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A simple technique for obtaining reactive differential cross sections from highly-averaged trajectory data
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
5 February 1982
A SIMPLE TECHNIQUE FOR OBTAINING REACTIVE DIFFERENTIAL CROSS SECTIONS FROM HIGHLY-AVERAGED TRAJECTORY DATA Howard R. MAYNE Lash Miller Chemical Laboratory, University of Toronto, Toronto, Ontario, Canada M5S 1A1
Received 25 September 1981; in final form 5 November 1981
A technique requiring only the mean reactive scattering angle and opacity, both as a function of impact parameter, is proposed to calculate reactive differential cross sections. The method was tested for H + H 2 and D + HBr. It is fairly sensitive to initial conditions even with very few reactive trajectories.
1. Introduction One of the aims of performing quasiclassical trajectory calculations is to ascertain the sensitivity of various reaction attributes (such as product vibration, rotation and angular distribution) to initial conditions. For a t o m - d i a t o m reactions, much progress has been made in determining the relative roles played by reagent vibrational, rotational and translational energy in enhancing reaction, and their effect on the product attributes [ 1 - 3 ]. One final attribute which has so far eluded satisfactory elucidation, however, is the reactive differential cross section (DCS). One reason may be that few experimental results are available for comparison. Another, though, is more practical: they are difficult to compute. Since one requires many histogrammic angular intervals to obtain reasonable detail in the DCS, one necessarily has few trajectories going into each interval. The results are, then, noisy due to large Monte Carlo error [4]. Computation of DCSs using trajectories has been discussed in a recent review [5]. Procedures for overcoming the difficulties inherent in the traditional histogram method [5 ] have been proposed recently. One involves the expansion o f the DCS in a series of Legendre polynomials [6]. There are, however, problems in knowing just when to truncate this series. Another group [7,8] has suggested expanding the DCS in a Fourier sine series. Although in principle extremely accurate, this technique seems to require an
inordinately large number of trajectories. Particularly if one is interested in the behavior of the DCS for a series of initial conditions, one obviously cannot afford to run many thousands of trajectories for each case. The problem is further compounded if one wishes to investigate a reaction which has a pathologically low reactivity. In this letter we propose a new technique, which seems capable of determining DCSs for direct A + BC type reactions to reasonable accuracy using averaged trajectory data. Since it requires fairly few trajectories, it is cheap (or, if the system is very unreactive, at least feasible) and well suited to investigating the effect of changes in initial conditions. This technique is based on the well-known fact that trajectory methods are more accurate for highlyaveraged quantities than for highly-specific ones [5,9, 10]. Recent calculations on the H + H 2 reaction have shown that the average reactive scattering angle, e), is strikingly well correlated with the impact parameter, b, by a straight line [11,12]. In fig. 1 we show a plot of three e)(b) for two systems, obtained from trajectory data: one for H + H 2 ~ H 2 + H on the P o r t e r Karplus [13] surface; two for D + HBr ~ DH + Br (abstraction) on the Parr-Kuppermann [ 14] surface. The straight line fit property is common to all of them, within the statistics. Thus, although the trajectory distribution about the mean may be somewhat "ragged", the mean itself is a smooth function o r b . The aim o f the present
0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
33
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
spread randomly between b t and bl+1 [15] . ) N trajectories are run for each I . N m scatter into the mth angular box. The probability,pR(®m) (for given l) of scattering into final angle ®m is then given by
180
;ii
L
P~l (Om) Nm - N-
b[bohr]
D+HBr
(b
I
gO --
05
I0
15
20
25
10
35
il
0 b[bohr]
Fig. 1. Centre-of-mass mean reactive scattering angle, (9, versus impact parameter, b, for (a) H + H2 at Etran s = 0.43 eV, (b) D + HBr at Etran s = 0.10 eV (*) and 0.25 (A). Error bars are one standard deviation. The results for (a) represent a total of 1098 reactive trajectories; those for (b) represent 29 (0.10 eV) and 50 (0.25 eV) reactive trajectories respectively. technique is to smooth these distributions, and use the smoothed results to construct the DCS. We shall do this by constructing physically reasonable distributions about ~)(b) for several values o f b .
