Analysis of the reaction He++He+He→He+ 2+He. II. Improved samplingJ. E. Russell Citation: The Journal of Chemical Physics 84, 4394 (1986); doi: 10.1063/1.450010 View online: http://dx.doi.org/10.1063/1.450010 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/84/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in PIV Analysis of Cavitation Flow Characteristics of He II AIP Conf. Proc. 823, 1677 (2006); 10.1063/1.2202594 Multiple transition states in chemical reactions. II. The effect of angular momentum in variational studies ofHO2 and HeH+ 2 systems J. Chem. Phys. 93, 5751 (1990); 10.1063/1.459569 Analysis of the reaction He++He+He→He+ 2+He. III. Effects of resonant quasibound states J. Chem. Phys. 91, 1015 (1989); 10.1063/1.457226 Analysis of the reaction He++He+He→He+ 2+He J. Chem. Phys. 83, 3363 (1985); 10.1063/1.449197 Dynamics of the Collinear H+H2 Reaction. II. Energy Analysis J. Chem. Phys. 54, 3592 (1971); 10.1063/1.1675385

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Analysis of the reaction He+ + He + He-+He2+ + He. II. Improved sampling J. E. Russell Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221

(Received 20 October 1985; accepted 3 December 1985)

The rate coefficient for He+ ions recombining in helium gas is computed using the Wigner-Keck Monte Carlo trajectory method. The present calculation is a refinement of a previous one. the principal difference between the two being that simpler. more reliable, and much less time consuming procedures are now employed to select the trajectories. The results of the new calculation are consistent with, and statistically more accurate than, those of the previous one. The improved sampling procedures. unlike those employed in the previous calculation, have the appearance of being easily applicable to other, similar three body association reactions.

I. INTRODUCTION

This is the second of what is intended to be a series of papers investigating theoretically a certain type of three body association reaction by using the Wigner-Keck Monte Carlo trajectory method. 1--6 In the first paper,7 this method was applied to the recombination ofHe+ ions in helium gas. The purpose ofthe present paper is to describe Monte Carlo sampling procedures for the He + problem that are far less cumbersome and apparently more reliable than those em­ployed in Ref. 7. It is expected that the improved sampling procedures can also be satisfactorily employed. possibly with minor modifications, in future papers that will deal with other, similar reactions in which each of the three reac­tants is an atom or an ion and in which at least two of the relevant two body potentials are attractive and relatively strong. In particular, it is intended to calculate the rate coef­ficient for formation of the molecular ion He (,uHe ) + in heli­um gas, which has recently been measured with relatively high accuracy at room temperature in an experiment at SIN.s

The computation of a rate coefficient using the Wigner­Keck Monte Carlo trajectory method requires the judicious selection of points on a suitably defined surface in phase space. This surface. which we shall call the Wigner-Keck surface, must be crossed an odd number of times by a trajec­tory if a molecule is to be formed. The points selected on this surface provide starting values for trajectory calculations that can be used to give an estimate of the equilibrium rate coefficient k. A detailed description of the method, as adapt­ed to the He+ problem, is given in Ref. 7 and will not be repeated here. The present paper is concerned almost entire­ly with the selection of points on the Wigner-Keck surface. Apart from the use of a somewhat different method to obtain an empirical fit to the temperature dependence of the com­puted values of k, all other features of the calculation de­scribed in Ref. 7 remain unchanged.

II. SAMPLING

In broad outline, the sampling procedures employed in Ref. 7 consist of two steps. The first step is the selection of a very large number of points on the Wigner-Keck surface. These points lead to an estimate ofthe absolute flux through the surface. The second step is the selection, from the very

large number of points, of a much smaller number to be used in trajectory calculations. The trajectory calculations lead to an estimate of the net flux in one direction through the sur­face. The net flux gives the equilibrium rate coefficient k. Both steps are modified in the present paper.

