A model study of symmetric light atom transfer reactions

Chemical Physics 99 (1985) 15-33 North-Holland, Amsterdam

15

A M O D E L STUDY OF S Y M M E T R I C L I G H T A T O M T R A N S F E R R E A C T I O N S Eli P O L L A K l Department of Chemistry, University of California, Berkeley, California 94720, USA

Michael BAER Applied Mathematics, Soreq Nuclear Research Center, Yavne, Israel 70600

and Najib ABU-SALBI and Donald J. K O U R I Department of Chemistry, University of Houston, Houston, Texas 77004, USA Received 19 December 1984

A collinear model for symmetric light atom transfer reactions is constructed and generalized analytically to 3D scattering via a sudden and an adiabatic approach. The model predicts oscillations in the energy dependence of the full 3D integral cross section in the threshold region. The conditions for oscillatory behaviour are a light atom transfer reaction which occurs on a collinearly dominated, low-barrier potential energy surface. At higher energies there is a switch from a loose to a tight transition state which prevents further oscillations in the cross section. For reactions with large activation energy, such as the CI+ HCI exchange reaction, already at threshold the transition state is tight so that one does not find an oscillatory integral cross section. For the C I + HCI reaction on the Last-Baer potential energy surface, the model rate constants are in good agreement with experiment. Implications of our study for future 3D quantal and classical computations are discussed in detail.

1. Introduction

One of the unsolved problems in reaction rate theory is understanding the temperature dependence of light atom transfer in a t o m - d i a t o m reactive collisions. Mei and Moore [1], and later Wiarzberg and Houston [2], have found experimentally that for reactions such as C1 + HI or F + HBr, the thermal rate of reaction is by no means a monotonic function of temperature. Thus for CI + HI, increasing the temperature above 300 K causes a decrease in the rate while for F + HBr, decreasing the temperature below 250 K causes an increase in the rate. This does not imply that for any light atom transfer reaction, curvature in the Arrhenius I n k versus 1/T plot should be observed. In fact, the C1 + HCI reaction rate has been measured twenty years ago by Klein et al. [3] and they found a simple Arrhenius-like temperature dependence with an activation energy of = 0.2 eV. Their results have been verified subsequently by a number of independent studies [4] and most recently by Kneba and Wolfrum [5]. In fact, the latter also measured the temperature dependence of the rate for vibrationally excited HCI and still find simple Arrhenius-like behavior. The measurements of Mei and Moore [1] for the CI + HBr reaction have been redone by Rubin and Persky [6], who find simple Arrhenius-like behavior in contradiction to ref. [1]. Thus the experimental picture is far from being completely clear. However, overall, it seems that for light atom transfer reactions,

1 Permanent address: Chemical Physics Department, Weizmann Institute of Science, Rehovot, 76100, Israel.

0301-0104/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

16

E. Pollak et al. / Symmetric light atom transfer reactions

Arrhenius-like rates, though possible, are not necessarily the rule. To the best of our knowledge, a theory which accounts for the extensive curvature in the Arrhenius plots does not exist. The theoretical picture is also quite intriguing. The first quantum mechanical treatment of light atom transfer reactions was given by Baer [7] a decade ago. However, slow convergence due to the h e a v y - l i g h t - h e a v y mass combination prevented a systematic study. Recently this problem has been solved through use of radial coordinates [8]. Quantally, Kaye and K u p p e r m a n n [9] and Manz and R6melt [10] found that the dependence of the collinear reaction probability on energy is oscillatory; The lighter the central atom, the more oscillations are observed. The oscillations have been observed both in symmetric [9-11] and asymmetric collinear light atom transfer computations [12]. They have also been observed in collinear quasiclassical trajectory computations [9,13]. The physics underlying this oscillatory behavior is by now well understood. The effect is in principle classical in nature. As shown by Pollak [14], the light atom j u m p s between the two heavy colliders an increasing number of times as the energy is increased. The collision outcome depends on whether an even or odd number of crossings occurs. If even, the reaction probability is zero; if odd, the probability is one. As energy is increased the number of crossings goes through the values 1,3,5 . . . . and so the reaction probability oscillates. Hiller et al. [15] have reached the same conclusion using semiclassical adiabatic arguments. Babamov et al. [16,17] have analyzed the oscillations in terms of a vibrationally adiabatic approximation. As shown by Hiller et al. [15] their analysis is identical with Pollak's in the classical limit. These oscillations are very suggestive. A working hypothesis would be that they exist also in the full 3D cross section and so are responsible for curvature in the Arrhenius plots. One obvious way of checking this hypothesis is via a 3D classical trajectory computation. Such studies have been undertaken by Polanyi and co-workers [18], Thompson and co-workers [19], and Smith [20], however with seemingly negative results. Cross sections and reaction rates increase monotonically with energy and temperature. Shirai et al. [21] studied the I H I system using a semiclassical impact parameter method. Although they do find some structure in the differential cross section, it is totally wiped out in the total cross section. Garrett et al. [22] have studied the C1HC1 system using transition state theory and find reasonable agreement with experiment: generically, transition state theory will give Arrhenius-like behavior. In summary, to date all 3D theories seem to indicate that the collinear oscillatory behavior is washed out and does not account for the curvature found in the experimental Arrhenius plots. The purpose of the present study is two-fold. Firstly, we would like to get a better understanding of the "smearing" that occurs as one goes from the quasiclassical collinear to the quasiclassical 3D world. Secondly, based on this understanding, we suggest a quantal mechanism which can cause greater oscillations in the quantal 3D world. To do this we construct in section 2 a collinear model for the reaction probability which is a hybrid of adiabatic transition state theory and the multiple reflection theories of Pollak [14] and Hiller et al. [15]. In section 3 we generalize the model to 3D via a reactive infinite-order sudden formalism [23,24] used previously to formulate a 3D sudden transition state theory [25,26]. In section 4 we generalize the model to 3D via an adiabatic bending formulation. The relationship between the sudden and adiab~atic models for the bend degree of freedom as well as the experimental relevance of the models is discussed in section 5.

2. A collinear model

The collinear model to be constructed in this section is a hybrid of Pollak's purely classical direct reaction theory [14] and Hiller et al.'s semiclassical theory [15], which we will review briefly. In a collinear symmetric exchange reaction, we denote the microcanonical flux at energy E of incoming reactants (or incoming products) as E~(E), and the microcanonical flux through the symmetric stretch in products (or

E. Pollak et al. / Svrnmetric light atom transfer reactions

17

reactants) direction ~ ( E ) . The reactant flux may be subdivided as

E~(E) = ~ Ed"(E),

(1)

n=0

where E~"(E) is the portion of E~ that crosses the symmetric stretch n times. The average number of crossings theorem [27] ensures us, provided that any trajectory originating on the symmetric stretch (barring sets of measure zero) makes it out to reactants or products that: ~(E)

= ~ nE~"(E).

