Angular momentum conserving statistical method for rate constant calculations.Application to N+(3 P)+H2(1Σ+ g ) →NH+(2Π)+H(2 S)Gunnar Nyman Citation: The Journal of Chemical Physics 96, 3603 (1992); doi: 10.1063/1.461913 View online: http://dx.doi.org/10.1063/1.461913 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Role of the HNO3↔NOH Isomerization in reactions (i) NH(3Σ−)+O(3 P) and (ii) N(4 S)+OH(2Π): Ab initiocalculations and quantum statistical Rice–Ramsperger–Kassel analysis of the potential energy surfaces J. Chem. Phys. 101, 3906 (1994); 10.1063/1.467508 A threedimensional potential energy surface for the reaction N+(3 P)+H2(1Σ+ g ) NH+ (X 2Π)+H(2 S) J. Chem. Phys. 96, 8202 (1992); 10.1063/1.462325 A lowenergy quasiclassical trajectory study of O(3 P)+OH(2Π)→O2(3Σ− g )+H(2 S). II. Rate constants andrecrossing, zeropoint energy effects J. Chem. Phys. 92, 2415 (1990); 10.1063/1.457985 Multichannel interactions in the (1σ g )2(1σ u ) n pλ(3Π g , 3Σ+ g ) Rydberg structures of He2 J. Chem. Phys. 81, 6013 (1984); 10.1063/1.447603 A distorted wave calculation for electronic energy transfer in molecular collisions. Application to the N2(A 3Σu +)+CO(X 1Σ+) →N2(X 1Σ g +) +CO(a 3Π) system J. Chem. Phys. 71, 2477 (1979); 10.1063/1.438655

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Angular momentum conserving statistical method for rate constant calculations. Application to N+(3P)+H2(1l:t)-.NH+(2n)+H(2S)

Gunnar Nyman Department of Physical Chemistry, The University of Goteborg and Chalmers University of Technology, S-41296 GOteborg, Sweden

(Received 1 April 1991; accepted 28 October 1991)

A quasiclassical Monte Carlo simulation procedure for calculating bimolecular rate constants is presented and applied to the reaction N+ ep) + H2 ( II +) -+NH+ ell) + HeS). Effective potential theory is used to generate a spherically symmetric potential energy surface from an ab initio. based global potential energy surface due to Wilhelmsson and Nyman. A complex is formed If the reactants can reach the strong coupling region. Based on the available phase space, a microcanonical procedure conserving total angular momentum is used to determine into which channel the complex decomposes. Complex formation cross sections, branching ratios, and reaction cross sections are presented. Good agreement with previous trajectory calculations is found.

I. INTRODUCTION

There exist a large number of statistical theories for cal­culating bimolecular gas-phase reaction rate constants. These may be divided into three main categories: I transition state, adiabatic capture, and Langevin-type theories.

In transition state theories, the canonical or microcan­onical flux across the transition state, which may be fixed or found variationally, is calculated. Both classical and quan­tum mechanical versions exist. Transition state theories have recently been reviewed by Hase and Wardlaw.2

In the adiabatic capture theories, which essentially are of quantum mechanical origin, capture effectively occurs when the centrifugal barrier is passed. A number of rotation­ally and vibrationally adiabatic potentials are calculated and capture is considered for each of them. More details regard­ing adiabatic capture theories may be found in recent work by Clary and Henshaw.3,4

The common requirement of the Langevin-type theo­ries, which are all classical in nature, is that a spherically symmetric interaction potential must be specified. The es­sential difference between the various Langevin-type theo­ries is in the method used to obtain the approximate, spheri­cally symmetric, potential. Once this potential has been determined, it is straightforward to calculate complex for­mation rate constants knowing that the orbital angular mo­mentum is conserved on any central field potential and con­sidering a complex to form if the centrifugal barrier is surmounted. Often it is assumed that the complex always decomposes reactively, whereby the reaction rate constant can be equated to the complex formation rate constant. Mar­kovic and Nordholm give a brief description of the more common Langevin-type theories in Ref. 1. There are, how­ever, at least two additional methods in the literature, name­ly, the extended Langevin model of Davidsson and NymanS

and the RRKM-SA method of Holm lid, Rynefors and Elof-u ' son. The last two methods have, however, only been im-

plemented for triatomic complexes. In the extended Langevin model an angle of approach is

chosen and the potential for this angle is used as if it applied to all angles. In this manner, cross sections are calculated for

various angles of approach and then averaged over all an­gles. A sine weighting is employed to account for the geomet­rical factor favoring perpendicular approach. The extended Langevin model is identical to the frozen rotor modelS ex­cept that the former allows the use of a general potential energy surface, while the latter is developed specifically for the asymptotic ion-dipole potential.

In the RRKM-SA method, a Monte Carlo procedure is used to average over the initial conditions. Conservation of the total angular momentum and redissociation into reac­tants are accounted for in approximate ways. The method is only implemented for a C6 11' potential, where the C6 pa­rameter is usually calculated from the atomic and molecular polarizabilities.

The method to be described in this paper is of Langevin­type origin. It uses effective potential theory9 to obtain a spherically symmetric potential. The criterion for complex formation is, however, not that the centrifugal barrier be surmounted, but that the reactants approaching each other reach a preselected center of mass separation, which is cho­sen to approximately represent a transition from weak to strong coupling between the reactants.

A recently developed angular momentum resolved mi­crocanonical sampling procedure iO is employed to deter­mine which products form as the complex decomposes. A Monte Carlo procedure is used to average over the initial conditions. The method now outlined will be referred to as the EFFEJ method as it uses effective potentials and an EJ, i.e., energy and angular momentum, resolved sampling tech­nique.

Section II details some of the ideas behind, and the theo­retical background to, the EFFEJ method. In Sec. III some particulars relevant to an implementation for the ion-mole­cule reaction

are given together with some recent trajectory results. Sec­tion IV contains the simulation results and Sec. V a discus­sion, while a summary and conclusions are presented in Sec. VI.

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3604 Gunnar Nyman: Statistical method for rates: N + + H2

II. THE EFFEJ METHOD

A. In general

The method to be presented here for the calculation of rate constants is based on Monte Carlo procedures. The cal­culations are performed much in the same way as a typical Monte Carlo trajectory study. First, initial conditions for a pair of reactants are selected. This pair is followed until its fate has been determined. Thereafter a new set of initial con­ditions is selected, and the fate of these reactants is deter­mined. This process is repeated until a representative selec­tion of reactants in the ensemble under study has been obtained. For each pair of reactants, it is determined whether a complex is formed or not. If a complex is formed, it is determined whether it decomposes reactively or unreac­tively. In this way, complex formation and reaction cross sections can be calculated just as in a trajectory study.

