J. Phys. Chem. 1993,97, 7034-7039
7034
Rapid Algorithms for Microcanonical Variational Rice-Ramsperger-Kassel-Marcus Theory Sean C. Smith7 Department of Chemistry, University of California at Berkeley, Berkeley, California 94720 Received: February 12, 1993
Rapid algorithms have been developed for the evaluation of sums and densities of states in the classical approximation for the application of microcanonical variational Rice-Ramsperger-Kassel-Marcus (RRKM) theory to reactions involving loose transition states with anisotropic interaction potentials. The algorithms implement recent theoretical advances which enable the analytic convolution of momentum states with rigorous imposition of the constraint of fixed total angular In this paper, the performance of the new algorithms is illustrated by application to the methyl radical recombination reaction. The general Monte Carlo algorithm for integration over the internal configuration space is exact to within the limits of statistical error and proves to be in excellent agreement with earlier more computationally intensive studies.
I. Introduction The development of fast and accurate algorithms for the prediction of rate coefficients is an important and challenging goal in chemical dynamics. The present paper is concerned with the development of efficient algorithms for the prediction of rate coefficients in unimolecular dissociation, recombination, and chemical activation reactions involving entrance or exit channels which possess no pronounced barrier to the recombination process. This type of reaction is important because it occurs so commonly in combustion and atmospheric processes. It is also problematic because of the difficulty of extrapolating experimental data to different temperature and pressure regimes on the one hand and the difficulty of ab initio prediction of temperature- and pressuredependent rate coefficients on the other. The temperature and pressure dependenceof the rate coefficient is obtained by solution of the master equation, e.g., for a unimolecular reaction,
RJ’J(E’,E) gJ(E)l - kJ(E) gJ(E) In eq 1, kuniis the thermal unimolecular rate coefficient, gJ(E) is the nonequilibrium population distribution of the unimolecular species, [MI is the concentration of bath gas, RJJ’(E,E’)is the rate coefficient for collisional transfer of energy and angular momentum, and %(E) is the microcanonical rate coefficient. Methods of solution of this two-dimensional master equation have been developed by a number of authors.’-3 In this paper, we focus on the calculation of the microcanonical rate coefficient ki(E). The microcanonical rate coefficient is given in general by
where E is the total energy, J is the total angular momentum, NAE) is the cumulative reaction probability,ll h is Planck‘s constant, and pXE) is the reactant density of states (for recombination, the density of states of the reactant pair). The difficulties associated with the calculation of microscopic rate coefficients for “barrierless” reactions such as radical-radical recombination or ion-molecule association and their reverse dissociations have been summarized in detail by several authors.&9 At low to moderate energies, vibrationallyexcited fragments make
t Addressafter September 1993: Department of Chemistry,The University of Queensland, Brisbane Qld 4072, Australia.
only a small contribution to the total reactive flux, and so the statistically important modes are the so-called “transitional modes”,1° i.e., the rotational degrees of freedom which become progressively more hindered as the fragments come closer together and ultimately transform into vibrations in the unimolecular species. Within the context of transition state theory (which is normally a very good approximation for such reactions12), the cumulative reaction probability is approximated by the sum of states, WAE), for the modes orthogonalto the reaction coordinate. Hence the Rice-Ramsperger-Kassel-Marcus theory microcanonical unimolecular rate coefficients is