A. First step

Apart from Euler angles specifying the orientation of the three body system as a whole, eight variables are required to specify a point on the Wigner-Keck surface, and the den­sity of flux through the surface at this point depends only on these eight variables. 2 The variables employed in the present paper are the same as in Ref. 7, where they are denoted by Zi'

i = 1,2, .... 8. In the present paper, as in Ref. 7, a very large number of points on a surface in phase space are first chosen by Monte Carlo selection using a random number density of the general form

8

density = II Pi (Zi ), Zi, min <'Zi <'Zi, max' (la) i=1

where the Pi are normalized functions satisfying the rela­tions

Pi (ZY~O, f"max Pi (Zi )dz; = 1. zi,min

(lb)

In most instances, a function Pi can be specified convenient­ly by the function Qi' which is defined by

Qi (z;> = {~inPi (Zi )dzi· (2)

The modification of the first step of the sampling procedures consists of employing a set offunctions Qi different from the ones employed in Ref. 7. However, thelimitszi• min andzi• max

appearing in Eqs. (1) and (2) are the same as before. It must be added that the surface specified by all points satisfying the inequalities appearing in Eq. (Ia) is not the Wigner-Keck surface. As explained in more detail in Ref. 7, the Wigner­Keck surface for the He + problem is composed symmetri­cally of parts of three such surfaces. For this reason, the selection of a set of values for the eight variables Zi by using the functions Qi does not guarantee the selection of a point on the Wigner-Keck surface. A test must be applied in every instance. In the present calculation, it was found that ap-

4394 J. Chern. Phys. 84 (8),15 April 1986 0021-9606/86/084394-07$02.10 @) 1986 American Institute of Physics

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J. E. Russell: The reaction He++He+He-..Hei+He.1I 4395

proximately 55% of the randomly selected sets of values of Zi

represent points on the Wigner-Keck surface. This is nearly twice as large as the fraction attained in the original calcula­tion.

In Ref. 7, each of the functions Qi was determined nu­merically by an iterative procedure. This procedure was complicated and tedious. The functions Q/ obtained in this way depend on the detailed behavior of the assumed three body potential used in their determination. Furthermore, the functions Qi obtained in Ref. 7 have the disquieting property of leading to a significantly more efficient Monte Carlo calculation of k when employed with certain three body potentials different from the one used in their own nu­merical determination. This is possibly a consequence of the Q/ having been determined by requiring an efficient calcula­tion of the absolute flux through the Wigner-Keck surface rather than the net flux. In the present calculation, each of the Qi is assumed to have a functional form that is simple and, except at a very few points in some instances, analytic. Although the forms chosen for the Qi in the present paper are rough-in one instance, exceedingly rough-approxi­mations to the functions determined numerically in Ref. 7, it cannot be said that any of them are very dependent upon the detailed behavior of any reasonable assumed form of the He3+ potential. As explained in more detail in Sec. III, the new functions were employed successfully to compute rate coefficients for several different assumed forms of the He3+ potential, with about the same degree of efficiency in each instance. Because there was no need to tabulate the Qi before using them to select values of Zt, the total time and effort required was substantially less than for the calculation de­scribed in Ref. 7, even though about four times as many trajectories were C?mputed in the present calculation.

Each of the functions Q2(Z2)' Q6(Z6) , Q7(Z7)' and Qg(Zg) employed in the original calculation varies more or less linearly throughout the entire range of its respective de­pendent variable. For this reason, each ofthese functions is given a precisely linear dependence in the present calcula­tion. This is equivalent to employing crude Monte Carlo se­lection for Z2' Z6' Z7' and Zg.

Slightly more elaborate forms are necessary for QI (ZI)

and Qs (zs)' The variable Z I is the height of the rotational barrier in units of kT, and Zs is a dimensionless quantity appearing in the Boltzmann factor. The functions QI and Qs

employed in the original calculation both vary more or less linearly with the logarithms of their respective dependent variables. But because these roughly linear variations do not extend throughout the entire relevant ranges ofln(zl) and In(zs), it was decided to give QI and Qs in the present calcu­lation the general form

Qi (Zi) = Qi,J - I + (Qt, J - Qi, J - I )

X [In(Zi) -In(Zi,J_1 )]I[ln(zi,J) -In(Zi,J_1 n, (3a)

where

i = 1 or 5, J = 1, 2, or 3,

z/, J _ I <Zi <Zt, J'

Zi.O =z;,min' Zi,3 =zi,max'

(3b)

(3c)

(3d)

0= Qi,O <Qt, I <Q .. ,2 <Qi,3 = 1. (3e)

The numerical values assigned to Zt, min and z/, max and to the other parameters in Eq. (3) in order to obtain rough ap­proximations to the functions QI and Qs employed in Ref. 7 are

(4a)

Zl,max = (maximum possible height of rotational barrier)/

(kn, (4b)

ZI,I =e- 2/(T/SO), ZI,2 =e+ 2/(T/SO), (4c)

QI,I = 0.25, QI.2 = 0.90, (4d)

and

ZS,min =e- 4.S

, zS,max =e+ 2.2

,

zs, I = e - 1.2, ZS,2 = e + 1.3,

Qs, I = 0.10, QS.2 = 0.95.