The concept of a centrifugal barrier is only well defined for a central field potential, which is generated using effec­tive potential theory in the microcanonical ensemble. A flux weighting is included in the effective potential since it is the flux across the transition state which is of interest, and not the density at it. This is further discussed in Sec. II B.

The criterion for complex formation is that the reac­tants approach to a preselected value of their center of mass separation. This separation defines a surface, referred to as the transition state surface, which is assumed to represent a transition from weak to strong coupling between the reac­tants. Strong coupling is taken to mean that only the total energy and total angular momentum are conserved, while other quantities are statistically redistributed. In the weak coupling regions also the angular momenta associated with the rotational motion of the diatom and with the orbital mo­tion are conserved. The centrifugal barrier is located either in the weak coupling or in the strong coupling region, de­pending on the initial conditions. If it is located in the weak coupling region, it has to be possible to classically surmount it for a complex to form. If the centrifugal barrier is located in the strong coupling region on the other hand, a complex is considered to form if the reactants can surmount the maxi­mum value of the potential, including the centrifugal energy, outside the transition state surface.

Of course, there exists no uniquely specifiable distance where a transition from weak to strong coupling occurs, and even if it did, its numerical value would depend on the initial conditions. On the other hand, Markovic and Nordholm have, in trajectory calculations, found that in their applica­tion to OeD) + R2 and K + NaCV I an acceptable ap­proximation was to specify a single distance for the transi­tion between the strong and the weak coupling regions. For the De P) + OR reaction, however, Markovic et al. \2 noted that the definition of a weak coupling and a strong coupling region is not as clear cut. Judging from the shapes of the potential energy surfaces employed in the 0 + OR study and in the present study, the transition from weak to strong coupling ought to be sharper for the system presented here.

Once the effective potential has been obtained, maxi­mum impact parameters for complex formation, as a func­tion of relative translational energy, are determined. From these, it is immediately seen whether a given set of initial

conditions, generated by Monte Carlo sampling, leads to complex formation or not. If a complex is formed, an angu­lar momentum resolved microcanonical sampling proce­dure is used to determine what products form on decomposi­tion. The complex decomposition part of the simulation procedure, which conserves the total energy and total angu­lar momentum exactly, is discussed in detail in Sec. II C.

B. Effective potential theory

The formation of a complex is studied on a spherically symmetric potential energy surface, obtained using micro­canonical effective potential theory. Effective potential methods have been used on various occasions (see Ref. 9 and references therein). Nordholm and co-workers have, for several model systems, successfully applied effective poten­tial theory in the canonical ensemble to describe the complex formation part of a molecular collision. I

•9.12 Markovic and

Nordholm have also discussed the extension to the micro­canonical ensemble,9 which is immediately relevant to the present work.

The point of using an effective potential is to eliminate certain degrees of freedom, which are judged to be less im­portant than the remaining ones. Usually, the angular de­pendence of the potential energy surface is eliminated, such that a spherically symmetric potential is obtained, which is also the objective in this work. In the microcanonical ensem­ble the total energy E t is a constant. Thus in order for the free energy A = E, - TS to remain unchanged as the potential is symmetrized, the entropy S has to remain unchanged, as the temperature Tis fixed. The entropy is, in the microcanonical ensemble, related to the molecular density of states accord­ing to the expression

Sa:ln(p), (2)

where P is the total density of states of a molecule.9 The total density of states can be obtained by integrating the momen­tum density of states over the spatial coordinates. In the present application, the density of states of the active mo­menta, i.e., the momentum coordinates (strongly) coupled to the motion along the reaction coordinate,9 is of interest. This density of states is given by9.10

Pa(E)=CnE(n-2)/2, (3)

where Cn is a constant, and E is the total kinetic energy of the n active degrees of freedom of the molecule.9 With E the total energy of the active degrees of freedom, we have a

E=Ea-V(Re,O), (4)

where the potential energy, V(Re,O), as usual does not in­clude the vibration of the diatom. The reaction coordinate Re is the distance between the centers of mass of the two reactants. The potential considered here is for atom-diatom reactions, where 0 is the angle between the diatomic internu­clear axis and a line from the atom to the diatomic center of mass, see Fig. 1. The diatomic internuclear separation is fixed at its equilibrium value when calculating the spherical­ly symmetric interaction potential.

The only spatial coordinate to be removed is O. It is therefore sufficient that the result on integratingpa (E) over o is the same for the full and the reduced potentials. We have

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Gunnar Nyman: Statistical method for rates: N+ +H2 3605

c

A --Li-------B

FIG. I. Illustration of the symbols used. The cross marks the diatomic cen­ter of mass. Rab is the diatomic internuclear separation. Rc is the distance between the diatomic center of mass and the atom. The symbols Rac and Rbc used in the text are the AC and BC distances, respectively. Note that the atoms A, B, and C are different for different channels.

I'" de sin e H(Ea - V(Rc,e» [Ea - V(Rc,e)] (n - 2)12

=2[Ea - U<lrCE;Rc)J<n-2)/\ (5)

where H is the Heaviside step function and Ueff is the effec­tive potential. The factor 2 on the right-hand side comes from the angular integration.

The aim is to calculate cross sections and rate constants. Therefore, the flow through the transition state surface, not the density on it, should be the same with the full and the reduced potentials. Markovic and Nordholm9 have shown that with this requirement, the microcanonical effective po­tential can be calculated using the equation

117 de sin e H(Ea - V(Rc,e»[ Ea - V(Rc,e) rn -1)/2

=2[Ea - Ueff (E;R c )](n-I)12, (6)

which is readily done. The only difference to Eq. (5) is that the exponent is changed from (n - 2)/2 to (n - 1 )/2.

