Opacity analysis of steric requirements in elementary chemical reactions

CHEhlICAL

Volume 1 OS. number 5

OPACITY R.D.

ANALYSIS

30 March 1984

PHYSICS LETTERS

OF STERIC REQUIREMENTS

IN ELEMENTARY

CHEMICAL

REACTIONS

LEVINE

Fritz Haber Research Center

for Molecdar

Dynamics.

The Hebrew

University, Jerudem

91904. Ixmd

and R-B BERNSTEIN Department

of GTremirtry. University of Califomlo. Lox Angeles. California 90024. USA

Received 27 January

1984; in final form 3 February

1984

The orientation dependence of the barrier to reaction 1s incorporated into the classical line+f%entcrs model to obtain the dependence of the “cone of acceptance” upon impact parameter and colhsmn cnerg). Simple expressions for the orientation-avcragcd opacity function. the reaction cross section and the steric factor are derived Comparison is made with recent classical trajectory calculations by Blais and Truhlar for the H + D, reaction.

I _ Introduction

P(b)=( i d(cos

d-l

d(cos

y)f’@. Y))( j

With the advent of more extensive experiments on oriented molecule reactions [l-3], it is desirable to develop appropriate theoretical techniques for their interpretation. As a preliminary step in thus direction, it seems reasonable to examine simpler models of the type that have well served molecular reaction dynamics

while averaging over b at fLyed y yields the partial

since the early days [4]

we wish to gen-

reaction

eralize the venerable “line-of-centers” model [L-7] by incorporating in the model the dependence of the barrier to reaction upon the mutual orientation of the reagents. Our approach makes use of some of the qualitative ideas of Karplus et al. [S] on the “sterlc requirements” of a reaction. In the past, we have considered the opacity function or reaction probability I@), averaged over all orientations at a given impact parameter b, for fixed translational energy ET [9] _Here, we develop a line-

namely

of-centers model for I’@, -y), the opacity function at a grven orientation angle y_ Averaging over y at fured

model for P(b, y). In section 3, we apply rhe model IO a realistic system, comparing calculated results with recent classical trajectory computations [lo] on the

_Specifically,

b recovers the usual opacity function

-1

-1 -:I-3

l

-1

d(cos y)P(b,

cross

uR(Y) = /

section

03.00 0 Elsevier Science Publishers BY. Physics Publishing Divlsion)

for specified

d(-rrb2)PIb.

angle of attack

Y?

(2)

Y) -

0 The total

OR(&)

reaction

= n i -1

cross section

d(cos 7) /

at ET 1s then

db bP(b. Y) ,

(3

0

which specifrcally allows for the steric requirements In section 2, we introduce a simplistic line-of-centers

well-characterized

0 009-2614/84/S (North-Holland

(1)

Y) _

-1

H, potential

energy surface [ 1l.lZ]_

467

2. Angledependent

line-of-centers

fib, ET) = $; d(cosY) ~(E,Cl - b2/d2)- EoW) ,

model

Our objective is to develop an expression for I@. Y) compatible with the line-of-centers assumptron, i.e. that reaction occurs with unit probably if, at some critical separation d, the relative kinetic energy along the line of centers (i.e. ET(1 - b2/d2), see ref. [4]) exceeds some threshold energy of barrier Elo_ The . range of impact parameters contributing to reaction is limited by ET{1

- b2/d2) - E. > 0 .

(4)

(In the simplest rigid-sphere model, d is the mean di-

ameter and (1 - b2/#)‘/2 is the cosine of the lioe-ofcenters angle $0. See fig. 3.3 of ref. [4] -) Here, we make use of the fact that the barrier height depends strongly upon the angle of attack 7, ia_ E. K E,,(y)_ Fig. 1 shows such an angle dependence, E. versus cos y, taken from min~um energy path calculations [ 121 for the best-known ab initio potential surface, namely that for the H3 system- We thus modify eq (4) by introducing the angledependent barrier, which deterrnmes the “cone of acceptance” b and ET: ET(I - b2/d2) - l&(y) 3 0 _

30 March 1984

CHEMXCALPHYSKS LETTERS

Volume 105, number S

-1 (6) where H(x) is the umt step function, If(x) = 1 for xZO,H(x)=Oforx
for &.(I - b2/d2) > E. (and zero otherwise), to a reaction cross section u&5+.)