(2)

n=O

Thus at energy E, the average number of times that reactants cross the symmetric stretch ((n(E))) is known, without doing any dynamics: (,,(e))

=

"

(3)

n=0

Note that if we define a quantity N ( E ) as the number of oscillations at energy E of the light atom between its heavy partners then clearly,

N(E)=½(n(E)).

(4)

The theory of direct reactions then states [14] that if

(n(E))=m+c,

m

integer,

0~<,~<1,

(5)

then Pro=l-c,

p,,,+l=c,

P; = O,

i ~ m,m + 1,

(6)

where

P ' = ~I( E)/Ed( E ).

(7)

Since ( n ) is usually a monotonically increasing function of the energy, ( n ) wilt go through the values 0, 1, 2 . . . . . as energy is increased. Every time it goes through an odd integer the reaction probability is unity (an odd number of crossings of the symmetric stretch implies a reactive trajectory). Whenever ( n ) is even, the reaction probability is zero. Thus one obtains an oscillatory reaction probability. This result may be put in the equivalent form

P(E)=2IN(E.)-[N(E)+½]

1,

(8)

where here [ ] denotes the greatest integer function, N ( E ) is defined in eq. (4) and P(E) is the reaction probability. Eq. (8) is precisely eq. (3.6) of ref. [15]. Hiller et al. [15] have shown that the rhs of eq. (8) is excellently approximated as

P(.E ) = sin 2 ['rrN( E ) ] ,

(9)

which is just the semiclassical limit of the two-state adiabatic reaction probability. Our hybrid model is obtained by combining eqs. (3), (4) and (9) with adiabatic transition state theory. For reactants in vibrational state m, with asymptotic vibrational energy E,,, and with an adiabatic barrier energy E,,*, for reaction, adiabatic transition state theory implies that P,v,sr(E) = O( E - Eft,).

(10)

18

E. Pollak et al. / Symmetric light atom transfer reactions

Here O ( X ) is the unit step function and P , , ( E ) is the reaction probability from state m at energy E. The collinear model we propose is

/,,,MOB(E) = P,Ir,ST(E) sin2 (½"~
(11)

Thus the reaction probability is the probability of crossing the adiabatic barrier to reaction at energy E, which is then corrected for recrossing effects. Note though that the recrossing term is evaluated microcanonically, it is independent of the initial vibrational state. Finally, we note [14] that is reasonably well approximated, at all energies from the saddle-point energy E* to the three-body dissociation limit as

(n(E)> = (WAB/WSS)(I -- E*/E)O(E- E*).

(12)

Here ~0AB and ~0ss are the harmonic vibrational frequencies of reactants and the symmetric stretch respectively. (It should be stressed that the energy dependence of given in eq. (12) is similar to that derived from semiclassical considerations in ref. [15].) Thus our final expression for the reaction probability will be p,,Mor)( E ) = p,;rsT(E) sin 2 [ ½"rr(~OAn/~Oss)(1 - E * / E ) ] O( E - e * ) .

os°lip icc. 0.6

~I

I

'

CI+ DCI

I I

i

0.2 O.4

Ill

0

1

0

0.4

0.8

1.0

,

o.~

12

I

0

0.4 0.8

'

:, \o_i

|

tlm

I

'

r

1.2 '

CI +

HCI

i~111"

0.6 g 0.4

Q,.

!/11~ 0

,o 0.4

0.2

0.6

IIIl'/

i[itl,

i',] , 1.2 1.8 0 v

A

I

0.6 I

CI+

1.2

'

I

tl

IWI'jiIIMuC 'c,. 0.8 1.6' 2.4

E/eV

1.8

t I

I

00

/-

,i1%/ I

0.2 0

I

(13)

0

l

0 8l

I

I

1.6 2.4

E/eV

Fig. 1. Collinear model for symmetric light atom transfer reaction - application to the CI + HCI system. The dashed lines are the model predictions,the solid lines the exact quantal results adapted from ref. [11]. For further details see text.

E. Pollak et al. / Symmetric light atom transJer reactions

19

This result has the nice feature, that it easily predicts the effects of isotopic substitution. Using the Porter-Karplus [28] notation, if A~l is the harmonic symmetric stretch force constant (using normal mode coordinates) and kAa is the harmonic vibrational force constant of reactants (using the rAB coordinate) then it is easy to see that (0~AB/t.OSS) 2 : ( 2 k A B / A l l ) ( m

A ÷ mB)/m

B.

(14)

Here m A, m B are the masses of atoms A, B in the A + BA ~ AB + A symmetric exchange reaction. Note that for light atom transfer m B << mA, eq. (14) may be rewritten as WAB/O~SS= ( 4 k A B / A l l ) ' / 2 ( 1 / s i n fl),

(15)

where fl is the "skew" angle of the collinear surface plotted in mass scaled coordinates. To test the validity of the model we have applied it to the collinear Cl + HC1 reaction and its isotopic analogs with D and Mu. The surface used is the extended LEPS surface of ref. [11]. Note though that their collinear saddle point parameters (E* = 0.371 eV, ~0ss = 344 cm -1) [29] are almost identical to those of the collinear Last-Baer surface [30] (E* = 0.362 eV, ~0ss = 357 c m - I ) . In fig. 1 we compare our model [eq. (13)] with the exact quantal computations of Bondi et al. [11] for the three systems for the ground and first excited vibrational state reaction probabilities. Note the good qualitative agreement for all six probability curves. The model correctly accounts for the number of oscillations and their energy dependence. Note that our model predicts that the excited state reaction probabilities should be superimposable on the ground state reaction probabilities. This prediction is also in good agreement with the exact quantal results. The model, of course, does not deal with the resonances (note the spikes in fig. 1) apparent in the exact quantal results. These are ignored throughout this paper. Note also that we have ignored tunneling. This may be easily remedied by substituting a suitable tunneling correction for the step function appearing in eq. (10). Finally, note that our model is superior to that of Bondi et al. [29] [their eq. (8)] which predicts the qualitatively wrong behavior that the period of oscillation decreases with increasing energy. As is evident from fig. 1 and as noted by others [10,14,15] the period of oscillation increases with energy, and this is correctly accounted for in our model.