In the present application n is set to three; i.e., the three relative translational degrees of freedom are considered ac­tive in contributing energy to the reaction coordinate. Nord­holm and co-workers have also used n = 5, where, in addi­tion to relative translation, the diatomic rotation is treated as active. The true value of n varies with the separation of the reactants, where three should be the best choice at large sep­arations. It would be simple to extend the present calculation to the case where n = 5. This would only require changing the value of n in Eq. (6) and including the rotational energy in Ea'

C. Complex decomposition

Once a complex is formed it is characterized by its total energy and total angular momentum. The branching ratio, i.e., the fraction of all decays which is reactive, depends on both of these constants of motion. The relative number of

decompositions into each channel is determined from the number density of molecules on the corresponding transi­tion state surface, multiplied by the average molecular speed across it. The density of molecules on the transition state surface is in this statistical treatment proportional to the density of states on the transition state surface, which is ob­tained using the EJ sampling technique.

In the present study, decomposition into three separated atoms is energetically prohibited. Three decomposition channels are considered, each corresponding to one of the three atoms leaving the other two, the charge always remain­ing on the nitrogen atom. Two of the channels are of course identical.

The EJ sampling method has been described in detail elsewhere,1O and only a brief description of the principles behind it is given here. Some details, specific to the present purpose, are thoroughly explained.

The aim of the EJ method is to sample the phase space coordinates of the molecule is such a manner that a prese­lected total energy and total angular momentum is obtained. A molecular configuration is specified such that its potential energy is less than the desired total energy. From the given configuration and the desired total angular momentum, the molecular rotational energy is calculated. This energy, summed with the potential energy, must be smaller than the desired total energy, for the specified configuration to be acceptable.

A momentum density of states, where conservation of the total angular momentum is accounted for, is calculated for the specified configuration. This density of states deter­mines the importance of the configuration at hand, relative to other configurations. If required, momentum coordinates for each atom are sampled from Gaussian functions,1O i.e., from the canonical ensemble at an arbitrary temperature, and scaled to comply with the desired total angular momen­tum and total energy. There is no approximation, nor biasing of distributions, involved in the scaling. 1O When the phase space coordinates for a large number of molecules are ob­tained, these represent a microcanonical distribution with the constraint that the total angular momentum is fixed.

In the present implementation, the sampled configura­tion is given a weight of zero if it is energetically forbidden, meaning decomposition into the corresponding channel will not occur for such a configuration. If the configuration is allowed, however, momentum coordinates are sampled and from these a velocity of separation along the reaction coordi­nate, that is, the line connecting the centers of mass of the two fragments, is calculated.

The transition state surface is specified by a chosen val­ue of the center of mass separation between the diatom and the atom, Reo' for each decomposition channel. This leaves only two spatial coordinates to be specified to fully deter­mine the molecular configuration at the transition state. These were chosen to be the diatomic internuclear separa­tion Rab and the angle e between the diatomic internuclear axis and the line connecting the atom and the diatomic cen­ter of mass.

The momentum density of states (for nonlinear geome~ tries) is given by

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3606 Gunnar Nyman: Statistical method for rates: N + + H2

pp(E IR) = C3N(lJbIc) - 1/2 [ E - E, _ VCR)] (3N- 8)12,

(7)

where C3N is a constant, la' I b , and Ie are the principal mo­ments of inertia, E is the internal energy of the complex, E r is the rotational energy, VCR) is the full potential energy sur­face, R is a vector completely specifying the molecular geom­etry, and N is the number of atoms in the complex. The transition state surface is completely specified by Reo' mean­ing Rab and 0 can have any energetically allowed values. Thus to get the relative flux through each point on the three transition state surfaces, the density of states, multiplied by the velocity along the reaction coordinate, is integrated over the unspecified coordinates, Rab and O. In order to include all possible orientations of the diatom, integration over a second angle f/J, should be performed. The potential is, how­ever, independent of f/J, meaning this integration merely gives a factor of 21T, which is left out below. One-half of the total flux is directed outwards and given by

1 ~max (Rob,max(e>

fa:.- Jo_ JJ pp(EIR)(urc)pR~b sin o dRab dO, 2 8 min Rob,min(O)

(8)

where (u rc > p is the velocity through the transition state sur­face averaged over all momenta leading to separation of the reactants. Notice that (urc)p depends on R. The integration limits are chosen such that

(9)

Equation (8) gives the flux through each point on the transition state surface under consideration. All points on each transition state surface are equivalent, and therefore, the total flow F through each transition state surface is ob­tained simply by multiplying the flux by the area of the tran­sition state surface; i.e.,

F= 41TR ~J. (10)

For the microcanonical ensemble, analytic expressions for the average of u"" over the momenta have been given.13

For the angular momentum resolved microcanonical ensem­ble, analytical means to average Urc over the momenta are not available. Furthermore, individual values of v rc are need­ed for each complex, see below, and it is preferable to evalu­ate Eq. (10) by Monte Carlo integration. Then

(11)

where

A = (Rab.max - Rab,min) (Omax - 0min), (12)

and (Rab,max - Rab.min) and (Omax - 0min) are the ranges over which Rab and 0 are sampled. N. is the number of Monte Carlo samples and Ii is the value of the integrand in Eq. (8), for each sample, using an individual value for Urc

and not an average over the momenta. For sampled geome­tries where Eq. (9) is not fulfilled, Ii is set to zero.

The flow of molecules through each transition state sur­face depends on the specific values of E and J characterizing each complex. The probability of selecting a given decompo­sition channel, for a given complex, is proportional to the

flow through the corresponding transition state surface, ob­tained from Eq. (11) .14

The transition states are chosen on the borderline be­tween the strong and weak coupling regions. Therefore, if there is a centrifugal barrier, or any other potential barrier, outside the transition state, it is in the weak coupling region. For decomposition to occur, the molecule is required to be able to pass such a barrier classically. If decomposition in the selected channel is not allowed, the selection of decomposi­tion channel is redone. The procedure is repeated until de­composition occurs. Decomposition is allowed if the impact parameter for the decomposing complex is smaller than the maximum impact parameter which would lead to complex formation for the relevant conditions; i.e., for the relevant values of the total energy and the total angular momentum for the particular channel.

The centrifugal barrier test is of course performed on the effective potential as the centrifugal barrier is not well de­fined on the full potential. It could be argued that this test should be performed on a microcanonical effective potential not including flux weighting. In order to preserve the sym­metry of the EFFEJ procedure under time reversal, how­ever, the flux-weighted microcanonical effective potential was used.

For comparison with experiments, we depend on the decomposition of the complexes formed in the experiments to be fast for all occuring E and J values, on the time scale of the experiment. That is to say, no combination of E and J values should lead to the formation of complexes that are so long lived that they have not decomposed when the measure­ment is over.

