Role of migration in the dynamics of bromine atoms reacting with iodine molecules

Chemical Physics ELSEVIER

Chemical Physics 207

(I 996) 203-214

Role of migration in the dynamics of bromine atoms reacting with iodine molecules J.J. Wang, D.J. Smith, R. Grice Chemistry

Department,

Uniuer.sity r,f’Munchrstrr,

Manchester,

Ml3

9PL. UK

Received 20 July 1995

Abstract Differential cross sections have been measured for the reactive scattering of Br(*P,,,) atoms with I, molecules at initial translational energies E - 87 and 52 kJ molt ’ using a supersonic beam of Br atoms seeded in He and Ne buffer gases generated from a high pressure microwave discharge source. The centre-of-mass angular distributions of IBr scattering peak sharply in the forward direction with subsidiary backward peaks of relative height _ 0.55 and 0.60. The product translational energy distributions are broad, peaking at a fraction f = 0.30 k 0.05 of the total available energy but with the forward scattering being shifted to lower translational energy than the wide angle scattering. An extended phase space theory is proposed to model the observed reactive scattering without prior assumptions as to the geometry of the collision complex or its mode of dissociation. This provides a good description of the observed reactive scattering only when the effects of a potential energy barrier in the exit valley of the potential energy surface are absent from the calculation. This is attributed to the contribution of migratory trajectories at large impact parameters sampling the less stable IBrl configuration and yielding scattering which is sharply peaked in the forward direction, while smaller impact parameter collisions sample the more stable BrII configuration and yield scattering over a wide angular range. The photoinitiated reaction samples only the BrII configuration due to its constrained geometry and low initial translational energy.

1. Introduction The exchange reaction of bromine atoms with iodine molecules was amongst the first systems to be studied in cross beam reactive scattering experiments [l] using mass spectrometric detection and an effusive atom beam source. An overall picture of halogen atom molecule exchange reactions occurring via a quasilinear trihalogen intermediate with a stability determined mainly by the identity of the central atom was established [2], with the stability increasing as the electronegativity of the central atom decreases 0301-0104/96/$15.00 0 1996 Elsevier Science B.V. All rights resewed SSDI 0301.0104(95)00277-4

along the series Cl, Br, I. Recently, the situation for the Br + I, reaction has been put on a more quantitative basis by a femtosecond laser study [3,4] of the photoinitiated reaction in an HBr-I, van der Waals molecule. Remarkably long lifetimes r = 44 and 53 ps were found for the BrII collision complex in the photoinitiated reaction at low initial translational energies E = 1.7 and 3.5 kJ mol- ’ compared with a lifetime of only N 5 ps estimated in the cross beam experiment at E = 12 kJ mol-‘. This very strong dependence of the complex lifetime on initial translational energy was attributed 141 to the presence of a

J.J. Wang et al./

204

potential energy barrier to the dissociation BrII complex forming reaction products

Chemical

of the Br(He) + I,(N,) -> IBr + I

However reaction in the cross beam experiments also extends out to larger impact parameters [5] than the constrained geometry [4] of the photoinitiated reaction and the effect of this difference in reactant orientation remains obscure. In order to compare the dynamics of the Br + I, reaction under cross beam and photoinitiated conditions more thoroughly, further cross beam experiments have been undertaken at much higher initial translational energy E = 87 and 52 kJ mol- ’ using a supersonic beam of Br atoms seeded in He and Ne buffer gases.

2. Experimental The experiments reported here were undertaken with the same cross beam apparatus using mass spectrometric detection as that employed in a previous study [6] of the F + ICI reaction. The supersonic Br atom beam was produced from a high pressure microwave discharge source [7] with an alumina discharge tube using a dilute mixture - 1 mbar of Br,, sustained by a reservoir temperature - 50°C in - 100 mbar of He or Ne buffer gases. The velocity distributions of the Br atom beam were measured by a beam monitor mass spectrometer and the peak velocities uPL, full widths at half maximum intensity Uwd and Mach numbers M are listed in Table 1. The supersonic I, beam issued from a glass nozzle of diameter - 0.2 mm using a dilute mixture - 6 mbar of I,, sustained by a reservoir temperature _55”C, in - 50 mbar of N, buffer gas. The velocity distribution of the I, beam was measured by the rotatable mass spectrometer detector using pseudo-random

uDkCm s-

Br (He) Br (Ne) I, (N,)

1510 1050 790

I

(1)

Table 1 Beam velocity distributions: peak velocity upk maxtmum mtenstty u,,, and Mach number M

’1

,

A

1.2+

Br + I, + IBr + I.