2. Method The quasiclassical histogram method used here is as follows. Every reactive trajectory has a centre-ofmass product scattering angle, ®, which is the angle between the initial collision axis and the centre-ofmass motion of the product diatom. The interval 0 ~< ® <~ n is divided into n equal boxes, each of width AO centred at 0 m = n - (m + 1/2)AO, m = 0, ..., n -- 1. O is boxed to the nearest @m value. We now consider scattering from a fixed value of the initial orbital angular momentum quantum number, l [ 15 ]. (In fact, the impact parameter, b, is
Nm ~'Nm Y.Nm N "
(1)
Z N m/N is simply/~/, the probability of reaction for given l. This can be large or small depending on the reactivity of the system. The term Nrn / Y_,Nm =Pl(Om ) is typically considerably less than unity, n, however, has to be fairly large in order to obtain a sufficiently detailed DCS. Thus there is considerable statistical noise in the rather specific term,Pl(O m). Averaged quantities (first moments), on the other hand, are accurately given by trajectory calculations [5,9,10]. Thus we expect the mean angle, ~) (= ~,NmOm/~,Nrn ) to be much less noisy than the individual PI(Gm ). The r~ corresponding to O is given by ZNmm/~,Nm, or equivalently, by = (~ - ½AO)/AO.
(2)
Given the mean, I~, o f a noisy distribution,P(O m ), we now construct about this mean a less noisy distribution, say P(m). This treatment is analogous to the classical P-matrix method [ 16] or information theoret ic [17] approach to the synthesis of a discrete vibrational distribution from energy transfer moments [18]. The difference is that we do not use the noisy trajectory angular distribution itself as a starting point. We propose instead to approximate Pl(m) with a gaussian distribution: exp [ - ( m - rfi)2/2w2l
el(m ) =
34
5 February 1982
(27r)l12wt
(3)
This is physically reasonable, since we must average over initial phases and sum over final rotational and vibrational states to obtain the total reactive DCS. The additional parameter required, Wl, is easily found as the standard deviation about the mean from trajectory data. If fill is low, then the gaussian may have significant contribution from negative values o f m . This is meaningless here. In these cases we reduce the value of w I until 95% o f the distribution is accounted for with m > O.Pl(m ) is then normalised and has the required mean. Results using eq. (3) are compared in fig. 2 with actual trajectory data. The fit is very reasonable. It
Volume 86, n u m b e r 1
CHEMICAL PHYSICS LETTERS
5 February 1982
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Fig. 2. Probability distribution for given l o f angular b o x n u m b e r occupation, Pl(m), versus box n u m b e r , m. Points are trajectory results with typical error bars. T h e c o n t i n u o u s curve is from eq. (3). The arrow indicates t h e mean, m. (a) H + H 2 ; (b) D + HBr at Etran s = 0.25 eV. For H + H 2 t h e total n u m b e r o f reactive trajectories is 140 (l = 0) and 52 (l = 7) o u t o f a possible 400. For D + HBr the total n u m b e r o f reactive trajectories is 6 (l = 0 - 3 ) and 4 (l = 1 8 - 2 1 ) o u t o f a possible 80.
lies mostly within the error bars, particularly in those cases where the noise is low (fig. 2a). Where the noise is high (fig. 2b), the trajectory data are very uninformative, in "any case. The opacity function pR(b (l)), is also available from trajectory data. Sophisticated techniques are available for its efficient calculation [5,19]. For example, for H + H 2 in fig. 3a,pR(b(l)) is very smooth, and can be used as it stands. The D + HBrpR(b(I)) however (fig. 3b) is very noisy, and could be smoothed, by some form of least-square fit, for instance. If the situation were even worse, and, say, only bma x were known, a step function could be assumed. Reactive DCSs are given by [15]
Fig. 3. Probability of reaction p R , versus impact parameter, b. Points are trajectory results with typical error bars. (a) H + H z ; (b) D + HBr at Etran s = 0.25 eV. Results for H + H z represent a total o f 1098 reactive trajectories o u t o f a possible 4 4 0 0 trajectories run. Results for D + HBr represent a total o f 50 reactive trajectories o u t o f a possible 1040 run.
do(Ore) dm
/i2
1
2/aEttan s 2AO sin O m
Imax X ~
(21 + 1)pR (O m)
(4)
l=0
where/a is the initial reduced translational mass, Etran s is the translational energy, and lmax is the maximum value of l which yields reaction. The present approximation to the exact quasiclassical reactive DCS consists in replacing pR(o m) in eq. (4) withPt(m ) from eq. (3) multiplied by the l~l(b(l)) from trajectory data [see eq. (1)].