3. Generalization to 3 D - the sudden approach

For direct reactions, o n e of the easiest and most reliable ways of generalizing a collinear result to three dimensions developed by Baer and Kouri and co-workers [23,26] and Bowman and Lee [24] is via the infinite-order sudden approximation. Here, for each value of a frozen bend angle 7 one solves an angular-momentum-dependent collinear-like problem. One then sums over all angular momentum and 7 contributions, with suitable weights to obtain the 3D cross section [31]. To simplify, we will always assume that reagents are in their ground rotational state. The angle 7 is defined as the angle between the diatom vector rAB and the v e c t o r RA_ BA from the atom to the center of mass of the diatom. Within the sudden formalism, the angle-dependent cross section am(E-r, Y) from reactants in the ruth vibrational state and with initial translational energy E v is given as [23,26,31] ,lTh 2

o¢

o£,(Ev, 7) = 2/tE----~y~=0( 2 J + 1)P""J(Ex' 7).

(16)

Here Pm,j(Ey, 7) is the collinear-like reaction probability at fixed 7 and total angular m o m e n t u m J. ~ is a reduced mass, /.t = m A ( m A + mu)/(2m A + roB). The total cross section o,,(E-r) is the average over 7 [23]: o,,,(Ev) = ½

dTsinTo,,(Ev, 7).

(17)

20

E. Pollak et al. / Symmetric light atom transfer reactions

One very simple method of evaluating the lOS cross section is via a vibrationally adiabatic transition state theory. We assume that the rnth vibrationally adiabatic barrier height at angle y and total angular momentum J may be approximated as E,~(y, J)

=

E,~ + V("(1 - cos y) + B,~J 2

(18)

Here Vv'" is a force constant and B,~ = h2/21, where I is the moment of inertia of the triatomic system at the saddle point. As noted by Garrett and Truhlar [32], the bend potential must usually be expanded to quadratic terms in (1 - cos "/); however, for simplicity's sake we retain only the linear term. This usually will cause a slight overestimation of the cross section away from threshold. A comparison of cross sections obtained using eq. (18) and a more exact analysis is given in ref. [33]. In principle, the rotational constant is also "¢ dependent, but only slightly [33], so this may also be neglected. An IOS transition state theory may now be formulated as [26,31]

P,',,°sxsv( E v, y) = O[ E - E ~ ( y , J ) ] .

(19)

This provides a maximal angular m o m e n t u m Jm,x(ET, Y) for which reaction is allowed

J~2,~x(Ev, V) = (1/B~)[ E - E,*,,(y, 0)] O[ E - E,*,,(~,, 0)].

(20)

Insertion of eqs. (19) and (20) into eqs. (16) and (17) gives

O,I~'S'TST( E1, 3')= (~h2/ZbtET)J,~x( ET, Y)

(21)

O,I~'S'SST(E x ) = ( ~rh2/81~Ev )( E - E,~ )2/B,~ V(".

(22)

and

As may be seen from figs. 5, 7 and 8 of ref. [33] this expression gives cross sections which are in good agreement with quasiclassical trajectory results. Levine and Bernstein [34] have recently rederived the same type of theory and reach a similar conclusion. The thermal rate constant at temperature T from the mth vibrational state k,,,(T) is given as [35]

k,,,( T ) = N(8/~kt )'/2 fl3/2 f o ~ d E T ETe-I'E'O,,,( ET ).

(23)

Here N is Avogadro's number and fl = 1 / k B T where ku is Boltzmann's constant. Using the IOS TST result [eq. (22)] we find

k,,,(T) = U(S/~rtx)'/2(~rh 2/411B,,,V~, * "' )fl 3/2 exp[-fl(E,*,,-E,,,)].

(24)

For the ground state C1 + HC1 and CI + DC1 reactions on the L a s t - B a e r potential energy surface [30] we find that eq. (24~ gives very reasonable agreement with experimental results [36]. The parameters used for fig. 2 are Vv° = 1.489 eV and B0~ = 0.106 c m - ~. For ground state C1 + HC1 and CI + DC1, the adiabatic barrier heights are 0.199 and 0.253 eV respectively. The main purpose of this section is, however, to understand the effect of the oscillatory reaction probability on the 3D rate. In the collinear case we have seen that the frequency of the oscillations is critically dependent on the mass of the central atom. Increasing the mass of the central atom increases the "skewing" angle, causing the surface to "open up". As a result the frequency of the oscillations.decreases [cf. eqs. (12)-(15)]. Consider now t h e IOS formalism. Here one solves the scattering problem at a fixed bend angle (Yx) in the entrance channel and at a fixed bend angle (y~) in the exit channel and " m a t c h e s " the solutions at a well defined boundary in the interaction region (the symmetric stretch line in a symmetric exchange reaction). As shown in appendix A for non-collinear bend angles this procedure implies that.the

E. Pollak et al. / Symmetric light atom transfer reactions -30,

~

~

15

Fig. 2. Thermal rate constants for the C1 + HCI and CI + DCI exchange reactions. The experimental results [3-6] are adapted from table 1 of ref. [22]. The solid lines are the lOS TST rates obtained from eq. (24) and the Last-Baer [30] potential energy surface. The dashed line is the model IOS result for this system based on eq. (38) and the cross section shown in fig. 7.

..5 =

-40

2.0

21

2.5

3.5

3.O

4.0

IO00/T

angle-dependent potential energy surface has an effective skewing angle that increases as one goes to larger bend angles [cf. eq. (A.15)]. The result is that within the IOS a p p r o x i m a t i o n we expect that for non-collinear bend angles the scattering amplitude will be less oscillatory. This will contribute towards smearing the 1D oscillations. N o t e that this smearing has nothing to do with the details of the potential energy surface in the interaction region. A second effect in 3D is dependent on the potential energy surface. For a collinearly dominated system we know that the potential barrier height increases with increasing angle. In analogy with the vibrationally adiabatic barrier to reaction [eq. (17)], we will assume that the ~,, J dependence of the symmetric stretch saddle-point energy is E*(y, J)=E*+

Vv(1-cos 7)+B*J

z.

(25)

In addition we have to determine the dependence of the symmetric stretch frequency on the angle T- This is done in appendix B. The final result for the h e a v y - l i g h t - h e a v y mass c o m b i n a t i o n (and for T values not too far from collinear) is: ~0ss (T) -- ~0ss/COS V.

(B.9)

N o t e that the symmetric stretch frequency becomes larger as the angle increases. The larger the frequency the smaller the average n u m b e r of crossings [cf. eq. (12)] and so the less oscillations. This is exactly what we expect from the lOS formalism. In fact, in appendix A we also show that eq. (B.9) m a y be approximately derived by defining a bend-angle-dependent skewing angle. Although the smearing of the oscillations as a result of the angle dependence of the frequency should not be neglected in quantitative work we proceed to do so in this p a p e r for a simple practical reason ~:. This is the price we pay to obtain analytical formulae for the cross section. With these simplifications, we find that for fixed T, J the average n u m b e r of crossings will be, in analogy with eq. (12):

= (¢°AB/~SS)[] -- E * ( T , J ) / E ] O[ E - E * ( y , J ) ] .

