value of unity.

Finally, we use Eq.~29! to expressCa in terms ofBut

and regroup factors in Eq.~1! for the S-matrix to obtain theresult

Sn2n15

i

2p E0

2pS m

2ig1PuD 1/2

ButGuteiLut /\ du, ~32!

where

Gut51

Axut

e2 i [F2(r ut ,Ju)1qut(J22Ju)]/\, ~33!

and

Lut52ia

2 S J2 lnJu

J21J22JuD1mv

Rut

Pu~J22Ju !

1F2~r u0 ,J1!1P1R12PuRut

1E0

tS put8

]r ut8]t8

1Put8

]Rut8]t8 Ddt8. ~34!

In Eqs.~32!–~34!, the limit t→` is to be implicitly under-stood.

Equation~32! is the desired expression for theS-matrix.Apart from arbitrary terms defining the irrelevant overallphase of the fullS-matrix, the phase of each factor in theintegrand is defined unambiguously. The functionLut is de-fined uniquely by Eq.~34!. The phase ofBut can be obtainedby requiring this function to be continuous with respect touat the initial time t50 and continuous with respect totafterwards. Finally, it can be proven that the factorGut de-pends onqut only via qut(mod2p) so that it can be calcu-lated uniquely from the values ofput and r ut as t→`. Toshow this, we recall that Eqs.~26!–~36! of Ref. 28 establishthe identity

Gut5~2\!21/2e2a0[J2 ln(Ju /J2)1J22Ju]/2\

3E2`

`

drcn2* ~r !e2 ~g2 /\!(r 2r ut)21 ~ i /\! p

u¯

t(r 2r ut),

~35!

for the case of a harmonic oscillator fragment and for thespecial choiceg25mv/2, consistent witha051. In this ex-pression,cn2

is the wave function for the oscillator withmassm in staten2 . It is clear that the right-hand side of Eq.~35! involves only quantities that depend uniquely on(put ,r ut). Although this expression is no longer exact formore general oscillators or other choices ofg2 , it remainsvalid as a semiclassical approximation that can be proven bysubstituting the WKB expression forcn2

and evaluating theintegral by the stationary phase method. This establishes thatthe functionGut and the entire integrand of Eq.~32! is freeof phase ambiguities, as desired.

In its present form, Eq.~32! is appropriate for the treat-ment of nonreactive collisions. To apply to reactive colli-

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sions, it is necessary to replace certain occurrences of reac-tant channel Jacobi variables (Rut ,Put ,r ut ,put) in thisexpression with the corresponding product channel Jacobivariables (Ru t

8 ,Pu t8 ,r u t

8 ,pu t8 ). Likewise, it is necessary to

replace particular occurrences of the action-angle variables(Ju t ,qu t), for the reactant diatomic, with the variables(Ju t

8 ,qu t8 ), describing the product fragment. Although cer-

tain quantities in Eq.~32! can be expressed in the same formwith equal validity in terms of either set of variables, it isconvenient to describe the changes needed to treat reactivecollisions by the following prescription:~a! the variables inthe quantity F2(r u0 ,J1)1P1R1 , appearing in Eq.~34!,should remain in their present form as reactant variables, and~b! all other variables~including Pu and Ju! appearing inEqs. ~10! and ~32!–~34! should be replaced by the corre-sponding product~primed! variables.

III. APPLICATION TO THE COLLINEAR H ¿H2COLLISION MODEL

We apply the treatment developed above to the collinearH1H2 reaction on the PK49 surface. To calculate theS-matrix using Eq.~32!, we need to choose values for theparametersa0 anda. As discussed above,a0 can be chosenarbitrarily, subject only to the condition 0,Rea0,Rea,since the final expression forS is formally independent ofthis parameter. The actual value used in our calculations isa050.1. The final expression for theS-matrix is, however,dependent on the parametera. Generally, this quantity is acomplex-valued function ofu, but we restrict it to be au-independent complex constant in this work. Although it ispossible to obtain qualitatively correct transition probabili-ties using almost arbitrary values ofa, it is useful to choosethis parameter more carefully so as to optimize the accuracyof the resultingS-matrix. In our previous paper,28 we pro-posed two methods that can be used to accomplish this with-out performing quantum calculations. The first method,based on the condition that the pre-exponential factor notvary strongly withu, does not appear to be easily applicableto the present system since the pre-exponential factor is adiscontinuous and rapidly-varying function for reactive andchaotic scattering. We therefore apply the second methodsuggested in Ref. 28 which is based on the formal propertiesof the S-matrix.