(5a)

(5b)

(5c)

The variablez3 is the ratio r 3/r I2, where r l2 is the separa­tion between the two nuclei associated with the rotational barrier, and r3 is the separation between the remaining nu­cleus and the c.m. ofthe first two. The function Q3(Z3) em­ployed in Ref. 7 has a very complicated behavior. In general, its variation from zero to unity occurs over a narrow range of values of Z3' with the location of this range depending sensi­tively on both the height of the rotational barrier and the separation r 12. In the present calculation, the function Q3 (Z3) is assigned the simple form

Q3(Z3) = Q3,J-I + (Q3,J - Q3,J-I)

X (Z3 - Z3,J-I )/(Z3,J - Z3.J _ I), (6a)

where

and

J= 1, 2, or 3,

Z3,J_I <Z3<Z3,J'

Z3, min = 10-6, Z3, max = S.O,

Z3, I = 3.0, Z3,2 = 4.5,

Q3, I = 0.95, Q3,2 = 0.99.

(6b)

(6c)

(6d)

(6e)

(6f)

(6g)

(6h)

The approximation represented by Eq. (6)-if indeed it really deserves to be called an approximation-is very sweeping. Its use nearly amounts to the use of crude Monte Carlo selection for all values of Z3' We believe that using this representation of Q3 is, by far, the most inefficient feature of the present calculation. However, the numerical results re­ported in Sec. III indicate that the inefficiency introduced by the use of this function is more or less offset by the effects of other modifications in the sampling procedures. In fact, it seems fair to state that one of the advantages of the new sampling procedures is that this very simple form for Q3 can be used successfully.

The remaining variable, Z4' is the cosine of the angle a I between the vector separations r l2 and r3• In the present cal­culation, the function P4(Z4) is given the form

P4(Z4) = c[ (IZ41 + d)/(l + d) ]2p, (7a)

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4396 J. E. Russell: The reaction He++He+He-+He:+He.1I

where

d = dolz3, P = POZ3'

do = 0.025, Po = La, 1O-6 <;z3<;S'O'

c = normalization factor.

(7b)

(7c)

(7d)

This is the only instance in the present calculation where a function P j depends explicitly not only on Zj but also on one of the other seven variables. The dependence of P4 on the variable Z3 causes the selected values of cos a I to be uniform­ly distributed as Z3 becomes quite small, but it causes the larger values of Icos all to be favored as Z3 becomes relative­ly large. Both of these features were found to enhance the efficiency of the computation. However, the numerical val­ues assigned in Eq. (7c) to do and Po were chosen after only a modest amount of not very systematic experimentation, and it is possible that the selection of another pair of values could have resulted in an even more efficient computation.

B. Second step

As explained below, the overall organization of the pres­ent calculation involves three parameters, NlO Nz, and te ,

with the latter two determining some details of the second step of the sampling procedures. The particular numerical values assigned to these parameters were chosen largely be­cause of the way the entire calculation had been planned. It is, therefore, appropriate to include a description of a few general features of the entire calculation in the description of the second step.

When the present calculation was planned, it was decid­ed to devise a uniform procedure for a single evaluation of k that, if necessary in order to attain a desired degree of statis­tical accuracy, could be repeated one or more times, each time with the same number NI of sets of Zj generated in the first step, but with different sequences of random numbers in both the first and the second steps. The numerical value as­signed to NI is

NI = 1.6x 105• (S)