Beam

Physics 207 (1996) 203-214

1.0

6 E

0.8

I

0.6:

Fig. 1. Laboratory angular distribution (number density) for IBr reactive scattering from Br+I, at an initial translational energy E=87 U molI’. Solid line shows the lit of the stochastic kinematic analysis.

cross-correlation time-of-flight analysis [8] and the corresponding velocity parameters are also listed in Table 1.

3. Results Angular distribution measurements of IBr reactive scattering yield - 60 and 40 counts s- ’ against backgrounds - 120 and 60 counts s- ’ for Br atoms

I-

I

I,

Br(Ne) + I+N>) -> IBr + I 1.2i

i I

full width at half

uwd(m s- ‘1

M

360 370 190

7.5 5.0 8.0

Lab Angle 0 /degrees

Fig. 2. Laboratory angular distribution for IBr reactive scattering from Br + I, at an initial translational energy E = 52 kJ molt ’

J.J. Wang et al./ Chemical Physics 207 11996) 203-214

205

(4 / Q 0.CL020

t-gytt-r~-- i-~-11.0 LAB VELOCITY/IOOms~’

LAB VELOCITY/IDOms

3k+-&-e-i 7.0

--t-9.0

II.0

13.0

t

i 15.0

+--.

13.0

15.0

17.0

’

I.

17.0

LAB VELOCITY/ICKhns-’

Fig. 3. Laboratory velocity distributions (flux density) of reactively scattered IBr from Br + I, at an initial translational mol-‘. Solid lines show the fit of the stochastic kinematic analysis.

energy E = 87 W

206

J.J. Wang et al./Chemical

seeded in He and Ne buffer gases. The laboratory angular distributions of IBr number density in Figs. 1 and 2 both show peaks at intermediate laboratory

Physics 207 (1996) 203-214 Br(He) + $(N,)

-> 1Br + I

0.8

0.6

o.oA 0

20

40

60

80

100

120

__i 140

160

I80

CM Angle, ‘3

LAB VELOCITY/lOthns-’

Trans.

Energy.

E’,‘kJ mot-’

Fig. 5. Angular function T(B) and translational energy distributions P( E’) for Br + I, at an initial translational energy E = 87 kJ mol-‘. Left-most product translational energy distribution corresponds to /3 = 0”. right-most distribution to 8 = 180”. Dashed curve shows the distribution of initial translational energy.

LAB VELOCITY/lOOms-’

Fig. 4. Laboratory velocity distributions of reactively scattered IBr from Br+ I, at an initial translational energy E = 52 kl mall ’

scattering angles with shoulders extending to wider angles close to the I, beam. The laboratory velocity distributions of IBr flux shown in Figs. 3 and 4 were measured using integration times - 3000 s to gain signal-to-noise ratios - 12 at the peaks of the distributions. Kinematic analysis of these data has been carried out using the stochastic method [9] with the differential reaction cross section expressed as a product of an angular function T(B) and a velocity function U(u, 13) which is parametrically dependent on centre-of-mass scattering angle 8 r,,(e,

u)=

qqq~,

0).

(2)

J.J. Wang et al./Chemical Br(Nc)

+ I,(NJ

-> IBr

+ I

Physics 207 (1996) 203-214

207

Table 2 Reaction energetics: reactant translational energy E, peak product translational energy Ebs, average product translational energy Ei,, reaction exoergicity AD, E (kJ mol- ‘1 Ebk &.I mol- ‘) EA, (kJ mol- ‘) AD, (kJ mol- ‘) 0=0”90” 87 52

:t

L 0

20

40

60 80 100 CM Angle. 0

120

140

160

180

33 17

180” 6=0”90”

41 41 22 26

42 26

180”

46 46 28 30

26.5 26.5

in Figs. l-4. The peak Ebk and average EL, product translational energies are listed in Table 2 together with the initial translational energies E and the reaction exoergicity AD,, calculated from the IBr and I, bond energies of Huber and Herzberg [lo].