This technique involves two steps. First, for each initialstaten at a given energyE, matrix elementsSmn(E) arecalculated using a variety of trial values fora. Each suchvalue is assigned a scoreQ0 defined by

Q05S (m50

Nc21

Pmn21D 2

, ~36!

wherePmn5uSmn(E)u2 are transition probabilities andNc isthe number of open channels. The besta for each initial staten is identified as that which minimizesQ0 . Clearly, a smallvalue ofQ0 implies that the sum of probabilities for all tran-sitions from staten is close to unity. Next, an attempt ismade to improve the resultingS-matrix by optimizinga foreach individual transition (m,n). Beginning with the values

of a for eachn obtained in the previous step, thea used tocalculate each elementSmn is varied separately to minimizethe quantity

Q15 (n50

Nc21 F S (m50

Nc21

Pmn21D 2

1S (m50

Nc21

Pnm21D 2

1 (m.n

Nc21

~Pnm2Pmn!2G , ~37!

and the procedure is iterated to convergence. A small valueof Q1 implies that the sum of probabilities for transitions toeach final state~as well as from each initial state! is close tounity and that microscopic reversibility, in the formPnm

5Pmn , is accurately obeyed.Minimization of Q1 may yield more accurate transition

probabilities than those obtained by minimizingQ0 but itrequires a calculation of the wholeS-matrix, even if transi-tions from only a particular initial state are of interest. Sinceit may sometimes be computationally advantageous to deter-mine the parametera by minimizing Q0 alone, we investi-gate this procedure as well as the full minimization ofQ1 inthis work.

The trial complexa’s used should have real parts in therange between the two nonuniform limits:28 Rea50 ~whereour expression reduces to the MM-IVR expression! andRea→` ~where it reduces to the PSC limit!. In practice, wevaried the values of Rea from 0.1 to 6.1~which is effec-tively near the PSC limit!, in steps of 1.0, and the values ofIm a from 215.0 to 15.0, in steps of 0.5. For almost allenergies, it was found sufficient to restrict the real part ofato the constant value 2.1 and to vary only the imaginary part.This simplification is possible here since the convergedPmn

are insensitive to Rea as long as this quantity is far enoughfrom the MM-IVR and PSC limits. The particular valueRea52.1 is found to be numerically advantageous since ityields relatively rapid convergence of the transition prob-abilities with respect to the number of trajectories. We notethat extreme choices for this parameter adversely affect theconvergence since they lead to insufficient damping of theintegrand’s oscillations~small Rea! or cause contributions tothe integral to be localized in very narrow but unknownranges ofu ~large Rea!.

We now briefly describe the computational procedure forthe calculation ofSn2n1

(E) for a particular total energyEand initial staten1 . This state defines the value of the initialvibrational actionvia the equationJu05J15(n111/2)\. Asdetailed below, a number of values are chosen for the conju-gate initial angleu so that the numerical integration in Eq.~32! can be performed. For each (Ju0 ,u) @and, thus,(r u0 ,pu0)# a trajectory is run with initial interfragment dis-tanceR15Ru059.6 ~in the asymptotic region!, and with ini-tial translational momentumP15Pu05A@2m(E2e1)#,wheree1 is the initial vibrational energy. Hamilton’s equa-tions for the reactant channel Jacobi variables(Rut ,Put ,r ut ,put) ~used for nonreactive trajectories! or theproduct channel Jacobi variables (R

ut8 ,P

ut8 ,r

ut8 ,p

ut8 ) ~used

for reactive trajectories! as well as linearized equations for

10581J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Semiclassical reactive collisions

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their derivatives with respect tou, and an equation for thetime-derivative of the classical action, are integrated overtime t, until the collision is over and the distanceRut ~for anonreactive trajectory! or R

ut8 ~for a reactive trajectory! takes

on the value 9.6. The phase of the pre-exponential factorBut

is chosen so that it is a continuous function ofu at t50 anda continuous function oft for t.0. To allow efficient cal-culation of theS-matrix for a variety of trial values fora, thepre-exponential factor for each trajectory is computed usingarbitrary constant values ofg1 andg2 . Towards the end ofthe trajectory, the values of these parameters are slowlychanged to those corresponding to the desireda’s, while thephase of the pre-exponential factor is kept continuous. Thefinal angle-action variables (qut ,Ju) or (q

ut8 ,J

u8) are calcu-

lated from (r ut ,put) or (rut8 ,p

ut8 ) at the end of the trajectory.

