Calculated rate constants for the reaction ClO + O → Cl + O2 between 220 and 1000 k

Chemical Physics 40 (1979) 185-206 0 North-Holland Publishing Company

CALCULATED RATE CONSTANTS FOR THE REACTION Cl0 + 0 + Cl + 0, BETWEEN 220 AND 1000 K Richard

L. JAFFE*

NASA Ames Research Center, Moffett Field, California 94035, USA

Received 30 May 1978

The results of classical trajectory calculations are presented for the reaction Cl0 •i- 0 + Cl -I- 0,. This reaction is an important step in the chlorine-catalyzed destruction of ozone which is thought to occur in the stratosphere. Rate constants have been calculated for temperatures between 220 and 1000 K. The calculated rate constant is 4.36 x IO-” exp (- 191/T) cm3 molecule-’ s-l. Its value at 300 K is (2.3 + 0.2) x 10” cm3 molecule-’ S-I, which is within a factor OFtwo of the recent experimental measurements. The calculated activation energy of 0.38 kcal/mole is in excellent agreement with the experimental value of 0.44 kcal/mole determined by Clyne and Nip and provides support for their measurements. The calculations were performed using both the quasi-classical and phase space trajectory sampling methods. The empirical potential energy surface used in the calculations was constructed to lit experimental data for CIO, O2 and Cl00 molecules. Other important ticures of this potential surface, such as the barrier to reaction, were varied systematically and calculations were performed for a range of conditions to determine the “best” theoretical rate constants. The present results demonstrate how classical trajectory methods can be applied to the determination of activation energies and other kinetic data for important atmospheric reactions.

altitudes (40-50 km), the ozone depletion of the cyclic process:

1. Introduction The potential threat of chlorine-catalyzed depletion of the stratospheric ozone layer has received considerable attention since Molina and Rowland [I, 21 suggested that anthropogenic chlorofluorocarbons may provide a signilicant source of atomic chlorine in the stratosphere’. In fact, this proposal [I J has stimulated a tremendous research effort in aeronomy [4-73 and chemical

Cl + 0, Cl0

*Cl0

is a result

+ o,,

(RI)

+ 0 + Cl + 02.

(R2)

Since atomic oxygen and ozone are in equilibrium, the removal of either odd oxygen species is importank At lower altitudes, the concentration of atomic oxygen is low enough that reaction (R2) is in

competition

kinetics [S, 91. Previously, a CIX cycle had been proposed as a possible sink for stratospheric ozone [lO-123 but no major sources of chlorine had been identilied. Since that time calculations have indicated that the chlorofluorocarbons may cause serious depletion of the stratospheric ozone layer [13]. The principal active chlorine species in the stratospheric ClX cycle are Cl and ClO. At high

with

Cl0 + NO + Cl + NO1, which is the dominant Cl0 reactions, Cl0 + co Cl0 5

* NASA/NRC Research Associate 1974-1976. + CFCI, has been detected in the stratosphere at altitudes between IO and 40 km, see ref. [3].

process below 20 km. Other

+ Cl + coz,

(R4)

Cl + 0,

(W

Cl0 + Cl0 + products,

UW

Cl0

(R7)

+ O,d

products,

are possible, but less important 185

(R3)

at stratospheric

186

R.L. Juffe/Rute

~~n~tant~for the reaction CIO + 0 4 Cl + O2

conditions owjng to lower rate constants or reactant concentrations. Clearly the chemistry of ClO, and particularly the Cl0 + 0 reaction, plays an important role in the overall CIX cycle. An accurate determination of the relative amounts of Cl0 consumed by each of these processes requires knowledge of all the rate constants for reactions (RZ)-(R7) at the ambient temperatures, which range from 210 to 270 K for the altitudes of interest (20-60 km) [2,6]. Unfortunately, complete reaction rate data are not available for these reactions. The rate constant for (R2) has been measured in several laboratories at 300 K using ultraviolet spectroscopy [ 14, 151 mass spectrometry [ 16, 171 and atomic resonance fluorescence detection methods [17]. The most widely accepted value [17] for that rate constant is (5.3 +_ 0.8) x 10-r’ cm3 molecule-’ s-’ with the earlier workers [14-161 giving values between 1 and 2 x IO-“. Two very recent measurements [lg. 201 have confirmed the work of Bemand et al. [17] yielding rate constants of(5.2 i 1.6) x 10-r’ and(4.4 + 0.9) x 10-r’ cm3 molecule-’ s-r, respectively, at 300 K. These studies [JS, 193 also provide the first experimental determinations of the rate constants between 220 and 426 K. While the observed temperature dependences are both small, the data are not in agreement since one study [19] yields a negative activation energy. In addition, Park [20] has measured the rate constant for (R2) between 1000 and 1500 K in shock tube experiments and reports an averaged result of (7 f 1.5) x IO-” cm3 molecule-’ s-r at 1250 K. Consequently, the variation of the experimental data between 300 K [17-191 and 1250 K [20] is extremely small. According to collision theory, the rate constant for reactions with zero activation energy should still exhibit a temperature dependence’*. However, the observed temperature dependence [ 17-201 is slightly less than the minimum expected on these grounds. In addition, the relatively large uncertainties in the measured

rate constants make it diflicult for one to extrapolate the rate constant data to lower temperatures with confidence. Reaction (R3) has been studied by Clyne and Watson [21] and Zahniser and Kaufman [ 191 who report rate constants of(1.7 f 0.2) x 10-l’ and (2.3 + 0.8) x IO-” cm3 molecule-’ s-r at 300 K, respectively. The amount of ozone depletion predicted by the stratospheric models is very sensitive to the ratio KI/K3 [13]. Thus, it is of extreme importance that the temperature dependence of these rate constants be accurately known. The rate constants for (R4), (R6) and (R7) have been measured at 300 K and are 3-5 orders of magnitude lower than the above [21,22-j. Photodissociation cross sections for Cl0 have been determined by Johnston et al. [23] between 2250 and 2800 A and predissociation is known to occur at longer wavelengths [M-26]. The ground state absorption probability is too small for photodissociation to contribute to the overall Cl0 chemistry in the stratosphere except at very high altitudes [3, 1 I]_ Predissociation is more rapid but it is still not important at altitudes between 20 and 50 km [27, 283. The present paper describes classical trajectory calculations of Cl0 + 0 collisions which have been performed to determine the reaction kinetics and dynamics of (R2). Previous studies on the Hz + D [29] and Hz + F [30-371 exchange reactions, among others, have shown that classical trajectory methods can be used to compute accurate rate constants, reaction cross sections, product vibrational and rotational energy distributions, etc. One requirement of these calculations is that the potential energy surface for the particular reaction system must be known. However, for most reactions of importance to stratospheric chemistry, this is not the case. In the present work, an attempt is made to * The collision theory expression for a bimolecular rate const;lnt equals S$3akT/~]*‘2, where S, is the hardsphere cross section

’ The temperature dependence

of the rate constant is defmed as - kd(ln K)/d(l/T). The activation energy E, equals this temperature dependence when defined as K = .4 exp( -&/AT). If K is written as ATqexp (-E&T) then the temperature dependence is equal to E,, + ykT.

and p is the reduced

mass. For the

general case this T”’ dependence will remain unless S, is an explicit function of temperature. As a result, there will always be a contribution of (#Tto the apparent activation energy of a gas phase bimolecular reaction which arises from the temperature dependence of the collision frequency.

R.L. Jafi/Rare

constamfor

construct a reasonable potential energy surface for (R2) based on experimental spectroscopic and thermodynamic data that are suitable for use in classical trajectory calculations. The development of reliable methods for constructing potential energy surfaces will permit the exten‘sion of classical trajectory calculations to a wide range of important chemical reactions. Section 2 of this paper describes certain physical properties of the Cl-O-O system based on the available experimental data. These properties are used as a guide in the construction of a family of empirical potential energy surfaces which are presented in section 3. The set of surfaces spans a reasonable range of topographies for ClO?, and should bracket the true potential surface which is not known. In section 4, we give a brief description of the classical trajectory method used in this work. Section 5 contains a discussion of the results of the reaction cross section and rate constant calculations performed for reaction (R2) for the temperature range 220-1000 K. The calculated rate constants are compared with the experimental measurements and used to describe the temperature dependence of this reaction. Finally, in section 6 we summarize these calculations and brielly discuss the implications of the present study on the chemistry of Cl0 in the stratosphere.