(3) WJ(E) is evaluated at the transition state, which is located ~ariationally.*~~J3 The position of the transition state and the calculated microcanonical rate coefficients are sensitive to the total energy, the total angular momentum, and the radial dependence and anisotropy of the assumed interaction potential. Unfortunately, the fact that all of the transitional modes are typically coupled at the energies of interest by the anisotropy of the potential makes the calculation of the necessary sums of states a difficult task, the more so when the constraint of fixed total angular momentum is imposed. Recent years have seen the development of exact numerical solutions to this problem within the classical approximation by Monte Carlo evaluation of the accessiblevolume in phase space, subject to the constraints of fixed total energy and angular moment~m.5.6JOJ+1~These approaches involve numerical integration over the dimensionality of the transitional-moderotational phase space. In this paper we illustrate the implementation of a newly developed theory which avoids numerical integration over the momentum space by analytically convoluting the momentum states at a given configuration with exact accounting for the constraintsof fixed total energy and angular momentum.7Js The exact implementation of the new techniquerequiresnumerical integration over only those internal angles in the loose transition state upon which the interaction potential is dependent, a maximum of five dimensions for two nonlinear fragments. The expression for the available momentum-space volume also leads to approximate analytical formulas for the transitional-modesum of states,’ which together with the exact numerical solution have been implemented in a microcanonical variational RRKM code which is an outgrowth of the UNIMOL package of Gilbert, Jordan, and Smith.19 We have chosen for a model system the methyl radical recombination and the reverse ethane dissociation reactions, for
0022-365419312097-7034$04.00/0 63 1993 American Chemical Society
Rapid Algorithms for RRKM Theory which the more numerically intensive phase-space integrations have been carried out previously by Wardlaw and co-workers.10-20 Since our focus in this paper is the improvement of the efficiency of calculation of microcanonical rate coefficients, we will not present pressure-dependent calculations. Comparison of the results of our exact numerical implementation with the numbers produced by the earlier calculations produces, as expected, excellent agreement. The present algorithm is fast, yielding converged high-pressure-limitingthermal rate coefficientsat 300 K for the methyl radical recombination reaction from the exact microcanonical variational RRKM treatment in 35 min on an IBM RS 6000/320 machine. In section 11, the theoretical approach is summarized, and in section I11 the results of the exact implementation of the method for the methyl radical recombination reaction are presented. In addition, we present in the appendix an improved method of imposing angular momentum resolution in the calculation of the density of states pAE) for unimolecular species by direct count which does not appear previously to have been presented. The densities of states and the microcanonical variational sums of states are utilized to generate unimolecular rate coefficients kAE) for ethane dissociation. 11. Summary of Theory
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7035 of E and J ,
where dI' = dqdp/hn (n being the number of transitional modes) if conventional Euler angles and momenta are usedu and d r = dpdql(27r)" if action-angle variables are 6 1 and uz are the symmetry numbers of the fragments, and HTMis the Hamiltonian for the transitional modes. The Heaviside step function S(EHTM)ensures that only classical states with energy less than or equal to E arecounted, and the Dirac delta function S(J-j) imposes a fixed value J on the total system angular momentum j in the integral. Beginning with eq 7 in standard Euler angles and of the conjugate momenta, the momenta (P~I,P~I,P*~,P~~,P~*,P*~) fragments and {P@,P& of the orbital rotation are first transformed to the corresponding principal-axis angular momentum components in units of h,' J5
ha sin 81sin 02 sin 8 c
v l x ~ l y ~ l r ~ z * ~ z y ~ 2 5(8) ,~x,~y)