This number was chosen mainly because attempting to select a significantly larger number of points in a single run would have resulted in scheduling difficulties at the computer. This choice of NI results in the selection of approximately 9 X 104

points on the Wigner-Keck surface. The second step of the sampling procedures provides the

basis for an approximate evaluation of the net flux associated with the very large number of points selected on the Wigner­Keck surface in the first step. The most thorough possible processing of these points would require the computation of a trajectory for each of them. While such an undertaking would be prohibitively lengthy, the results can be formally expressed as a sum St of terms proportional to the corrected flux density at each of the points divided by the random number density defined in Eq. (la). The corrected flux den­sity takes into account the effect of multiple crossings of the Wigner-Keck surface and also, in those instances where the trajectory crosses the surface an odd number of times, the effect of outward tunneling through the rotational barrier after the molecular ion has been formed. But because the random number density specified by Eq. (la) is not, as it

ideally should be, proportional to the correted flux density, the terms in St are not all of comparable magnitude. In fact, despite efforts to make the first step of the sampling proce­dures as efficient as reasonably possible, they span several orders of magnitude. Therefore, the accuracy of any esti­mate of this sum depends on how reliably the larger terms are taken into account. In the present calculation, the sum St is evaluated approximately by dividing it into two sums, Sl and Sz, with Sl evaluated exactly by computing both the trajectory and, if need be, the tunneling correction for each of the terms involved, and with Sz evaluated approximately by performing the same sort of computations for representa­tive terms chosen by Monte Carlo selection. By definition, the sum Sl is over all terms in St for which the uncorrected absolute flux density divided by the random number density is greater than the parameter te' The Monte Carlo evaluation of S2 is accomplished by sampling a relatively small number N2 of the remaining terms. This procedure is different from the one followed in Ref. 7, where the entire sum St was eval­uated approximately by the Monte Carlo method. As in Ref. 7, the sampling involved in the second step is called the sec­ond sampling, and the random number density used in this sampling is chosen to be proportional to the uncorrected absolute flux density at each of the points involved divided by the random number density specified by Eq. (la).

When the present calculation was planned, it was decid­ed, somewhat arbitrarily, that a single evaluation of k would involve the selection, for trajectory calculations, of a total number of points on the Wigner-Keck surface roughly equal to the number selected in the second step of the original calculation, which was between 1200 and 1600. The values chosen for NlO Nz, and te are such that this number is roughly 1250. The value chosen for te is

te = 50 times the average value of the uncorrected abso­lute flux density divided by the random number density specified by Eq. (1a), as determined for all NI points selected in the first step, but with the uncorrected flux density replaced by zero for each point not lying on the Wigner-Keck surface.

(9)

This value of te , which was chosen after only a very small amount of experimentation, causes Sl usually to consist of between 250 and 450 terms. The value assigned to N2 is

N z = 1000, (10)

but the actual number of trajectories that must be computed in the estimate of Sz is, in every instance, only about 900 because the second sampling selects some points more than once. It must be acknowledged that no systematic attempt was made to determine, for a given amount of computer time, optimum values for N I , Nz, and te , even though the statistical error in k should depend to some extent on these parameters. However, it was generally found that Sl ac­counts for most of the estimated statistical error in k, even though it accounts for somewhat less than half of the esti­mated value of k.

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J. E. Russell: The reaction He++He+He--+Hei+He.1I 4397

III. NUMERICAL RESULTS AND DISCUSSION

As before, computations were performed for four differ­ent assumed forms of the relevant He3+ potential surface. These three body potentials, which all have the same general form, are specified by sets of parameters labeled A, B, C, and D. Detailed descriptions of these potentials are given in Ref. 7. We only repeat here that the potentials specified by pa­rameter sets A and B are very nearly identical except for configurations in which at least two of the internuclear sepa­rations are less than approximately 4.0 a.u. As before, no very significant contribution to any computed value of the rate coefficient k was found to be associated with trajectories for which the second largest momentary separation was ever less than approximately 4.0 a.u. For this reason, the results for parameter sets A and B are combined in the present pa­per, as they were in Ref. 7.

A. Analysis of results

Most of the results of the present calculation are shown in Fig. 1, where values of k are plotted with respect to the temperature T. Results for parameter sets A and Bare shown in Fig. 1 (a), while those for sets C and D are shown separately in Figs. 1 (b) and 1 ( c ). Each point in these figures represents a single evaluation of k obtained using the compu­tational parametersNI , N2, and tc specified by Eqs. (8), (9), and (10). In each of these figures, there are four points for each of 13 equally spaced temperatures. It was decided to make four separate evaluations of k at each temperature in order to attain a level of accuracy somewhat higher than attained in Ref. 7. The spread in the four values of k at a given temperature is a rough indication of the statistical er­ror, which obviously does not depend sensitively on the pa­rameter set. In each of these figures, a curve of the form

kilt = k300(T /300) -x (11)