187

fhe reaction Cl0 + 0 + Cl -I- Oz

2. Physical characteristics

of the Cl-O-O

system

In this section we summarize those properties of the ClO-O-O system which are used in the construction of the potential energy surface for reaction

(R3) described

in section

3. First the

reactant (Cl0 + 0) and product (Cl + OJ limits are discussed and then the properties of the stable triatomic species are presented along with the adiabatic correlations between the triatomic molecules and the atom-diatom limits. 2.1. The Cl0 + 0 limit The ground state of the Cl0 molecule X’fl,,, is known to dissociate to ground state atoms (Cl”P,, and 03PJ [24,33-3.51. The dissociation energy (63.427 _I 0.009 kcal/mole) has been measured by Coxon and Ramsey [26] and by Durie and Ramsay [24]. Until recently, the detailed shape of the ground state potential curve was not well known since progressions had been observed only for v < 2 in absorption [26,35] and for 5 G u < 9 in emission [25]. However, accurate ab initio MCSCF + CI potential curves have been obtained for the X’fI and A*n states of Cl0 by Arnold et al. [36]. Their calculated ground state dissociation energy is within 2 kcai/mole of the above experimental value. At 300 K, only 1.8 % of the X’II Cl0 molecules is in excited vibrational states (this decreases to 0.2 y0 at 220 K). However, the rotational spacings [33] are -small and a broad thermal rotational distribution

Table I Physical constants for Cl0 and 0,

0. (kcal/mole) r, (A) w, (cm-‘) 0,x, (cm - ‘) Et.=, (kcal/mole) AE(o = 1 + 0) (kcal/mole) B, (cm- ‘) E, = 20 (kcal/mole)

c10(2n)“’

0,(32,)“’

OZ(’A$’

W IV+ kg )bl

63.3 I” (63.427) 1.546d’(1.56965) 859 (853.8) 6.8 (5.5) 1.223 2.417 0 643” (0.62345) 0.767”

117.97 1.20739” 1580.211 Il.99 2.250 4.449 1.445572 1.724

95.43 1.2157 1509.3 12.9 2.148 1.4264 _

80.45 1.22685 1432.6874 13.9500s 2.038 1.4004 _

” From Ref. [3S]. unless specified otherwise. Values in parentheses are from Coxon et al. [25] and were not availrtblc until the present calculations were completed. ” From ref. [34], unless specified otherwise. ” From ref. [24]. d’ From ref. [34]. e, From ref. [37]. ” From ref. [33].

R.L. J&/Rote

188

constuntsfor the reuction Cl0 + 0 -+ Cl + 0,

exists, even at 220 K. The peak at 220 K occurs for

J * 12. and significant populations exist for .f k 30 (1.77 %). These data for Cl0 are summarized in table I, based on refs. [24,33-35-J Basco and Morse [35] have determined the ‘II3,1 + ‘l-I,,, spin-orbit coupling constant to be 318 cm-’ (AE= I.364 kcaI/ mole), resulting in 90.8 % of all Cl0 being in the X’II,,, state at 300 K (95.8% at 220 K). Thus, a thermal ensemble of Cl0 at stratospheric temperatures consists mainly of molecules with excited rotational levels within the ground vibrational and electronic state manifolds. At low temperatures only ground state oxygen atoms (‘I’,) exist in thermal systems. In the stratosphere less than 0.1 0Aof the oxygen atoms is eiectronically excited and these are considered to be separate chemical species in most treatments of

stratospheric chemistry. There are three fine structure states for 0 (3P,) [33,38]: 3P,, the ground state, 3P, (0.453 kcal/mole above the ground state) and the 3P, state (0.648 kcaljmole higher in energy than 3PZ). At 220 K, 79.5 % of a thermal sample will be in the ground fine structure state. There are six possible fine structure-state combinations for ClO(‘II) + O(3P) collisions and their relative thermal collision frequencies are given in table 2. At temperatures up to 1000 K the ground state (2113,r + 3PZ) is by far the most probable. However, the (‘r13,2 + ‘P,) combination also occurs with appreciable probability temperature thermal collisions.

Table 2 Relative

collision

frequencies

in low-

for CIO(‘lI)

f

0(3P,) fine

StCUCtLlre states T=

-

0.761

K T=

300

K T= lo&) K

0.674 0.189 0.045

0.410 0.196 0.059

0.958

0.908

0.765

0.033 0.007 0.002

0.068 0.019 0.005

0.207 0.099 0.030

0.042

0.092

0.336

f 0(3P,) + 0(3P,) -!- 0(3P,)

0.162 0.035

c10(?13,1) + OCP,) CIO(VI,,,) t O(‘Pz) CIO(W,,,) + 0(3P,) clo(‘Il,,z) + 0(3P,) ClO(‘rI,,,) i- 0(3P*)

cIo(-r13,z)

cIo(‘rI,,2) clo(‘rl,,z)

220

2.2. The O2 + Cl hit In contrast to ClO. the lower electronic states of O2 have been extremely well characterized. Three

of these states are energetically accessible as products of reaction (R2): ‘xi, ‘4 (T, = 22.64 kcal/ mole) and ‘Zz (T. = 37.73 kcaljmole) [34]. The ground state reaction is highly exothermic [33]: ClO(W) + O(3PJ + Cl(‘P”) + O,(%,

u’ < 13),

A@?,, = - 54.76 kcal/mole,

(R2a)

as are the reactions which form the first two excited states of 02, ClO(‘I-I) i- O(3P,) + Cl(‘P,) AH&,

+ O,(‘A,

u’ 6 7),

= - 32.26 kcaljmole,

ClO(3-I) + o(3Pg) + cl(zP”) + ogq 0 Akss = - 17.29 kcal/mole.

(R2b)

u’ d 4), (R2c)

However, there is presently no positive expe:imental evidence for singlet oxygen formation [i.e., reaction

(R2b) or (R~c)]. Lipscomb, Norrish, and Thrush [39] and Basco and coworkers [15,40,41] have observed vibrationally excited 0, produced by reaction (R2a). They found that the u’ = 2 to u’ = 7 levels (primed quantum numbers refer to product molecules and unprimed quantum numbers refer to reactants) were equally populated, and that the population gradually increased from the u’ = 8 to o’ = 13 levels. Small populations were observed for d = 414 and there was a large rate of formation for O2 u’ = 1. They found that approximately 45 % of the heat of reaction was channeled into 0, vibration [41]. Even though large uncertainties exist in the measured 02. vibrational distribution, these data demonstrate that the major products of reaction (R2) are vibrationally excited O,(3X; ) and CI(‘P,). Based on these measurements [41] and the work of Polanyi and coworkers [42,43], who have correlated the nature of the “energy release” on potential energy surfaces with product vibrational energy distriburions, one can conclude that reaction (R2a) proceeds along a predominately “attractive and mixed” potential energy surface. The splittings between the tine-structure states of

0,(3C;)

are suficiently small (z 2 cm-‘) to be

R.L. JuffefRate

constmtsfor

the reuctiorl Cl0 + 0 -P C! + Oz

considered zero. Some of the important physical constants for O2 are listed in table I _The ground state of the chlorine atom has ‘P, symmetry with a spin orbit splitting between the ‘P,,, and ‘P,,? states of 2.523 kcdljmole [38]. There is no experimental evidence for preferential formation of one of the fine structure states in reaction (R2). At low temperatures, no other electronic states of Cl can be formed in this reaction.

Table 3 Physical constants for Cl00

2.3. Trintoruic CIO, There are two distinct isomers of ClO,. One (OClO) is a chemically stable free radical at room temperature whose ground state has C,, symmetry. The other, a peroxy-like radical (ClOO) of C, symmetry, is highly reactive and has been observed only at low temperatures. Cl0 + 0 col!isions must pass through intermediate geometries resembling the structure Cl00 in order to form Cl + O1 products. The existence of an unstable Cl00 species was first proposed by Benson and Buss [44] in order to account for certain observations of chlorine chain reactions. They estimated that Cl00 has a lower total energy than OCIO and that its high reactivity is due to a very weak Cl-0 bond. Confirmation of their hypothesis has come from matrix IR [45] and ESR [46] experiments. In the former, Arkell and Schwager have determined the equilibrium geometry and force constants. They found Cl00 to have a bond angle of I lOJ, a weak and elongated Cl-0 bond, and an O-O bond similar to that found in 0,. Restricted Hartree-Fock calculations for Cl00 have identified the symmetries of the two lowest electronic states as ‘A” and ‘A’ (T, < I eV) [47,48]. Johnston and co-workers 1231, Wagman and Garvin [49], and, most recent!y, Clyne et al. [‘I] have revised the values for the heat of formation of Cl0 and ClOO. The most recent estimates of the Cl-0 and O-O bond energies are 7.7 and 62.7 kcal/mole, respectively, and Cl00 is thermodynamically more stable than OClO by 3.1 kcdl/ mole. Some of the physical constants for the electronic ground states of Cl00 and OClO are listed in table 3. Fig. 1 is a schematic energy diagram of the low-lying states of C102.

