Pattern prediction in collinear atom-diatom trajectories

Volume 69, number 2

CHEMICALPHYSICS LEMERS

PATTERN PREDICTION IN COLLINEAR

ATOM-DIATOM

LS

ianuary

L980

TRAJECTORIES*

Richard T. PENLY, John S. HUTCHINSON and Robert E. WYATT of Ckmntry. Umerstty of Texus, Atmitt. Texas 78712. USA

Department

Received22 August 1979; m final form 15 October 1979

trajectory ensembles are predwed, usually wthout integration, by cornpar& Tra~ectones are numerially mtegrated only If the stochasw impulseevceectsa value. The technrque E applied to three chemtcai reactxons The accuracy ISeuceiknc m aU cases.

The final states of large atom-&atom

them to a small set of trammg trajectones ~lmurn

1. Introduction The determimsm of classical mechamcs lends a certain predictability to clasncal trajectory studres, given a tixed set of mitral conditions, Ham&on’s equatrons always generate the same final conditions. What IS falsely imphed ISthat simdar sets of 1~~~1 condltlons WrIIproduce predrctably simiiar sets of fin& conhtrons. In fact, a great deaI of research on the stabrhty propertres of classrcal trajectorres has shown that imtraUy srrular, or “nerghboring”, trajectones may lead to dramatrcally drfferent final condrtrons [ I]. This instabrhty, or “chattering”. has been shown to produce ergodrc and statrstrcal behavtor m trajectory ensembles. As an off-shoot of these studres. we have recently proposed [a] the apphcatron of the stabrhty properties of trajectones to pattern recognition methods m chemical dynamics. Attempts to predict the outcome of a random trajectory by drscermng a pattern m a small “traming” set of such trajectones have been made prevtously, as a means to bypass trme-consummg mtegratron of large trajectory ensembles [3]. What has been Iackmg is the abrlrty to Judge, in advance, the validrty of the pattern. By studying the stability or mstability of the set of trammg trajectories, we will show tbat rt is possrble to correctly predict the pattern of the ensemble. In tEus study, we seek to predict tire results ofcollinear atom-diatom trajectories by means * Supportedin part by researchgrants from the Nationai Science Foundauon and the Robeit A Welch Foundation.

of the minimum stochastrc impulse cnterion

12 J _

2. Theory and technique 2. I _Patternpredictim in chemical reaction studies, Ham&on’s equations piay the role of a very complicated function by which one may feed in imttal con~gurations and energies of the various atoms and then grind out fiial configuratrons and energies. For collinear atom-dratom reactrons at constant total and initial vibrational energies, the imtial conditions may be specrfied by a single variable, the initial phase of the diatom, #,,. Similarly. the fInal state of the reaction system may be specified by the final vibrational action u’, which is the quasiclassical vrbratronal quantum number. Trajectory integration [2] then provides u’ as a function of the variable go. Frovrded chat this function is smooth and CORtinuous, we might hope to predict U’ for a particular ‘pe by comparrson wrtb the results of a training set of trajectories spanning the entire range of @a values, This forms the basis of the “pattern recognition” technique [3] - The pattern of the u’ function is determined by a‘ limited number of trajectory integrations, and the results of all trajectories are assumed to obey this smooth pattern. Indeed, for the so-called ‘%iireclt” region of the action-ache function, pattern recog& tron should be fairly accurate. However, in the cbattering region, the smoothness proviso is invalid, and 255

Volume

69, number

CHEMICAL

2

PHYSICS

pattern recogmtron will fall rruserably. We have recently employed a new techmque of “pattern pre&ctlon”. utrlizmg the stochastic Impulse as a means of Identifying chattering traJectoncs m the trammg set. For those trajectories which have a stochastic impulse above a certam muumum criterion. pattern recognttion is abandoned and the traJectory IS integrated. We thus generate “smart” training sets m that the sets identify the limits of their own apphcablhty. An alternative to Integration of the “bad” traJectorles, once they are Identified by comparison with the pattern, IS to employ a st3tlstlcal theory. Th~s optlon IS not explored here. 2.2. 7&e tttitzittmttz srochasric