(26)

T h e IOS model reaction probability, will also be, in analogy with eq. (13): P,,M,jOD"OS(ET, y ) = sin2(½v(n( E, 7, J)>)PI,,°S'TST( ET, y ) ,

(27)

where P,I,OS'TST(ET, T) has been defined in eq. (19). As usual, one must now sum over J values. It is obvious from eqs. (19) and (26) that for fixed T; Jm2,x( ET, 3') = m i n { [ E -

E*(T, 0)1/B*,[ E-

E,,*,(T, 0)]/B,,*, }.

'~ We have verified that in fact adding the (cos 7) x term ~toes not alter any of the qualitative features.

(28)

E. Pollak et al. / Symmetric light atom transfer reactions

22

Consider a reaction with a sizable static barrier to reaction. For the lowest lying vibrational state, the location of the adiabatic barrier and the symmetric stretch will often coincide so that for such a system, to an excellent approximation, V° = Vv and B0* = B*. However, as the vibrational state increases, we know that the vibrationally adiabatic barrier migrates toward the entrance and exit channels. Here the three-body interactions are usually smaller than in the saddle-point region, one would expect a "looser" transition state and so W" < Vr" Also, the "size" of the system at the vibrationally adiabatic barrier will be larger than at the symmetric stretch so that B~ < B*. However, Eft, > E* since we are assuming that for almost all energies the symmetric stretch behaves as a well. All this implies, that as energy and angle 7 are increased, eq. (28) predicts a j u m p of the bottleneck to reaction from the adiabatic barrier in the entrance or exit channel to the symmetric stretch. This effect is similar to the switching from a loose transition state to a tight transition state as energy is increased in ion-molecule reactions [37]. The summation over J values is straightforward: pmMOD'IOS(ET, "y) =fog":'"~d(J:) ½[1 - c o s " ~ n ( E, ~, J ) ) ] 0[ E - E ~ ( ~ , 0)] = { Jm2a,(ET, 3`)/2 + -sin

(E/2~Bt)(tOss/CoAB)[sin "~(n(E, y, Jm~x))

"~(n( E, 3', 0))1 } O[ E - E*,,( ~,, 0)].

(29)

This result has an interesting interpretation. Note that J2max(E T, ~) would be the lOS TST result if the oscillatory sin2½v(n) term had not been added into the model. The 1 / 2 multiplying Jm~x(ET, 2 3') in eq. (29) is a direct result of the oscillatory behavior. In fact, in the statistical limit, the reaction probability would be exactly J2m,x/2 since once having crossed the barrier there would be equal probability for being transmitted or reflected. Thus within our model the statistical result is modulated by an oscillatory term. Here the reaction is direct and the oscillations are a result of the light atom transfer mechanism. To gauge the effect of these oscillations we consider two reactions. The first is the ground state CI + HC1 reaction on the L a s t - B a e r potential energy surface [30], for which the relevant parameters are the same as for figs. 1 and 2. In fig. 3 we plot the energy dependence of the cross section for different bend angles. Here the location of the m = 0 barrier coincides with the symmetric stretch because of the relatively large saddle-point energy. Note that the graphs do show some structure; however, it is evident that by the time we average over angles even this structure will be wiped out. Note how the different angles interfere destructively, so that when summed over they will give a monotonically increasing function of energy. The C1 + HCI (m = 0) reaction is, however, qualitatively different from the light atom transfer reactions which exhibit curvature in the Arrhenius plots. The latter have negligible activation energy. Therefore we chose to study also a prototype for a reaction with no activation energy. Here one might expect [31] that already for the ground state the adiabatic barrier will be in the entrance (and exit) channel and will not coincide with the symmetric stretch. Specifically, in fig. 4, we plot the energy and angle dependence of the model cross section for a system with the same masses as CIHC1 but having E* = 0, E0* = 0.2 eV. Thus we assume that the ~ground state reaction has an adiabatic barrier in the entrance channel whose collinear height is 0.015 eV. The, parameters W, B* are left as in the previous model, however V° = ½ V~, B0* = ½B*. This implies, that for all angles 3', the bottleneck switches from the adiabatic barrier to the symmetric stretch at E = 0.4 eV. Note the dramatic effect this switchover has on the cross section. Below the switching energy, the cross section rises rapidly and exhibits noticeable oscillations. Above this energy, the energy dependence becomes much smoother and similar to that shown in fig. 3 (note specially the ~, = 30 ° curve in this range). Clearly, the activation energy serves to suppress any oscillatory behavior. It is only in the region below the switchover where the system effectively encounters a well in the interaction region that not only do oscillations persist but they stay more or less in phase for different angles. This is a first indication that for zero activation energy light atom transfer reactions, oscillations can be found in the 3D integral cross section.

E. Pollak et aL/ Symmetric light atom

23

transfer reactions

Instead of first summing eq. (27) over the total angular momentum, one may first average over angles to look at the E T, J dependence of the cross section. Specifically, ["~ P ' ~MOD,IOS 'J (ET) = ½J0

dy s i n ypM°D'I°S(ET, "" '

V J "y) = ½ /".......

Jo

nMOD,IOS,"

tEr -, y).

(30)

Here, y = 1 - cos ~' and

Ym~x= min[(

E - E*- B*j2)/Vv,( E - E~- B~J2)/V~'].

(31)

The integration in eq. (30) is carried out in the same manner as in eq. (29) with a similarly looking result MOD,IOS P2,.j (ET)

=

(Ymax/4-F(E//4'rrVv)(tOSS/tOAB)[sin "~
Ymax, J ) >

-

sin

~r(n(E, O, J)>]}

E~(0, J ) ] .

(32)

Experimentally, at least in principle, the differential cross section can be measured as well as the integral cross section. Even if oscillations are averaged over in the integral cross section they may be observable in angular distributions. Such oscillations, if they exist, will usually show up as oscillatory behavior in the opacity function [38,391. Since, in our treatment the initial rotational state is j = 0, any total angular momentum J is of course also the initial orbital angular m o m e n t u m l. Thus eq. (32) immediately gives us the energy and impact-parameter (b) dependence of the opacity function. In figs. 5 and 6 we plot the J dependence of the opacity function [eq. (32)] for the two C1HC1 models. Note again that when the saddle point energy is substantial (fig. 5) the plots show relatively little structure. On the other hand, when the saddle point energy is negligible one finds a large number of undulations (fig. 6) in the opacity functions. The next step in the model is to obtain an expression for the integral cross section. Thus either eq. (29) is to be averaged over angles or eq. (32) must be summed over angular momentum. To simplify the expression we will assume that

V~/B*.