189

and OCIO OCIO

CIOO”

R,,,(A)

1.83

Root&

1.23

x

E(0,0,O)(kcal/mole) tr),(cm-‘) w&m- ‘) w&m- ‘)

1.47b’ , ,&’ 3.55” 945.3°’ .4,$7_3”

110: 3.18 i440.8”’ 407”’ 3734

1109”

d’ Data from ref. 45. ” Data from: ref. [72]. ” Data from ref. [;;I_ ‘) The corresponding force constants are:foo = 9.65 mdyne/A.J,, = I.29 mdync/A&,,, = I.04 mdyne A. = 0.07 mdynt./A. /oc,-ooc, = 0.54 mdyne.fO,_,,

2.4. Adiubutic rlectrorlic stotr correlutions In order to connect the atom-diatomic and triatomic regions of the CIOz potential surface, one must consider the adiabatic etectronic state corretations under the appropriate symmetry restrictions. Shuler [SO] outlined the basic method for determining the symmetries of atomic and diatomic (C,, and D,,) fragments under lower point groups (e.g., C, or C1, for nonlinear triatomic molecules). For the present case, six doublet and six quartet states arise from the combination of a ‘P, atom

O+O+cQ

t64.55 CQO+O , 00 . CQ +

\ \ \ \

O,('c;)

-18.10

O*(‘&$ -33.19

CR+

\ \

’

CQ+ O2(3X4) -55.83

Fig. 1. Energy diagram for the CIO, system. The energies are in kcal/mole relative to separated Cl0 (‘H3,J f OPP,).

190

R.L. JaffelRate

constantsfor

the reaction Cl0 + 0 ---t Cl + 0,

with a ‘fI diatomic. The doublet surfaces are the

only ones of importance in the present work, since the quartet surfaces correlate with high energy states of Cl00 [48]. Of the six possible states in C, symmetry. three are A’ and three are A”_ The first ‘A” state correlates with the Cl00 ground electronic state and the first ‘A’ state correlates with the low lying excited state of Cl00 discussed above. The energies of the four other states of Cl00 that dissociate to ground state Cl0 + 0 are not known. Similarly, 01( ‘CJ + CI(‘P,) correlates to *A’ + 2’A” (and 3 quartets). The resulting correlation between reactants, intermediates and products for (RZ) is shown in fig. 2. Only the 3Xp and ‘As oxygen products correlate to ground state reactants in C, symmetry. Since the 1’A’ and L’A” states of CIOO correlate to the ground state of Cl + O2 (only =: 8 kcal/mole above 1 ‘A”), they are most likely not bound. Fine structure effects can also be estimated by the use of extended point groups [SI, 521 to combine the orbital and spin symmetries. All polyatomic states (‘A’ and ‘A”) become 2E1,2 symmetry and a one-to-one correspondence between reactant and product fine structure states can be made by invoking the noncrossing rule* (see fig. 7). The (‘H3,1 + 3P2), (‘H,,, -I- 3P,), and (‘IT,,, + 3P,) reactant combinations are most probable in thermal systems. The first two correlate with (‘Pyz + 32,) products through the 1‘A” and 1 ‘A’ states of Cl00 and the third correlates with (‘P3,1 + ‘A,) products. Of course these correlations hold only within the adiabatic representatica. Nonadiabatic effects, such as surface hopping or electronic transitions, could lead to the formation of Oz(rZ+)t_ These thermodynamic and lpectroscopic data combined with the electronic correlations provide a good qualitative picture of the Cl-O-O system. Much of this information is used in the construction of a set empirical potential energy surfaces which is described in the next section.

of

* Strictly speaking the noncrossing rule applies only to diaromic systems. However, the allowed intersection of states of the same symmetry is still very improbable in polyatomic systems. For a discussion of this point see reT. [53]. ’

For a review of the theory of nonadiabatic collisions see ref. [I%]_

Fig. 2. Adiabatic electronic correlations between low-lying states of Cl0 -I- 0, CIOO. and Cl i O1. Electronic states of Cl00 denoted by dashed lines have not been observed experimentally or theoretically.

3. The potentia1 energy surface Most calculations of collisional phenomena require as input the potential energy of the system for a large volume of configuration space. Since accurate ab initio calculations of the potential energy are not usually available, empirical and semi-empirical valence-bond methods (such as LEPS 1142,551)have been devised to represent potential surfaces. In the present case, an accurate ab initio potential surface is unavailable and the LEPS formulation is not suitable, since LEPS potential surfaces always favor collinear triatomic geometries. Empirical angular dependencies have been artilicially introduced into the LEPS framework by either supplying an additional angle dependent term [56] or by modifying the standard repulsive potentials for the diatomic fragments C57]. However, these approaches do not yield satisfactory results for CIOO. In this section an empirical model for the Cl00 potential energy surface is described based on the physical properties of the constituent species presented in section 2. The underlying rationale employed here should be generalIy applicable to the

191

R.L. JaffejRate constuntsjbr the reaction Cl0 + 0 -+ Cl + 0,

construction of potential surfaces for other chemical systems for which only limited experimental and/or

theoretical data exist. The potential energy surface for the ground state of ClO, has a deep well corresponding to the Cl00 equilibrium geometry and troughs corresponding to separated atom-diatom geometries+. For reaction (R2) to occur, the oxygen atom must impinge on the oxygen end of the Cl0 molecule_ The reaction coordinate for process (R2a) (i.e., the minimum energy path connecting separated Cl0 + 0 and 0, + Cl geometries) must traverse the stable Cl00 region of configuration space and the favored direc-

tion of approach for the colliding reactant species will be highly nonlinear (the Cl-O-O angle would be approximately 1IO”). For the present study, the ground electronic state potential surface has been divided into live regions, as shown in Iig. 3: two separated atom-diatom regions for Cl0 + 0 and Cl + 0,, labeled I and V, respectively; a triatomic region for Cl00 (III) and transition regions between the atom-diatom and triatomic configurations (II and IV). Mathematical expressions are given for the potential energy in regions I, III, and V and an interpolation scheme is presented to combine these formulae in the transition regions. In the separated atom-diatom regions (I and V), the only contribution to the potential energy arises from the diatomic potential curve. For computational convenience, Morse functions (&,,) are used to represent the C10(2113,1) and O#.Z;) diatomic molecules. The potential curves are constructed from the experimental well depth (D,), equilibrium bond length (R,) and vibrational constant (wJ according to [37]: V,(R) = 0, {I - exp[-fi(R

- R,)])‘,

(3.1)

where the parameter fi is given by fi = ;2n’~~/D,h}“~ rv,, h, c and p are Plan&s

(3.2) constant, the speed of light

’ The observed chemicalstabilityof OCIO means that Cl00 and OCIO must be separatedby a sizeablepotential energy barrier. Since processes involving OCIO are

not of interest in the present work the empirical potential energy surface constructed for the study of reaction (R2a) does not include the well corresponding to a stable OCIO species.

,L I

f 2

I

I

-3

I 4

Roo (Al

Fig. 3. Schematic partitioning ofrhe ground stlttc Cl00 potential energy surface for 2 = I,. Regions I and V correspond to separared atom-diatom conligurations and region I11 corresponds 10 triatomic configurations. The transition regions are II and IV. VAis determined from V; for R, > 2.2A and VBis determined by an imerpolation between v,, and V, for R, < 3.0 a. The potential in region IV is specilied by V, for 3.1 c Ro_no -G 3.7 A and the poteniial in region I1 is specified by an interpolarion between V, and VB.The reaction coordinate for Cl0 c 0 -+ Cl + O1 is also shown. The barrier is located at 0 and the Cl00 equilibrium geometry is located at 0.

and the diatomic molecule reduced mass, respec-\ tively. Morse potentials are generally accurate near R, but they can deviate considerabiy from the exact potential curves at other internuclear separations.

The calculations of Arnold, Whiting and Langhoff [36] indicate that the true CiO potential curve rises more steeply at large R. However, only very low vibrational levels of Cl0 are populated under the conditions of the present study, and the fitted potential curve is expected to be adequate. On the other hand, the Morse function for OX exhibits a potential well that is considerably narrower than the RKR curve [%I. In order to properly describe the reaction dynamics, the empirical potential surface should accurately reproduce the O-O interaction within the range of vibrational levels that might be

populated by reaction (RZa)(1.00~ f R d 1.62A for IJ’,( 14). A better fit (within 0.04& to the RKR curve in this range is obtained by setting fi equal to 70 % of the value given in eq. (3.2). The triatomic potential region III is represented

192

R.L. Juffe/Rure constantsfor the reaction

by Morse ftinctions for the CL-0 and O-O separations (Rclo and R,,, respective!y) and a double minimum “harmonic” potential for the Cl-O-O bond angle (a). The Morse parameters are determined from the experimental force constants (f) [45] according

to [37]

/3 = (f/2&)“‘.