f 21

ittiprdse cri~etioti

Our method of ldentlfymg the chattering trajectories IS based on the observation that these traJectories exhibit unstable behanor Specifically, this instability IS manIFested m exponential separation m phase space of two mitlally slmllar trajectories [ I] The extent of tlus e\ponentlatlon may be prcdlcted by a variational equation technique developed by Brumer and Duff [4,5] and Toda [6] (BDT theory) One analyzes the elgenvalues of a stability matnk S. which For colhnear trajectories In mass scaled coordlnates, has the form0

s=

l/P

0

0

0

0

I/P

-a2 vfaqf

-aWaq,aq2

0

0

1

O1

-aWaqlaq2

0 -a2v/aqs The elgenbalues of S come m pairs, where h, = -h, = (x1/p)*/2. and h, = -h, = (x2/p)t/‘, Jo IS the reduced mass and xt and x2 are the eigenvalues of the 2 X 2 matnx of second derivatives Typically, one of the elgenvalues, h?, IS always lmagmary at low energres, whrle h, can be either pure real or pure Imaginary [2,5] Exponential separation IS predicted when Re(Xt) >O. Moreover, we have shown [2] that the extent of exponential separation may be predicted by evaluatmg the integral

LETTERS

15 January

1986

where I IS the stochastic impulse. For chattering traJe&_ tories, I rises to charactenstlcally tugh values Moreover, utzlrke u’. I IS a smooth function of Q. so that pattern recogmtlon may be reliably applied to find the: stochastic impulse of a random traJectory from a trar& ing set.

3. Results We have apphed the technique of pattern prediction to three dlstmctly different types of collinear chemical reaction systems (1) the F + H2 exothermlc reactIon,(2) the I + Hz endothermic reactron,and (3) the He + H?; reaction, whose potential surface contarns a shallow well Just before the endothermic barrier For each system, a smart training set of 100 trajectories (uniformly mcremented m Qo) was integrated, and u’ and 1 were calculated for each traJectory_ Then an ensemble of 500 trajectories wth mtzdom initial vlbratlonal phase was “run”. For each traJectory. the value of I was Interpolated From the two nearest neighbors m the training set, and, IF the value exceeded a minimum cnterlon (typically I > 71, the traJectory was mtegrated to find u’. IF I Falled to exceed the criterion, the trajectory was assumed to belong to the direct region of the action-angle Function, and u’ was mterpolated from the nearest neighbors. OF course, the traJectory was then integrated to check the accuracy of the prediction 31.F+H2 The potential energy surface used was the semiLEPS surface of Muckerman * We have previously studled the stability properties of traJectories on this surface using the BDT theory, and contours of the eigenvalues of S For the F + H, surface may be Found m our prevrous paper [2] A crrterlon of I > 7 was required for trajectory mtegratlon. From the ensemble of 500 trajectories run at a total energy oF0.375 eV and mnial u = 0, only 86 (17%) required mtegratlon. The accuracy of predictions for the remainder of the traJectorles IS dIsplayed as an error tustogram m fig. 1. Here we have plotted the fraction emplncal

‘f I=

s 0

2.56

Re[hl(ql(fI.

q2W)l

dr ,

* The form of the F + Hz potential the parameters

are from ref

181.

IS from hIuckerman

[7];

Volume 69, number 2 Oi---2,

~----

CHEMICAL PHYSICS LETTERS

--

_- --__-__

I

-F+

-l

H2

E=0375

!

eV

v-o

-0.20

1s January 1980

I

500

TRAJ

BIN

SlZE=O.Ol

0

0.20

V’(PREDICTED)-V’(ACTU~LL)

I?g 1 Error hatogram for F + Hz system, showing the prol.~ atxhty of a trajectory having a predlcted u’ m error from the actual U’ by an amount u’(predicted) - u’(actua1)

of trajectcnes

versus Au = u’@re&cted) - u’(actua1). It is easily seen that no trajectones have error greater than +O 10, and that the vast maJorlty (91%) have

error less than +O 01. The accuracy tion IS thus excellent.

