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The calculation and interpretation of average collisional energy transfer parameters
Volume 152, number 4,5
CHEMICAL
THE CALCULATION
PHYSICS LETTERS
IS November
1988
AND INTERPRETATION
OF AVERAGE COLLISIONAL
ENERGY TRANSFER
PARAMETERS
Andrew R. WHYTE, Kieran F. LIM ‘, Robert G. GILBERT Department of Theoretical Chemistry, University of Sydney, Sydney, NSW 2006, Australia
and William L. HASE Department of Chemistry, Wayne State University, Detroit, MI 48202. USA Received
10 August 1988
It is pointed out that the proper comparison to make between theoretical and experimental results on collisional energy transfer in highly cxcitcd molecules involves the rate of energy transfer (or its moments) rather than the average energy transfer per colhsion. This rate can be computed from trajectory studies without the need to define a “collision” or to omit arbitrary defined elastic trajectories. Illustrative calculations for CH,CH,CI/Ne collisions are presented,
1. Introduction
In this Letter we consider the correlation of theoretical and experimental results in collision-induced energy transfer for highly excited polyatomic molecules in low-density bath gases. The following point, although straightforward, does not appear to have been generally appreciated. Experimental results for such systems are usually reported as the average energy transferred per collision, (A@, or any other equivalent quantity such as (A.!?). To compute such a quantity apparently requires one to define a collision. However, it is well known that (except for hard spheres) no such definition is possible. The dynamical observables of a system can be obtained directly, e.g. from trajectory calculations [ 13, as an appropriate ensemble average over a sufficiently large number and range of initial conditions (impact parameter, relative translational energy, etc.). A number of recent trajectory studies of collisional energy transfer processes [ 2-5 ] have nonetheless relied upon ad hoc definitions of a collision in order to eliminate the contributions to energy transfer parameters from elastic or near-elastic collisions which predominate in the limit of large impact parameters (classically, no trajectory is exactly elastic in collisions in a system with internal degrees of freedom). However, the establishment of criteria for excluding these elastic events is problematic [ 5] and, since it cannot be rigorously defined, is likely to lead to significant errors in the derived energy transfer parameters. Here we note that the observed quantity in such energy transfer experiments is in fact the average rate of energy transfer or some other equivalent quantity such as the root-mean-square (rms) rate of energy transfer. We present an explicit definition of the energy transfer rate distribution and its moments, which arc the quantities appropriate for proper comparisons among the results of different experimental and theoretical studies in this area. The derivations of the equations given here w,ill be presented in a future paper [ 61, Our approach is illustrated in a comparison of calculated and experimental results for the chloroethane/neon system; this example also shows that our method provides a reliable and economical means of determining the maximum ’ Present address: Department
0 009-2614/88/$ ( North-Holland
ofchemistry,
Stanford
University,
03.50 0 Elsevier Science Publishers Physics Publishing Division )
Stanford,
B.V.
CA 94305, USA.
377
Volume 152, number 43
CHEMICAL
PHYSICS
I8 November
LETTERS
198 8
impact parameter for use in a trajectory study of this type. We note that various comments relating to these issues have been made previously [2-5,7-121, and indeed have been properly taken into account by some workers [ 81, Our intention hcrc is to clarify some apparent areas of confusion as well as to propose some new methods.
2. Theory Consider a system consisting of an excited polyatomic molecule A which may undergo unimolecular reaction as a result of collisions with a bath gas M. The competing processes of collisional activation/deactivation and reaction may be described by the integral master equation:
Q(E) -=Wlj: dt
(1)
[R(E,E')g(E')-R(E',E)g(E)ldE'-k(E)g(E),
0
where g(E) is the population of A molecules at an internal energy level E,R(E,E') is the rate coefficient (which has the dimensions of (energy x concentration x time ) - ’ ) for the transfer by collisions of A molecules from initial energy E' to final energy E, [M] is the concentration of bath gas, and /c(E) is the microscopic rate coefficient for reaction at energy E.For notational simplicity in this paper, we consider only the master equation in terms of E, although extensions to include the total angular momentum are straightforward. Eq. ( 1) describes a range of different types of experiment on collisional energy transfer, including both “direct” [9-l 1 ] (based on observations related to the time evolution of the average energy) and “indirect” measurements [ 13,141 (based on observations of the reaction rate)_ It is often convenient for visualization purposes to rewrite the rate coefficient R (E,E')in terms of an energy transfer probability function P( E,E')together with some reference collision frequency w, or collision number Z, viz.