III. APPLICATION TO N+ +H2

Reaction ( 1 ) has been proposed as a possible first step in the gas phase synthesis of ammonia in dense interstellar clouds. IS It is also of considerable interest from a theoretical point of view I 6 and has been widely studied experimental­ly17-24 and theoretically.16,25-3s In this section, the potential energy surface used for reaction ( I ), and results from a pre­vious trajectory study16 on the same potential, are briefly described. Thereafter, some details of the implementation of the EFFEJ method are given.

A. The potential energy surface

The potential energy surface used is the one obtained by Wilhelmsson and Nyman, including explicit consideration of the long range electrostatic ion-induced dipole and ion­quadrupole interactions in the entrance channel, referred to as PES III. 16 This potential is based on the ab initio points obtained by Wilhelmsson et a/.25 from CASSCF and MR­CCI calculations. The ab initio points were fitted to an ana­lytic form, using a many body expansion technique, which includes both positive and negative powers of the internu­clear separations. 16 The procedure is similar to that original­ly suggested by Sorbie and Murrell.36 In the entrance chan­nel at reactant center of mass separations of at least 7.5 A, the fit was, however, not to ab initio points, but to points calculated from the classical electrostatic interactions men-

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Gunnar Nyman: Statistical method for rates: N + + H2 3607

tioned above. Details of the ab initio calculations and the fitting procedure are given elsewhere16

•25 and will not be

repeated here. PES III is exothermic by 17 meV, as measured between

the quantum mechanical ground states of the reactants and the products. The present calculation is, however, classical in nature and due to the larger zero point energy ofH2 than ofNH +, the reaction is endothermic by 63 meV, according to the energy difference between the classical ground states.

B. Previous trajectory results

Complex formation cross sections, branching ratios, and reaction cross sections from the present statistical calcu­lations are compared with the corresponding results of a re­cent quasiclassical trajectory study16 on the same potential energy surface. In the trajectory calculations a complex was defined to have formed when one minimum distance ex­change37 had occurred, which is not the same criterion as used here. In practice, however, the two criteria can be ex­pected to yield similar results; (a) hardly any minimum dis­tance exchange would occur without the centrifugal barrier, or Rco ' being passed, and (b) in most cases when the centri­fugal barrier, or Reo' is passed, a minimum distance ex­change occurs. At the highest relative translational energies used, there is, however, a noticeable fraction of the trajector­ies which pass the centrifugal barrier but do not have a mini­mum distance exchange. The trajectory determined com­plex formation cross sections are therefore expected to be somewhat smaller than the ones obtained from the statistical simulations.

The initial conditions for the quasiclassical trajectory calculations were chosen so that comparisons with the ex­perimental results of Ervin and Armentrout,22 would be possible. Therefore, the direction in space of the N + transla­tion was fixed while that of H2 was sampled isotropically from a 300 K thermal density distribution. A rejection pro­cedure favoring high relative velocity of the reactants was included. 16 Other details of the trajectory calculations can be found elsewhere. 16 Complex formation and reaction cross sections from the trajectory calculations are presented in Table I for initial diatomic rotational quantum numbers of zero and one. The initial translational energy of N + in a laboratory reference frame Et•N + is 0.3, 1.0, or 3.0 eV.

TABLE I. Complex fonnation U c and reaction u, cross sections and branching ratios b, obtained in the quasiclassical trajectory calculations of Ref. 16.

/) j E',.N+ (eV)' uc(A') u,(A') b,

0 0 0.3 69.2( 1.8)b 40.7( 1.5) 0.588 0 I 0.3 61.6( 1.7) 38.0( 1.4) 0.616 0 I 1.0 48.0( 1.6) 30.0( 1.4) 0.626 0 I 3.0 33.2(0.6) 17.3(0.5) 0.520

'Translational energy ofN+ in a laboratory reference frame. The transla­tional energy ofH2 was sampled from a thennal population at 300 K.

bThe values in parentheses give one standard deviation.

c. The EFFEJ calculations

The EFFEJ calculations were performed so as to be di­rectly comparable with the quasiclassical trajectory calcula­tions presented in Ref. 16. In all runs, ensembles of 10 000 samples were produced. The initial diatomic rotational quantum number, the translational energy, and the positions of the transition state surfaces were varied between runs. The relative translational energy was obtained as described in Sec. III B. For each run, the impact parameter was sampled to give a triangular distribution. The maximum impact pa­rameter bmax was always 10 A. The complex formation cross section U c was calculated from the number of complex form­ing systems Ne using

U c = 1Tb~axNJN" (13)

where Nt = 10 000. Reaction cross sections were calculated analogously, but with Ne replaced by NT' the number oftra­jectories leading to reaction. The branching ratio by was found from

(14)

IV. RESULTS

The dependence of the cross sections and branching ra­tios on the Reo values has been investigated. In Fig. 2, com­plex formation cross sections as a function of the entrance channel Reo value are depicted. Results for j = 0 and 1 are presented for three choices of translational energy. There is a statistical uncertainty (as defined in the heading of Table II) of approximately 2 % in the cross sections throughout. Com­parison with complex formation cross sections from the qua­siclassical trajectory calculations16 is made. The simulated complex formation cross sections generally increase with in­creasing entrance channel Reo value, but for Reo values not larger than 3-4 A, the complex formation cross section is

80 T J, " • " c;'

i i ~

-S 60 " " 0

tl'" Q

• ~ j Ii.N' Ii 0 0 o 1 0.3 0 • 0 0.3 o 1 1.0

40 ~ • 0 1.0 ¢ o 1 3.0

~ • 0 3.0 0

2 3 4 5

Entrance channel Rco value (A.)

FIG. 2. Complex fonnation cross section as a function of entrance channel Reo value. H, was initiated with vibrational zero-point energy. Filled sym­bols are for j = 0 and open symbols for j = I. Squares are for a translational energy ofN + of 0.3 eV in a laboratory reference frame. Circles are for 1.0 eV and diamonds for 3.0 eV. The translational energy of the H, gas was thennal at 300 K. Results from a previous quasic1assical trajectory study (Ref. 16) are shown to the left, and indicated by the letter T.