(3.3)

For a nonlinear

triatomic

molecule

the bending

potential has a maximum at a = 180” (the so-called inversion barrier) and two minima at the equilibrium bond angles a, and (2a - a,). One expression for such a bending potential is Y = AX’

+ B/(C + X2) + D,

where X = a - R This function X = 0 and minima at x = +x0

= +[-c

(3.4) has a maximum

+ (B/A)“‘]“‘.

at

(3.5)

The four parameters in eq. (3.4) can be determined from a,, the bending force constantfa, the height of the inversion barrier E*, and the requirement that Y(X,) = 0. The derivation is given in the appendix.

In summary, the potential energy in each of the above three regions is given by: Y = %(R,,o).

(3.6a)

Y,, = IG.,&,o) + G(Roo) + Y(x) + &a,,

(3.6b)

and v, = URoo)

Cl0

W(x) =

bEF2,

i(x - p) +i, + (x - p)’

Ix-fI=G%,

(3.7)

which’switches between the two forms of the potential energy. x is Roe in region II and R,-,_oo in

region IV. p and 1 are parameters which fix the location of the center of the interpolation region and its width, respectively. W varies between 0 and 1 and dW/dx = 0 at x = p + 1.. While this form of interpolation is continuous, it can introduce spurious local extrema into the potential energy surface. To prevent this, an extra term,

v,,,,

= 4V,W(X) [ 1 - W(x)].

(3.8)

has been added in each transition region to smooth the potential surface. The use of VP,,, also permits the placement of adjustable barriers in the entrance and exit channels of the potential surface. For l&, VP = 0.5 kcaljmole for the zero barrier case when (y.= CZ= (the only one used) and for I&, VP = 3.0,2.5, and 2.0 kcal/mole for EaARR = 0 (no barrier), 0.5 and 1.0 kcal/mole, respectively, when a = a,.

The entire interpolation scheme is shown in fig. 3 and summarized below. The potential energy function is first partitioned into two parts V, = q and v, = Yu.

R,_,,

< 3.1 A;

V, = Cl - W(Rcr-oo)1V;u + W(RCI-oo)V, +

(3.9a) YPERT,

3.1 1\ G Rc,_oo c 3.7 A; V, = V,,

+

__ A-

+ 0 --t Cl + O2

R,,-,,

(3.9b)

2 3.7A.

(3.9c)

(3.6~)

where V, and V,. symbolize Morse potentials parameterized for the diatomic and triatomic systems, respectively. V,,,, and VREF2are the constants required to give the three potentials the common reference point as shown in fig. 1 (separated reactants with Cl0 at R=). For this case V REFl = -63.59 kcal/mole and v&2 = -55.83 kcal/mole. The transition regions are defined by 2.2A < R,, c 3.OA for V;,and 3.1 A
Then V, and Vaare connected by a similar interpolation to complete the construction of the potential surface: v = v,,

Roe < 2.2 A;

(3.10a)

V = W(Roo)V, + Cl - W(Roo)IV, + VPERT, 2.2 8, < Roe < 3.0 A; v = v,,

R,,

2 3.0A.

(3.10b) (3.1Oc)

Eqs. (3.10b) and (3.9b) represent the potent&l in the transition regions I& and I&, respectively_ Contour plots of the resulting potential surfaces for 01= 90, 110 and 130” are shown in fig. 4. The parameters used in the calculation of the potential energy are given in table 4. The empirical potential energy function described

R.L. Jaffelate

constantsfor

the readon

Cl0

Table 4 Potential surface parameters D,(kcaI/mole)

Cl0 ( 6) 02(&r) 00 in Cl00

( Y,,) Cl0 in Cl00 (v;,,)

a(A-‘)

R,(A)

64.553 120.217

2.31 I53 2.62747

1.546 1.20739

63.592

3.3132

1.23

7.765

3.4577

1.83

E*(kcal/mole)

-

193

+ 0 + Cl + O2

-

&(mdyne A) I,

Cl00 bend -

(v,,,)

c;l VIV

70.0

1.04

110”

P. (A)

i.(A)

V,(kcal/mole)

2.6 3.4

0.4 0.3

3.0,2.5,2.0”’ 0.5

” The values of V, correspond to E,,,, kcal/mole. respectively.

a = 130”

= 0,0.5. and 1.0

above should provide a good description of the actual potential surface in regions I, III and V. A qualitative assessment of this potential can be made in terms of the nature of the energy release by using the reaction coordinate criterion of Polanyi and co-workers [42]. The formation of Cl00 from Cl0 + 0 involves only mixed energy release since R,, decreases and Rcro increases as Cl0 and 0 approach. Likewise, the formation of Cl00 from Cl + O2 involves attractive energy release as the O-O bond length remains nearly constant during this process. Overall, the potential surface for reaction (R2) is classified as “mixed” and one would expect a high degree of vibrational excitation in the products [42,43] in agreement with experiment c411-

While this potential energy surface seems to be very reasonable for reaction (R2a), the calculated reaction cross sections and rate constants will be most sensitive to the topography ofthe potential

I

(f)

2 Roe

D

(A)

3

Fig. 4. Contour plots of the ground state Cl00 potential energy surface for E aARK= 0. The energy contours are in kcal/mole relative to separated Cl0 + 0. The bond angle a is fixed as follows for each plot: (a) 2 = 90”, (b) 01= .z= = 1IO”, and (c) ix = 130”. The Cl00 equilibrium geometry is located at 0 in (b).

R.L. JaffefRate

194

constantsfor

surface in transition region II which is largely unknown. Topographical features like the height of the energy barrier, the O-O separation at the onset of ClO-0 attraction in C!O + 0 collisions and the variation of the potential with a near the energy barrier will control the fraction of collisions which reach geometries corresponding to transient Cl00 complexes and continue to products. Reasonable estimates based on experimental observations or a range of values have been used for these quantities in constructing the potential surfaces. It is hoped that future ab initio calculations of the potential energies will identify which of these choices is most reasonable. The magnitude of the energy barrier has been varied between 0 and 1.0 kcal/mole through the choice of V,. The onset of the O-O attraction is governed in part by the shape of the O-O Morse potential in x,, and by the location of the transition region II. The potential surface parameters given in table 4 result in an O-O attraction of 1 kcal/mole at R,, = 2.5 A for the zero-barrier surface. Great uncertainty lies in the selection of the rate of attenuation of the bending potential Y(a) as R,, and R,;, are increased. In the above formulation, the variation of the potential energy with a in region II is given by v, = [I - W&J]

W1.

(3.1 I)

The magnitude of V, at a particular value of Roe can be varied through the choice of boundaries for that transition region or through the choicef, and E*. It must be noted that the latter affects the accuracy of the potential in region III and could result in erroneous product angular and energy distributions.

4. Classical trajectory

calculations

The trajectory of a reactive collision corresponds to a unique path in phase space, proceeding from some product contiguration. For electronically adiabatic collisions this path can usually be determined from the laws of classical mechanics’, provided that the potential energy surface is known The classical motion of a system of N atoms is + For comparisons between quantum and classical calculations olchemical reaction dynamics see rel![59].

the reaction. Cl0 + 0 + Cl f O2

described by 6.X coupled first-order differential equations, dQj/dt = cW/5Pi, dP,/dt = -ZZ/ZQi,

(4.la) j = I, 2, ___,3X_

(4.1 b)

The Qj and Pi are the 3N coordinates and conjugate momenta, respectively, needed to describe the system and .%’is the classical hamiltonian (sum of kinetic and potential energies). This hamiltonian, relative to the center-of-mass, is given by:

+ V(Q~,QZ,---rQci)r

(4.2)

for a triatomic system, A + BC. The coordinates (Qr, Qz, QJ are the Cartesian components of the vector connecting atoms B and C and (Q4, Qs, Q6) are the components of the vector between A and the center-of-mass of BC. Pee and pA-cC are the reduced masses of atoms B and C and or atom A with diatomic BC, respectively. For .N > 3, numerical integration methods must be used to compute the trajectories. In the present work a modified version of Muckerman’s quasiclassical trajectory code [60] was used. This computer program contains an extremely accurate numerical integrator (1 lth order predictor-l 1th order corrector) which is required to prevent the accumulation of truncation errors 1611 when long-lived trajectories are computed. Individual classical trajectories are subject to the constraints of conservation of total energy E,,, and angular momentum L which are used to provide internal checks on the accuracy of the numerical integration. In the present work the changes in E,,, and L were typically less than one part in lo5 from the start to the finish of a trajectory. In addition, back integration of selected trajectories to return the initial coordinates constituted another check of the computational procedure. Microcanonical or canonical ensembles of colliding atoms and molecules can be simulated by calculating large numbers of trajectories with initial values for Qj and Pj randomly selected from suitable distributions. There are several well-defined methods for selecting these initial conditions. Two of these, denoted the quasiclassical trajectory (QCT)