R(E,E')=w(E)P(E, E’)/[M]=Z(E) If ~1)(or 2) is independent
d&E)
dt=W
P(E,E').
(2)
of energy this leads to the more familiar form of the master equation:
m s[f’(E, E’) g(E’ ) -P(E’, E) g(E) 1do’ -40
g(E) .
(3)
0
The requirement
that the probability
Z=co,,M]=jR(E,Er)dE. 0
distribution
P(E,
E')be normalized
leads to
(4)
Note that eq. (2) defines a total (inelastic) collision rate. Given R(E,E') (or P(E,E')and o) and the microscopic rate coefficients k(E) (obtainable from e.g. RRKM theory [ 15,16]), then solution of eqs. ( 1) or (3) will allow the full dynamical behaviour of the A-M system to be calculated. In fact, a knowledge of the detailed form of R(E,E') (or P(E,E')) itself is not usually necessary - it is by now well-established that, to an excellent approximation, the falloff curve (“indirect” measurements) and the time dependence of the average internal energy (“direct” measurements), resulting from solution of eqs. ( 1) or (3), can be specified completely in terms of the first (or a higher) moment of the distribution in question, or of some related measure [ 171. In what follows, we present expressions for the rate distribution R(E,E')and its moments, based on the assumption that classical mechanics is applicable (i.e. that the system is executing chaotic motion and is in the quasicontinuous region of its internal state distribution), The principal point of this Letter is that each of these moments of R(E,E') is independent of the collision number to within a multiplicative factor which 378
Volume 152, number 4,s
CHEMICAL
PHYSICS LETTERS
I8
November I988
turns out to be just the corresponding moment of P(E, E’ ). Further, the first moment of R(E, E’ ) is measurable in “direct” experiments and is trivially calculable from the results of “indirect” ones; thus it is the quantity of choice for comparisons of the results from different experimental studies of a given system. Similarly, the second moment of R(E, E’ ) can be obtained (indirectly) from experimental studies, and may also be calculated very conveniently from a small number of classical trajectory simulations, rendering it suitable for both theory-experiment and theory-theory comparisons. This will be seen to obviate the problems of defining a collision and elimination of “anomalous elastic” trajectories. Elementary considerations from classical mechanics lead to the following expression for the rate coefficient R(E, E’ ) for transfer of a molecule A from an initial state at energy E’ to a final state at E, if A is in translational equilibrium with a bath gas M at temperature T:
J’j
R(E, E’ )= (8,xp)‘/2(kgT)3/2
E,exp(
-E,jksT)
B(E, E’; E,, b) 2xb dbdE,
0 n 7E,.exp(-E,ik~T)B(E,E’;E,,b)~dbdEr. (k H:, Here Z(d) = (8nkgT/p)“2d2 duced mass, and B(E, E’; E,, initial impact parameter b and ( 5) may be identified as P(E,
is a gas kinetic collision number, k, is Boltmann’s constant, p is the A-M reb) is the probability that A will undergo a transition from E’ to E for a given initial relative translational energy E,. From eq. (2) the double integral in eq. E’ )_ We define the nth moment of R( E, E’ ) as
ixI
RI;,,,=
s 0
(E-E’)“R(E,
=)EW)
j [
E’) dE
1VW”&B
(6a)
E, exp( -E,/k,
T) B(E, E’; E,., b) 3
db dE:, dE
(6b)
0 0 0