V~"/B~ =

(33)

IO0 40

l

i

I

CI + HCI (m = O)

CI + HCl ( m = O )

E :I:

OeV

I =

E ::1: = 0 . 3 6 2 eV

o

80

60,

o }o b

I0

d o ~o b

///oo/ 20 °

0

Ill

0.3

0.4

/

I

0.5

4o

20

,,/I

0.6

I

0.7

E (eV)

Fig. 3. Energy and angle dependence of the model lOS cross section for light atom transfer with a large activation energy. For further details see text.

0

/i

I I 0.4 0.6 0.8 E (eV) Fig. 4. Energy and angle dependence of the model lOS cross section for light atom transfer with a small activation energy. For further details see text. 0

I 0.2

!

24

E. Pollak et a L / Symmetric light atom transfer reactions

T h i s c o n d i t i o n e n s u r e s us t h a t Jm2ax(ET, y ) = [ E -

e~(y,

0)]/B,~,

E < ~ E ~*.

= [ E - e*(~, o)]/B*,

(34)

E>~E*.

Here

(35)

e,,*, = ( 8+e~ - B~e* )/( 8 * - ~ )

Eq. (34) e n s u r e s us that the s w i t c h i n g o f t r a n s i t i o n states is a f u n c t i o n of e n e r g y only. If we d e n o t e y = 1 - cos 7 t h e n y is l i m i t e d b y the c o n d i t i o n s y ....

=(E-e~)/vy, e<.e:,;

=(E-E*)/w, E>~E*. I n t e g r a t i o n of

eq. (29)

(36)

is slightly t e d i o u s b u t s t r a i g h t f o r w a r d , one finds

o, MOD'IOS ( E T )

1 IOS,TST/w \ - 20'n

=

l

]~2 (¢"0SSI 2 E _ . . ~2

~'LT) ~- 4 v g \ ~OAB ]

{.~+oa.[2(l_~_)

V, sin 5 +ss l ~

E-E,~

~ WAB si n ,~*°AB 1 ~Oss [ ~Oss

B$ET V¢ZE 2 -~

B,,*,

E-

V7 (1_ E___~)]}sin['2-~+ss COAB V+(i E~)]}O(E_E,..); Vv"' E ~< E *m

O MOD'IOS( E T ) =

"rrh 2

8#B*ETVv~

CI + HCI

0.06

I

2

.+2

+0Aa]

[ g +OS----~. 1 -- --E

r

0.08

E$ = 0.562 eV

_

_.w

+

\

~a. o-

'

(37a)

" E >/e.*,.

(37b)

f (E - E*) 2

i I

1)

1-E

i

E = 0.55 ~ 0.50 O.

r

i

CI + HCI E:t: = 0 eV

0.06 0.35

g -, 0.04

' 0.6 o

0.02 0~0.4 0

0

0.5 i 40

80 120 160 J Fig. 5. Model lOS opacity functions at different total energies for light atom transfer with a large activation energy. Energies are in eV relative to the bottom of the asymptotic HCl well.

0

0

40

80

120 160 200 J Fig. 6. Model IOS opacity functions at different total energies for light atom transfer with negligible activation energy. Other details are as in fig. 5.

25

E. Pollak et al. / Symmetric light atom transfer reactions

N o t e that if Vv" = Vv, B~ = B* then E* = oo, that is one must use eq. (37a) for all energies. In this case it is easy to see that o,MOD.,OS( ET ) = , _,OS.TST, ET ) +

~o.,

t

-sin

["n" (,dAB (1 2 ~ss \

1 --

4.~I~B,VyE T

,rr ~0A..._BBsin "n ~0AB E,~ -- E* "~ 00ss 00ss ~-

- E* + E~--E*)].['rr~AB(sm E-

E

1 -- ~ ) ] } O ( E

- E~),

2 Wss \

B,,*,=B*). (38)

(Vv'=Vv;

The structure of the integral cross section is similar to that of the y - d e p e n d e n t cross section. The first term is the usual lOS T S T cross section multiplied by 1 / 2 because of equal probability for transmission or back reflection. Added to that is the oscillatory term that reflects the direct nature of the reaction. As usual, we have applied eqs. (37) and (38) to the two C1 + H C I models. In fig. 7 we show the integral cross section for C1 + HC1 with activation energy, and c o m p a r e it to ½0~OS'TST(ET). We note that the oscillations have virtually disappeared. In fig. 8 we plot the cross section for the model C1 + HC1 reaction with negligible activation energy also c o m p a r i n g with 1 o 0IOS.TST, ,- T)., Here, the oscillations appearing in fig. [/7. 4 persist, and there is a one-to-one correspondence between the oscillations in the two figures. H o w e v e r the deviation from the IOS T S T expression is quite small. N o t e that above the switching energy E* = 0.4 eV, all oscillations disappear and on the scale of this plot the model cross sections coincide with half the IOS T S T cross section. T h e final question to be answered is whether any structure remains after thermal averaging. We were not able to obtain an analytic expression for the rate constant, thus we had to resort to numerical integration. For the C1 + HC1 reaction with activation energy this is shown as the dashed line in fig. 2. As expected the Arrhenius plot is linear, the rate is only slightly smaller than the IOS T S T prediction. As can be seen from IO 2,0

1.5

I

[ CI + HCI

I

8 /

.

%MOD,I0S

/

"..///

~• i.o

.C o

/

o

z

o

4

2

0 0.45

0.6

I-,I

/~p-.....~ o.IOS,TST

// //

/7

/

///

~2

0.35

0.4

E:I:: 0

½

--=6

0

I

I

E:I: = 0.362 eV

0.5

I

CI + HCI

I

I

0.55

0.65

E (eV) Fig. 7. Energy dependence of the integral cross section of CI + HC1 with activation energy. The solid line is based on eq. (38), the dashed line on eq. (22). For further details see text.

0.2

0.8

E (eV) Fig. 8. Energy dependence of the integral cross section for CI + HCI with negligible activation energy, based on eq. (37). Note the oscillations and their termination at the transition state switching energy E* = 0.4 eV. The dashed line is half the lOS TST cross section for this model.

26

E. Pollak et a L / Symmetric light atom transfer reactions

fig. 7, in the threshold region the model cross section is almost equal to twice the dashed line and so roughly equal to the lOS cross section. In fact, since the symmetric stretch frequency is small, E~ = E *, so that eq. (37b) is a good approximation to the cross section at all energies. It is easy to see that in the threshold region ( E --- E ¢), eq. (37b) simply gives the lOS TST cross section. In fig. 9 we plot the temperature dependence of the thermal rate for the model CI + HC! reaction with negligible activation energy. Here, the model and half the lOS TST rate coincide on the scale of the plot. Clearly, by the time one averages the cross section thermally, all the oscillations disappear and one obtains a typical Arrhenius-like behavior.