S-matrix elements for all energetically allowed transitionsare calculated for 26 energies in range 0.35 eV to 1.60 eV.

Unfortunately, integration overu is not straightforward

for the present system. The range2p<u<p can be dividedinto continuous ‘‘zones’’ where all trajectories are either re-active or nonreactive. If both reactive and nonreactive zonesexist at a particular energy, the integrand will be discontinu-ous at the zone borders. Enlargement of a border region mayreveal a gap containing chaotic trajectories. We call such agap a chattering zone. Within a chattering zone, the reactive/nonreactive nature of the trajectories~as well as the final

action-angle values! varies rapidly as a function ofu, andintegration becomes very difficult. Although chattering zonesfor our system are typically small, they cannot, in general,simply be neglected, since the chaotic trajectories are crucialfor the description of resonances.41,53 Due to the highly non-uniform structure of the integrand, Monte Carlo integration

techniques that involve uniform sampling of the entireurange converge very slowly and are of limited practical usehere.

We introduce an integration strategy that is numericallysimple and yields relatively quick convergence. The tech-nique adapts the choice of integration points to the fractalstructure of the chattering zones,47,51,53–56and is based on ananalysis of these zones presented by Stine and Marcus.41 Onebegins by noting that, at the energies considered here, theoverall range2p<u<p consists of either a single reactive~or nonreactive! zone or a large reactive zone plus a largenonreactive zone~note that a zone ‘‘beginning’’ atu52p isusually a continuation of one ‘‘ending’’ atu51p!. In thelatter case, there are two reactive/nonreactive boundaries atwhich chattering zones may exist. Upon enlargement, eachsuch chattering zone is again found to contain one relativelylarge reactive region and one relatively large nonreactive re-gion. For example, a chattering zone located between a re-active and a nonreactive region~proceeding from left toright! will contain a nonreactive region on the left and areactive region on the right. Thus, such a chattering zoneitself contains three zone boundaries: one at the extreme leftedge of the chattering zone, one between the relatively largereactive and nonreactive regions, and one at the extremeright edge of the chattering zone. Further magnification may

reveal that each such a boundary is actually a gap containinga chattering zone having the same qualitative structure de-scribed above~i.e., it contains one relatively large reactiveregion and one relatively large nonreactive region!. Continu-ing this patternad infinitumproduces a fractal structure.

Thus, the following procedure is used to perform the

integration overu. A certain, relatively small numberN0 ofequally spaced points is placed along the overall range2p

, u<p, which is called level 0. A trajectory is run fromeach such point, and the largest reactive and nonreactivezones are identified. The region between the last point of onesuch zone and the first point of the next is identified as apotential chattering zone. The internal structure of such achattering zone is said to belong to level 1.N1 equallyspaced points are then placed across this chattering zone and,again, smaller reactive, nonreactive, and potential chatteringzones are identified. The structure in each new chatteringzone is said to belong to level 2 and is subsequently spannedby N2 equally spaced points. Although, in principle, this re-cursive procedure can be repeated indefinitely, the number ofzones at each succeeding level grows exponentially, so welimit our present calculations to three levels. The reactive~nonreactive! S-matrix is evaluated by adding the contribu-tions from all the reactive~nonreactive! zones in levelsl50,1,2, where each contribution is calculated by trapezoidrule integration over that portion of theNl points that actu-ally fall within a particular reactive~nonreactive! zone.