is fitted to the numerical results. The values of k300 and x shown in Fig. 1 are obtained by a weighted least squares fit to the weighted averages of the four Monte Carlo estimates of k computed for each temperature. The weighting factors em­ployed in determining the average of a set of four points are proportional to the squared reciprocals of Monte Carlo esti­mates of the standard deviation for each of the estimates of k, and the weighting factors employed in the least squares fits are proportional to the squared reciprocals of the estimated standard deviations associated with the weighted averages of the sets of four points. The curve labeled kexp in each of these figures is a fit to the experimental data of Johnsen et al.9 As before, the experimentally determined temperature depen­dence of the rate coefficient is reproduced rather satisfactori­ly by the calculations for each of the parameter sets, but the computed values of k are 10% to 20% lower than the experi­mental values, with the largest discrepancies being associat­ed with parameter sets C and D.

Some typical estimated statistical errors are shown in Figs. 2 and 3. The error bars in these figures represent stan­dard deviations. Figure 2 shows, for each ofthe 13 tempera­tures considered, a representative Monte Carlo estimate of k and its statistical error, as obtained for parameter set C. For the purpose of comparison, the weighted least squares fit

shown for parameter set C in Fig. 1 (b) is reproduced in Fig. 2 together with the fit to the experimental data. The larger error bars in Fig. 2 are generally associated with estimates of k substantially larger than kilt. But even in these instances, the errors are such that the estimates of k are by no means incompatible with k fit. This is generally true of all of the Monte Carlo estimates of k for each of the parameter sets, not just those shown in Fig. 2. It must be emphasized that the curve labeled k fit in Fig. 2 is a fit, not to the 13 points shown, but rather to weighted averages of four sets of such points. These weighted averages and their estimated errors are shown in Fig. 3, which also shows the same curves shown in Figs. 1 (b) and 2. The results shown in Fig. 3 indicate that a fairly high level of statistical accuracy is being claimed in the present calculation.

The procedure followed to obtain the values of k300 and x shown in Fig. 1 is different from the one used in Ref. 7, where an unweighted least squares fit was employed in every instance. Using a least squares procedure with weights pro­portional to the squared reciprocals of the errors is welljusti­fied if the errors are known fairly reliably. 10 But it must be acknowledged that the statistical errors in the present calcu­lation, especially the larger ones, are not known very accu­rately. Nevertheless, we believe that a weighted fit is more appropriate than an unweighted one. However, for the pur­pose of comparison, Table I lists values of k300 and x ob­tained from results of the trajectory calculations by using both types of least squares fit, weighted and unweighted, with the unweighted fits being made to sets of unweighted averages off our estimates of k. Values of k300 and x obtained with both methods are listed also for the original calculation. In the case of the original calculation, the only weighting that was done was in the curve fitting procedure, since only one determination of k had been made for a given tempera­ture and parameter set. It seems fairly clear that the analyses of both the present and the original calculations are not espe­cially sensitive to the type of curve fitting employed, though the temperature dependences of the two calculations are, for a given parameter set, generally in better agreement for the weighted fits.

Table I also lists estimated statistical errors in the values of k300 and x. These errors, which are standard deviations, are obtained by well established procedures either from the estimated errors in k (in the case of a weighted least squares fit) or from the differences between the estimate of k and the value of k fit (in the case of an unweighted least squares fit). II Table I also lists values of the rm ( weighted) s deviation of the computed values of k from k fit' with the weighting fac­tors, or the lack of them, for the squared deviations being the same as those used in the least squares fit. In interpreting the statistical errors and rm(weighted)s deviations listed in Ta­ble I, it must be recalled that in the original calculation, unlike the present one, the combined results for sets A and B were obtained by computing about twice as many trajector­ies as for either set C or set D. No matter which method of curve fitting is considered, the numbers presented in Table I, especially the rm(weighted)s deviations, indicate that the present calculation is more accurate than the original one, and they also indicate that the two calculations are, within

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4398

u CD 1/1 ....... "' E U

;;; I 0 ::. .. c:: . ! U ;: ... CD 0 U CD .. ca

II:

u G) 1/1 ;;--E u

2.0

1.0

0.5

2.0

~ 1.5

.... -.. c:: CD ·u

!E G) 1.0 o u ! ca II:

0.5

(a)

SetA _

Set B •

(b)

J. E. Russell: The reaction He++He+He-+Hei+He.1I

-

,I-kexp = 1 .10 IT /300)-0·38

.' , -, , -, ',.. ... .............