4. G e n e r a l i z a t i o n

to 3D - the adiabatic approach

A different, well-known method for generalizing collinear results to 3D, developed by Bowman and co-workers [40], Walker and Hayes [41], and others [33,42], comes from treating the bend motion adiabatically. Here one assumes [41] that the bend motion is not too strongly coupled to the reaction coordinate so that as the reaction proceeds the system remains in its ground bending state if reactants were initiated in the ground rotational state. If the zero point bend energy at the symmetric stretch is Ea~ then in analogy with eq. (25) we can assume that at the symmetric stretch

E*(J) = E* + E~a+ B*J 2.

(39)

Similarly, if the zero point bend energy at the mth vibrationally adiabatic barrier is E~a.,, then in analogy with eq. (18) we may assume that

E~(J) = E~ + EL.,, + B~J 2.

(40)

One now obtains 3D cross sections by following the same scheme as in the sudden approach, but with one important difference - there is no need to average over angles - this has been taken care of through the adiabatic treatment of the bend. The TST adiabatic reaction probability is now, in analogy with eq. (19) ADB.TST( E ) =0[ E- E ~ ( J ) ] . P£.j

(41)

The average number of crossings at energy E and total angular m o m e n t u m J is in analogy with eq. (26)

(n(E, J ) > = (~%B/~0SS)[1

V

-24

I

I

I

0

-~ O

-25

_~~.~

-

E*(J)/E] O[E- E*(J)].

I

-21

CI + HCI E$: 0

-22

~ADE

-25

E E

O

-26

(42)

5o O

.-7

~-

-27

0

It)

Fig. 9. Thermal rate constant for CI + HCI with negligible

H

activation energy. The solid line denoted IOS is the model rate

o

d

-25

-28

o ~o

-~ C

-29

I 2

I 3

J 4

IO00/T

I 5

-26

6

based on numerical integration of eq. (37). The dashed line coinciding with the solid line is half the IOS TST rate. T h e solid line denoted by A D B is the adiabatic model rate obtained by numerical integration of eq. (46). The dashed A D B line is half the A D B TST rate.

E. Pollak et al. / Symmetric light atom transfer reactions

27

Thus, the adiabatic model reaction probability is in analogy with eq. (27) MOD,ADB ( E ) P/,,,j

= sin2½v(n ( E ,

~11"/]\pADB'TST/E),,,,J ~. .

(43)

It is of interest to compare this result, with the opacity function obtained via the sudden approach - eq. (32). Clearly eq. (43) gives a highly oscillatory opacity function since in the adiabatic limit, implicitly, all bend angles are oscillating in phase with one another. In the sudden approach, the oscillations differ at different bend angles giving rise to destructive interference. The opacity functions for the adiabatic model eq. (43), are plotted in fig. 10 for a number of energies. In the top half we plot the opacity function for the C1 + HC1 reaction with activation energy. The zero point bend energy at the saddle point is 0.0125 eV. In the bottom half of the figure, the model is Cl + HC1 but E*(J = 0) = 0 and the adiabatic barrier height including the bend zero point energy is 0.2 eV. As usual B0* = ½B* so that the switching of transition states is at E* = 0.4 eV. Qualitatively, fig. 10 is very similar to figs. 5 and 6. The number of oscillations are greater and they stay in phase longer for the E* = 0 case. Of course, the probabilities are much larger since, in an adiabatic bend model all asymptotic angles contribute to the reaction probability [33,43]. This is not so in the sudden limit. To obtain the integral cross section one must sum eq. (43) over angular momentum. As usual, in principle one may have a switching of transition states

E - E*(O)]/B*,[ E - E*,,(O)]/B*,,}.

J2ax ( E ) = min{[

The switch occurs at the energy E.* = [ B*E.*,(O)-

I

(44)

E.*;

B*oe*(o)]/( B*- B*.,).

i

i

(45)

I

E =0.45

40

0.4

0.6

I

I

CI + HCI

E (ev) 0.8

1.0

I E ~= 0.362

eV

CI +HCI

20

o.6 ~./ / \

, ~

I

~.--E=0.65

o ._~

0

120

E.I: 0 ~ ~ ~

,

''z •

l

'

~

I

Ct + HCI

d 0

%o 80

1.0

/

o. il;

40

0.2

I

0

0.2 0

E ::l: = 0

40

80

120

160

J

Fig. 10. Model adiabatic opacity functions at different total energies (in eV) for the two C1 + HC1 model reactions. For further details see text.

I

I

0.4

0.6

0.8

E (eV)

Fig. 11. Energy dependence of the model adiabatic integral cross section for the two Cl + HCl model reactions. The solid lines are the model [eq. (46)] results. The dashed lines are 1 / 2 the A D B TST cross section [eq. (47)]. For further details see text.

E. Pollak et al. / Svmmetric light atom transfer reactions

28

The summation over J values is straightforward, one finds o,~OD.AD"(E) =

(vh2/2#Er){ j2 ( E ) / 2 + (E/2"~B*)(COss/~oA,) × [sin

"~(n(E, Jm~x)) -- sin ",(n ( E , 0 ) ) ] ) O[E - E ~ ( 0 ) ] .

(46)

This is just the adiabatic analog of eq. (29), except that there is no need to further average over bend angles. In fig. 11 we plot the integral cross section for the two models. Comparison with figs. 3 and 4 shows that the adiabatic integral cross section behaves qualitatively like the sudden 3' = 0 cross section. Thus as usual, there are pronounced oscillations when E* = 0 and much weaker ones when the activation energy is substantial. Note that overall the adiabatic integral cross sections are an order of magnitude larger than the sudden cross sections. As has been noted by Pollak and Wyatt [33,43] this is characteristic of the two approximations and is a result of the absence of a steric effect within the adiabatic framework. In the figure we also compare the model cross section to the adiabatic transition state theory cross section given as o0Ar)r~"rsr( E ) = ( vh2/2t~ E-r) J2a, ( E ).

(47)

As in the sudden case, the oscillations provide a modulation around 1 / 2 of the TST cross section - the factor of 1 / 2 arising from a statistical treatment of the multiple crossings of the light atom. Finally, we have evaluated the thermal rate constant for the adiabatic model. For the E* = 0 case, this is shown as the solid line in fig. 9 denoted ADB. In addition we also plot (the dashed line, denoted ADB) the thermal rate obtained when the oscillations are neglected - that is by thermal averaging of eq. (47). Note that the oscillatory term is more substantial than in the sudden analysis. However, it does not bring about qualitative changes in the temperature dependence of the thermal rate. The thermal rate of C1 + HCI with activation energy also shows Arrhenius-like behavior but the magnitude is -- 50 times greater than the IOS and experimental rates plotted in fig. 2.