3. Results and discussion The present method was applied to two systems: H + H2(o = 0,] = 0) at Etran s = 0.43 eV on the P o r t e r Karplus [13] surface, and D + H B r ( u = 0 , i = 0) at Etran s = 0.10 and 0.25 on the P a r r - K u p p e r m a n n [14] surface.We used the technique as follows. ~)(b(l)) was fitted from the trajectory data using least squares to give the best straight line passing through ® = n. 35
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
This fit was then used to obtain nql [eq. (2)] and o l. AO was 5 °, whence n was 36.P/R was taken directly from trajectory data, without smoothing for the H + H 2 case. Since the D + H B r data are so p o o r , w e used an average standard deviation from the mean, ~ , rather than the/-specific w I. In addition, the pR(b(1)) function was fitted by least squares and the trajectory DCS for D + HBr were smoothed by a simple technique [20] to reduce noise. In fig. 4 we compare the DCS from trajectory data with the present method. In fig. 4a, where there are many reactive trajectories, the agreement is excellent. It should be stressed that the same curve would have been obtained from far fewer trajectories - so long as
H+H 2 0A3 . e V r i,r,-t .os ,~ REACtiVET~,~JECrOR~r . ( ~ T~ o~
(a)
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TRAJECTORIES
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o,
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0 [o] Fig. 4. Reactive differential cross sections as a function of centre-of-mass scattering angle, O. Points are trajectory results with typical error bars. The continuous curve is the present calculation. (a) H + Hz; (b) D + HBr at Etran s = 0.10 eV; (c) D + HBr at Etran s = 0.25 eV. 36
they yielded the same O(b) slope, Wl, and a similar H + H2, however, is a fairly reactive system. The D + HBr abstraction reaction has a much lower p R ( b ( I ) ) (see fig. 3). We have therefore far fewer reactive trajectories to distribute among the angular boxes. In figs. 4b and 4c we compare the present results with the trajectory DCSs. The trajectory DCSs are shapeless: there is no information available at this noise level. To reduce the noise by a factor f w o u l d require r u n n i n g f 2 times as many trajectories. Since over a thousand have been run to produce even this level of information, this is evidently impracticable. The present results however do give a reasonable interpolation. Comparison of figs. 4b and 4c shows that even with so few reactive trajectories, the method is capable of revealing significant differences between the two DCSs. The method yields a dip in the DCS near O = 180 °. This feature has been seen in the Fourier sine series fits to the DCS for H + H 2 using many more trajectories [7]. Thus the method seems capable o f fair accuracy (fig. 4a), but also o f discerning small differences between DCSs even when very few trajectories are reactive. It is clear that the success of this method depends on the presence of a simple (preferably linear) relationship between ~)and b. We believe (although this remains to be tested) that this is a general feature of allreactions which occur through a preferred collinear configuration. The method should, then, be applicable to all reactions which proceed via a direct mechanism on surfaces of the LEPS [ 1 ] type.
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5 February 1982
4. Conclusion A new technique for calculating reactive DCSs has been proposed and tested. The method utilises a straight line relationship between mean reactive scattering angle and impact parameter. At each impact parameter a reasonable angular distribution about this mean is synthesised, by assuming gaussian behaviour. These distributions are then convoluted with the opacity'functionP~(b), to produce the DCS. The results appear to be in good agreement with trajectory data, yet sensitive to small changes, even when only a few reactive trajectories are available.
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
Acknowledgement The a u t h o r is grateful to Drs. Mark Keil and Ralph Wolf for their criticisms o f the manuscript, and gratefully acknowledges the generous hospitality o f Professor J.C. Polanyi during the course o f this w o r k .