......... -... • I. -; ..........

kfit = 0.95 IT/300) -0.34

Parameter Sets A & B

100 200

Temperature (K)

•

, , , "', •

' ... ........ .......

300

• ... ----•

kilt = 0.86 IT/300) -0.41

Parameter Set C

100 200 300

Temperature (K)

(c)

2.0

U CD 1/1 ;;--E u ;;; 1.5

6 .... -.. c:: CD 'u ;:

1.0 -CD 0 U

CD .. ca II:

0.5

•

• kexp = 1 .10 (T /300)-0·38

\/ . " , , . " . ... ".......... .

.... ... ... ~ . ....... . -...

• kfjt = 0.88 (T /300) -0.38

Parameter Set 0

100 200 300

Temperature (K)

FIG. 1. (a)-(c) The equilibrium rate coefficient k for the reaction He+ + He + He-He2+ + He, as computed for parameter sets A, B, C, and D.

Each point is an estimate obtained by first sampling approximately 9 X lcr points on the Wigner-Keck surface and then selecting from them a much smaller number, roughly 1250, for further investigation by means of trajec­tory calculations. In each instance, the solid curve is a weighted least squares fit to the weighted averages of the four points computed for each of the 13 temperatures considered, while the dashed curve is a fit to the experi­mental data of Johnsen et al.

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J. E. Russell: The reaction He++He+He-.Hei+He.1I 4399

u := ;0-

2.0

~ 1.5

~ ... --c ., 'u 1.0

:= GI o o ., -II m:

0.5

k.xp = 1 .1 0 (T /300)-0·38

" \ , ,

/ kilt = 0.86 (T /300) -0.41

Parameter Set C

100 200 300

Temperature (K)

FIG. 2. The equilibrium rate coefficient k for the reaction He+ + He + He-Het + He, as computed for parameter set C. Each point is

an estimate obtained by first sampling approximately 9 X 1at points on the Wigner-Keck surface and then selecting from them a much smaller num­ber, roughly 1250, for further investigation by means oftrajectory calcula­tions. The error bars represent Monte Carlo estimates of the standard devi­ation. The solid curve is a weighted least squares fit, not to the set of points shown in this figure, but rather to the weighted averages offour points com­puted for each ofthe 13 temperatures considered. The dashed curve is a fit to the experimental data of Johnsen et al.

the estimated errors, in agreement with one another. Finally, it must be noted that each of the values of x listed in Table I is

2.0

k.xp = 1 .1 0 (T /300)-0·38

,I U \ , GI , III ....... , ... " E 1.5 " U " " ;;; ... ..... I ..... 0 ......... ... ..... ....... -- -... c .... -GI 1.0 'u ~ -GI 0 klil = 0.86 (T/300) -0.41 0 GI -II m: 0.5

Parameter Set C

100 200 300

Temperature (K)

FIG. 3. The equilibrium rate coefficient k for the reaction He+ + He + He-He2+ + He, as computed for parameter set C. Each point is a

weighted average of four separate computations. The error bars represent standard deviations. The solid curve is a weighted least squares fit to the points shown in this figure. The dashed curve is a fit to the experimental data of Johnsen et al.

compatible with the experimentally determined value of this parameter, which is x = 0.38 ± 0.06.9

TABLE I. Values of k300 and x obtained by fitting the expression k300( T /300) - x to computed values of the rate coefficient k. Results are presented for both the present and the original calculation, for both weighted and unweighted least squares fits, and for parameters sets A, B, C, and D, with the results for sets A and B combined in every instance. The errors listed for k300 and x are estimates ofthe standard deviation. Values are also given for the rm(weighted)s deviation of the computed values of k from values of k300( T /300) - x •

Least rm(weighted)s squares k300 deviation

Calculation fit Set (10-31 cm6/s) x (10-31 cm6/s)

Present weighted A,B 0.95 ±0.02 0.34 ±0.03 0.037 C 0.86 ±0.02 0.41 ±0.03 0.036 D 0.88 ±0.02 0.38 ±0.03 0.039

unweighted A,B 0.96 ± 0.02 0.4D±0.03 0.058 C 0.91 ±0.03 0.37 ±0.04 0.061 D 0.90 ± 0.03 0.44±0.05 0.078

weighted A,B 0.97 ±0.02 0.35 ±0.04 0.075 C 0.86± 0.03 0.4D ± 0.06 0.057 D 0.92 ± 0.03 0.35 ± 0.05 0.053

unweighted A,B 0.97 ±0.04 0.36±0.06 0.098 C 0.86 ±0.03 0.44 ± 0.04 0.071 D 0.93 ± 0.03 0.34±0.04 0.061

• Reference 7.