where Jcis the Jacobian for the transformation. The sum of states is then cast in the form
The high-pressure-limiting thermal recombination rate coefficient is obtained by averaging eq 3 over the Maxwell-Boltzmann population distribution of reactants as (see, for example, ref 1Oc)
In eq 4, k~ is the Boltzmann constant, T is the temperature, Qr(Tj is the partition function for reactants (center-of-mass motion excluded), and g, is an electronic degeneracy factor which will be 1/4under the usual assumption that recombination of the methyl radicals occurs only on the singlet electronic surface.'& The assumptions involved in the model for the loose transition state have been elaborated in detail elsewhere (see, for example, refs 5-7, 10, and 15) and will not be considered at length here. The reaction coordinate used in this study is the separation R between the centers of mass of the two fragments. As has been shown by Klippenstein recently,6J6 an alternative reaction coordinate such as the separation between the bonding atoms can provide a better approximation to the rate coefficient for reactions involving fragments wherein the bonding atoms are not close to the centers of mass. This is not expected to be a problem in the methyl radical recombination reaction, though, since the centers of mass of the fragments lie so close to the bonding carbon atoms. In the loose transition state, it is convenient to assume separability between the internal vibrations of the fragments and the transitional modes for the purposes of state counting, thus the sum of states may be written as a convolution of the transitionalmode sum of states WTM(E)and the vibrational density of states pvib(E),
-
P o 1 gP41*p*1 'P8Z~P~2~~*Z,P8,P+~
sin dl sin 8, sin 0 O,(E,q)
= (@,(E,q) (9) where N = 2856 is the normalizing factor for the angular integrals, the vector q represents the angular configuration, and @AE,q) is the classically available momentum-space volume? O,(E,q) = l / . ~ ~ J . . . J d j , dj, dl S(E*-H,,)
S(J-j)
In eq 9, the sum of states has been explicitly written as a normalized average of @XE,q)over the configurationspace. The convenience of this form is that any cyclic angles will integrate out to unity and hence need not be sampled during the Monte Carlo integration. It is convenient to transform to a set of external angles {0,4,+) (which the potential does not depend on) and bodyfixed internal angles (Bl'r+l',4t,B2'r421),7,10b1c thus reducing the configuration-space integration to at most five dimensions,
WTM(E) =
&$J ...J
de', d q 1 d4t de', d q , @,(E,q) (11)
where the normalization factor N'= Fa3. In eq 11, 4t is the torsional angle between the two fragments and (81',+1',021,+21} are body-fixed Euler angles for the fragments, the body-fixed z-axis being the line joining the centers of mass. Making useof the fact that the kinetic energy of overall rotation is instantaneously (i.e., at a given configuration) separable from the kinetic energy of internal motion,21J8 the expression for the momentum space in eq 10 is reduced to18
where E , & ) is the minimum energy required to generate the angular momentum J and In eq 6, V,in(R) is the potential at the minimum-energy configuration for the center-of-mass separation R. WAE) may alternatively be obtained by convoluting the transitional-mode density of states with the vibrational sum of states.1° The transitional-mode sum of states is evaluated using the classical approximation as the classically accessible volume in phase space (in units of Planck's constant) for the given values
(10)
B(q)J(J and
+ 1) IE* - V(q) I A(q)J(J + 1)
Smith
1036 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE I: k L Calculated by pVTST(4J) temp/K 300 500
A(q)J(J
a
+ 1) 5 E* - V(q)
k,", "/ 1O-II cm3 s-' ref 1Oc 7.0 7.0 5.9 6.1
+ l)]'[E*
1
- V(q) - A(q)J(J + 1)](@-1)/2)-v
+
ref 1Oc
loo0 2000
4.0 1.8
3.9 1.8
&&a / 10-11 cm3 s-I
RmiJA
temp/K
this work
ref 1Oc
this work
ref 1Oc
300 500 1000 2000
4.2 3.7 3.4 2.9
4.2 3.8 3.4 3.0
8.2 6.9 4.7 2.3
8.4 7.3 4.6 2.2
a
Statistical uncertainty in both sets of numbers is &3%.
TABLE IIk Sums of States Calculated by Microcanonical Variation. ~~
E/kcal mol-' 0.13 0.44 1.18 2.36 4.13
9.53
19.55
B(q)J(J
this work