[8] [9] [10] [11] [12]
References [13] [ 1] J.C. Polanyi and J.L. Schreiber, in: Physical chemistry an advanced treatise, Vol. 6A, eds. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1974) p. 383. [2] R.B. Bernstein, ACS Symposium Series, 56, eds. P.R. Brooks and E.F. Hayes (American Chemical Society, Washington, 1977) p. 3. [3] M. Kneba and J. Wolfrum, Ann. Rev. Phys. Chem. 31 (1980) 47. [4] M. Karplus, R.N. Porter and R.D. Sharma, J. Chem. Phys. 43 (1965) 3259. [5] D.G. Truhlar and J.T. Muckerman, in: Atom-molecule collision theory: a guide for the experimentalist, ed. R.B. Bernstein (Plenum Press, New York, 1979) p. 505. [6] D.G. Truhlar and N.C. Blais, J. Chem. Phys. 67 (1977) 1532; N.C. Blais and D.G. Truhlar, J. Chem. Phys. 67 (1977) 1540. [7] A.D. Jorgensen, E.A. Hillenbrand and E.A. Gislason, in: Potential energy surfaces and dynamics calculations for chemical reactions and energy transfer collisions, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 421 ;
[14]
-
[15]
[16]
[17]
[18] [19] [20]
5 February 1982
E.A. Gislason, E.A. Hillenbrand and A.D. Jorgensen, XII International Conference on the Physics of Electronic and Atomic Collisions, Gatlinburg, 1981, Book of Abstracts, ed. S. Datz, p. 1013. A. Kosmas, E.A. Gislason and A.D. Jorgensen, J. Chem. Phys. 75 (1981) 2884. F.E. Heiddch, K.R, Wilson and D. Rapp, J. Chem. Phys. 54 (1971) 3885. W.H. Miller, Advan. Chem. Phys. 30 (1975) 77. H.R. Mayne, J. Chem. Phys. 73 (1980) 217. G.D. Barg, H.R. Mayne and J.P. Toennies, J. Chem. Phys. 74 (1981) 1017. R.N. Porter and M. Karplus, J. Chem. Phys. 40 (1964) 1105. C.A. Parr, Ph.D. Thesis, California Institute of Technology (1969); C.A. Parr and D.G. Truhlar, J. Phys. Chem. 75 (1971) 1844. G.D. Barg, G.M. Kendall and J.P. Toennies, Chem. Phys. 16 (1976) 243; G.-D. Barg, Diplom-Thesis, Max-Planck-Institut fllr StrSmungsforschung, GSttingen, Bericht 124 (1973). D.G. Truhlar and J,W. Duff, Chem. Phys. Letters 36 (1975) 551; D.G. Truhlar, Intern. J. Quantum Chem. 10S (1976) 239. R.D. Levine and R.B. Bernstein, in: Dynamics of molecular collisions, Part. B, ed. W.H. Miller (Plenum Press, New York, 1976) p. 323. U. Halavee and R.D. Levine, Chem. Phys. Letters 46 (1977) 35. M.B. Faist, J.T. Muckerman and F.E. Schubert, J. Chem. Phys. 69 (1978) 4087. M. Baer, H.R. Mayne, V. Khare and D.J. Kouri, Chem. Phys. Letters 72 (1980) 269.
37
CHEMICAL PHYSICS LETTERS
5 February 1982
A SIMPLE TECHNIQUE FOR OBTAINING REACTIVE DIFFERENTIAL CROSS SECTIONS FROM HIGHLY-AVERAGED TRAJECTORY DATA Howard R. MAYNE Lash Miller Chemical Laboratory, University of Toronto, Toronto, Ontario, Canada M5S 1A1
Received 25 September 1981; in final form 5 November 1981
A technique requiring only the mean reactive scattering angle and opacity, both as a function of impact parameter, is proposed to calculate reactive differential cross sections. The method was tested for H + H 2 and D + HBr. It is fairly sensitive to initial conditions even with very few reactive trajectories.