5. Discussion

The analytic model presented in this paper is based on a hybridization of conventional transition state theory and a recently formulated theory of direct reactions [14]. As such, it is superior to a TST approach the model does not resort to sophisticated reaction paths however it does reproduce the oscillatory behaviour found in exact quantal collinear computations for light atom transfer reactions. This is not saying that the model has no deficiencies. In order to obtain analytic expressions we have resorted to a separable, harmonic, collinearly dominated hamiltonian. It is though possible to do away with these simplifying assumptions, using numerical integration within the same framework. Thus one can, using y-dependent periodic orbit dividing surfaces [31], obtain the " t r u e " 7-dependent average number of crossings and rotational constants. Thus the formulation of the model is of much more general value then used at present. However within the severe limitations mentioned, the model in its analytic form is instructive in showing qualitatively the "smearing" that occurs as one goes from collinear to 3D mechanics. The sudden and adiabatic models analyzed in this paper have several qualitative features in common. Foremost, for light atom transfer reactions, on the integral cross section or rate constant level one should expect that a statistical approximation to the rate will give reasonable results. Although the mechanism is not statistical - the reaction is direct in the sense that the two heavy particles will go through only one encounter with each other - the averaging involved in obtaining integral cross sections and rates makes a statistical approach reasonable. In different terms, although the heavy atom motion is not statistical, the light atom motion is statistical, thus justifying a statistical approach. A second conclusion is that for light atom transfer reactions with considerable activation energy, the

E. Pollak et aL / Symmetric light atom tran6Jer reactions

29

threshold behavior is dominant and so effectively does away with any underlying oscillatory phenomena. Thirdly, at the integral a n d differential cross section levels, both the sudden and adiabatic models predict that oscillations are possible provided that the activation energy is small and of course, the mass of the atom transferred is small in comparison with the mass of the colliders. It is well known that the reactive IOS approximation usually gives cross sections that are in very good agreement with quasiclassical trajectory calculations [44 46]. Furthermore, the IOS TST approach has also been shown to compare well with quasiclassical trajectory computations [25,26,31,33,34,43]. Therefore, we believe, that the sudden model prediction of oscillations in opacity functions and integral cross sections should be observable also in quasiclassical trajectory computations. Probably, to date, these have not been found because of the Monte Carlo nature of the quasiclassical trajectory technique. The noise is easily big enough to smear out the oscillations shown in fig. 8. All this suggests construction of a potential energy surface for which our model predicts considerable oscillations and then carefully undertaking a trajectory computation. As stressed in section 4, quantally, one must seriously consider the problem of whether the bend motion should be treated adiabatically. If yes, this gives rise to increased quantal oscillations which cannot be observed in a quasiclassical trajectory or sudden approach. However, Pollak and Wyatt [43] have shown, that the bend should be treated adiabatically only in the near threshold region. At moderate energies the system will move fast enough to j u s t i f y a sudden approximation. The transition from adiabatic to sudden dynamics will have an important effect on the oscillatory structure of the integral cross section. Unfortunately, this transition is to date not very well understood and so is not incorporated in our present model. Two additional quantal effects were disregarded in our analysis. Tunneling may have an important effect at low temperature. In principle, it is easy to incorporate tunneling within our model. All that needs to be done is change the step functions in eqs. (10), (12), (19), (20), (26), (41) and (42) into harmonic tunneling probabilities [47]. In this paper, though, we were interested in understanding the averaging process as one goes from 1D to 3D without added complications. Adding tunneling corrections would not yield the simple analytic expressions given in the present paper and so is left for future study. As for resonances, we do not have a simple way of incorporating them. However, exact quantal reactive IOS computations will, of course, give the correct resonance behavior. Such computations are presently being undertaken [48]. From an experimental point of view, our study leads us to the conclusion, that just as in exoergic ion molecule reactions, curvature in the Arrhenius plots [49,50] is probably due to the switching from a loose to a tight transition state, as has been suggested in the early experimental papers, rather [1,2] than to the oscillations of the light atom. However, there is a .good chance that oscillations in the differential cross sections will be observable experimentally in a molecular beam type experiment.

Acknowledgement EP would like to thank Professor W.H. Miller for stimulating discussions and for his hospitality. This work was partially supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under Contract No. DE-AC03-76SF00098 and by a grant of the US-Israel Binational Science Foundation. Support of N A and D J K by the R.A. Welch Foundation under Grant E-608 is gratefully acknowledged.

Appendix A: The ~, dependence of the skew angle fl(~'x) The skew angle fl('~x) is defined (in the symmetric case) as: =

(A.1)

E. Pollak et al. / Symmetric light atom transferreactions

30

where ~/(Ta) is the polar coordinate of the line along which the Ya plane and the ~,, plane intersect, or in other words along which the matching of the solutions is carried out. If R x and r x are the mass-scaled cartesian coordinates in the )~ arrangement then the equation of this line can be shown (see also fig. 12) to be: R x = r x cotg~/( Yx )"

(A.2)

The two coordinates R x and r x are related to the ordinary translational and vibrational coordinates in the following way: R x = a x R ' x,

r x = a xlrf,,

(A.3)

where a x is defined as: ax=

B+mc)]

[mA(m B+mc)2/mBmc(mA+m

W4.

(m.4)

To determine ~/(Yx) the triangle shown in fig. 13 is considered. Employing the cosine law with respect to the angle Yx the following expression in terms of mass-scaled coordinates is obtained: r~2 = R~ sin2fl0 + r~ cos2fl0 - rxR x cos Yx sin 2/30.

(A.5)

Here flo is the collinear skew angle namely COS flO

=

[mAmc/(mA + m.)(mc

".1 1/2

+ mB)J

(A.6)

,

and r v is the mass-scaled vibrational coordinate in the u channel. The corresponding scaling factor a, is given in the form

a~= [ m c ( m B + mA)2/mBmA(rnA + mB + mc)] 1/4

(A.7)

Assuming that the transition from the 2, arrangement to the u arrangement takes place whenever r~ = rx, A

Ru/

(A.8)

~ channel

, r.Irb

rx

X channel

Rh

,.

Fig. 12. The X and the p arrangement channels for a fixed Yx value. F is the intersection line of the "Yx plane and the "~ plane. In the symmetric case 7~ = "~- Yx, and ~ox = ~o~. The 3D skew angle fl(Tx) is defined as fl(Yx) = ~ox + ~o~.

C

rk

B

Fig. 13. The three-body system. Channel )~ is the A + BC arrangement and channel p is the C + AB arrangement.

E. Pollak et a L / Symmetric light atom transfer reactions

31

we find that eqs. (A.2) and (A.8) are compatible if and only if cotg ~(Yx) fulfills the equation cotg 2 r/(yx) sin2fl0 - cotg "O(Yx) sin 2flo cos Vx - sin2fl0 = 0.