R.L. Jaffe/Rote

constantsfor

and phase space trajectory (PST) sampling methods, have been used in the present work and are described brielly below. 4.1. Quasiclassical trajectory sampling method In the standard QCT sampling method the initial conditions are chosen to correspond to separated reactants with some relative momentum towards each other. The reactants are initially given classical vibration and rotation energies that correspond to quantum states of the isolated molecules. As eqs. (4.1) are numerically integrated, the species will lirst approach each other and then separate as products or the original reactants. The quasiclassical trajectory sampling method (QCT) has been discussed in detail by Karplus et al. [29] and in a recent review by Porter [62]. The initial values of the coordinates Qy for the atom A and diatomic molecule BC in the (u, J) state are illustrated in fig. 5. The three atoms lie in the X-Y plane of an arbitrary space-fixed coordinate system with Rut equal to one of the vibrational turning points of the (u, .I) state and the angle of orientation, <, of BC randomly selected from a uniform distribution. Atom A is located a distance p0 from the center of mass of BC and a distance b (the impact parameter) from the x-axis. p. is a constant chosen such that the atom-diatom interaction potential is essentially zero at this separation. In the present work pc = 4.5 A, which results in asymptotic Cl0 + 0 and Ci + O2 interaction potentials of less than 0.01 kcal/mole. The square

195

the reaction Cl0 + 0 + Cl + O2

of the impact parameter is selected randomly from a

uniform distribution with a cutoff at b < b,, where 6, is chosen such that the probability of reaction is less than 0.01 trajectories with b > b,. The standard error associated with this statistical sampling method is minimized if 6, is set at its minimum possible value (b, = 2.5 A in the present work). Qz is incremented by sQ8, the product of a random fraction of the vibrational period r(u) and the relative collision velocity. This ensures that the vibrational phase will be randomly chosen even though all trajectories are started at the vibrational turning points. The Pp are chosen so that the rotational angular momentum vector of BC has random orientation and-magnitude to the Jth equal to J(J + l)fi”/R&, corresponding quantum level. The relative momentum between A and BC has magnitude (2,~,,-,&~,,)~~~, where E,,,, is the collision energy and is directed along the x-axis (Pt = Pg = 0). Thus, the initial conditions are specitied for fixed IJ, J, E_,,,_ The numerical integration is terminated when one of the interatomic distances IS greater than p0 + R&.

The total reaction cross section, S,, for fixed u, 3, and Eeoll is defined by

where N is the total number of trajectories and N, is the number of reactive trajectories. The standard error (one standard deviation) in this statistical sampling scheme is e = WL,,,,

n, J) [(N - NWN,]“‘.

(4.4)

Typically, N, < O.lN, so that at least 250 trajectories are needed to reduce the sampling error to less than 20 %. Thermal rate coefficients K(T) are computed from the thermally weighted reaction cross sections: K(T) = Q~;-(KR~-~~)-~‘~ (2/k7J3”

x exp(- .L/k~)Ec,l, dLl, Fig. 5. The parameters used to specify the initial coordinates for the quasi-classical trajectory sampling method. SQZ is the displacement of atom A along the x-axis to effect randomization of the initial vibrational phase of BC.

(4.5)

where Qo,, is the exact vibration-rotation partition function, E,, is the internal energy of BC (u, J) and k is Boltzmann’s constant. The electronic degeneracy

factor [Sl] has not been included in eq. (4.5) and.is

R.L. Jaffe/Rate comtuntsfor the reaction Cl0 f 0 -+ Cl •t O2

196

discussed in section 5. The summation over u and J can generally be truncated when E,., 2 4kT without introducing significant error. For most non-hydride diatomic molecules the rotational spacings are small and QCT cross sections must be computed for many rotational levels. e.g., for Cl0 rotational levels up to J = 40 have appreciable population at 300 K. In an alternate approach, a single set of trajectories can be determined for each value of LJand Eco,,, but with J randomly chosen from a thermai distribution for a given rotational temperature TR [63]. Using this procedure, the computed reaction cross sections are

rotationally

averaged and can be defined as

S,(E collr~,TR) = O;‘cW J

+ l)exp(--E,/kT,) (4.6)

x S,EC,,,, 03J). Similarly,

the rate constant

is now given by

K(T) = Q‘~1(~~A-0c)-1’2(2/kT)3’1

zexp(--EJkT)

IO x

S,(E cou9u, T,) exp (-L,,lkT)L

d&r,.

(4-7)

I 0

This sampling method assumes that vibration and rotatron are separable, I.e., &, = E.. -I- E, and Qr., .= Q,Q,, which should be valid at low temperatures. In the present work, rotationally averaged cross sections have been computed using 500 trajectories for each set of initial conditions (EEo,,, v, TR).For a given value of v and TR, these data are then lit to a third-order polynomial in Eco,, and the integrals in eq. (4.7) are numerically evaluated using Simpson’s rule. The error bounds for the rate constants are determined by using $(E,,,,, u, TR) f E and reevaluating K(T). The main advantage in using the QCT sampling

an4 as a result, a typical rate constant calculation requires z 10000 trajectories. A much more elTrcient procedure for thermal systems, which is also used in the present study, is discussed below. 4.2. Phase space trajec!ory sampling aerhod The phase space trajectory (PST) sampling method [&l-67] is based on Keck’s phase space theory [68]. This approach can be much more eflicient than the QCT sampling method because the calculation of many nonreactive trajectories is eliminated_ In the PST method, trajectories are started in the interaction region of configuration space, where all atoms are in close proximity, and integrated both backward in time to separated reactants, and forward in time to the linal species which result from the collision. Most nonreactive trajectories are never examined because they do not reach this interaction region. The details of the method are given in refs. [64] and [6SJ. The 6-N dimensional phase space, fi of a chemical reaction system contains distinct regions that can be identified as reactants fia and products R,. If a (6X - I)-dimensional surface S is constructed to completely separate the reactants and products regions of phase space, then the flux of trajectories leauittg the reactants region constitutes an upper bound to the rate constant K,. The actual rate constant is obtained by eliminating the contribution to the flux from any nonreactive trajectories which return to the reactants region. The upper bound rate constant is calculated with the assumption that the system is in thermochemical equilibrium, i.e., thermal equilibrium exists throughout the entire phase space, according to K,(T)

= 0,’

exp( - L/k-T) KY S==O Y”>O

method is the control one has in selecting the initial conditions. The computed cross sections can be weighted by suitable distribution functions to simulate a wide range of experimental conditions, e.g., thermal system, molecular beam experiment, etc However, a major disadvantage of the QCT approach is that only a small fraction of the computed trajectories (usually less than 0.1) are reactive

(4.8) where V, is the velocity component normal to S and 7 = IVS//(dS/dq,) relates the dilTerentia1 surface area dS to the position coordinates and momenta. QR is the reactants classical partition function per unit volume, ^v, and is delined as

R.L. Jalfelute

QR

=

$‘--I

s exp

constmtsfor

3.\‘-3 ( -

EmIkT)

fl

dPj

(4.9)

@j-

j=l

RR

For the three-body case A + BC it is convenient to use spherical polar coordinates R,,, &. &, and O,_,, and to choose the dividing R,-,c, 4A-m surface as S = q, - qf, where q1 is R,, or RAmBC (in the present study q1 = Roe). Eq. (4.8) can be simplified to x

K,(T) = (2nkT/po0)

“’ qf’

do,,-oo cl

-z dR,-,-,,

sin Oc,-ooR&oo

d&,,Rf,,exp

[ - ~,I(&d/~~l

x

exp

I RC.o=ql l-7 x 0

the rwcrion Cl0 + 0 + Cl + O=

197

trajectories with the elimination of any extra contributions from those trajectories that make multiple crossings of S). Based on this definition, E is a multiplicative correction to K, [66] and the thermal rate constant, K, is given by (4.1 I)

K(T) = S( T)K,( T).