J. Chern. Phys., Vol. 96, No.5, 1 March 1992

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3608 Gunnar Nyman: Statistical method for rates: N + + H2

1.0

0 ..... 0.8 1d

'"' be T !=:

,l, ;.a t)

~ 0.6 8 • '"' !Xl <>

0.4 2

~ 0

• 0 <> e

j fi.N' o 1 0.3 • 0 0.3 o 1 1.0

• 0 1.0 <> 1 3.0 • 0 3.0

3 4

Entrance channel Rco value (A)

5

FIG. 3. Branching ratio as a function of entrance channel Reo value. The corresponding exit channel Reo values were 5.05, 5.35, and 5.0 A, respec­tively, from left to right. Otherwise as in Fig. 2.

relatively constant. The best choice seems to be Reo = 3.1 A, which will become clear below. The agreement with the tra­jectory results is most satisfying. Two specific differences are, however, observed. First, at the highest translational energy studied, the complex formation cross section of the trajectory study16 is smaller than those from the statistical sjmulations, the difference being about 15% for Reo = 3.1 A. As was noted above, the different definitions of a complex used in the trajectory study16 and in the present work, is probably largely responsible for this.

Second, at the lowest energy studied, the trajectory complex formation cross section for the ground rotational state is larger than the one obtained using the EFFEJ proce­dure and the best choice of transition state location. This probably results from the fact that when the diatom is initial­ly nonrotating, the reactants can partly align themselves, thereby searching out a low energy path in the trajectory study. Such an effect was previously found by Davidsson and Nyman in a quasiclassical trajectory study of the 0 + OH reaction.38 Also in their work, the corresponding complex

0.8 0 T i ~ ~ J. a ~

be 0

] 0.6 8 • 0

<> t) • ~ <> I'l j fi.N'

• o 1 0.3

!Xl • 0 0.3 <> o 1 1.0

0.4 • • 0 1.0 <> 1 3.0

• 0 3.D

3.0 3.5 4.0 4.5 Exit channel Rco value (A)

FIG. 4. Branching ratio as a function of exit channel Reo value. The en­trance channel Reo value was 3.5 A. Otherwise as in Fig. 2.

1.0 I I I

~ j fi.N' 0 o 1 0.3

• • o 0.3 0 <> 0 1 1.0

e • 0 1.0

I- <> 1 3.0 -i • 0

3.0 0.8 0 a

~ " • 0 0 0 • 0 0 <> • • -0.6 ~ • • ~ <> •

0.4 I I I

T A B c D E F G H

Rco category

FIG. 5. Branching ratio for eight combinations of Reo values. The letter T to the left indicates trajectory results (Ref. 16). The other letters refer to the Reo values used in the statistical calculations. Quoting entrance/exit chan­nel Reo values in A; A = 2.5/5.05, B = 4.0/5.35, C = 3.5/4.2, D = 3.1/3.5, E = 3.5/3.88, F = 4.5/4.5, G = 5.0/5.0, H = 3.5/3.4. Other­wise as in Fig. 2.

formation cross section was larger for the lowest rotational state than for the other rotational states. 12

In Fig. 3, branching ratios for three positions of the en­trance channel transition state surface are shown. The corre­sponding exit channel Reo values were similar to each other. The variation in branching ratio thus mainly stems from the variation in entrance channel Reo value. The branching ratio is seen to decrease when the entrance channel Reo value is increased. The observed behavior results from the flow through each transition state surface having an explicit R ~o dependence, Eq. (10), combined with the decrease in Pp which occurs when Reo is increased.

In Fig. 4, branching ratios are presented as a function of exit channel Reo value, with the entrance channel value fixed at 3.5 A. The branching ratio is seen to increase with in­creased exit channel Reo value. The explanation for this is

II 60 - -

~ 0

iii 0

~ 0 • • g •

D 1:1 g ~ 0 • <> -

c> ~ 0

I'l <)

• j fi.N' <> • I- o 1 0.3 ~ ~

• 0 0.3 20

o 1 1.0

• a 1.0

<> 1 3.0

• 0 3.0

o T A B c D E F G H

Rco category

FIG. 6. Reaction cross sections for eight combinations of Reo values. Other­wise as in Fig. 5.

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Gunnar Nyman: Statistical method for rates: N + + H2 3609

I I 0

60 - • -,-.. ~

"$ 0

• s:: 0 40 f- 0 ..... -..., • 0 <:) Q) • 00 0 00 00 0 20 l-I-< 0 -

C) •

0 I I

0 1 2 3

~,N + (eV)

FIG. 7. Complex formation (squares) and reaction (circles) cross sections as a function of the N + translational energy. The H2 translational energy was thermal at 300 K. H2 was initiated with vibrational zero-point energy in the first excited rotational state. Filled symbols are trajectory results and unfilled symbols are from EFFEJ calculations using entrance and exit Reo values of 3.1 and 3.5 A, respectively.

completely analogous to the one given in regard to Fig. 3. In Fig. 5, branching ratios are shown for the eight com­

binations of Rca values employed in this study and compared with the quasi classical trajectory results. Rca values of 3.1 and 3.5 A for the entrance and exit channels, respectively, are seen to give branching ratios in good agreement with the quasiclassical trajectory results, the deviation always being less than 8%.

In Fig. 6, reaction cross sections are shown for the same eight combinations of Rco 0 values used in Fig. 5. As above, Reo values of 3.1 and 3.5 A for the entrance and exit chan­nels, respectively, were found to give the best agreement with the quasiclassical trajectory results. The reaction cross sec­tion for the highest relative translational energy tends to be overestimated in the EFFEJ simulation, which is related to

TABLE II. Cross sections and branching ratios obtained with the EFFEJ method for various rotational quantum numbers of H2. Uncertainties are not specified, but are in the range 0.5-1.5 A2 for the cross sections, when calculated from <7. [(Nt - N. )/(NtN.)j 112, where x = rore. Thetrans1a-tiona! energy ofN + is. 0.3 eV and Reo is 3.1 and 3.5 A for the entrance and exit channels, respectively. The symbols are explained in the text.

<7e U r j Ne Nr (A2) br (A2)

0 1983 1251 62.3 0.63 39.3 1 2024 1315 63.6 0.65 41.3 2 1953 1386 61.4 0.71 43.5 3 1964 1491 61.7 0.76 46.8 4 1920 1505 60.3 0.78 47.3 5 1943 1624 61.0 0.84 51.0 6 2022 1705 63.5 0.84 53.6 7 1943 1658 61.0 0.85 52.1 8 1957 1695 61.5 0.87 53.2 9 1984 1718 62.3 0.87 54.0

10 2043 1774 64.2 0.87 55.7 15 1998 1755 62.8 0.&8 55.1 20 1983 1756 62.3 0.89 55.2

the dynamics of the trajectories. Some of them are not long­lived enough for complete energy redistribution to occur, and preferentially decompose unreactively.