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4400 J. E. Russell: The reaction He+ + He+ He-.He;- + He. II

B. Other reactions

Before discussing the applicability of the new sampling procedures to other reactions, it is convenient to discuss the quantity kmax , which is defined in Ref. 7 and which is pro­portional to the absolute flux through the Wigner-Keck sur­face. As reported in Ref. 7, it is about an order of magnitude larger than k, which is proportional to the net flux in one direction. The quantity kmax is, of course, much more easily calculated than k because no trajectories are required. Be­cause of this, the determination of the precise form of the function P4 in the present calculation proceeded, as it did in a much more elaborate fashion for each of the Qj in the origi­nal calculation, by trying to achieve an efficient computation of kmax • The determination of the Qi in the original calcula­tion was accomplished by requiring kmax to be given rather accurately, to within less than 5% in most instances. The form of P4 in the present calculation was obtained without placing such a stringent requirement on the statistical accu­racy of kmax • In fact, it was ultimately found in the present calculation that estimates of kmax , especially for low tem­peratures, are generally less accurate than those of k. In oth­er words-and for reasons that are admittedly not really understood-the sampling procedures in the present calcu­lation are, in contrast to those employed in Ref. 7, somewhat more efficient in singling out representative trajectories that cross the surface an odd number of times and contribute to the net flux in one direction than they are in selecting repre­sentative trajectories that cross an even number of times and contribute only to the absolute flux.

It was asserted at the beginning of this paper that the new sampling procedures should be easily applicable to oth­er, similar three body association reactions. One reason for this assertion is the fact that, in contrast to the original calcu­lation, the statistical accuracy of k in the present calculation does not appear to depend sensitively on the assumed form of

the Het interaction, even though, as discussed in Ref. 7, the differences between the potentials obtained with parameter sets B, C, and D are not negligible. Another reason is that our own experience in devising both the old and the new sampling procedures suggests that, apart from the param­eters specifying Q3' there are only a few of the parameters in the definitions of the new Qj for which reasonable adjust­ments would seem likely to have an appreciable effect, one way or the other, on the efficiency of a rate calculation. Be­cause the position of the rotational barrier as a function of angular momentum is one of the quantities most likely to be significantly affected by a change in the potential, and also because the position of this barrier defines a boundary of the Wigner-Keck surface, we believe that the most likely candi­dates for further adjustment in a rate calculation for another reaction are the parameterszl , I andzl ,2 in QI' The param­eters do andpo inP4 andzs, I andzs, 2 in Qs are also possibili­ties. We also believe that suitable adjustments could be readi­ly made, as they were for do and Po in the present calculation, by requiring the calculation of kmax to be only moderately efficient, especially at low temperatures.

IE. P. Wigner, J. Chern. Phys. 5, 720 (1937). 2J. C. Keck, J. Chern. Phys. 32, 1035 (1960). 3J. C. Keck, Discuss. Faraday Soc. 33,173 (1962). 4B. J. Woznick, Res. Rep. No. 223 AVCO-Everett Research Lab., Everett, Mass. (1965).

5J. C. Keck, Adv. Chern. Phys. 13, 85 (1967). 6J. C. Keck, Adv. At. Mol. Phys. 8, 39 (1972). 7J. E. Russell, J. Chern. Phys. 83, 3363 (1985). 8H. P. von Arb, F. Dittus, H. Heeb, H. Hofer, F. Kottrnann, S. Niggli, R. Schaeren, D. Taqqu, J. Untemahrer, and P. Egelhof, Phys. Lett. B 136, 232 (1984).

9R. Johnsen, A. Chen, and M. A. Biondi, J. Chern. Phys. 73,1717 (1980). lOSee, for example, N. Arley and K. R. Buch, Introduction to the Theory of

Probability and Statistics (Wiley, New York, 1950), pp. 181-3. liSee, for example, A. A. Clifford, Multivariate Error Analysis (Wiley, New

York, 1973), pp. 47-58.

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