= $( 1 - cos r*)?&(l

_

(7)

In this special case, the “steric factor” is simply f(1 - cos yo), a constant scaling down the lme-ofcenters cross section at all ET_

In the more general case, the an~e~ependent tial cross section, eq. (Z), is given by Q&r) = xd* t f - EO(T)IE-rl

par-

ET 2 E&Y) ,

7

so that the orientation-averaged section, eq (3), becomes

at a given

(8)

total reaction cross

1 OR@+)=

rfd

21

(5)

From eq. (l), we obtain

- EO/ET)

leading

x H(ET

3 1 -1

- EO(y))

d(cos

y%l

- &(y)IE,l

-

(9)

One can express eq. (9) in simpler terms: OR(&)

= ud2

i(l -

cos rmax)(l

- (~o(7)>ET/Kr)

_ 00)

Here, ym,(ET) is the largest angle of attack for which reaction is allowed, 1-e. E(y,,) = ET, or 1 :_(I -

cos y max) = 5 f(, d(cos

yFW+

- Eo(Y))

I)

(11) and (Eg(y)>~~ is the cosine-weighted average barrier height (averaged over the restncted angular range E&Y) =GET)’ Fig. 1. The barrier height Eo versuscos r for the ab initio Ha potential energy surface [11,12]_ The inser defies Y for the reaction A + BC - AB + C The right-hand ordinate is the translational energy ET taken from the lloor of the potential surface in the asymptotic region. Arrows indzcate tbeE~ values at which comparisons are made with trajectory computations (diScussed in section 3). 468

(E cy))E = I”, d(cos r)Q,Ccos r)cr(E, - EC,(~)) 0 T $5, d(cos Y)n(E, - Et&Y)) (12)

Note that eqs. (11) and (6) yield a sunple result for the opacity at zero impact parameter (at E$)_

Volume 105. number 5

p(“, +)

CHEMICAL

= ; [ 1 - =os Trna,(&)]

Thus, eq. (10) can be rewritten

PHYSICS

(13)

-

in terms of the hmit-

ing opacity ($@#rd’

= P(O, +)[

1 - {~O(Y)}~~/~T]

-

(14)

The notatzon indicates explicitly that the tirmting opacity and the effective barrier he& ET +The orientation-averaged

depend upon

opacity function at a

given ET can be readily obtained formed mto the more convenient

via eq. (6), transform

P(b. ET) = $( 1 - cos 5) ,

11%

where cos r is the b-dependent

solution

of

f+( 1 - b’/d2) = E,,(cos 5) _

(16)

(The U-ISof eq_ (16) determines Eo: then cos 7 obtains from fig I_) Eq. (15) is applicable over the entire range oiET over which the Eo(~T) curve IS known. However, at energies in the post-threshold region, it may be possible to approximate Eo(cos y) by a linear function of cos y, 5-e. Eo(cos

-y) =

E, + E;(l

Then we obtain an

- cos y)

explicit

P(b, ET) = (1/2E;))[ET(1

eXpreSSiOn

(17) for p(b, ET):

- b2,‘d2) - Eo] _

(16

Indeed, in general, the slope Eh can be a useful measure of the energy dependence of the cone of acceptance_ Next, we proceed to apply the model numerically to the H + D, reaction, for which the dynamics have been simulated via the quasi-classical trajectory method by Blats and Truhlar [lo] _

* Only at such hish ET (ti ever) that reaction is possibleat all orientations will the integration in eq. (9) span the entire range or cos 7, yielding oR/nd' = 1 - (Eo(l))/E-f, where tEo(y)) is now tbe unrestricted average barrier height, cEo(yI) = fr?, d(cos r)Eo(cos y) Note also that, througbout the discussion of the model. it has been splicing assumed &at the barrier is lowest for the collinear approach (CDS-~= 1). as For the H3 system (fip_ 1). The necessary changes in the equations for the case of. say, perpendicutar (Insertion) reactions are minimal.