4. Discussion

Trans. Energy. E’/kJ mol.’ Fig. 6. Angular function T(6) and translational energy distributions P(F) for Br+ I, at initial translational energy E = 52 kJ mol-‘. Left-most product translational energy distribution corresponds to 8 = O”, centre distribution to 0 = 90”. right-most distribution to 0 = 180”.

The resulting centre-of-mass angular distributions and product translational energy distributions P(E’) are shown in Figs. 5 and 6 for Br atoms seeded in He and Ne. In both cases the angular distributions peak sharply in the forward direction with a subsidiary backward peak and almost isotropic scattering at intermediate scattering angles. The product translational energy distributions are shifted to lower translational energy for the forward scattering than the wide angle scattering, the effect being more prominent for Br atoms seeded in Ne. The backfits of the stochastic transformation are shown by solid curves

The potential energy surface proposed by Sims et al. [4] has a potential energy minimum in the BrII configuration, which is accessible to the reactants without a potential energy barrier but involves a barrier to dissociation to reaction products as shown in Fig. 7. In order to test the compatibility of this description with the reactive scattering data observed in these experiments at much higher initial translational energy, a phase space description was adopted since this makes no assumptions as to the preferred geometry of the collision complex or its mode of dissociation. The calculation of angular distributions of reactive scattering is extended in order to accommodate the sharp forward and backward peaking observed in these experiments within a phase space description. Phase space theory [ 11,121 assigns equal probability to all accessible quantum states of the

Br+l2 Brl-I

Fig. 7. Potential energy profile for the Br+ I, reaction in the BrII and IBrI configurations. Energies in kJ mol- ’

J.J. Wang

208

et al./Chemical

Physics

“i

ia’

!

(3)

where J’ denotes the IBr product rotational angular momentum. Previously [ 131 the probability distribution has been expressed as a three term polynomial expansion with the expansion coefficients being determined by polynomial least squares fitting to the numerical results of the phase space calculation. P( sin cr ) = c0 + c,si&

203-214

LO-

reaction products subject only to the conservation of energy and total angular momentum. Consequently the probability distribution for the cosine of the angle n/2 - (Y lying between the initial L and final L’ orbital angular momenta may be calculated by summing over angular momentum coupling diagrams as shown in Fig. 8a c0s(lr/2-a)=(L*+~*-.V2)/2LL’,

207 (1996)

+ c,sin4cu.

0.04 0.0

02

0.4

0.6

0 8

0.6

0.x

sina

(4)

However for many reactions the phase space distribution increases rapidly as sin (Y+ 1 so that the expansion of Eq. (4) is poorly adapted to fitting the numerical results. An alternative procedure is proposed here, whereby a part of the range of sin (Y is fitted by least squares to the first two terms of Eq. (4) and the remaining part of the range close to

1.0 (b)

0.8 r

0.6.

t

(a):

0.0i 0.0

02

0.4

i:o

\l”cx

(b):

Fig. 9. Fitting of Eq. (5) to the &in (Y) distributions calculated from phase space theory at initial translational energies E = 87 and 52 kJ mol- ’ in panels (a) and (b). Starred points are fitted by the two term expansion, open circles by the three term expansion.