Of the three chattering zones in levell , obtained by en-larging a particular chattering zone of levell 21, the ‘‘cen-tral’’ zone ~i.e., that separating the relatively large reactiveand nonreactive zones of levell ! is found to yield a negli-gible contribution to theS-matrix and is thus neglected. As aresult, the maximum total number of reactive or nonreactivezones per energy and initial state considered in this work isseven~11214 from levels 0, 1, and 2!. However, for sev-eral energies, the number of such zones is actually substan-tially smaller. This situation arises since level 0 is sometimesfound to consist of only a single reactive or nonreactive zone~so that there is no chattering or need to define subsequentzones! and since potential chattering zones are sometimesfound apparently to contain no chattering trajectories.

Even with the limitation to three levels, the rapid varia-tion of the integrand’s phase and the large values of the pre-exponential factor in chaotic zones make it extremely diffi-cult to achieve convergence of the integral in Eq.~32! unlessadditional measures are taken. The technique we adopt herefollows Refs. 6, 11, and 12 and involves simply discardingcontributions to the integral from highly chaotic trajectories.Such trajectories are identified by values ofuButu that are inexcess of a pre-determined cutoffK.

Figure 1 compares probabilities obtained by our tech-nique to those obtained quantum mechanically for the 0→0 reactive transition. For the present calculations we chosethe trajectory numbersNl to be N05N15N25100 and thechaos cutoff parameterK to be 20. We examined optimiza-tion of a both by the minimization ofQ1 and by the mini-mization ofQ0 alone, and Table I gives the resulting ‘‘best’’values of this parameter obtained using these methods. Asdiscussed above, Rea was taken to have the constant value

10582 J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Y. Elran and K. G. Kay

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2.1 for most energies. However, the value Rea50.1 wasused at energy 0.60 eV where the results were found to bemore sensitive to variations in this quantity and convergencewith respect to the number of trajectories was rapid despitethe small value chosen. Another special case occurs atE50.87 eV where only the optimization ofQ0 was performedfor reasons discussed below. Using the minimization ofQ0 ,the number of trajectories per energy used to calculate thetransition probabilities shown in this figure ranged fromN0

5100 toN012N114N25700.

It is seen that both methods for choosinga produce rea-sonably accurate 0→0 reactive transition probabilities. Inparticular, the rapid variations in the transition probabilitynear the quantum mechanical resonances at 0.87 and 1.31 eVare clearly described, even though a rather modest chaoscutoff K is chosen~the effect of varyingK is discussed be-low, in connection with Fig. 7!. However, the gradual rise inthe quantum transition probability with energy in the reactivetunneling region (E&0.5 eV) is not properly reproduced byour calculations. As in the case of the MM-IVR method, thepresent technique yields a reactive transition probability ofzero when there are no classically allowed reactive trajecto-ries.

In Figs. 2 and 3 we present results for other reactive andnonreactive transitions. The parameters used for these calcu-lations were the same as those for Fig. 1 except thatN2 waschosen to be 150 for transitions fromn151 due to slowerconvergence in this case. Close examination reveals that theminimization of Q1 sometimes yields more accurate resultsthan minimization ofQ0 , but the differences are usuallysmall. Generally, our IVR results are again in good agree-ment with quantum calculations for all transitions, apart fromenergies in the reactive tunneling regions and near the clas-sical threshold energy for reaction from the first excited vi-brational state ('0.90 eV). Quantum-semiclassical discrep-ancies at this latter energy are especially prominent fortransitions originating fromn151, shown in Fig. 3, but arealso noticeable for other transitions shown in Fig. 2. Themechanism causing these discrepancies may also be respon-sible for the apparent broadening of the semiclassical reso-nance nearE50.90 eV, observed in Fig. 1. These inaccura-cies appear to be related to the behavior shown in Fig. 4,indicating thatuButu becomes very small during the course of

FIG. 1. 0→0 reactive transition probabilities for the PK system vs energy.Solid line: quantum results, dotted line: IVR treatment using the minimiza-tion of Q0 ; dashed line: IVR treatment using minimization ofQ1 .

TABLE I. Best values for Ima for the reactive 0→0 transition using theminimization ofQ1 andQ0 . Unless otherwise indicated, Rea52.1.