The determination of the conversion coefficient compensates for the assumption of complete equilibrium invoked in eq. (4.8) [66] by removing all contributions to the flux which are only present at chemical equilibrium (contribution (3) above). There m-e no trajectories present in the nonrquilibriunl cuse which ure absent ut chanical equilibrium. The set of “successful” trajectories is equivalent to the set of reactive trajectories which would be obtained in a conventional trajectory calculation, if the initial conditions are sampled from classical thermal distributions (i.e. no zero-point vibrational energy or other features of the QCT sampling method). The value of K(T), as determined by eq. (4.1 l), does not depend on the choice of S. Thus, minimization of K,(T) results in maximization of E, and minimization of the computational effort required for the calculation of N, “successful” trajectories. In other words, most of the nonreactive trajectories do not cross the optimal dividing surface especially if S intersects the crest of a barrier on the reaction coordinate. For many systems E > 0.5 and values as large as 0.9 have been computed [64,65]. As a result, rate constants and thermal reactant and product distributions can be calculated from only 500 to 1000 total trajectories. The PST method has undergone extensive critical exahination and

(-V/.kT) 1

1 -1

,

(4.10)

and the integrals in eq. (4.10) can be evaluated numerically using a trapezoidal rule algorithm. The dividing surface is positioned to intersect the reactants’ channel of the potential energy surface at R,, = 2.65 A, the value of q: for which K, is a minimum at T = 300 K. K, is an upper bound to the rate constant because it includes (1) contributions from those nonreactive trajectories which cross the dividing surface, (2) extra contributions from reactive trajectories which make multiple crossings of S and (3) contributions from trajectories which originate in the products’ region of phase space. To correct K,, classical trajectories are computed with initial conditions corresponding randomly selected points on the dividing surface+ and integrated forward in time towards products and backward toward reactants according to the scheme given in ref. [64]. The conversion coefficient X(T) is the fraction of “successful” trajectories (i.e., the fraction of reactive The initial coordinates and momenta are chosen with a weighring that reflects the magnitude ofthe integrand in eq. (4.10) with R,, = qt. This means each pair %-00. U,,_,) ~ has a statistical weight given by [sin Uc,_ooR~-,, exp ( - V//CT)], the other coordinates are chosen from uniform distributions and Ihe momenta are selected from Maxwellian distributions.

been shown to be equivalent to the conventional classical trajectory method [66,67]. However, for certain systems with large reactant vibrational and rotational spacings the QCT method should be used to correctly account for the effect of quantized initial conditions. The Cl0 + 0 system is ideally suited for

the PST method because the Cl0 vibrational spacing is fairly small, and the rotational energy distribution temperature

is nearly continuous thermal conditions.

under low-

198

R.L. JafjrlRutr

constatztsfor

5. Results In this section the results of the trajectory calculations using both the QCT and PST sampling methods for reaction (R2a) are presented. The QCT data are for a single Cl0 rotational temperature of 300 K. Thermal reaction rate constants, reaction cross sections and activation energies are reported. The results of the PST calculations provide rate constants for the temperature range of 220 to 1000 K and complement the QCT data. Reagent and product energy distributions have also been determined for reaction (R2a) and these results will be reported separately.

the reuction

Cl0 + 0 + Cl + Oz

E ban- = 0, OS, and 1.0 kcal/mole. The initial conditions were specified for values of E_,,, between 0.03 and 6.0 kcal/mole, u = 0, 1 and J chosen from the thermal Cl0 rotational distribution for 300 K. The rotationally averaged reaction cross sections, rate constants, and average reactant and product energies and scattering angle are summarized in table 5. The variation of bond lengths and bond angle with time for a typical trajectory is shown in ftg. 6.

150 “2

Q,

deg

60

XI. Quasiclassical trajectory calculations In the present work, more than 36000 quasiclassical trajectories have been computed for reaction (R2a). These calculations used the potential energy surfaces described in section III with Table 5 Summary of QCT results a) E BARR= 0

N

8917

N,

811 0.092, (1.27f0.18) X lo-” 0.50 17 98” 7.5 19.36 (33 %) 32.43 (55 %) 6.72(12x)

pT

K EAb)

:;;> G%zY a) Energies

are in kcai/mole,

E,,,

= 0.5

o

E BARR= I.0

(kcal/mole)

(kcal/mole)

12486 753 0.054 (0.78+0.13) X lo-” 0.60 1s lOl0 7.6 22.66 (38 %) 3 1.23 (53 “/.) 5.19 (9 %)

10444 411 0.030 (0.47+0.11) X 10-r* 0.77 ND” ND ND ND ND ND

bcde t

Rclo

rate constants in cm’

molecule-' s-’ and angles in degrees. All other quantities are dimensionless. All data is for 300 K. w Determined from rate constants calculated using eq. (4.7) for T = 280 to 320 K. This assumes that the averaged cross sections are independent of rotational temperature. ct Quantities marked ND were not determined. dt Quantities in parentheses are percentages of the total energy. cJ The experimentally determined value for (E;) is 25 kc&/mole (45 %), as given in rsf. [41].

I

2

3 Roe

Fig. 6. Typical reactive trajectory. (a) Variation of bond lengths and bond angle with time for a trajectory with E _,, = 1.0 kcal/mole, E,,,, = 0. The encounter time is 4.2 x IO-l3 s. (b) Path of this trajectory in (R,,,, R,) space superimposed on the potential surface with r* = cr,. Selected points labeled a through e are shown in both (a) and (b).

R.L. Ja&/Rate

constantsfor

The rotationally averaged cross sections, %(E eoll, u, TR = 300 K), were determined for u = 0 at numerous collision energies between the apparent threshold’ and 6 kcal/mole and for u = 1 at E coil= 1,2,3, and 4 kcaljmole. These data are shown in fig. 7. The translational energy thresholds for u = 0 are zero for Ebarr = 0 and 0.5 kcal/mole and less than 0.1 kcal/mole for the 1.0 kcal/mole barrier case. Examination of the trajectory data indicates that reactant rotation provides all the additional energy necessary for reaction at low collision energies (i.e., E, > Ebarr - EeO,,for all reactive trajectories), even though the vibrational zero-point energy (1.223 kcal/mole) could promote the reaction. Undoubtedly the classical description of the threshold region is not completely accurate owing to the neglect of quantum mechanical phenomena such as tunnelling [59]_ However, the most impotant contributions to the rate constants come from cross sections with 0.4 < EC,,, f 1.5 kcal/mole, well above the observed threshold values. This is seen by examining plots of the integrand of eq. (4.7) which are equivalent to the therma! translational energy distributions for reactive collisions (fig. 8). The maxima occur at EC,,,,z 1 kcaljmole and the magnitudes of the integrands nearer the thresholds are considerably smaller. Since the cross sections for this energy range are large (0.4 to 1.5 A*), the neglect of quantum threshold effects should not introduce significant errors into the calculated rate constants at temperatures between 200 and 1000 K. The cross

the rruction CIO + 0 --t Cl + O2

199

2.5

I

1

0

I

I

I

6

I

I

I

2 3 4 5 ECOLL (kcal/mole)

3.0 “7

(b) 2.5

0

0

1234567

5, (&

ECOLL(kcol/mole) 3.0

f The apparent threshold energy in a trajectory

calculation is usually defined as the minimum collision energy for which NJN > 0. Since N is generally fess than 1000

2.5

the reaction probability at the apparent threshold is greater than 10m3. This apparent threshold approaches the true threshold energy as N becomes very large. In the present work a lower limit of0.03 kcal/mole has been imposed on the apparent threshold energy.

2.0 S, @I 1.5

1.0 Fig. 7. Rotationally averaged reactive cross sections for Cl0 + 0 + Cl + O2 at 300 K. The points denoted by 0

and 0 are for u = 0 and o = 1, respectively. The smooth curves are polynomial least square tits to the u = 0 data. Statistical error limits are shown for the u = 0 data. The results are for: (a) Ebrrr = 0, (b) E,,,, = 0.5, and(c) E ba,r = 1.0 kcal/mole.