1. Introduction One of the aims of performing quasiclassical trajectory calculations is to ascertain the sensitivity of various reaction attributes (such as product vibration, rotation and angular distribution) to initial conditions. For a t o m - d i a t o m reactions, much progress has been made in determining the relative roles played by reagent vibrational, rotational and translational energy in enhancing reaction, and their effect on the product attributes [ 1 - 3 ]. One final attribute which has so far eluded satisfactory elucidation, however, is the reactive differential cross section (DCS). One reason may be that few experimental results are available for comparison. Another, though, is more practical: they are difficult to compute. Since one requires many histogrammic angular intervals to obtain reasonable detail in the DCS, one necessarily has few trajectories going into each interval. The results are, then, noisy due to large Monte Carlo error [4]. Computation of DCSs using trajectories has been discussed in a recent review [5]. Procedures for overcoming the difficulties inherent in the traditional histogram method [5 ] have been proposed recently. One involves the expansion o f the DCS in a series of Legendre polynomials [6]. There are, however, problems in knowing just when to truncate this series. Another group [7,8] has suggested expanding the DCS in a Fourier sine series. Although in principle extremely accurate, this technique seems to require an
inordinately large number of trajectories. Particularly if one is interested in the behavior of the DCS for a series of initial conditions, one obviously cannot afford to run many thousands of trajectories for each case. The problem is further compounded if one wishes to investigate a reaction which has a pathologically low reactivity. In this letter we propose a new technique, which seems capable of determining DCSs for direct A + BC type reactions to reasonable accuracy using averaged trajectory data. Since it requires fairly few trajectories, it is cheap (or, if the system is very unreactive, at least feasible) and well suited to investigating the effect of changes in initial conditions. This technique is based on the well-known fact that trajectory methods are more accurate for highlyaveraged quantities than for highly-specific ones [5,9, 10]. Recent calculations on the H + H 2 reaction have shown that the average reactive scattering angle, e), is strikingly well correlated with the impact parameter, b, by a straight line [11,12]. In fig. 1 we show a plot of three e)(b) for two systems, obtained from trajectory data: one for H + H 2 ~ H 2 + H on the P o r t e r Karplus [13] surface; two for D + HBr ~ DH + Br (abstraction) on the Parr-Kuppermann [ 14] surface. The straight line fit property is common to all of them, within the statistics. Thus, although the trajectory distribution about the mean may be somewhat "ragged", the mean itself is a smooth function o r b . The aim o f the present
0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
33
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
spread randomly between b t and bl+1 [15] . ) N trajectories are run for each I . N m scatter into the mth angular box. The probability,pR(®m) (for given l) of scattering into final angle ®m is then given by
180
;ii
L
P~l (Om) Nm - N-
b[bohr]
D+HBr
(b
I
gO --
05
I0
15
20
25
10
35
il
0 b[bohr]
Fig. 1. Centre-of-mass mean reactive scattering angle, (9, versus impact parameter, b, for (a) H + H2 at Etran s = 0.43 eV, (b) D + HBr at Etran s = 0.10 eV (*) and 0.25 (A). Error bars are one standard deviation. The results for (a) represent a total of 1098 reactive trajectories; those for (b) represent 29 (0.10 eV) and 50 (0.25 eV) reactive trajectories respectively. technique is to smooth these distributions, and use the smoothed results to construct the DCS. We shall do this by constructing physically reasonable distributions about ~)(b) for several values o f b .
2. Method The quasiclassical histogram method used here is as follows. Every reactive trajectory has a centre-ofmass product scattering angle, ®, which is the angle between the initial collision axis and the centre-ofmass motion of the product diatom. The interval 0 ~< ® <~ n is divided into n equal boxes, each of width AO centred at 0 m = n - (m + 1/2)AO, m = 0, ..., n -- 1. O is boxed to the nearest @m value. We now consider scattering from a fixed value of the initial orbital angular momentum quantum number, l [ 15 ]. (In fact, the impact parameter, b, is
Nm ~'Nm Y.Nm N "
(1)
Z N m/N is simply/~/, the probability of reaction for given l. This can be large or small depending on the reactivity of the system. The term Nrn / Y_,Nm =Pl(Om ) is typically considerably less than unity, n, however, has to be fairly large in order to obtain a sufficiently detailed DCS. Thus there is considerable statistical noise in the rather specific term,Pl(O m). Averaged quantities (first moments), on the other hand, are accurately given by trajectory calculations [5,9,10]. Thus we expect the mean angle, ~) (= ~,NmOm/~,Nrn ) to be much less noisy than the individual PI(Gm ). The r~ corresponding to O is given by ZNmm/~,Nm, or equivalently, by = (~ - ½AO)/AO.