of pattern

predic-

W2 employed the LEPS surface of Patter&l and Pclanyl 193, which has an endothermicity of 1.48 eV and barrier height of 1.53 eV (see fig. 2a). The standard behavior is observed- X2 remams Imaginary at ail energies for cur study, and h, IS the elgenvalue of interest The real part of the elgenvalue X, is contoured in fig. 2b as a function of the mass scaled coordinates q1 and CJ?. Again, the mmimum crltencn was set at I > 7 for Integration. The batch of 500 trajectcrles was run at a translational energy of 0.500 eV and an uutlal IJ = 2. The results of the prechctlons were once more excellent. Only 68 of 500 trajectories required mtegraticn, and 88% of the ensemble had an error less than iO.01. Only 1% of the trajectcnes had error greater than %.lO.

Re (XI)

OS 5

-I IO

qi

Rg. 2 I + Hz system. (a) Potential eneey contours in mass scaled coordinates. Energres in eV. (b) Contours of the real part of the At ergenvalue of S. Scale as in (a). Units are LOI’ s-1

3-R He +@ The semi-empirical DIM potential surface of Ku&z [IO] was stuhed. This surface has an endothermictty 257

CHLLIICAL

Volume 69, number 2

I

PHYSICS

15 January

LETTERS

1980

the stabrhty analysis. The eigenvalue, Al, of the stabtlrty matnx is displayed m fig. 3b. In thrs case, the above behavior IS nor observed. The most consprcuous difference is the arm of Re(Xt) values protruding to small values of q2, thrs is due to negattve curvature as the surface falls off into the well. The more rnterestmg difference is the existence of Re(ht) values at low rnergres lymg along the reactant and product channels. Thus feature IS due to the presence of a “seam” (or avoided crossmg) m the A, and X2 surfaces along which the -‘drabatrc” dragonal elements are equal

(a)

I-&

+

In the typtcai case, hz may be asymptotically assoctated wrth the vrbrattonal degree of freedom and Xt wrtl1 the transldttonal degree of freedom * Tlus IS convcment because the vrbratronal etgenvalue IS known not to caus2 exponenttation [ 113 and hence may be discarded Thn 1s not posstble for the present case because of the schrzophren1a of the etgcnvalues on opposite srdes of the seam, X, and AZ switch rdentrtres. The result of thus comphcatton IS the accumulatron of a larger value of I for highly vtbratronally e\crted products. as tllustrated for a typrcal trajectory by the “bhps” in the A: versus t plot tn fig 4. Methods to deal wrth thus problem are drscussed below. An ensemble of 500 traJectorres was run at a trans-

“;

(b)

t AS>mptotrc assrgnment of erpenvahrcs to degrees of freedom rs drscusscd m appendrx

A of ref [S]

1: 2-

I -

fr_r 3 He + Hi system (a) scaled coordmatcs Energws part of the ,\t crgenvalue of s-t Brohen hnes rsprcsent

Potcnttal energy contours in mass m eV (b) Contours of the real S Scale .rs m (a) Unrts are IO’= the drabatlc s&am (set text)

of0.75 eV wrth a well depth of about 0 2 eV as seen m fig 33. Srnce BDT theory IS prrmarrly a study of negatrve curvature of the potenttal surface, the well was expected to produce mterestmg new aspects to 258

nr: I

t

2

FIN 4. XT rersus ttme for a partrcular He + Hz trajectory leadmg to a hrgh vrbratronat energy product Notrce the unceasmg perrodtc blips of A: > 0 fate tn the trajectory These bhps are the source of the ‘kxcess” burld-up of I A: umts are IO” s-*_ Trme uruts are IO-t3 s (ror a more complete descrtptton of X: versus t plots, see ref [?I )