---ALLV

Fig. 8. (a) Coupling of final orbital angular momentum C and product rotational angular momentum J’ when the total angular momentum f may be approximated by the initial orbital angular momentum L. (b) Effect of reactant rotational angular momentum J on tilting the conserved total angular momentum _Y away from the initial orbital angular momentum L, which is perpendicular to the initial relative velocity u.

sin (Y= 1 includes a third term involving an additional parameter a2 as suggested previously [ 141 by microcanonical theory P( sin a)

for

= co + c2sin2cu,

0Gsin2crG

1 -(l/u’)

= co + c,sin2ff + c4a sin cz( 1 - a2cos2~) for

1 -(l/a*)

< sin*a<

1,

312

(5)

,

J.J. Wang et d/Chemical

where the value of a2 is determined by visual inspection and the normalisation P(l) = 1 requires c0 + c2 + cqa = 1. Transformation to the distribution of the cosine of the helicity angle (Y’ lying between the final relative velocity u’ and the initial orbital angular momentum involves [15] performing the integration P(cos

cy’) = ?/“I”

a

1 P(sin

CY)d sin (Y (6)

Physics 207 (1996) 203-214

The differential I( B)sin

209

cross section given by

8 do = const dp,

then becomes I( 0) = l/( sin28 - cos*y - cos2ff’ +2 cos y cos cr’ cos e)‘/‘.

P(cos for

Eqn(go) then(zlzs

- sin2~ ) “* ’

1 - (l/a2),

= c0 + +c2sin2& + j?jcq( 1 - a*cos*~~)*, for

1 - (l/a*)

< sin2a’<

1.

(7)

The principal effect of reactant rotational angular momentum J on the observed angular distribution of reactive scattering arises from the angular momentum coupling of J with the initial orbital angular momentum L, so that the conserved total angular momentum x no longer lies perpendicular to the initial relative velocity v as is the case when J = 0. As illustrated in Fig. 8b, the reactant angular momentum J may be assumed to be isotropically distributed and will have a range of magnitudes determined by the Boltzmann distribution of rotational energy. However attention will be confined to the typical experimental situation where the most probable magnitude of the reactant rotational angular momentum is considerably less than the maximum initial orbital angular momentum J_, K L,. In this case it will suffice to consider the representative coupling diagram shown in Fig. 8b where the cosine of the helicity angle y lying between the total angular momentum dp and the initial relative velocity v is given by cos y = J,,/L,

.

The final relative velocity uniformly distributed about gle CY’lying between /and 8 lying between v and u’ is angle cp of v’ about f formula

a’)

d cos

a’

(12)

cos2y + 2 cos y cos R cos a’ - cosv)“2

cd) = co + +c2sin2cu’, OG sin*&<

(11)

The observed angular distribution of reactive scattering is then obtained by integration P(cos

Employing

(10)

where min = cos(6 + y) or - l/a which ever is the greater and max = cos(8 - y) or l/a which ever is the less. The range of integration is shown in the appendix together with the analytical formulae appropriate for the terms in Eq. (7). The P(sin cu) distributions calculated by phase space theory with the maximum initial b, and final bk impact parameters listed in Table 3 are shown in Fig. 9 together with the fits of Eq. (5) which yield the parameters cO, cZ, cq and u* also given in Table 3. The resulting angular distributions of reactive scattering calculated from the equations given in the appendix multiplied by the decay function exp( - e/0 * ) according to the osculating complex model [16], show a good general agreement with the experimental results in Fig. 10 though with less sharp forward peaking. The product translational energy distributions calculated from phase space theory give the good agreement with the experimental results shown in Fig. 11 only when there is no effect of a potential energy barrier in the exit valley of the potential energy surface increasing the product translational energy [4]. In the present experiments the very large initial orbital angular momentum L, = 350-4OO?i greatly exceeds the rotational angular

(8) v’ now lies on a cone _Y with the helicity anv’. The scattering angle related to the precession by the dihedral angle

cos 0 = cos y cos ff’ + sin y sin ff’ cos cp.

(9)

Table 3 Extended phase space theory: initial translational energy E, maximum initial b,,, and final bk impact parameters used in phase space calculation with fitting parameters cO, c2, cd, u2 used in Eq. (5) and decay parameter 0 ’

* (“)

E (kJ mol-‘)

b,,, (.&) bk (A)

co

c2

cq

a2

0

87 52

2.5 2.8

0.09 0.08

0.35 0.36

0.35 0.36

2.5 2.5

290 310

4.0 5.0

J.J. Wang et d/Chemical

210

0.0

0

20

40

60

80

loo

120

140

160

Physics 207 (19961203-214

I80

CM angle BP

._ 50 Trans.