(2)
Given the mean, I~, o f a noisy distribution,P(O m ), we now construct about this mean a less noisy distribution, say P(m). This treatment is analogous to the classical P-matrix method [ 16] or information theoret ic [17] approach to the synthesis of a discrete vibrational distribution from energy transfer moments [18]. The difference is that we do not use the noisy trajectory angular distribution itself as a starting point. We propose instead to approximate Pl(m) with a gaussian distribution: exp [ - ( m - rfi)2/2w2l
el(m ) =
34
5 February 1982
(27r)l12wt
(3)
This is physically reasonable, since we must average over initial phases and sum over final rotational and vibrational states to obtain the total reactive DCS. The additional parameter required, Wl, is easily found as the standard deviation about the mean from trajectory data. If fill is low, then the gaussian may have significant contribution from negative values o f m . This is meaningless here. In these cases we reduce the value of w I until 95% o f the distribution is accounted for with m > O.Pl(m ) is then normalised and has the required mean. Results using eq. (3) are compared in fig. 2 with actual trajectory data. The fit is very reasonable. It
Volume 86, n u m b e r 1
CHEMICAL PHYSICS LETTERS
5 February 1982
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lies mostly within the error bars, particularly in those cases where the noise is low (fig. 2a). Where the noise is high (fig. 2b), the trajectory data are very uninformative, in "any case. The opacity function pR(b (l)), is also available from trajectory data. Sophisticated techniques are available for its efficient calculation [5,19]. For example, for H + H 2 in fig. 3a,pR(b(l)) is very smooth, and can be used as it stands. The D + HBrpR(b(I)) however (fig. 3b) is very noisy, and could be smoothed, by some form of least-square fit, for instance. If the situation were even worse, and, say, only bma x were known, a step function could be assumed. Reactive DCSs are given by [15]
Fig. 3. Probability of reaction p R , versus impact parameter, b. Points are trajectory results with typical error bars. (a) H + H z ; (b) D + HBr at Etran s = 0.25 eV. Results for H + H z represent a total o f 1098 reactive trajectories o u t o f a possible 4 4 0 0 trajectories run. Results for D + HBr represent a total o f 50 reactive trajectories o u t o f a possible 1040 run.
do(Ore) dm
/i2
1
2/aEttan s 2AO sin O m
Imax X ~
(21 + 1)pR (O m)
(4)
l=0
where/a is the initial reduced translational mass, Etran s is the translational energy, and lmax is the maximum value of l which yields reaction. The present approximation to the exact quasiclassical reactive DCS consists in replacing pR(o m) in eq. (4) withPt(m ) from eq. (3) multiplied by the l~l(b(l)) from trajectory data [see eq. (1)].
3. Results and discussion The present method was applied to two systems: H + H2(o = 0,] = 0) at Etran s = 0.43 eV on the P o r t e r Karplus [13] surface, and D + H B r ( u = 0 , i = 0) at Etran s = 0.10 and 0.25 on the P a r r - K u p p e r m a n n [14] surface.We used the technique as follows. ~)(b(l)) was fitted from the trajectory data using least squares to give the best straight line passing through ® = n. 35
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
This fit was then used to obtain nql [eq. (2)] and o l. AO was 5 °, whence n was 36.P/R was taken directly from trajectory data, without smoothing for the H + H 2 case. Since the D + H B r data are so p o o r , w e used an average standard deviation from the mean, ~ , rather than the/-specific w I. In addition, the pR(b(1)) function was fitted by least squares and the trajectory DCS for D + HBr were smoothed by a simple technique [20] to reduce noise. In fig. 4 we compare the DCS from trajectory data with the present method. In fig. 4a, where there are many reactive trajectories, the agreement is excellent. It should be stressed that the same curve would have been obtained from far fewer trajectories - so long as
H+H 2 0A3 . e V r i,r,-t .os ,~ REACtiVET~,~JECrOR~r . ( ~ T~ o~
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t ~°
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•
'
o,
12'0
08
I~3
180
0 [o] Fig. 4. Reactive differential cross sections as a function of centre-of-mass scattering angle, O. Points are trajectory results with typical error bars. The continuous curve is the present calculation. (a) H + Hz; (b) D + HBr at Etran s = 0.10 eV; (c) D + HBr at Etran s = 0.25 eV. 36
they yielded the same O(b) slope, Wl, and a similar H + H2, however, is a fairly reactive system. The D + HBr abstraction reaction has a much lower p R ( b ( I ) ) (see fig. 3). We have therefore far fewer reactive trajectories to distribute among the angular boxes. In figs. 4b and 4c we compare the present results with the trajectory DCSs. The trajectory DCSs are shapeless: there is no information available at this noise level. To reduce the noise by a factor f w o u l d require r u n n i n g f 2 times as many trajectories. Since over a thousand have been run to produce even this level of information, this is evidently impracticable. The present results however do give a reasonable interpolation. Comparison of figs. 4b and 4c shows that even with so few reactive trajectories, the method is capable of revealing significant differences between the two DCSs. The method yields a dip in the DCS near O = 180 °. This feature has been seen in the Fourier sine series fits to the DCS for H + H 2 using many more trajectories [7]. Thus the method seems capable o f fair accuracy (fig. 4a), but also o f discerning small differences between DCSs even when very few trajectories are reactive. It is clear that the success of this method depends on the presence of a simple (preferably linear) relationship between ~)and b. We believe (although this remains to be tested) that this is a general feature of allreactions which occur through a preferred collinear configuration. The method should, then, be applicable to all reactions which proceed via a direct mechanism on surfaces of the LEPS [ 1 ] type.