Energy~eV! Minimization of Q1 Minimization of Q0

0.40 215.0 215.00.45 215.0 215.00.50 26.0 26.00.55 27.5 26.00.60a 2.0 2.00.65 2.5 2.50.70 23.0 23.00.75 24.0 24.00.80 210.0 210.00.85 212.0 210.50.87b – 13.50.90 210.5 14.00.92 12.5 11.50.95 25.5 8.51.10 21.5 22.51.20 3.5 4.01.25 2.0 3.01.28 24.5 22.51.30 211.5 24.51.31 2.5 5.01.32 0.5 25.51.33 8.5 7.01.35 24.0 23.01.40 21.0 23.01.50 4.0 22.51.65 3.0 15.0

aFor this energy Rea is taken to be 0.1. See the text.bThe minimization ofQ1 is not performed for this energy. See the text.

FIG. 2. Probability vs energy for transitions fromn150 to ~a! n250, non-reactive;~b! n251, reactive;~c! n251, nonreactive. Solid lines: quantumresults; dotted lines: IVR treatment using the minimization ofQ0 ; dashedlines: IVR treatment using the minimization ofQ1 .

10583J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Semiclassical reactive collisions

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the collision for certain trajectories at this energy. It has beenestablished that, if the pre-exponential factorBut strictlyvanishes for anyu or t, the IVR semiclassical approximationfor the scattering wave function loses its uniform asymptoticproperties.57,58Since ourS-matrix expression is derived fromthis wave function,28 such behavior would cause our presentresults to become inaccurate.

As described in the Appendix, the near vanishing ofBut

at the effective classical threshold for reaction fromn151 isapparently due to the rather extensive vibrational adiabaticityof the dynamics near the entrance channel of the PKsystem,59 as evidenced by the large discrepancies betweenthe energetically allowed and actual classical threshold ener-

gies~e.g., 0.79 eV vs 0.90 eV for reaction fromn151!. Thisapproximate separation of variables effectively reduces thedimensionality of our system and permits the pre-exponentialfactor to become small along certain trajectories, as it doesfor one-dimensional systems at energies corresponding tobarrier maxima.57

The vibrational adiabaticity also complicates optimiza-tion of a at the energy 0.87 eV. In that case, the 0→1 reac-tive transition is energetically allowed and both the semiclas-sical and quantum calculations yield nonzero transitionprobabilities~see Fig. 2!. However, vibrational adiabaticityprevents reaction from the initial excited staten151 at thisenergy so that the semiclassical 1→0 reactive transitionprobability is zero~see Fig. 3!. Thus, while the microrevers-ibility condition P015P10 is, of course, exactly obeyed bythe quantum probabilities, it is not even approximatelyobeyed by the semiclassical probabilities. Clearly, this prob-lem arises from the dynamics of the system and cannot beremedied by changinga. In this case, then, it makes no senseto determinea by minimizing Q1 , which assumes that agood choice of this parameter is signaled by the satisfactionof microreversibility. We therefore do not attempt to opti-mize a by minimizing Q1 for this energy but instead obtainthe best value for this parameter by minimizingQ0 alone.

Figure 5 verifies the convergence of our results with re-spect to the number of trajectoriesN2 that are run at level2—our deepest recursive level. Figure 6 shows the conver-gence of our method with respect to the number of recursivelevels used to expand chattering zones. Although, for someenergy ranges, convergence is already obtained using only asingle level, for other ranges, especially near the resonances,

FIG. 3. Probability vs energy for transitions fromn151 to ~a! n250, reac-tive; ~b! n250, nonreactive;~c! n251, reactive;~d! n251, nonreactive.Solid lines: quantum results; dotted lines; IVR treatment using the minimi-zation ofQ0 ; dashed lines: IVR treatment using the minimization ofQ1 .

FIG. 4. The minimum value ofuButu with respect tou andt vs energyE forlevel 0 reactive trajectories with initial conditions corresponding ton151.The arrow marks the effective classical threshold energy for the reactionfrom n151.

FIG. 5. IVR transition probabilities vs energy in the PK system forN0

5N15100 but varying numbersN2 of trajectories in the final recursivelevel. a is determined by minimizingQ1 . ~a! 1→0 reactive transition,~b!0→0 nonreactive transition, and~c! 0→0 reactive transition. For~a! and~b!, solid line:N2550, dotted line:N25100, dashed line:N25500. For~c!,solid line: N25100, dotted line:N25150, dashed line:N25500.