/

Energy,

100 E’/kJ mol-’

(b)

1.0

Fig. 10. Angular distributions of reactive scattering calculated from the extended phase space theory with an exponential decay term (solid curves) compared with experimental distributions (broken curves) at initial translational energies E = 87 and 52 kJ mol- ’ in panels (a) and (b).

momentum of the reactant I, molecule JmP = 35A. and the correspondingly low value of cos y= = 0.1 has only a very small effect on the Jm,/Ll predicted angular distributions in Fig. 10. The lifetimes T of the BrII complex may be estimated using the RRKM formula [17] T =

v- ’[ E,,,/(

Et,, -

En)].?-’,

(13)

% 0.4 a 2 E 0.2

0.0

0

20 Trans.

40 Energy,

60 E’/kJ mold1

Fig. 11. Product translational energy distributions calculated from phase space theory (broken curves) compared with experimental distributions (solid curves) at initial translational energies E = 87 and 52 kJ mol-’ in panels (a) and (b).

JJ. Wang et al./Chemica/

Physics 207 (1996) 203-214

cosa’

where E,,, denotes the total energy of the complex, E, the energy required for dissociation, V the mean vibrational frequency and s the number of internal vibrational degrees of freedom. The lifetimes observed in the photoinitiated experiments both correspond to a mean vibrational frequency V = 35 cm-’ when the number of internal vibrational degrees of freedom s = 4 appropriate to a quasilinear complex is employed. This is somewhat less than the estimated bending mode - 55 cm-’ and stretching ’ frequencies [4], which would be in mode -9Ocmline with an increase in these frequencies in the product transition state weighting the quotient [17] which determines the effective average frequency. Employing this value in Eq. (13) yields estimates of the lifetime Q-- 5 and 3 ps for the BrII complex in the present experiments at initial translational energies E = 52 and 87 kJ mol- ‘, which are similar to the rotational periods of the complex calculated from 7, = 2TrI */L,,

Fig. 12. Range of integration for Eq. (12) with a2 = 2.5. The ordering of the formulae in the appendix is indicated.

(14)

where I * denotes the moment of inertia of the BrII complex. The trajectory calculations of Sims et al. [4] over the BrII potential energy surface illustrated in Fig. 7 show that the potential energy of the barrier in the exit valley is disposed into additional product translational energy. The observation that the experimental product translational energy distributions, particularly in the case of Br atoms seeded in Ne, are not compatible with the inclusion of such a potential energy barrier in the phase space calculation, suggests that reaction trajectories do not necessarily sample only the BrII potential energy well under cross beam conditions. Indeed the trajectory calculations under photoinitiated conditions show a small proportion of reaction trajectories sampling the alternative and less stable IBrI configuration, which does not have an exit valley barrier as shown schematically in Fig. 7. This view is supported by the variation of the product translational energy distribution

Appendix.

Integration

When P(cos I(R)=l-$3

211

with scattering angle whereby trajectories arising from large impact parameter collisions undergo migration [18] via the IBrI configuration and scatter mainly in the forward direction with lower product translational energy while trajectories arising from small impact parameters sample the BrII potential energy well and scatter over a wide range of scattering angles with higher product translational energy. The distinction is more apparent at the lower initial translational energy E = 52 kJ mol-’ where the differences in the stabilities of the IBrI and BrII configurations have a greater effect than at the higher initial translational energy E = 87 kJ mol-‘. Evidently the elongated isosceles triangular BrI, configuration on the lowest potential energy surface is stabilised by electronic configuration and spin-orbit interactions [ 19-221, thus allowing access to the IBrI configuration. However trajectories sample essentially only the BrII configuration at the very low initial translational energies and confined geometries of the photoinitiated experiment.