Calculated by CVTST
In obtaining eqs 12 and 13, the assumption has been made that,
+
temp/K
Statistical errors associated with both sets of numbers are &3%.
TABLE II: :k
at any given configuration q of the loose transition state, the overall body may be approximated as a prolate top. Thus, A(q) is the unique (largest) rotational constant of the system at configuration q, and B(q) is the geometrical average of the two small rotational constants. This approximation can be avoided without difficulty by the evaluation of a one-dimensional integration,'* but since the error incurred is generally smaller than the statistical error of our Monte Carlo integration (Le., less than 2%), it was considered acceptable. In eqs 12 and 13, r(x) is the gamma function, s is the total number of rotational degrees of freedom of the fragments (e.g., s = 4 for the linear + linear case; s = 6 for the nonlinear nonlinear case), a n d p = 2 if one of the fragments is monatomic, otherwise p = 4. BO is the rotational constant for relativeorbital motion of the two fragments at the center-of-mass separation R. The (hi}are the rotational constants of the fragments. Note that for symmetric fragments, the degenerate rotational constants must be repeated the appropriate number of times in the product (e.g., this gives Bl if a fragment is linear, or B3/*if it is a spherical top). V(q) is the transitional-mode potential for the given angular configuration q. In eq 13, G(E,J,q) has the following forms for even and odd values of s, respectively:
k L a / 1O-II cm3 s-l
this work
39.10
J/h 0 0 25 0 25 50 0 25 50 0 25 50 100 0 25 50 100 0 25 50 100 150 0 25 50 100 150
W(E,J)b 1.1(0) 1.1(1) 1.6(3) 1.1(2) 4.4(4) 1.0(4) 6.3(2) 3.9(5) 2.5(5) 4.6(3) 4.1(6) 3.9(6) 1.3(5) 6.3(4) 7.8(7) 9.3(7) 1.7(7) 2.4(6) 3.7(9) 4.9(9) 2.0(9) 1.1(8) 2.4(8) 4.4( 11) 6.0(11) 4.2( 11) 6.8( 10)
W(E,J)c W(E,J)d k(E,J)/s-I 0.5(0) e 8.5(3) e 0.9(1) 8.4(4) 1.4(3) 1.4(3) 6.2(3) e 7.7(5) 1.1(2) 1.5(5) 4.4(4) 434) 2.0(4) 1.1(4) 1.1(4) 3.7(6) 6.4(2) e 4.3(5) l.l(6) 4.4(5) 4.3(5) 2.8(5) 2.7(5) 2.0(7) e 4.3(3) 8.7(6) 4.2(6) 4.3(6) 4.9(6) 4.1(6) 4.2(6) 1.4(5) 6.3(5) 1.5(5) 1.5(8) 5.8(4) e 9.1(7) 7.1(7) 7.5(7) 6.2(7) 8.8(7) 9.4(7) 1.7(7) 4.0(7) 1.7(7) 2.1(6) 1.8(9) e 3.4(9) 1.3(9) 3.1(9) 4.9(9) 9.4(8) 5.0(9) 1.8(9) 1.2(9) 1.9(9) 1.O(8 ) 1.5(9) 1.1(8) 1.9(8) 2.3(10) e 2.0(10) 3.6(11) 3.7( 11) 6.7( 11) 1.3(10) 6.9(11) 2.1 (10) 4.1(11) 4.2( 11) 6.7(10) 4.9( 10) 6.4( 10)
Numbers in parentheses are the exponents. This work. Reference 1Oc. Reference 15c. Reference 15c does not present numbers for J = 0.
111. Application to Methyl Radical Recombination and Discussion
and
B(q)J(J + l)]'[E* - V(q) - A(q)J(J
I
+ 1)]((8-1)/2)-"
f o r s o d d (15) The notation (x;n;u) in eqs 14 and 15 has the following meaning:
Note that (x;n;O) is defined to be unity.
For application to the methyl radical recombination reaction, we have chosen the same parameters and the same model interaction potential as those of Wardlaw and Marcus.lo Table I shows the high-pressure thermal recombination rate coefficient calculated by microcanonical variational transition state theory with energy and angular momentum resolution [pVTST(E,J)] for temperatures in the range 300-2000 K, together with those of Wardlaw and Marcus. Table 11 gives a similar comparison for the high-pressure rate coefficients calculated by canonical variational transition state theory (CVTST), wherein only a single transition state is utilized but its position is optimized at a given temperature to give the minimum rate coefficient. At both levels of the theory, the agreement with the earlier calculations is excellent. In Table I11 the sums of states calculated by microcanonical variation are presented and compared with those calculated previously by Wardlaw and Marcus1& and by Klippenstein and Also included in Table I11 are the corresponding microcanonical rate coefficients for unimolecular
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7037
Rapid Algorithms for RRKM Theory i
0
40
80
120
160
200
&/kcal
Figure 1. Plot of log [ p ( E , J ) ]for ethane as a function of the nonfixed energy z = E - BJ(J + 1) at J = 0 (-), J = 10 (- -), J = 25 (- - -), and J = 50 (- - -).