.5

0

I

2 3 4 5 ache,_ tkcal/mole)

6

I 7

R.L. JafiJRate constuntsfor t8e reuctiotlCl0 + 0 + Cl -I-O2

200

(5.1) where N, and N are determined as in eq. (4.3). At 300 K 9.2 % of the Cl0 + 0 collisions are reactive for the zero barrier case. This is in good agreement with the experimental estimate of 10% [17]. 0

I

2

3

4

5

ECOL~(kcal/mole)

Fig. 8. Translational energy distributions for reactive trajectories. F(E,,,,)S,(E,,,,. c = 0. T = 300 K). The poIynomiaI least squares fits to rhc rotationally averaged cross sections were used.The dashed curve is the distribution for all collisions,F(E,,,,).

sections computed for ClO(u = 1) are somewhat

higher than the corresponding cross sections for o = 0. However, less than 2 ‘A of the Cl0 molecules are in excited vibrational levels at 300 K and reactive collisions involving vibrationally excited Cl0 can make only small contributions to the lowtemperature thermal rate constants. The calculated rate constants range from 1.26 to 0.47 x lo-” cm3 mole -I s-l as E bnrr is varied

between zero and 1 kcal/mole. The statistical sampling method introduces an error ofabout & 15 %. These values are a factor of 3 to 5 lower than the recent experimental rate constants for reaction (R2) [17-191. The temperature dependence of the rate constant has been extracted from the 300 K cross section data by an Arrhenius fit of rate constants calculated at 280 and 320 K. These values for E, are only approximate since they are based on the assumption that the cross sections are independent of rotational temperature. The activation energies estimated from the QCT data are greater than 0.5 kcaljmole for all values of EbJr_ and are in fair agreement with the experimental value of 0.38 kcaljmole determined by Clyne and Nip [IS]. The overall thermal reaction probability, P,, is g&en by

X2. Phase space trajectory cahtiatiotts

Thermal rate constants and energy distributions been determined for reaction (R2a) using the phase space trajectory sampling method. In all, over 14000 phase space trajectories have been computed for T = 220,300,500 and loo0 K. These calculations employed potential energy surfaces with E barr = O,OS,and 1 kcal/mole and various forms of V,, the angular dependence. For each of 21 sets of initial conditions (combinations of T and the nature of the potential energy surface) between 50 and 88 % of the trajectories are “successful” as compared with less than 10 % for typical QCT data. This results in have

statistical sampling errors of f2-5 y0 which are considerably smaller than in the QCT calculations because of the greater fraction of reactive trajectories. The calculated PST rate constants for temperatures between 220 and 1000 K are shown in table 6 and fig. 9. The Arrhenius activation energy increases with increasing E,,_ and temperature according to

EA 2 Ebrr. + +k(T, + T,).

(5.2)

The second term in eq. (5.2) arises both from the hard sphere temperature dependence $T and from the variation of the potential energy with a in the transition region. Comparison of the QCT and PST rate constants at 300 K shows good agreement, especially for the larger values of Eb__ Exact agreement between the two methods is not expected because of the differences in selecting the initial conditions. The

QCT rate constants are calculated for reactants with

R.L. Jafle/Ratr constulttsfor Table G PST rate constant data W’K)

x

K

ri0 300 500 1000

EBARR= 0” 10" cm3 molecule-‘s-’ 2.007 2.828 4.593 10.116

E UARR-10 - .

EsARR= 0.5 (kcal/mole)”

(kcal/mole)b’

0.685 I .286 3.050 7.986

0.234 0.585 1.901 6.305

0.824 0.815 0.795 0.706

0.874 OS50 0.810 0.692

-

.!Z

220

300 500 1000

0.747 0.757 0.759 0.679

K x 10” cm3 molecule-‘s-’ 220 1.499 f 0.028 0.565 _+0.00s 300 2.141 +_0.038 1.048 i: 0.027 500 3.724 f 0.066 2.425 f 0.039 1000 6.869 _+0.149 5.638 i_ 0.115

0.205 _+0.003 0.497 r 0.016 1.540 & 0.033 4.363 -+ 0.130

‘) Data based on samples of 1000 trajectories.

‘I Data based on samples of 500 rrajectories.

01234567 IOOO/T,OK-’

I I I locm!5oo. 300

I 220

T, “K Fig. 9. Arrhenius plot of the rate constant data for Cl0 + 0 + Cl + Oz. The solid curves are calculated PST rate constants. The dashed curve is the “best” PST result for E,,,, = 0 and f; = fs/2. The points represent experimerd measurements: (a) Bemand [17], (b) Clyne and Nip [IS] (_ ._), (c) Zahniser and Kaufman [19] (- . -), (d) Freeman and Phillips [16], (e) Basco and Dogra [lSJ, and (f) Park [20].

the reaction Cl0 f 0 + Cl + O2

201

thermal quantized rotation and collision energy distributions and fixed vibrational energy (1.223 kcal/mole, corresponding to IJ = 0) while the PST rate constants are determined for fully classical thermal distributions of ET,E,,and ER.The average translation and rotation energies are in close agreement (PST = 1.55; ,, = 0.69 and PST= 0.80kcal/mole. for Ebrrr = 0), but the agreement between vibrational energies is not quite as good: ( Ev>ocT = 1.223 and (Ev),, = 0.80 k&/mole. The overall agreement between the 300 K QCT and PST data justifies the exclusive use of the PST sampling method in all subsequent calculations, which has considerably reduced the computational effort. Rate constan?; have also been calculated for potential surfaces with different angular dependencies ( V,) in the transition region (l&). These potential energy functions have different values of the bending force constant and the height of the inversion barrier E* designated& and E*’ (see the appendix and table 4). Modifications offd and E*’ change the angular width and shape of the reaction channel (the range of bond angles and I?,,, that can lead to reaction at low energies) as shown in fig. IO. As& is decreased by a factor of four the angular width of the channel to reaction is doubled (the crosssectional area for V c I kcal/mole with R,, = 2.65 A increases from 2.00 to 3.96 A degree). When E*’ is varied with fixed& this channel merely shifts to encompass a different range of at. Table 7 shows the sensitivity of the PST calculations to the strength of the angular variation of the potential energy surface with Ebnrr = 0. The largest calculated rate constant is K, = 4.86 x IO-” exp (- 156/T) cm3 molecule-’ s-’ = 2.89 x IO-” cm3 molecule-’ smt at 300 K for the case E*’ = E* and f; =/a/4. The magnitude of the Arrhenius activation energy decreases asf; is reduced and L

t Under these modifications the potential energy surface is not accurate in the triatomic region (F,,). This will not affect the rate constants since any phase space trajectories which reach region III will certainly be reactive. However. the product energy distributions depend on the breakup of the Cloo complex and will be aRcted by changes m E*’ and&. In fact, reduction of the bending rorce constant to fd4 results in a 15 to 20 % increase in the amount orenergy channeled into product vibration.

202

R.L. J&e/Rate

constmtsfor

the reaction Cl0 + 0 -, Cl f O2 Table 7 Variation of the angular energy surface”.bl

dependence

T(K) E*’ (kca,,mo,e~

$n:

of the potenlial

1o’L

molecules-1)

z

;;

’

loLL

molecuie-’ s-r)

230

300

i I __--I

------

cz 5

IO

I I I

i 90

k

_Ar

2.008 2.007 2.835 4.00 1 2.830

0.736 0.747 0.644 0.597 0.680

70

/a

2.828

0.757

70

AJ2

3.993

0.575

70

fd4

5.633

0.513

10.139 10.116 14.309 20.605

0.686 0.679 0.502 0.527

;;:

1000 40 70 70 70 -

1

(b) 60

&O ------.

:

lGJ_

2.0

I

---

40 70 70 70 40

I I20

t I50

a.deg

k $$

I.478 f 0.040 1.499 I 0.028 1.826 + 0.061 2.389_ 2 0.113 1.924 + 0.059 (0.43)“’ 2.141 * 0.038 (0.58)d' 2.30s r 0.088 (0.3SP 2.890 _+ 0.162 (0.3 l)d’ 6.956 + 0.210 6.869 & 0.149 7.183 & 0.320 10.852 L- 0.594

a’ Data based on samples-of 500 trajectories except E*’ = 7O.f; =Jn case (1000) and E*’ = 70,f; = fJ4 case

(3tW. ” EaaiR= 0.

Fig. 10. Contours of equal potential energy as a function of Rclo and x with R,, = q,* - 2.65A for Ebarr = 0. The contours are for V = 1,5 and 10 kcal/mole. The dashed lines are for R,,, = R, and in = ‘LC.(a) E* = 70 kcal/mole and f; = fn. (b) E* = 70 kcaljmole and f; = fJ4.

=’ The values of& are given in terms off. = 1.04 mdyne AI the experimental bendins force constant for ClOO. ‘I The values in parentheses are the activation energies in kcal/mole as determihed from the 720 and 300” data. HThe lower limit (hard sphere case) is 0.26 kcaljmole.

approaches the limiting value of 0.26 kcal/mole between 220 and 300 K for the case of aVJ8a = 0. The PST rate constants discussed above have been calculated using potential energy surfaces which were parameterized to exhibit a range of barriers to reaction and bending potentials. A third important feature which has not been varied in the present work is the range of the O-O attraction_ Modifications of this parameter should not affect the calculated temperature dependence of reaction (R2), but would definitely affect the magnitude of the calculated rate constants.

tally adiabatic and nonadiabatic the excited fine structure states

5.3. Multiple surjace efects in the Cl0 + 0 reaction The rate constants discussed above do not account for possible contributions from electroni-

collisions involving (multiple potential

energy surfaces) [Sl, 54,69-711. Adiabatic effects include processes like the reaction of ClO(‘II& + 0f3P,) on the 1*A’ potential surface, while nonadiabatic effects can be reactive, e.g., CIO(zl-I,,l) + 0(3Pz) 4 Cl(‘P,,,)

+ O,(‘A,),

(an electronic transition from the 1‘A” to the 3’A’ potential surface [see fig. 23) or nonreactive, e.g., ClO(‘l&)

+ 0(3P,) + ClO[‘rI,,l)

+ O(3Pz).