where ( AEn> is the nth moment of P(E, E’ ) with AE=E-E’. In evaluating Rr.;,, from classical trajectories, there is no need to exclude “elastic” trajectories [ 2-4 ] (which contribute with measure zero to the integrand, as E+E’ ), and there is no need to define a collision, since the integral is evaluated in the limit as d+m. Classical trajectory simulations amount simply to a means of evaluating (by a Monte Carlo technique) the conditional energy transfer probability B(E, E’; E,, b) [61. In a calculation of this type, the triple integral in eq. (6b) is replaced by a sum over N trajectories of (pi)“, where AE,=E-E’ for the ith trajectory, and it is assumed that there exists some value of b, b,,, say, which is sufficiently large that all inelastic events will be included in this sum, the convergence of which will be guaranteed provided that N is also sufficiently large. In previous studies [ 25-7 ] b,,, has been found from trial calculations as the minimum value for which some criterion for ( AE) is fulfilled. Alternatively it has been suggested [ 121 that choosing b&, = u?LJ52(2.2)*(where qLLJis the Lennard-Jones collision radius and s;Z(2,2)*is a reduced collision integral which depends on the Lennard-Jones well depth t,_, ) is sufficiently accurate for most purposes. We note here that the second moment of the rate of energy transfer R k.,2, - or (AE2), is a more convenient parameter than RI:-., or (AE) on which to carry out the computation, due to the smoother and more rapid convergence of the former quantities with N [ 181. Details of the algorithms for these calculations will be given later [ 6 1; for the present the utility of our approach is clearly demonstrated in the following example. Table 1 shows the results of a set of 120 trajectory simulations of collisions of chloroethane with neon bath gas at 975 K. These calculations were performed using a modified version of the MERCURY program [ 191. 379
18November 1988
CHEMICAL PHYSICS LETTERS
Volume 152, number 4,5
Table 1 Convergence of the second moment of R(E, E’) for a set of 120 chloroethane-neon trajectories (initial energy=activation energy for decomposition [Zl], T=975 K, ,u= 15.37 amu b,,,= 6.0 A). N is the number of trajectories having b in the indicated range b(A)
N
b (A)
&. L 104Z(bm,,)
N
0 4 2 7 9 8
0 0.012 0.228 0.511 1.923 2.164
(Am
(cm”s-‘)(cm-‘)’
(cm3s-‘)(cm-‘)2 o-o.5 0.5-1.0 1.0-1s 1.5-2.0 2.0-2.5 2.5-3.0
R A.2 1O”Z(L,,)
cm2>
3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0 5.0-5.5 5.5-6.0
6 9 13 18 20 24
2.183 2.446 2.464 2.466 2.467 2.461
The trajectories have been sorted according to their initial impact parameters into “bins” of width 0.5 8. The tabulated second moments of R (E, E’ ) represent cumulative averages of the (A& ) 2 over these bins, the contribution from the ith bin being weighted according to the associated incremental impact area [ 61. It can be seen that the second moment converges rapidly with increasing b, even for such a small number of trajectories; this convergence is much more rapid than that found for the first moment, where individual (hEi) values have many sign changes. The second moment converges for values of 0 greater than about 4 A; this is close to the Lennard-Jones radius g= 3.85 A.