dissociation of ethane. In calculating the microcanonical rate coefficients, we have utilized a convenient method of imposing J resolution on the density of states of the ethane molecule calculated via direct count as summarized in the Appendix (see also Figure 1). The present method is considerably faster than the previous exact implementations. For example, the calculation generating the converged thermal rate coefficient a t 300 Kin Table I required 35 min on an IBM RS 6000/320 machine. Convergence of the Monte Carlo integrations becomes harder as the anisotropy in the potential becomes greater in comparison with the available energy. Hence, for a fixed separation R and therefore a fixed anisotropy, convergence of the integrations a t low energies requires the greatest number of Monte Carlo points. Thenumber of points selected a t low energies varies from typically 700 at 5.5-8, separation to 25 000 a t 3.0-A separation. As a crude estimate of the average time required for the calculation of a single WJ(E,R) (i.e., the sum of states a t a given center-of-mass separation R ) , one may divide the running time by the total number of Monte Carlo integrations (this includes all of the additional processing, but the Monte Carlo integrals are the dominant factor in determining the computational time of the program). This yields an average time for computing WJ(E,R)of 2.4 s. Note that the time required to converge any given WJ(E,R)varies considerably, depending on the energy and the separation between the fragments. The average time required to calculate thevariationally optimized WJ(E)increases linearly with the number of separations used in the variational calculation. The number of separations required depends in turn on how great the range of energies and angular momenta for which one requires converged microcanonical rates is. For the calculation a t 300 K, we used 14 separations between 3.6 and 5.7 A, whereas for the calculation at 2000 K we used 17 separations between 2.4 and 5.1 8,. Convergence of W;M(E) in the range of thermally significant energies was achieved to within f 3 % at all separations. The results for WXE) of Table I11 were obtained using 19 evenly-spaced separations between 2.1 and 5.7
a selected set of energies by integration in phase space will typically involve a certain amount of rejection of randomly selected phasespace points which do not correspond to one of thedesired energies. Since the momentum-space volume is already evaluated as an analytic function, eqs 12 and 13, a sparse and irregular grid of energy points presents no such problem. For each separation R, the positions of the points in the energy grid are optimized in order to minimize the number of energy points while still satisfying a predefined accuracy limit (typically 2%) for interpolation of the sums of states to intermediate energies. In practice, we carry out this optimization procedure for the energy grid before entering into the Monte Carlo integration routine by using analytic sums of states based on a simple model ani~otropy,~ since these are very fast to evaluate and have the same qualitative functional dependence on the energy as the final numerically evaluated sums ofstates. (4) Repeatedcalculationofthesumofstatesisnecessary only up to a certain value of J; thereafter, increasing the angular momentum simply shifts the effective potential but does not change the sum of states profile. To see this, note that eqs 12 and 13 display the low-energy and high-energy forms of the momentumspace volume. These correspond to the typical "high-J" (eq 12) and "low-J" (eq 13) regimes which were utilized previously within the context of phase space theory4922-24 (PST). The low-energy form of QJ(E,q),eq 12, can be written in terms of the available kinetic energy e = E* - B(q)J(J+ 1). The utility of this becomes apparent when one recognizes that the energy for switching between the two forms, E*,, = V(q) + A(q)J(J + l ) , rapidly increases with increasing J . Once J has become sufficiently large so that E*,, is greater than the maximum energy required for convergence of the thermal rate, only the low-energy form of the integrand in eq 11 is involved and hence WTM(E) can be written in terms of the same energy e, (17) Thus, at high J values the sum of states is a function only of the energy e and need not be recalculated for each new J value. In the case of the methyl radical recombination, for example, repeated calculation of the sum of states is unnecessary for J > 40 at 300 K (energy ceiling 5000 cm-l above threshold), for J > 50 at 500 K (energy ceiling 8000 cm-l above threshold), and for J > 70 a t 1000 K (energy ceiling 17 000 cm-l above threshold). It should be stressed that the interaction potential V(q) is necessary input data for the entire calculation, and reliable a b initio interaction potentials are of crucial importance to the reliability of rate coefficient predictions. Detailed calculations of the interaction potential for the methyl radical recombination have been carried However, for our purposes, the simple model interaction potential chosen by Wardlaw and Marcus in their initial calculations suffices to illustrate the performance of the new algorithm.
A.
IV. Conclusion
There are four major computational advantages brought about by the use of eqs 12 and 13: (1) Numerical integration over the momentum space is avoided. (2) The integration may be carried out in terms of body-fixed, internal angles of the loose transition state, thereby minimizing the number of coordinate transformations that must becarriedout within the Monte Carlo procedure in order to evaluate the potential a t each randomly selected configuration. (3) Thenumber of energies at which onecomputes the sums of states by Monte Carlo integration can be conveniently minimized in the following manner. Since the computing time is dominated by the Monte Carlo integration, the grid of energy points should be as sparse as possible, so long as sums of states a t intermediate energies can be evaluated suitably accurately with an interpolating routine. Evaluation of the sum of states for
The new algorithm for the implementation of microcanonical variational RRKM theory illustrated in this paper represents a substantial advance in the efficiency of such calculations. Modeling of the rates of unimolecular, recombination and chemical activation reactions involving one or more arrangement channels without a saddle point in the potential surface will as a result become achievable with significantly less investment of time and computational resources. Acknowledgment. The author gratefully acknowledges the support of Professor C. Bradley Moore and funding of this work by the U.S. National Science Foundation (Grant No. CHE8816552).