10584 J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Y. Elran and K. G. Kay

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all three levels are required in order to obtain an adequatedescription of the transition probability, and the convergence,even with three levels, is not certain. Figure 7 shows theconvergence of our results with respect to the chaos cutoffKand suggests that the choiceK520 used in Figs. 1–3 issufficiently large to yield effective convergence at most en-ergies. In addition, this figure illustrates that it is possible toimprove the semiclassical description of the resonances, tosome extent, by choosing a larger value forK, but the priceis a substantial increase in the number of trajectories neededfor convergence.

IV. DISCUSSION AND SUMMARY

In this paper we have derived an IVR expression for theS-matrix that can be used to treat chaotic nonreactive andreactive scattering, and we have applied it to the H1H2 col-

linear exchange reaction on the PK surface. Our expressionis a uniform semiclassical approximation and is defined un-ambiguously. Notwithstanding the failure to treat reactivetunneling and some reduction in accuracy at reactive thresh-olds, we find that our method generally gives very goodagreement with accurate quantum calculations for all transi-tions. We can compare our method to the early MM-IVRtechnique29–34 ~which is a special case of our present treat-ment! and to the more recent IVR propagator methods.22–27

The MM-IVR method is a nonuniform semiclassical ap-proximation and is not well-defined due to phase ambigu-ities. It cannot be applied to treat transitions at all energies inreactive systems such as H1H2.35 The more recent propa-gator methods, in contrast, are uniform approximations thatcan be applied to treat reactive systems at all energies. Theyare capable of yielding good agreement with quantum calcu-lations, but require a large number of trajectories, since theyinvolve two integrations for each degree of freedom~exceptperhaps translation!. This problem becomes increasingly se-vere as the systems become larger. Since our treatment re-quires only one integral for each degree of freedom, manyfewer trajectories are needed and it is easier to tailor theintegration method to the fractal nature of the integrand forchaotic collisions and to achieve greater control over conver-gence. As a consequence, our method may be more readilyapplicable to large systems. Nevertheless, in contrast topropagator-based IVR methods, our treatment requires aseparate computation of theS-matrix at each energy. There-fore, our method is most advantageous when theS-matrix isneeded at only a limited number of energies.

A more formal problem associated with the existing IVRpropagator methods is that the computed transition probabili-ties generally depend on the initial and final interfragmentdistances used for trajectory propagation, unless certain pa-rameters, analogous to ourg1 andg2 , are chosen carefully.As is clear from the derivation of our expressions here and inRef. 28, our specific definition of these parameters@Eqs.~17!, ~26!, and ~28!# results in anS-matrix expression thathas the desired property of not depending on initial/final dis-tances. The only independent parameter actually occurring inour calculations isa, and we have presented methods for itsoptimization that are well-defined and do not require runningadditional trajectories.

On the other hand, the present IVR theory has a formalflaw that does not arise in the propagator-based IVRapproaches:22–27 it yields transition probabilities that do notsatisfy the condition of microscopic reversibility exactly.However, we have shown that this feature provides a crite-rion for optimizinga and thus can be used to our advantageas a means of improving the accuracy of the calculations.With few exceptions, such as those noted in the previoussection, even the transition probabilities that are not obtainedby imposing microscopic reversibility~i.e., those calculatedby minimizing Q0 alone! nevertheless obey microreversibil-ity to a good approximation.

The accuracy of our results, including our description ofthe resonances, compares very favorably with that obtainedin previous IVR treatments22–25 of collinear H1H2 scatter-ing on the Wall–Porter potential. If we are willing to pay a

FIG. 6. 0→0 reactive transition probabilities for the PK system vs energy~calculated usingN05N15N25100 and the minimization ofQ1! for vary-ing numbers of recursive levels. Thick solid line: quantum results. Dottedline: IVR with level 0 only; dashed line: IVR with levels 0 and 1; thin solidline: IVR with levels 0, 1, and 2.

FIG. 7. 0→0 reactive transition probabilities for the PK system vs energycalculated using the minimization ofQ1 with varying values for the chaoscutoff K and numbers of trajectoriesNl for zones at each levell . Solid line:K510, N05N15N25100; dotted line:K520, N05N15N25100; dashedline: K560, N05N15N25900.