~0
• 9~
5 February 1982
4. Conclusion A new technique for calculating reactive DCSs has been proposed and tested. The method utilises a straight line relationship between mean reactive scattering angle and impact parameter. At each impact parameter a reasonable angular distribution about this mean is synthesised, by assuming gaussian behaviour. These distributions are then convoluted with the opacity'functionP~(b), to produce the DCS. The results appear to be in good agreement with trajectory data, yet sensitive to small changes, even when only a few reactive trajectories are available.
Volume 86, number 1
CHEMICAL PHYSICS LETTERS
Acknowledgement The a u t h o r is grateful to Drs. Mark Keil and Ralph Wolf for their criticisms o f the manuscript, and gratefully acknowledges the generous hospitality o f Professor J.C. Polanyi during the course o f this w o r k .
[8] [9] [10] [11] [12]
References [13] [ 1] J.C. Polanyi and J.L. Schreiber, in: Physical chemistry an advanced treatise, Vol. 6A, eds. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1974) p. 383. [2] R.B. Bernstein, ACS Symposium Series, 56, eds. P.R. Brooks and E.F. Hayes (American Chemical Society, Washington, 1977) p. 3. [3] M. Kneba and J. Wolfrum, Ann. Rev. Phys. Chem. 31 (1980) 47. [4] M. Karplus, R.N. Porter and R.D. Sharma, J. Chem. Phys. 43 (1965) 3259. [5] D.G. Truhlar and J.T. Muckerman, in: Atom-molecule collision theory: a guide for the experimentalist, ed. R.B. Bernstein (Plenum Press, New York, 1979) p. 505. [6] D.G. Truhlar and N.C. Blais, J. Chem. Phys. 67 (1977) 1532; N.C. Blais and D.G. Truhlar, J. Chem. Phys. 67 (1977) 1540. [7] A.D. Jorgensen, E.A. Hillenbrand and E.A. Gislason, in: Potential energy surfaces and dynamics calculations for chemical reactions and energy transfer collisions, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 421 ;
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5 February 1982
E.A. Gislason, E.A. Hillenbrand and A.D. Jorgensen, XII International Conference on the Physics of Electronic and Atomic Collisions, Gatlinburg, 1981, Book of Abstracts, ed. S. Datz, p. 1013. A. Kosmas, E.A. Gislason and A.D. Jorgensen, J. Chem. Phys. 75 (1981) 2884. F.E. Heiddch, K.R, Wilson and D. Rapp, J. Chem. Phys. 54 (1971) 3885. W.H. Miller, Advan. Chem. Phys. 30 (1975) 77. H.R. Mayne, J. Chem. Phys. 73 (1980) 217. G.D. Barg, H.R. Mayne and J.P. Toennies, J. Chem. Phys. 74 (1981) 1017. R.N. Porter and M. Karplus, J. Chem. Phys. 40 (1964) 1105. C.A. Parr, Ph.D. Thesis, California Institute of Technology (1969); C.A. Parr and D.G. Truhlar, J. Phys. Chem. 75 (1971) 1844. G.D. Barg, G.M. Kendall and J.P. Toennies, Chem. Phys. 16 (1976) 243; G.-D. Barg, Diplom-Thesis, Max-Planck-Institut fllr StrSmungsforschung, GSttingen, Bericht 124 (1973). D.G. Truhlar and J,W. Duff, Chem. Phys. Letters 36 (1975) 551; D.G. Truhlar, Intern. J. Quantum Chem. 10S (1976) 239. R.D. Levine and R.B. Bernstein, in: Dynamics of molecular collisions, Part. B, ed. W.H. Miller (Plenum Press, New York, 1976) p. 323. U. Halavee and R.D. Levine, Chem. Phys. Letters 46 (1977) 35. M.B. Faist, J.T. Muckerman and F.E. Schubert, J. Chem. Phys. 69 (1978) 4087. M. Baer, H.R. Mayne, V. Khare and D.J. Kouri, Chem. Phys. Letters 72 (1980) 269.
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