Acknowledgement Support of this work by EPSRC and the European Commission is gratefully acknowledged.

of Eq. (12) over the ranges shown in Fig. 12 yields

a’) = 1 - u2cos2cr’: cos’y cos*0 + sin% - cos*y

),

for l/a>cos(8-y)>cos(6+y)>

-l/a,

JJ. Wang et al./Chemical

212

cos y cos O- l/a

Z(0) =

1 ( q--arcsin

sin y sin 0

Physics 207 (1996) 203-214

a2 1 - ~(3

)I(

cos’y cos*O + sin% - cos2y) 1

2 a2 1 + -y-y ; + 3 cos y cos 0 sin*8 - cos*y + -cos a Ii (

1 y cos 0 - a2 1

I/2 ,

forcos(O-y)al/a>cos(8+Y)>-l/a, cos y cos 8+ l/a I( f3) =

arcsin [

sin y sin

i a2 X 1 - --(3 i

e-

cos y cos - arcsin

e

1

sin y sin 0

(

2 sin*8 - cos*y + -cos a i

+

1 - - 3 cos y cos i a

0

iI

) I a2

cos*y cos28 + sin*8 - cos2y)

X

l/a

+ T

1 ; + 3 cos y cos 0 1

1 l/2 y cos 0 - a2 1

2 sin*0 - cos2y - -cos a Ii

1 I/* y cos 0 - , a2 ) I

forcos(O-y)>l/a>-l/a>cos(O+y), cos y cos O+ l/a

++arcsin

Z(8)=

[

a2 1 + z a - 3 cos y cos i for l/a>cos(O-

1 - + (3 cos*y cos*O + sin20 - cos*y)

sin y sin 8

i

e

2

I

)I(

2 sin20 - cos*y - -cos a ii

y) > -l/a>cos(8+

1 l/2 y cos 8 - , a2 1

y).

In the limit a2 + 1 this reduces to Z(O) = i[l + cos*y + (1 - 3 cos*y)cos*~] term of the expansion for I( 19) given in the Appendix of Ref. [13]. When P(cos a’) = (1 - u2cos2Q1’)*: Z(O) = 1 - a*(3 cos2y cos*O + sin20 - cos*y + f

)

[7 cos2y cos20(5 cos2y cos*O + 3(sin28 - cos*y))

+3(sin*8 for l/a>cos(O--

- cos2y)(3

1 ( t-arcsin

+ G(7

cos* y cos*fI + sin28 - cos2y)] ,

y) >Cos(O+y) cos y

Z(O)=

which agrees with the second

> -l/a,

cos O- l/a

sin y sin 0

1 - a*(3 cos*y cos*6 + sin28 - cos*y) )I(

cos*y cos*8[5 cos2y cos28 + 3(sin28 - cos2y)]

X (3 cos*y cos28 + sin28 - cos2y)}

+ i

a”( l/a (

+ 3(sin28 - cos2y)

+ 3 cos y cos 0)

JJ. Wang et a/./Chemical

--

Physics 207 (1996) 203-214

1 5 1 - + 7 cos y cos 0 3a2 + zcos 4 [ a3

a4

(

+ +(sin2e -

cos2y)( l/a

y cos 0 + $os’y

+ 3 cos y Cos e)

=

arcsin [

l/a

cos y cos e- arcsin

sin y sin e

( X

- cos2y) 1 l/2 ’

cos(e+y)>--l/a, cos y cos e+

Z(e)

cos20 + f(sin2e

2 1 cos2y + - cos y cos 8 - a a2 i

sin2e IN

forcos(e-y)>l/a>

213

1

l/a

sin y sin 0

i

)

1 - a’(3 cos’y cos2e + sin28 - cos’y

11 a4 + s (7 cos2y cos2e [5 cos2y cos2e

( + 3( sin28 - cos’y

)]

)( 3 cos2y cos2e + sin28 - cos2y)}

+ 3( sin28 - cos*y

1 1 5 ---Q+--&COSycose

1 a4 1 -;;+7cosycose + a2 - + 3 cos y cos 8 - a 4 [a 1 i ( +$os2y

cos20 + 2(sin2e - cos’y)

i

+ z(sin2e - cos2y) i

sin28 -

X

i --

a4

2 cos2y + -cos a

l/2

1 y Cos 9 - a2

1

- - 7 cos y cos e $ 4 [ a3 (

f + 3 cos y cos e i

111

1 + a2 - - 3 cos y cos 0 a i i(

) 5 - -& cos y cos 0 + ~cos2y cos20 + +(sin2e - cos’y ) 1

1 + +(sin2e - cos2y) - - 3 cos y cos e i a N forcos(ey) > l/a> -l/a> cos (e+ y),

2 sin28 - cos2y - -cos a

1

l/2

y cos e - a2 1

,

cos y cos e + l/a Z(e)