10585J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Semiclassical reactive collisions

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higher computational price, it may be possible to further im-prove our treatment of the resonances by descending todeeper recursive levels in the chattering zones and usinghigher chaos cutoffs. As is stands, our method is unable totreat reactive tunneling below the classical reaction thresh-old. However, the treatment of deep tunneling using real-valued trajectories is very tricky7,60,61and the success of theIVR techniques presented in Refs. 22–25 in describing thisphenomenon is not certain.

As already noted, the accuracy of our semiclassical ap-proximation tends to decrease near quantized threshold ener-gies for reaction. The mechanism for this behavior appears tobe related to the vibrational adiabaticity of the classical mo-tion along certain portions of the PK potential energy sur-face. It is not obvious from the published results22–25

whether similar inaccuracies occur in treatments of theS-matrix based on the semiclassical IVR propagator. Furthernumerical calculations should help clarify this matter.

There are several extensions and improvements of ourmethod that should be investigated. The idea of optimizingaby variationalS-matrix techniques seems worthy of consid-eration. It may be possible to modify our expressions to treatreactive tunneling by reinstating the integral overt or Rut

that appears in the derivation of Eq.~1! @see, e.g., Eq.~56! ofRef. 28# and using imaginary values ofg1 .7 The treatment ofchattering in arbitrary systems will probably require a gen-eralization and fuller automation of the recursive descentinto chaotic regions described here. For larger systems, itmay also require Monte Carlo integration over the individualzones. To improve the accuracy of our treatment of chaoticscattering and allow calculations for systems with more ex-tensive chaos, it may be possible to exploit the scaling prop-erties of the chaotic fractals to sum the chattering contribu-tions to our IVR expression to all orders~with a properaccounting of phases!, in a manner related to that of Tiyapanand Jaffe´47 and Guanteset al.56 Finally, although our expres-sion was derived for the special case of collinear scattering,it can be generalized to allow a calculation of theS-matrix orthe differential cross-section62 for three-dimensional poly-atomic scattering. Work along these lines is in progress.

ACKNOWLEDGMENTS

This research was supported by the Israel Science Foun-dation ~Grant No. 530/99!. We wish to thank Professor B.Ramachandran for supplying the quantum transition prob-abilities quoted in this paper. Y. E. wishes to thank ProfessorD. Tannor for helpful discussions and hospitality at the Weiz-mann Institute.

APPENDIX: VIBRATIONAL ADIABATICITY ATTHRESHOLD ENERGIES AND NONUNIFORMITY

Here we describe how vibrational adiabaticity can causethe pre-exponential factorBut to become small along certaintrajectories so that ourS-matrix expression may lose accu-racy. For this purpose, it is convenient to introduce collisioncoordinatesu andv that are directed, respectively, along and

orthogonal to a suitably-defined reaction path.63 The Jacobicoordinates for the reactant arrangement channel can be rep-resented as functions of these variables:

Rut5Rut~u,v !,~A1!

r ut5r ut~u,v !,

while the Jacobi momenta can be expressed as functions ofthese coordinates and their conjugate momenta:

Put5Put~u,v,pu ,pv!,~A2!

put5put~u,v,pu ,pv!.

Similar expressions can be presented for the primed Jacobivariables for the product arrangement channel.

Consider scattering at the effective classical thresholdenergy for reaction from staten151 ('0.90 eV). In linewith previous studies,59 we assume that the motion is vibra-tionally adiabatic for all times until the trajectories encountera barrier in the adiabatic potentialVad(u) along the reactioncoordinate, coinciding with an outer periodic orbit dividingsurface ~PODS!64 located near the entrance arrangementchannel. Such adiabaticity implies that the translational vari-ablesu and pu at the PODS do not depend on the anglevariable u that distinguishes between different trajectories.At the threshold energy, the motion alongu stops at thebarrier maximum where]Vad(u)/]u50, so that the condi-tions

]u/]t5]pu /]t5]u/]u5]pu /]u50 ~A3!

are obeyed at that point.The vibrational variablesv and pv depend onu and t

only through the angle variable

qu,t5 u1vmR1 /P11Vt ~A4!

that evolves fromu at t50 @cf. Eqs. ~2! and ~4!#. In thisexpression,V5V@u(t)# is the local frequency for vibra-tional motion alongv; it varies with the coordinateu and hasthe valuev in the asymptotic regions. From Eq.~A3!, itfollows that ]V/]t50 at the PODS, so that Eq.~A4! im-plies

]v/]u5V21 ]v/]t,~A5!