(m.9)

Solving for cotg */(Yx) it is found that: cotg ~(yx) = [cos Yx cos fl0 + ( 1 -

sin 2 Yx c°s2 flo)1/2]/sin flo.

(A.10)

Returning now to the symmetric case for which eq. (A.1) holds we get that

c o t g ( f l ( y x ) / 2 ) = [cos Yx cos/30 + ( 1 - s i n 2 Yx cos2 flo)1/2]/sin flo.

(A.11)

A more explicit expression may be obtained for Yx not too close to ~r/2 and for the heavy-light-heavy mass combination for which fl0 is close to zero. Expanding eq. (A.11) in a Taylor series in/30 and keeping only terms up to the third power we find: c o t g ( f l ( y x ) / 2 ) = (2 cos yx/fl0)[1 - ¼fig(½ - tg 2 Yx)].

(A.12)

In most applications the more relevant expression is for sin fl(y). Thus, since sin f l ( y ) = 2 c o t g ( f l ( y ) / 2 ) / [ 1 + c o t g 2 ( f l ( y ) / 2 ) ] ,

(A.13)

we obtain that sin fl(Yx) = (/3o/c°s Yx)[ 1 - ( f l 0 2 / 6 ) ( 1 + sin2 Yx)/cos2 Yx]-

(A.14)

Eq. (A.14) clearly demonstrates that (consider the condition that Yx << ~/2) fl(Yx) increases as Yx becomes larger. Having an expression for fl(y) we can now determine the dependence of the symmetric stretch frequency ~0ss on bend angle YxAssuming the validity of eq. (15) given in the text for both 1D and 3D and ignoring the dependence of All on Yx the following relation is obtained: (A.15)

(~Oss)3 D = (~Oss)l D sin f l ( y ) / s i n / 3 0 . For/3o small enough we find that

(A.16)

(tdSS)3D = (6OSS)ID(COS "~,X.)- l

which is identical with eq. (B.9).

Appendix B: The "t dependence of the symmetric stretch harmonic frequency At a fixed value of the angle y we assume that the hamiltonian of the system may be written as H

1~-~m(p)+p~)+-~-~m1(1- ~ + - ~ 1) p : +

1 1 .j2 V(r, y 2m r : + R 2 + R, ).

(B.1)

Here r is the scaled vector r = c(r a - r e ) and R the scaled vector C-I[rg- ( m a r s + m c r c ) / ( m a + mc) ]. The constants ra and c are defined as

¢n2 = m a m s m c / ( m A + m s + m c ) ,

C4 =

m s m c ( m A + m a + m c ) / m A ( m s + m c ) 2.

(B.2)

E. Pollak et al. / Symmetric light atom transfer reactions

32

T h e angle y is defined as the angle between the vectors r a n d R. T o o b t a i n the s y m m e t r i c stretch frequency one defines the s y m m e t r i c a n d a n t i s y m m e t r i c stretch c o o r d i n a t e s q~= ½[(rA--ru)+(r,--rc)],

qa= ½[(rA--rB)--(rB--rc)].

(B.3)

W e assume that the p o t e n t i a l in the vicinity of the s y m m e t r i c stretch is h a r m o n i c and s e p a r a b l e in these coordinates:

V(q~,q~, y ) =

~( A 11q~2 + A 3 3 q 2 ) + Vr(1 - cos 3').

(B.4)

T h e a r d u o u s task is now to explicitly t r a n s f o r m the kinetic energy a p p e a r i n g in eq. (A.1) so that the m o m e n t a (p~, p~) conjugate to q~ a n d q~ a p p e a r explicitly. Thus one defines the generating function

F = r(q~, qa, Y)Pr + R(q~, q~, 3')PR + YPr"

(B.5)

T h e d e p e n d e n c e of R on q~, qa a n d 3' m a y be inferred from the cosine law c o n n e c t i n g R with rAB, rBc a n d 3' a n d eqs. (B.3). The t r a n s f o r m a t i o n is s t r a i g h t f o r w a r d but tedious, one finds that for p u r e l y s y m m e t r i c m o t i o n (q~ = P a = 0, py = 0, rn A = m c )

H~= ( 1 / 8 , , n c 2 ) { [ 1 + cos 2 fl sin 2 y - cos/3 cos y ( c o s 2/3 cos 2 3' + sin 2 / 3 ) ' / 2 ] 2 + s i n 2/3(cos 2/3 cos 2 y + sin 2 f l ) ) p f + BJ 2 + V.

(B.6)

H e r e / 3 is the " s k e w angle" [cf. eq. (A.6)]. F o r light a t o m transfer r e a c t i o n s / 3 is very small. E x p a n s i o n of eq. (B.6) keeping only terms of the o r d e r of/32 gives H~ = ( 1 / 2 ¢ n c 2 ) [ s i n 4 3' +/32(¼ + j sin 2 3' - 2 sin 4 "/)] + BJ 2 + V.

(B.7)

If one deals with a collinearly d o m i n a t e d reaction, one m a y further expand, k e e p i n g terms only of the o r d e r .y2, to find //~, = ( 1 / 8 ~ c 2 ) ( 1 + sin2 T)P~ + ~Allq~ ' ~ +

v~(1

-

cos

Y) + B j 2 .

(B.8)

It is then easy to see that ~0ss(~') = ~ss/COS 3'-

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[1] [2] {3] [4] [5] [6] [7] [8] [9]

,,

C.C. Mei and C.B. Moore, J. Chem. Phys. 67 (1977) 3936; 70 (1979) 1759. E. Wi~rzbergand P.L. Houston, J. Chem. Phys. 72 (1980) 5915. F.S. Klein, A. Persky and R.E. Weston, J. Chem. Phys. 41 (1964) 1799. J.H. Lee, J.V. Michael, W.A. Payne, L.J. Stief and D.A. Whytock, J. Chem. Soc. Faraday Trans. 1 73 (1977) 1530; A. Persky and F.S. Klein, J. Chem. Phys. 44 (1966) 3617. M. Kneba and J. Wolfrum, J. Phys. Chem. 83 (1979) 69. R. Rubin and A. Persky, J. Chem. Phys. 79 (1983) 4310. M. Baer, J. Chem. Phys. 62 (1975) 305. G. Hauke, J. Manz and J. R6melt, J. Chem. Phys. 73 (1980) 5040; A. Kuppermann, J.A. Kaye and J.P. Dwyer, Chem. Phys. Letters 74 (1980) 263. J.A. Kaye and A. Kuppermann, Chem. Phys. Letters 77 (1981) 573.

(B.9)

E. Pollak et al. / Symmetric light atom transfer reactions

[10] [11] [12] [13] [14] [15] [16] [17] [18]

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