In Fig. 7, complex formation and reaction cross sections from the EFFEJ simulations are presented for Rca values of 3.1 and 3.5 A for the entrance and exit channels, respective­ly, and compared with the quasiclassical trajectory results. The trajectory cross sections are generally smaller than the simulation results, the difference being largest at the highest energy, which is expected from the arguments given above. Overall, the agreement is seen to be good, in particular since the differences seem to be related to the specific dynamics of the reaction under study and to the definitions of a complex used here and in the quasiclassical trajectory calculations. 16

In Table II, cross sections and branching ratios are pre­sented as a function of diatomic rotational quantum number for Et,N + =0.3 eV and Rco = 3.1 and 3.5 A for the entrance and exit channels, respectively; i.e., the choice of Reo values giving the best agreement with the trajectory results. The initial rotational state is varied fromj = 0 to 20. The com­plex formation cross sections are independent of the initial rotational state ofH2 , the small variations seen being statis­tical fluctuations resulting from the Monte Carlo sampling of the initial conditions. In a recent quasiclassical trajectory study on 0 + OR, it was found that the complex formation cross section depends only weakly on the initial rotational state of the diatom. 12,38 If reaction (1) behaves similarly, this would support the use of only three active degrees of freedom when calculating the effective potential, which makes it independent of rotational state.

From Table II, it is seen that the branching ratio in­creases with increasing rotational quantum number, in the range shown, which is also true for the reaction cross sec­tions. This was expected as the total energy increases with increasingj, and the reaction is endothermic when measured between the classical ground states of the reactants and the products. Furthermore, the angular momentum restrictions probably become less severe with increased j, in the range shown. This is discussed in Sec. V B.

v. DISCUSSION

A. Choosing Rco

Clearly, the statistical method outlined above is sensi­tive to the values used for the center of mass separation, where the switch from strong to weak coupling is considered to occur for the three channels. It is to be expected that at low relative velocities between the reactants, the centrifugal bar­rier will fall outside a given Rco value, whereby the specific value of Reo becomes unimportant for the formation of a complex. At high relative velocities, the centrifugal barrier is more likely to fall inside R co ' whereby the entrance channel Rco value partly determines the complex formation cross section. For the case where Et,N+ = 0.3 eV, approximately 80% of the samples have the centrifugal barrier falling in the range 3.3-4.2 A. For the cases where E N+ = 1.0 or 3.0 eV t,

the corresponding ranges are 3.0--3.8 and 2.5-3.1 A, respec-tively. These numbers directly relate to where in Fig. 2 the complex formation cross sections appear independent of Rco ' and where they vary with Reo'

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3610 Gunnar Nyman: Statistical method for rates: N + + H2

TABLE III. Reo values for the entrance and exit channels. These values were obtained using the criterion that the Morse potential term for respec­tive channel, at Reo, should correspond to the given fraction of the dissocia­tion energy for that Morse term. Here the Morse potential was chosen to be zero at infinite separation of the two atoms involved.

Reo (A) Reo (A)

Entrance Exit Fraction channel channel

0.01 3.50 3.88 0.02 3.10 3.50 0.05 2.62 3.00 0.10 2.27 2.65

A possible way to determine the best value of Reo would be to run trajectories to investigate where energy exchange between the reactants sets in, using, for instance, the meth­ods of Markovic et al. 11

,12 This should be possible for both the exothermic and the endothermic direction of a reaction as there is no need to find reactive trajectories or await the decomposition of a formed complex. It would, however, be advantageous to have a criterion for choosing Reo without resorting to trajectory calculations.

One way of choosing Reo is to investigate where the di­atomic potential terms, here Morse functions, begin to de­viate significantly from their values at infinity. Table III lists four sets of Reo values obtained by setting Reo equal to the internuclear separation at which the deviation reaches a giv­en fraction of the dissociation energy for that Morse term. This procedure is slightly inappropriate as Reo is not defined in terms of an internuclear separation, but as a center of mass separation. The first two sets of Reo values shown were used to produce some of the results which have been presented, the second set giving the best agreement with data from qua­siclassical trajectory calculations.

On a spherically symmetric potential energy surface, the orbital and rotational angular momenta are conserved, whereby energy transfer between the reactants should be harder to achieve than on a nonspherical potential. There­fore, it is expected that the transition from strong to weak coupling should depend on where the angular dependence of the potential becomes significant. It seems plausible that when the angular variation of the potential (measured as the difference between the maximum and the minimum in the potential energy on sweeping the angle (j through 0 to 21T for a fixed value of Rc ) becomes comparable to kT, the absolute values of the orbital angular momentum III and the rota­tional angular momentum Ij I may begin to change signifi­cantly. This suggests that Reo values may be obtained by determining distances, for the entrance and exit channels separately, at which the angular variation of the potential becomes for instance 25 or 50 meV, which approximately corresponds to kT and 2kT, respectively, at room tempera­ture. Using these criteria for choosing Reo gave 2.5 and 4.3 A for the entrance channel, and 4.3 and 4.5 A for the exit. chan­nels, for 50 and 25 meV, respectively. Results from such calculations add no additional insight, however, and are not presented.

A criterion giving unique Reo values would be to set Reo equal to the location of the minimum flux as given by a vari­ational transition state calculation. Although there is no theoretical basis for proceeding in this way to find the weak to strong coupling transition, it could prove to be a practical­ly useful approach.

A definite way of determining Reo values has not yet been established. From Figs. 2, 5, 6, and 7 it is, however, seen that by choosing Reo = 3.1 A for the entrance channel and 3.5 A for the exit channel, good agreement is obtained be­tween all presented trajectory data and the statistical simula­tions. The differences seen are not large and do not depend strongly on the Reo value over a limited range. The results suggest that the EFFEJ procedure should be useful in future calculations of thermal rate constants.

B. Branching ratios

Each complex decomposes to products or back to reac­tants. The relative values of Reo for the entrance and exit channels affect the branching ratio, as do the total energy, total angular momentum, and the absolute value of Reo for the entrance channel. The branching ratio is generally large when the exit channel Reo value is large and/or the entrance channel Reo value is small, which was explained above.