]pv /]u5V21 ]pv /]t,

at that point. Finally, the substitution of Eqs.~A3! and ~A5!into Eqs. ~A1! and ~A2! shows that each Jacobi variableZut5Rut ,r ut ,Put ,put obeys a relation of the form

]Zut /]u5~]Zut /]v !V21~]v/]t!

1~]Zut /]pv!V21~]pv /]t!

5V21Zut /]t ~A6!

at the PODS. Thus, when the trajectory reaches this point,the two rows in the determinant appearing in Eq.~10! be-come linearly dependent and the pre-exponential factorBut

vanishes. If these conditions are strictly obeyed, the semi-

10586 J. Chem. Phys., Vol. 116, No. 24, 22 June 2002 Y. Elran and K. G. Kay

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classical expression for the scattering wave function, used toobtain ourS-matrix for n151, becomes nonuniform~and,thus, inaccurate! at the classical threshold energy for reactivescattering from this channel.

There exists a related mechanism that may causeBut tobecome small along certain trajectories from arbitrary initialstates. Although it applies to scattering at all energies, sta-tionary phase arguments suggest that it should seriously af-fect the accuracy of theS-matrix only for energies coincidingwith thresholds for the reaction to final quantized vibrationalstates. To illustrate this mechanism, consider scattering frominitial staten150 at the effectiven251 reactive thresholdenergy. We assume that trajectories ending up withn251pass very slowly across an outer PODS, located near theexitarrangement channel, and thereafter undergo adiabatic mo-tion until the scattering is complete. Thus, Eq.~A3! is againobeyed at the PODS and the actionJut of the trajectories hasthe constant valueJ25(3/2)\ during and after passageacross this surface. We parametrize the behavior for this timerange using variablesq u @see Eq.~5!# and t, where t5t ut

5Tu2t is the time along the trajectory measured from thetime Tu at which Rut5R2 , whereR2 is a particular, arbi-trary, ‘‘final’’ asymptotic interfragment distance for the tra-jectories.

The symmetry between definitions of the ‘‘initial’’ vari-ables (u,t) and ‘‘final’’ variables (q,t) suggests defining afunctionBqt in which the derivatives with respect to (u,t) atconstantJ1 , appearing in Eq.~10! for But , are replaced byderivatives with respect to (q,t) at constantJ2 . The argu-ments presented above can then be repeated to show thatBqt

vanishes at the PODS for trajectories withJut5(3/2)\ in thepresent case. Since the energy of interest corresponds to then251 threshold, the value (3/2)\ is the largest one possiblefor Jut at this energy, so that]Jut /]u50 for the trajectoriesdescribed above. Thus, the two functionsB are related by

But5@]~q,t !/]~ u,t!#Bqt ~A7!

for such trajectories. Apart from points where the IVR inte-grand is discontinuous, the Jacobian factor in Eq.~A7! isfinite and we thus conclude thatBut vanishes during thecourse of certain trajectories at then251 reactive threshold.

If vibrational adiabaticity were exactly obeyed in thesense needed for the above derivation, this argument couldbe generalized to imply a breakdown of the semiclassicalapproximation of Ref. 28 for allS-matrix elements at classi-cal thresholds for reaction to and from all quantized vibra-tional states. Indeed, our semiclassical results show relativelylarge errors for transitions originating fromn151 at thethreshold energy for the reaction from this state. In addition,it is possible that the inaccuracy in the semiclassical 0→0transition probability, observed at this energy, is caused bythe same mechanism. However, other transitions at this en-ergy and transitions at other threshold energies are less seri-ously affected by errors of this sort. This suggests that vibra-tional adiabaticity is not sufficiently complete to entirelydestroy the semiclassical approximation at thresholds for thepresent system.

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