The cross sections defined in section 4 and described above are for an adiabatic (single surface) process. If the cross section in eq. (4.3) is replaced by the general cross section averaged over all possible electronic states (potential surfaces) then the rate

R.L. Ja#e/Rare

~o~~~tant~jk

constant will include contributions from all possible adiabatic and nonadiabatic processes. This electronically averaged cross section is defined as follows: S:(J%~, ~3J) = QJ

’C i giexp( - EiIkT)

x I~,#%,I,, wJ,OS%,,,,u,J,4 + Cl- ~A(L,, 4 J,WTV,,,,, u,J,91,

(5.3)

where Q, is the electronic partition function and the index i runs over all electronic states with degeneracy yi and energy EP St(E,,,,, v, J, i) is the adiabatic cross section for reaction on surface i [the ground state cross sections delined in section 4 is then S:(Lu. u, J, l)] and PA(Ec,I,, v, J, i) is the probability that a collision on surface i with conditions specified by Eco,,, u and J will be adiabatic. Similarly, S,SA is the reaction cross section for nonadiabatic collisions originating on surface i &n Sy(E,,,,,

u, .I, i) = 2s

P~X(Ecollr u, J,

i, b)b

db,

(5.4)

I where PV” is the probabihty that a nonadiabatic collision will be reactive. If reaction occurs only via the adiabatic ground state pathway, PA = 1 and Sc(E,,,,,

u, J, i) =

0

for

i 1 1.

For this case WE,,,,, 0, J) = o;‘Y,S:(%?,,,

the reaction

Cl0

t

0 -

Cl l O2

203

An electronic degenemcy factor of l/3, which corresponds to the ratio of the doublet states degeneracies to the doublet-plus-quartet degeneraties, has been used in the present study. Since the percentage of nonadiabatic collisions is generally small [70, 711 (PA, therefore, is close to unity). This approximate treatment of multiple surface effects should not introduce serious errors. However, at higher temperatures (500-1000 K) a significant fraction of the Cl0 + 0 encounters occur on the higher excited surfaces as governed by the electronic partition functions for the line structure states (see table 2). The effect of multiple potential surface phenomena on these rate constants is not known.

6. Discussion and comparison with experiment In the absence of an accurate ab initio potential energy surface for reaction (R2a), the selection of the “best” empirical potential is somewhat arbitrary. In the present study, we have chosen the following parameters for the “best” potential surface: E barr = 0 for c( = q, E*’ = E* and& =fJ2. This surface yields the second largest rate constant at 300 K of all potential surfaces considered and an activation energy which is in excellent agreement with the results of the Clyne and Nip [lS] experiments as illustrated in fig. 9. This “best” rate constant, Kz = 4.36 x IO- l1 exp (- 191/T) cm3 molecule-1 s- I, has a magnitude of

“, J, I),

which is the product of the standard electronic degeneracy factor [S!] mentioned in section 4 and QCT reaction cross section. In the limit of very low temperatures only the ClO(‘II,,,) and O(3P2) levels are populated and Q;‘gr = 2/10. At very high temperatures the electronic degeneracy factor is 2/36. However, the 1‘A” and 1‘A’ potential energy surfaces appear to have similar topographies [47,48]. As a result, one expects that S?(E,,,,, u, J, 1) zz St(E,,,,, u, J, 2). At low temperatures, most of the collisions occur on the two lowest doublet and quartet potential energy surfaces (1 2A”, 1 2A’, 14M and 14A’) and

2.31 x lo-” cm3 molecule-’ s-1 at 300 K which is about a factor of two lower than the recent experimentally determined values [17-191. For comparison, Clyne and Nip [ 181 obtained Kz = .10.7 x lo-” exp (-224/T) while Zahniser and Kaufman’s result [19] is 3.38 x 10-t’ exp(75/T) in the same units. Since none of the potential surfaces considered yielded a rate constant with a negative activation energy, the results of the present theoretical study definitely favor the measurements of Clyne and Nip [18] over those OrZahniser and Kaufman [l9]_

The source of the disagreement between the calculated and measured rate constants is not clear. The approximate effect of increasing the range of the O-O attraction in the potential surface can be

204

R.L. Ja/fe/Rate constantsfor the reaction

determined by examining eq. (4.10). The effective range of the O-O attraction is given by q: and K cc q:” if all other potential surface features are held fixed. In the present work, the value of this parameter is 2.65 A which seems reasonable in light of the 0, RKR potential curve [%I. In order to increase K by a factor of two, qr would have to be 3.7 A, an unreasonably large value. Thus, it is likely that the deficiencies in the empirical potential energy surface are more subtle. For example, the Cl0 vibration frequency might be more rapidly attenuated as the oxygen atom approaches. This would also result in a greater probability of reaction at all temperatures. In the absence of more detailed knowledge of the Cl0 + 0 potential energy surface it is diflicult to determine the magnitude of Kz to better than a factor of two by theoretical methods since the rate constant is very sensitive to small modilications in the potential energy function. However, these potential surface features affect the preexponential factor much more than the activation energy. The resulting temperature dependence for reaction (R2) implies that the rate constant for this process is about 20 % lower at average stratospheric temperatures than at 300 K. According to the recent NAS report on stratospheric halocarbons [13], this would mean a 5% decrease in the predicted amount of odd-oxygen depletion than if a temperature independent value were used for Kz. The amounts of stratospheric Cl0 consumed by reactions (R2) and (R3) would also be slightly shifted. It seems safe to conclude that the body of data for the Cl0 + 0 reaction is sufliciently accurate to permit a reliable determination of the rate of stratospheric ozone depletion if all the other necessary kinetic data were available.

Cl0

+ 0 + Cl + O2

temperature dependence for these rate constants are in excellent agreement with the observations of Clyne and Nip [IS] and provide support for the accuracy of their measurements. The reaction rate constant at stratospheric temperatures is approximately 20 % lower than the room temperature value due to the activation energy of 0.38 kcdl/mo1e. The results demonstrate that classical trajectory calculations based on empirical potential energy surfaces can provide useful kinetic data for important atmospheric reactions.

Acknowledgement The author is grateful to many helpful discussions on chemistry of the stratosphere and E.E. Whiting for critical work and manuscript.

Dr. L.A. Capone for the physics and and to Drs. J.O. Arnold discussions on this

Appendix:

of the bending potential

Parameterization

The double minimum

function

in eq. (3.4),

Y = A(a - ?c)’ + B/CC + (a - II)‘] + D, is parameterized the equilibrium (a,

-

n)*

=

-C

64.1)

as foliows. Fixing the minima at bond angles a, and (2x - a,) implies +

(B/A)‘!‘,

(A.2)

and A(cr, - x))’ + B/[C + (a, - n)‘] + D = 0,

(A.3)

since Y(a,) = 0. The maximum occurs when CL= 7~ and Y = E* which yields a third equation: E* = BJC t D.

(A.4)

Finally the second derivative of Y at the minimum, equals the bending force constant&:

7. Conciusions Rate constants

Cl0

have been computed

+ 0 + Cl + 02.

for reaction (R2)

for the temperature range 220-1OOO K. The resulting rate constants, KZ = 4.36 x 10-l’ exp (- 191/T) cm3 molecule- L s- I , are in good agreement with experimental rate data for 300 K. The calculated

fa = 2A + 2B[3(ae - x))’ - C]/[C

+ (a, - ~$1~. (A.5)

Eqs. (A.2HA.5) comprise a set of four simultaneous algebraic equations in four unknowns (A, B, C, and D). The solution of these equations is given below:

R.L. JuJfi/Rntc constmtsfor A =

fAE*fl-,

64.6)

B = 64faE*3(a, - r#/r3.

64.7)

C =&(a.

(A.81

- 7d4/T.

and D =fnE*(a,

- 7+[f&,

where r is delined r = 8E* -_&(a,

- i-r)’ -

16E*]/r’;

(A-9)

by - a)‘.

(A.lO)

References [‘I M.J. Molina and F.S. Rowland.

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the reucth

C/O + 0 --t Cl f O2

105

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R.L. Jaffe/Rate constantsfor rhe reaction Cl0 t 0 + Cl c O2

206 [49]

[SO] [St] [52]

[53] [54]

[5S] [56] [57]

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[58]

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