It is interesting to note that when the exit channel Reo value is large, 5.05 or 5.35 A, and the entrance channel value is smaller [i.e., Reo categories (A) and (B) ], the branching ratio is largest for the highest collision energy, while the op­posite is true for the other cases shown in Fig. 5. The case where the high collision energy favors reactive decomposi­tion can at first be rationalized simply by noting that the reaction is classically endothermic, whereby the forward channel is favored by a larger total energy.

The case where low collision energy favors reaction can be rationalized by assuming that the orbital angular momen­tum I is approximately conserved in the collision. We have

III = flUre! b, (15)

where Ii is the reduced mass ofthe collision or decay process, Urel the relative velocity of the fragments, and b the impact parameter. Since fl is larger for the entrance than the exit channel, conservation of I would require band/or Ure1 to increase for reactive decay to occur. It is easier to obtain an increase in ure1 if it is small initially, which explains why a small collision energy can give a larger branching ratio than a high collision energy.

The smaller the ratio of the Reo values for the entrance to the exit channel, the more likely an increase in b becomes. In this case it is expected that a relatively small, if any, in­crease in Urel is needed in order to conserve III, whereby the energetics may be of larger importance in determining the branching ratio. Thus also after this more detailed consider­ation, it is plausible that for the case where the ratio of Reo values for the entrance to the exit channel is small, a high collision energy gives larger branching ratio than a small collision energy.

In Table II, it is seen that the branching ratio increases with increasing initial rotational quantum number. This is probably connected to the fact that the contribution of the

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Gunnar Nyman: Statistical method for rates: N+ + H2 3611

rotational angular momentum to the total angular momen­tum becomes comparable to or larger than the orbital angu­lar momentum as the rotational quantum number is in­creased. As the rotational and orbital angular momenta become comparable in size, the angular momentum restric­tions should become less severe. This should affect the reac­tive channels more than the unreactive channel, thereby in­creasing the branching ratio.

C. Comparison with other theories

There are several statistical theories available for calcu­lating bimolecular rate constants for attractive potential en­ergy surfaces. The more common ones include adiabatic cap­ture (AC) theory, the statistical adiabatic channel model (SACM), variational transition state theory (VTST), phase space theory (PST), average dipole orientation (ADO) the­ory, and average quadrupole orientation (AQO) theory. Each theory has its advantages and disadvantages, some of which are highlighted here and compared to the EFFEJ method.

The AC theories have the advantage of being of quan­tum mechanical origin. A disadvantage is, however, that the reaction rate is only obtained by equating it to the capture rate. In SACM, on the other hand, which is otherwise simi­lar to the AC theories, redissociation of the collision com­plex back to reactants is treated. In SACM, however, the adiabatic potentials have usually been obtained using a Morse potential and an exponential interpolation method to interpolate between the reactant and product states. This has produced incorrect long-range behavior of the potential and in recent work this approach has been changed.39 An advan­tage of the EFFEJ procedure compared to SACM and AC theories lies in the generality of employable potential energy functions and in the latter comparison also in the considera­tion of redissociation back to reactants. The EFFEJ method is, however, classical in nature.

The ADO and AQO theories have the same drawbacks as the AC theories. Further, they only allow a single inverse power in the potential energy function and they do not con­serve the total angular momentum, although this is approxi­mately done in angular momentum ADO (AADO). An ad­vantage of these three theories is that they result in analytic expressions for the thermal rate constant, which can be eval­uated using a hand calculator.

PST is due to Light and co-workers40--42 and is based on the evaluation ofthe phase space available to form a collision complex and to form products from the decomposing com­plex. Restrictions due to conservation of the total angular momentum are considered. A one-dimensional electronic potential is assumed,43 though in most applications a single inverse power is used. The method was initially developed for atoms or ions colliding with diatoms,40--42 but extensions have been made, for instance, by Chesnavich and Bowers who derived expressions for rigid rotor harmonic oscillator spherical topS.44 The EFFEJ procedure as implemented here is a method to evaluate the available phase space with the advantages of allowing exact densities of states and a general potential energy surface to be used.

Transition state theory is available in a variety of imple-

mentations. For reactions without potential energy barriers, the variational version of transition state theory is required. Here the transition state is located at the point along the reaction path where the flux is minimal, assuming local equi­librium throughout the collision process. The required den­sities of states or partition functions are most easily evaluat­ed assuming separability of vibrations and rotations and using the harmonic approximation. More elaborate ap­proaches are generally needed for reliable results, see the recent review by Hase and Wardlaw.2 The EFFEJ proce­dure does not suffer from these approximations. Instead there is the assumption that no relaxation occurs between the various degrees of freedom until the strong coupling re­gion is reached.

In several of the mentioned theories, passage of the cen­trifugal barrier defines capture, while transition from strong to weak coupling replaces that criterion in the EFFEJ proce­dure.

D. Future developments

A possible future development of the EFFEJ method would be to allow the effective potential to be calculated for a larger number of active degrees of freedom than three. This would make the effective potential more attractive, and de­pendent on the rotational state of the diatom. Neither theo­retical considerations, nor comparison with trajectory re­sults, motivate such an extension.

Another possible development would be to resolve the density of states expression into the various vibrational lev­els, whereby decomposition into quantum mechanical vibra­tional energy levels could be studied.

An important question, which has not been fully an­swered, is how the Reo values should be chosen. Hopefully, such insight could be gained by comparing simulation and trajectory results for a few representative systems.

Given the very reasonable accuracy obtained here, the most interesting step, however, would perhaps be to extend the EFFEJ method to make it applicable to polyatomics. Such an extension should be relatively straightforward as the principles involved are general.

v. SUMMARY AND CONCLUSIONS

A simulation procedure based on the use of an effective potential and Monte Carlo methods was presented and named the EFFEJ method. In this method, the reaction is divided into a complex formation step, and a subsequent step where the complex decomposes. The complex is formed on a microcanonical effective potential conserving the flux of molecules on the full potential. The decomposition step is performed using an energy and angular momentum resolved sampling technique. In this way, the total energy and total angular momentum are exactly conserved.

Complex formation cross sections, branching ratios and reaction cross sections obtained with the EFFEJ method were presented for the N + + H2 reaction on an ab initio based global potential energy surface with explicit consider­ation of the electrostatic interactions in the entrance chan­nel. The results were found to be in good agreement with

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3612 Gunnar Nyman: Statistical method for rates: N + + H2

results from a previous quasiclassical trajectory calculation employing the same potential energy surface.