Smith
1038 The Journal of Physical Chemistry, Vol. 97,No. 27, 1993
asymmetric top. Writing the nonfixed energy E - B*({)J(J 1) as e, pTlatC(E,{)is given by
Appendix. &Resolved Densities of States
In this appendix, we present a convenient means of imposing angular momentum resolution in the calculation of the density of states pXE) within the rigid-rotor/harmonic-oscillator approximation. The harmonic-oscillator states are convoluted exactly with the Beyerawinehart direct count. Anharmonic (but separable) oscillators are easily incorporated into the directcount procedure by use of the Stein-Rabinovitch method.26 The rotational state densities are evaluated by the classical approximation. We assume that Coriolis coupling facilitates exchange of vibrational and rotational energy on a time scale short compared to the lifetime of the excited unimolecular species, so that only E and J are resolved in the state count (Le,, all energeticallyaccessible K states subject to the constraint -K IJ IK are included). Within the framework of the above approximations, the density of states is written4
The convolution of eq 18 is carried out by entering the rotational density of states into the initial array for the Beyerawinehart direct count.9-27 Our taskis toevaluate theclassical approximation to the rotational density of states pyt(E). Consider the case where there are u internal free rotors having symmetry numbers ut and rotational constants Bt. The convoluted state density for the internal rotors is p z ( E ) . One then has
-'(E) (2J+ 1)
- A$"/2dffwdK
&{E - B*(r)J(J
+
pylate(e,S;) -
( 2 J + 1)
e I [A-B*(r)]JZ (24)
and
pyy e, 5)
-
( 2 J + 1)
where
+ 1) -
0
[ A - B * ( r ) l ~ 2 1(19)
In eq 19, u is the rotational symmetry number of the molecule, A is the largest rotational constant, and B*({) is given by
1) [ A- B*(53]1/2
~ * ( r=)ccos2 { + B sin' b
(20) where {is the projection of the angular momentum vector J onto the plane perpendicular to the A principal axis (see, for example, ref 18). The symmetry of B*({) is such that the {integration need only be evaluated from 0 to 7r/2. The internal rotational density of states is given by
sin-'
for u odd (26) 2 P
and
B*({)]J2)'u/2t'' and the maximum value K, principal axis is
of the projection of J onto the A
-- [ E - B * ( f ) J ( J+ l ) ] 'I2 [ A-
' ( E / A ) ' / 2IJ I [E/B*(5")]'/2(22)
Evaluation of the integral over K i n eq 19 yields
where p!m""'c(E,{) is the formula for the rotational density of states of a prolate top except that the B rotational constant, which is degenerate in the prolate top, is replaced by B*({) for the
I
for u even (27)
The notation (x;n;v) has the same meaning as in the text (eq 16). Limiting forms of eqs 23-27 above and 29-31 below have been presented previously.4JObJ8 The formulas of Troe4J* apply to prolate and oblate tops in the limits of low and high J, and the result of Wardlaw and Marcuslob applies to a prolate top and is valid for all J. Note that the equations above apply to molecules of arbitrary inertial symmetry and arevalid for all Jvalues. Figure 1 shows profiles of the density of states calculated for ethane at different J values using eqs 18-26. The frequencies for ethane are as in ref lob, except that we have treated the torsional mode as a free internal rotor. Since ethane is a prolate top, the average over f i n eq 23 is redundant and the formulae of eqs 24-27 apply directly with B*(f) = B. For completeness, the relevant formulae for the rotational density of states of an oblate top, wherein the largest rotational constant is not unique and so eq 19 does not hold, are also presented. The derivation proceeds analogously to that above. In this case, since A = B 1 C, the nonfixed energy e is given by e