30 March 1984

LEl-I-ERS

3. Application Stimulated

to the H + D, reaction by the recent

state-to-state

experiments

of the Zare [14,15] and Valentmi [ 161 groups on the reaction H + D2(u = 0) -+ kiD(u’, J’) -t D, Blais and Truhlar [lo] (hereafter BT) carried out quasi-classical tralectory (QCT) computations to simulate the experimental results, employing the best ab mitio potential surface for H, [ 11,12]_ For the present purpose it is not important, but the QCT results appeared to yield a reasonably good representation of the experimental rotational and vIbrationa state distribution of the HD products. Using the angle-dependent barrier function plotted in fig. 1, the total reactlon cross section uR(ET), in umts of rrd’, can be readily calculated via eq. (10). The cone angle ym,(ET) is read directly from fig. 1, while CJIT~(~>>~,

is evaluated by graphical mtegrationThe value of the size parameter d has been chosen somewhat arbitrarily to be 2-S “o for all the present calculations. The resulting curve is plotted in fig_ 2.. [Here, Ere, = ET - E (DZ) is used as the abcissa with the zero-point ei&gy taken as 0.193 eV (‘I 35

kcal mol-I).]

To be noted is the post-threshold

cur-

vature (not due, of course. to any tunneling) reflecting the opening up of the cone of acceptance with increasing energy The two QCT-calculated cross sections (at 0.55 and l-30 eV) are plotted on the curve for comparison. Since the cross section scales with I@, the range of “acceptable” d values is restricted within -510% by the fit to the BT calculations.

IO

E,,,

I5

20

23

kv)

Fig. 2. hiodel-calculated orientation-sieraxed reaction cross section o~(_Er~l) for H + D2 - HD + D, via eq (10). using EQ(COS-y) from fig. 1 and taking d = 2.5 no (discussed in section 3). The two points shown denote the quasi-zlassinl trajectory (QCT) computations by Blais and Tmhlar [ 101 on the same potential surface 469

CHEMICAL PHYSICS LETTERS

Volume 105, number 5

included

bfo,f

Fig. 3_ Onentatian-averaged opacity function P(b) for the H + D2 reaction at the two energies (f?rel = 055 and 1.30 eV) of the ET computations. Histograms are.the results of BT [IO], smooth curves from present model via eqs (15) and (16). with d = 25 a0 (discussed m section 3).

Next, we calculate opacity

function,

Exe1 for which

the orientation-averaged

using eq. (IS), at the two values of

the QCT

opacities

have

been

com-

puted by BT. First, merely by inspection of fig. 1, the values of cos ~max(ET) are read to be 0.50 and -0.12 at 0.55 and 1.30 eV, respectively, leading to P@ = 0) of 0.25 and 0.56, respectively, in good accord with fig_ 1 of BT. Next, consider the highest b value for

which reaction is Possible, given by the solution of ET( 1 -

b2/d2)

= E o, where the minimal

barrier

height (above the asymptotic “floor”) IS taken to be 9.8 kcal mol-1 (0.425 eV). phe cut-off b values are found to be 1.63 and 2.12 ao, respectively, in fair accord with BT. Fig. 3 shows the curves of p(6) at the two energies, calculated via eqs. (15) and (16), superimposed on the histogrammic representation of BT (their fig. 1). As a check on the model-calculated fib) curves, they were integrated numerically [/Znbp(b) db] at 0.55 and 1.30 eV to yield uR values that agreed, within the numeri@ uncertainty, with the corresponding values from the curve of fig. 2.

4. Considerations on experimental molecule reactions

studies of oriented

Analyzing the experimental results of Parker et al. [ 171 on the orientation dependence of the backscattered product RbI from the reaction of Rb with oriented CH,l, Stolte et al. [2] deduced the dependence upon -y of P(b = 0). Unfortunately, the accuracy of the data (available at only one energy) was insufficient to allow an unambiguous determination of the -y dependence. The range of acceptable functions 470