To perform the EFFEJ simulations, it is necessary to specify where the assumed transition from weak to strong coupling occurs. Even though no a priori procedure was giv­en for determining where this transition occurs, it was no­ticed that the results do not depend strongly on the exact location of the transition state, as long as it is physically reasonable. Furthermore, the EFFEJ procedure relies on the assumption of statistical behavior in general, whereby it is not essential that the choice of transition states is fully opti­mized, as a degree of uncertainty in the results will always be present.

The present study spans a reasonably large range of translational energies, 0.3 eV to 3.0 eV for N + . Therefore, it seems likely that it in gel.leral should be straightforward to obtain thermal rate constants with good accuracy using the EFFEJ simulation procedure, without extensive a priori knowledge of the dynamics of the system.

The EFFEJ method allows the use of a general potential energy surface, which gives it a wide range of possible appli­cations. It is presently implemented only for reactions in­volving three atoms, but the principles are general and it should be feasible to extend the method to polyatomics.

ACKNOWLEDGMENTS

The author is grateful to N. Markovic for helpful discus­sions and supplying several subroutines from his previous work on effective potential methods. Illuminating discus­sions with S. Nordholm and H. W. Schranz are also ac­knowledged.

I N. Markovic and S. Nordholm, J. Chem. Phys. 91, 6813 (1989). 2 w. L. Haseand D. M. Wardlaw, in Advances in Gas-Phase Photochemistry and Kinetics, edited by M. N. R. Ashfold and J. E. Baggott (The Royal Society of Chemistry, London, 1989), Chap. 4.

3D. C. Clary and J. P. Henshaw, Faraday Discuss. Chem. Soc. 84, ( 1987).

4D. C. Clary, Mol. Phys. 53, 3 (1984). 5J. Davidsson and G. Nyman, Chem. Phys. 125,171 (1988). 6 L. Holmlid and K. Rynefors, Chem. Phys. 60, 393 (1981). 7K. Rynefors, Comput. Phys. Commun. 27, 201 (1982). K. Rynefors, L. Holmlid, and P.-A. Elofson, Chem. Phys. 118 (1987) 417.

8 J. V. Dugan, Jr. and J. L. Magee, NASA Technical Note NASA TN-D-3229 (1966); J. V. Dugan, Jr., Chem. Phys. Lett. 21, 476 (1973).

9N. Markovic and S. Nordholm, Chem. Phys. 135, 109 (1989). lOG. Nyman, S. Nordholm, and H. W. Schranz, J. Chem. Phys. 93, 6767

(1990). II N. Markovic and S. Nordholm, Chern. Phys. 134, 69 (1989). 12N. Markovic, G. Nyman, and S. Nordholm, Chem. Phys. Lett. 159,435

(1989). 13 H. W. Schranz, L. M. Raft', and D. L. Thompson, Chem. Phys. Lett. 171,

68 (1990); J. Chem. Phys. 94, 4219 (1991). 14 In order to conserve computer time, two complementary procedures were

used to evaluate Eq. (11). Details of these procedures and the computer program are available from the author.

15E. Herbst and W. Klemperer, Astrophys. J. 185, 505 (1973). 16U. Wilhelmsson and G. Nyman, J. Chem. Phys. 96,1886 (1992). 17 J. A. Luine and G. H. Dunn, Astrophys. J. 299, L67 (1985). 18S. G. Hansen, J. M. Farrar, and B. H. Mahan, J. Chem. Phys. 73,3750

(1980). 19N. G. Adams and D. Smith, Chem. Phys. Lett. 117,67 (1985). 20J. B. Marquette, B. R. Rowe, G. Dupeyrat, and E. Roueft', Astron. As­

trophys.147, 115 (1985). 21 S. E. Barlow, J. A. Luine, and G. H. Dunn, Int. J. Mass Spectrom. Ion

Proc. 74, 97 (1986). (Compare with Ref. 15). 22K. M. Ervin and P. B. Armentrout, J. Chem. Phys. 86, 2659 (1987). 23 J. B. Marquette, C. Rebrion, and B. R. Rowe, J. Chem. Phys. 89, 2041

(1988). 241. Kusunoki and Ch. Ottinger, J. Chem. Phys. 70, 710 (1979). "u. Wi1helmsson, P. Siegbahn, and R. Schinke (to be published). 26 M. A. Gittins and D. M. Hirst, Chem. Phys. Lett. 35, 534 ( 1975). 27 M. A. Gittins, D. M. Hirst, and M. F. Guest, Faraday Discuss. Chem.

Soc. 62, 67 (1977). 28C. F. Bender, J. H. Meadows, and H. F. Schaefer Ill, Faraday Discuss.

Chem. Soc. 62, 59 (1977). 29S. D. Peyerimhoft' and R. J. Buenker, Chem. Phys. 42,167 (1979). 30D. M. Hirst, Mol. Phys. 35,1559 (1978). 31 D. Gerlich, J. Chem. Phys. 90, 3574 (1989). 32M. Gonzalez, A. Aguilar, and R. Sayos, Chem. Phys. 132, 137 (1989). 33M. Gonzalez, A. Aguilar, and Y. Fernandez, Chem. Phys. 104, 57

(1986). 34M. A. Gittins and D. M. Hirst, Chem. Phys. Lett. 65, 507 (1979). 35M. GonzilJ.ez, A. Aguilar, and J. Virgili, Chem. Phys. Lett. 113, 179

( 1985). 36K. S. Sorbie and J. N. Murrell, Mol. Phys. 29,1387 (1975). 37 D. Gerlich, U. Nowotny, Ch. Schlier, and E. Teloy, Chem. Phys. 47, 242

(1980). 38 J. Davidsson and G. Nyman, J. Chem. Phys. 92, 2407 (1990). 39 J. Troe, J. Chem. Phys. 87, 2773 (1987). 4OJ. C. Light, J. Chem. Phys. 40,3221 (1964). 41 P. Pechukas and J. C. Light, J. Chem. Phys. 42, 3281 (1965). "P. Pechukas, J. C. Light, and C. Rankin, J. Chem. Phys. 44, 794 (1966). 43 J. R. Beresford, G. Hancock, and A. J. MacRobert, Faraday Discuss.

Chem. Soc. 75, 211 (1983). "w. J. Chesnavich and M. T. Bowers, Prog. React. Kinet.11, 137 (1982).

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