=

t

farcsin

[

i

+ f

X

sin y sin e

1 - a2(3 cos2y cos2e + sin28 - cos2y) iii

(7 cos’y cos2e [5 cos2y cos’e + 3(sin2e - cos’y

(3 cos2y cos2e + sin’e - cos2y)]

+ i

--

a4

+ 3(sin2e - cos’y )

1 u2 - - 3 cos y cos 8 a 1 H

1

- - 7 cos y cos e --& - &co, 4 [ a3 ( 1 +:(sin2ecos’y) - - 3 cos y cos 8 ( a for l/a3cos(e-

)]

y) > -l/a>cos(e+

y).

y cos 8 + +os2y

cos2e + $(sin’e

ilic

2 1 l/2 sin20 - c0s2y - -COS y cos 8 - , a a2

In the limit a2 -+ 1 this reduces to I( e> = (1 - 3 cos~y)cos2e + :(sin2e

+ cos2y + Tcos”y

- cos2y)cos2y

cos4e

cos2e + +(sin2e - cos2y)‘,

in accord with the third term of the expansion

- cos’y)

for I(6) given in the Appendix

of Ref. [13].

214

J.J. Wang et d/Chemical

References [l] Y.T. Lee, J.D. McDonald, P.R. LeBreton and D.R. Herschbach, J. Chem. Phys. 49 (1968) 2447. [2] Y.T. Lee, P.R. LeBreton, J.D. McDonald and D.R. Herschbach, J. Chem. Phys. 51 (1969) 455. [3] M. Gruebele, I.R. Sims, E.D. Potter and A.H. Zewail, J. Chem. Phys. 95 (1991) 7763. [4] I.R. Sims, M. Gruebele, E.D. Potter and A.H. Zewail, J. Chem. Phys. 97 (1992) 4127. [5] R. Grice, Nature 359 (1992) 584. [6] R.D. Jarvis, R.W.P. White, Z.Z. Zhu, D.J. Smith and R. Grice, Mol. Phys. 75 (1992) 587. [7] P.A. Gorry and R. Grice, J. Phys. E 12 (1979) 857. [8] C.V. Nowikow and R. Grice, J. Phys. E 12 (1979) 515. [9] E.A. Entemann and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 289. [lo] K.P. Huber and G. Heraberg, Constants for diatomic molecules (Van Nostrand Reinhold, New York, 1979). [l I] P. Pechukas, J.C. Light and C. Rankin, J. Chem. Phys. 44 (1966) 794.

Physics 207 (1996) 203-214 [12] J. Lin and J.C. Light, J. Chem. Phys. 59 (1966) 2545. [I 31 J.J. Wang, Z.Z. Zhu, D.J. Smith and R. Grice, J. Phys. Chem. 98 (1994) 10787. 1141 R.D. Jarvis and R. Grice, Mol. Phys. 65 (1988) 1205. [15] R.D. Jarvis, J.J. Harkin, D.J. Smith and R. Grice, Chem. Phys. Letters 167 (1990) 90. [16] G.A. Fisk, J.D. McDonald and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 228. [17] R.W.P. White, D.J. Smith and R. Grice, J. Phys. Chem. 97 (1993) 2124. [18] Z.Z. Zhu, R.W.P. White, D.J. Smith and R. Grice, Can. J. Chem. 72 (1994) 523. [19] J.J. Duggan and R. Grice, J. Chem. Sot. Faraday Trans. II 80 (1984) 795. [20] J.J. Duggan and R. Grice, J. Chem. Sot. Faraday Trans. II 80 (1984) 809. [21] A.B. Samtigrahi and S.D. Peyerimhoff, Intern. J. Quantum Chem. 30 (1986) 413. [22] A. Viste and P. Pyykko, Intern. J. Quantum Chem. 25 (1984) 223.