30 March 1984

a step functions

However,

the best fit was

obtained assuming a smoothly declining cut-off as a function of-y, implying a “softening” of the step function (a diffuse or “gray zone” between the black and white regions of the model of ref. [ 131). Such behavior is remmiscent of that found in previous opacity analyses, e g ref. [9], where a “smoothed” step function in ffb) was called for_ In the present case of the -y dependence, it is possible that the reo~entation forces between the reagents at close separation lend to smear out the initially prepared, relatively narrow distribution of angles of attack, accounting for the softening of the step function. Experiments tion dependence

by Parker et al. [2] on the orientaover a wide range of scattering

angles, not yet analyzed in detail, suggest that the r dependence of p(6.y) persists out to fairly large impact parameters, in qualitative accord with the present model. Clearly required are oriented molecule experiments wried out over a range of energies that wrll enable us to obsente the openmg-up of the cone of acceptance with increasing collision energy. If measurements of the fatal reaction cross section for oriented reagents could be made as a function of 7, the data could be inverted directly using eq. (8) to yield &(T)- In a subsequent study, we explore the feasibility of ~ve~~g the ~~ependent deferential cross section data (such as that of ref. 121).

5. Conch.lding remarks

The orientation dependence of the pot~nlial energy barrier has been incorporated into the classical line-ofcenters model to account, at the most simplistic level, for the steric requirements of elementary chemical reactions. The model provides simple expressions for the cone of acceptance as a function of impact parameter and energy, and for the orientationaveraged opacity and reaction cross section versus energy, and gives a quantitative interpretation of the steric factor. Comparison with classical trajectory computations suggests that the model warrants serious consideration as a physically motivated, ~miquantitative representation of the steric effect upon reactivity:

Volume 105. number 5

CHEMICAL PHYSICS LETTERS

Acknowldgement This work received partial financial support from the following sources: US Air Force Office of Scientific Research, grant AFOSR-81-0030 (RDL), the US National Science Foundation, grant CHE8316205 (RBB); the US-Israel Bmational Science Foundation (RDL and RBB), and the Chemistry Department of UCLA. The Fritz Haber Research Center is supported by the Mmerva Gesellschaft fur die Forschung, mbH. Munich, FRG_

References frl S. Stolte. Ber. Bunscnges. Physik. Chem. 86 (1982) 413. I21S. Stolte. K-K. Chakravorty. R.B. Bernstein and D.H. Parker. Chem Phys. 71 (1982) 353; D-H. Parker, K-K. Chzdcravorty ;md R.B. Bernstein, Chem. Phys. Letters 86 (1982) 113. 131D. van den Ende, S. Stolte, J.B. Cross. G.H. Kwei and J J. Vatentini, J. Chem. Phys. 77 (1982) 2206. 141 R-D. Levine and R.B. Bernstein, Molecular reaction dynamics (Clarendon. Oxford, 1974). I51 R-C. Tohnan. Statistical mechanics with applications to physics and chemistry (Chemical Catalog Co , New York. 1927).

30 March 1984

[6] A-A. Frost and R-G. Pearson, Kinetics and mechanism (Wiley, New York, 1953). [7] R-D. Present, Proc. Natl. Acad. Sci. US 41 (1955) 415. [S] M. Karplus. R.N. Porter and R-D. Sharma, J. Chem. Phys. 43 (1965) 3259, table 3. [P] R-B. Bernstein and RD. Levine, J. Chem. Phys 49 (1968) 3872; R.D. Levine and R B. Bernstein, Israel J. Chem. 7 (1969) 314. [IO] N-C. Blais and D-G. Truhlar, Chcm. Phys. Letters 102 (1983) 120. [ 11) P. Siegbahn and B. Liu, J. Chem. Phys. 68 (1978) 2457. [ 12) D-G. TmhIar and C-l. Horowitz, J. Chem. Phys. 68 (1978) 2466 [ 131 R.J. Beuhler and R.B. Bernstein, J. Chcm. Phys 51 (1969) 5305. [ 141 C-T. Rettner. EE. Marinero and R-N Zare. in: Physics of Electronic and Atomic Collisions: Invited Papers from the XlIIth ICPEAC, Berlin, 1983. eds. J. Eichler, I-V_ Hertel and N. Stolterfoht (North-Holhnd, Amsterdam, 1983). [15] CT. Rettner. E.E. hiarincro and R.N. Zare. J. Chem. Phys 80 (1984). to be published [ 161 D-P. Gerrity and J-J. Valentmi. J Chem. Phys. 79 (1983) 5203. [ 171 D-H. Parker, K K. Chakravorty and R-B. Bernstem. J. Phys. Chem. 85 (1981) 466.

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