= E - [BJ(J
+ 1) + ( C - B) JZ]
One obtains for u odd, at all energies c 1 0,
(28)
Rapid Algorithms for RRKM Theory
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7039
References and Notes
+ + 1) ( B - c)'/'
2t1/* (23
In[
21t - ( B - C)Jz11/2
For u even,
t
1 ( B - c)J'
(31)
(1) Troe, J. J. Chem. Phys. 1977,66,4745,4758;Z . Phys. Chem. 1987, 154,73. (2) (a) Smith, S.C.; Gilbert, R. G. Int. J . Chem. K i m 1988.20, 307, 979. (b) Smith, S.C.; McEwan, M. J.; Gilbert, R. G. J . Chem. Phys. 1989, 90,1630,4265. (3) (a) Keck, J. C. J . Chem. Phys. 1%7,46,4211. (b) Borkevic, M.; J. Phys. Chem. 1985,89,3994. Berne, B. J. J. Chem. Phys. 1986,84,4327; (c) Robertson, S. H.; Shushin, A. I.; Wardlaw, D. M. J. Chem. Phys., in press. (4) Troe, J. J. Chem. Phys. 1983, 79,6017. (5) Wardlaw, D.M.; Marcus, R. A. Adv. Chem. Phys. 1988,70,231. (6) Klippenstein, S. J. J. Chem. Phys. 1991, 94,6469. (7) Smith, S. C. J . Chem. Phys. 1991,95,3404. (8) Hase, W. L. Acc. Chem. Res. 1983,16,258. (9) Gilbert, R. G.; Smith, S.C. Theory of Unimolecular and Recombination Reactions; Blackwell Scientific Publications: Cambridge, MA, 1990. (10) (a) Wardlaw, D.M.; Marcus, R. A. Chem. Phys. Lett. 1984, 110, 230;(b) J. Chem. Phys. 1985,83,3462;(c) J. Phys. Chem. 1986,90,5383. (1 1) Miller, W. H.J . Chem. Phys. 1974,61,1823. (12) Clary, D.C. Annu. Rev. Phys. Chem. 1990,41,61. (13) Truhlar, D.G.; Hase, W. L.; Hynes, C. J . Phys. Chem. 1983,87, 2264. (14) Aubanel, E. E.;Wardlaw, D. M. J. Phys. Chem. 1989, 93,3117. Aubanel, E. E.;Wardlaw, D. M.; Zhu, L.; Hase, W. L. h r . Rev. Phys. Chem. 1991,10,249.Aubanel, E.E.;Robertson, S. H.; Wardlaw, D. M. J . Chem. Soc., Faraday Trans. 1991,87,2291. (15) (a) Klippenstein, S.J.; Marcus, R. A. J. Chem. Phys. 1988,91,2280; (b) 1990,93,2418;(c) J. Phys. Chem. 1988,92,3105.(d) Klippcnstein, S. J.; Khunkkar, L. R.; Zewail, A. H.; Marcus, R. A. J . Chem. Phys. 1988,89, 4761. (16) Klippenstein, S.J. Chem. Phys. Lett. 1990,170,71;J. Chem. Phys. 1992, 96, 367. Yu, J.; Klippenstein, S. J. J. Phys. Chem. 1991, 95, 9882. (17) Song, K.;Chesnavich, W. J. J . Chem. Phys. 1990,93,5751. (18) Smith, S.C.J. Chem. Phys. 1992,97,2406. (19) Gilbert, R. G.; Jordan, M. J. To;Smith, S.C. UNIMOL program suite, available from R. G. Gilbert, School of Chemistry, Sydney University, N. S. W. 2006,Australia, (20) Wagner, A. F.; Wardlaw, D. M. J . Phys. Chem. 1988,92,2462. (21) Jellinek, J.; Li, D. H. Phys. Rev. Lett. 1989,62,241;Chem. Phys. Lett. 1990, 169,380. Li, D.H.; Jellinek, J. Z . Phys. D 1989, 12, 177. (22) Klots, C. E.J. Phys. Chem. 1971,75,1526;2.Naiurforsch. Teil A , 1971,27,553. (23) Chesnavich, W.J.; Bowers, M. T.J. Chem. Phys. 1977,66,2306. (24) The "low-J" F'STexpressions correspond to eq 12 only as J - . 0, and they do not apply to the full range of fragment inertial symmetries. Furthermore, and most importantly, the F'ST expressions apply only to the case of a central potential. (25) Hirst, D.M.; Wardlaw, D.M. Private communication. (26) Stein, S.E.;Rabinovitch, B. S . J. Chem. Phys. 1973, 58, 2438. (27) Astholz, D.C.; Troe, J.; Wieters, W. J . Chem. Phys. 1979,70,5107. (28) Troe,J. J . Phys. Chem. 1979,83, 114.