Volume 69, number 2

CHEMICAL

PHYSICS

latlonal energy of 0.900 eV and Initial u = 4. Applying th2 I > 7 critenon provides outstanding accuracy_ With only 25 mtegratlons, 96% of the traJectones m the ensemble had error less than or equal to +O 0 1. In fact, because of the larger I ralues mentionzd above, the value of 7 IS not sufficiently restrictive for tlus case. By employing an I > 9 criterion, only 24 integratlons are required and 95% of the trajectones have error less than +O 01 _ Thus, even with the above comphcatlons, the pattern predictlon techmquz IS highly accurate. The character of the stability elgenvalues for systems with wells may be approached m two manners, one pragmatic and the other analytic. The pragmatic approach is to simply discard the unwanted areas of Re(X1) > 0. This may be effected by drawing a bou around the reactive regon of the surface and restncting th2 integration of I to traJectory points wlthm the bos. We exammed one trammg set generated m this manner. The resulting I values are mu& smaller than the pnor ones, and a cntenon of I > 7 was found fo be excessively restnctlve so that great errors resulted. By lowermg the mmlmum to 5, we required only 6 integrations of the 500 traJectones, with 95% havmg error less than +O 01. The mathematical approach IS based on the critical point analysis of CerJan and Remhardt [ 121. They provide a systematic means of tiscardmg “secondorder instablhtres” based upon the etistenc2 of competmg critical pomts m the flow of the exponentiatlon. Unfortunately, their method, when applied to He + Hz, requires an analytic solution to an extremely complex system of algebraic equations, a solution which we were unable to find. We surmise that the cntlcal pomt analysis would assign second-order mstabditles m the asymptotic channels, and would thus have the result of telling us how to draw the “box” described above.

4. Conclusion fis study has clearly illustrated the utility of the technique of pattern pre&ctlon. Although we have not attempted a cost or time savmgs analysis, the adVantages should be ngniticant. With an I > 7 cnterion and a smart trammg set of 100 traJectories, this technique produces vutually flawless predictions on each

15 January

LETTERS

1980

of the three reaction systems wlule requiring only 1755, 1455, and 19%, respectively, of the total number of integrations. Extension of this study remains necessary. The complete apphcation of the critical point analysis to the HeG surface and to other similar systems with wells IS needed

to calculate

I in a less arbitrary

manner.

Moreover, we have made no study of pattern prediction in systems wth more than two degrees of freedom. We may conjecture that the extension to threedunennonal reactions is straightforward_ A similar techruque for predicting unstable trajectories, based on a post hoc calculation of exponential separation of neighbormg trajectories, has been successfully applied by Duff and Brumer 1131 to the H + ICI and K + NaCl three-dimensional systems. At any rate, the pattern predlctlcn technique appears to be readily and rehably applicable to collinear atom-diatom systems.

References [ 1] J Ford, Advan Chem. Ph)s. 24 (1973)

155; P. Brumer, Intramolecular Energy Transfer: Theories for the Onset of Statistical Behavior, Advaa Chem Phyr to be pubbshcd. [ 21 J-S Hutchmson and R E. Wyatt, J. Chem. Phys_ 70 (1979) 3509. [ 3 ] J-H. McCreery and G. Wolken, Chem. Phys. Letters 46 (1977) 469. [4] P. Brumer, J Comput. Phys. 14 (1973) 391; P Brumer and J-W. Duff, J. Chem. Phyr 65 (1976) 3566 15 ] J W. Duff and P. Brumer, J. Chem. Phys. 67 (1977)

4898 [6) M. Toda, Phys Letters A48 (1974)

335. [7] J.T. Muckerman, J. Chem. Phys. 56 (1972) 2997. [81 G C- Schatz. J.M. Bowman and A. Kuppcrmann. J. Chem Phys. 63 (1975) 674. [91 M.D. Pattengtll and J-C. Polanyi. Chem. Phys 3 (1974) 317. [lo] PI. Kuna, Chem Phys. Letters 16 (1972) 581. [ 11) G. Bennettm. R. Brambtlla and L. Galgani, Physica 87A (1977) 381. [ 121 C. Cejan and W P. Reinhardt, Critical Pomt Analyxis of Instabditres m Hamiltoman Systems: Classical MechanICSof Stochastic Intramolecular Energy Transfer, to be pubbshed. [ 131 J-W. Duff and P. Brumer, Exponentwting Trajectories and Statistical Behavror Three-Dimensional K + Nail and H + ICl, to be published.