3. Discussion and conclusion It is a relatively trivial matter to calculate average energy transfer parameters directly from the results of classical trajectory simulations via eq. (6b). However, we wish to stress that the significant quantity in a comparison of the results of two or more such studies is the product RE,, = Z( b,,, ) (Al?“) rather than (AI?‘) in isolation. Experimental results are reported as a value of RE,,, (for n = 1 or 2) divided by an “experimental” collision frequency w (or equivalently a collision number ZeXP) computed from e.g. the Lennard-Jones form [ 121. The relation between different measures of the energy transferred per collision ( ( AE) , ( A&_), (Al?> ‘I*) requires an assumed form of the distribution R (E, E’ ) or P(E, E’ ), although the interconversion is not strongly sensitive to the chosen functional form as long as it is physically realistic. For this purpose, we employ the displaced Gaussian form suggested by the biased random walk model [20], which is based on physically reasonable requirements of microscopic reversibility, energy conservation and randomness of the internal energy over a series of collisions. We emphasize that the choice of a Lennard-Jones, or any other, prescription for Z,,, in interpreting experimental data is solely a matter of convenience: further, given that the Lennard-Jones parameters for any system are likely to be uncertain and that there exist several different methods of calculating the Lennard-Jones collision frequency [ 9,111, this prescription is itself a potential source of uncertainty. It is therefore vital that any report of experimental (dE) (or (A&,,,, }) values includes details of the method used to calculate Z. We now compare the calculated and experimental values of the energy transfer parameters for the chloroethane/Ne system. This can be done directly through the second moment of the energy transfer rate, RF,.,. Alternatively (and admittedly, more conveniently for purposes of visualization) one can “renormalize” to the value of ( AE’), through the factor Zexpor Z( b,,,). The results (“renormalized” using a hard-sphere collision frequency calculated with dz4.2 A) are (AI?*) ‘/2= 621 cm-’ (experimental [ 2 11) and 620 cm ’ (calculated). This agreement is so good as to be fortuitous, particularly since the present calculations yield the total energy transfer rather than energy transfer in the active degrees of freedom (vibration plus active external rotor) as observed in the experiment. Nevertheless, the results show that the potential parameters employed in the calculation are quite acceptable. It is to be noted that the present method, wherein care has been taken to 380
Volume 152, number 4,5
CHEMICAL
PHYSICS LETTERS
18 November
1988
ensure convergence in all quantities used in the trajectory calculations (bmax+cc and N-a), represents an accurate classical evaluation of the second moment of the energy transfer rate. When expressed in finite difference terms, the double integral in eq. (5) is equivalent to the histogram distributions used by other workers in previous trajectory calculations of collisional energy transfer parameters [ 2-5 1. In these studies, in which attempts have been made to lit either P(E, E’ ) or the differential energy transfer cross section with some analytical function, the contributions of elastic events to derived energy transfer parameters have been eliminated by excluding the central (A,!?,% 0) peak in the fitting procedure. However, this “exclusion” approach is unnecessary (and liable to introduce arbitrary errors) for the derivation of average energy transfer parameters for comparison with experimental results, since the classification of trajectories as “anomalous” or otherwise [ 21 and the values of the resultant parameters depend on the arbitrary form assumed for the fitting function [ 3 1. Further, this procedure also alters the effective collision number (albeit in a well-defined manner) and, more importantly, may depend critically on the choice of bin width - use of too large an energy grain may result in the elimination of significant information if the true distribution is sharply peaked about AE= 0. The use of eq. (6) obviates the need for any such ad hoc definition of a collision and the concomitant arbitrary rejection of “anomalous” events. This approach is also consistent with the result noted above that fitting of falloff data is insensitive to the functional form of P(E, E’ ). The technique discussed here, which consists of comparing the energy transfer rate (or equivalent measure) obviates the apparent problem of comparing theory and experiment.
Acknowledgement The support of the Australian Research Grants Award (KFL) are gratefully acknowledged.
Scheme, and of a Commonwealth
Postgraduate
Research
References [ 1 ] R.N. Porter and L.M. Raff, in: Dynamics of molecular collisions, Part B, ed. W.H. Miller (Plenum 121 N.J. Brown and J.A. Miller, J. Chem. Phys. 80 ( 1984) 5568. [ 3 ] H. Hippler, H.W. Schranz and J. Troe, J. Phys. Chem. 90 (I 986) 6 158. [4] H.W. Schranz and J. Tree, J. Phys. Chem. 90 (1986) 6168. [ 5 ] X. Hu and W.L. Hasc, J. Phys. Chem. 92 ( 1988 ), to be published. [ 6 ] A.R. Whyte, K.F. Lim and R.G. Gilbert, to be published. [7] A.J. StaceandN.J. Murrell, J. Chem. Phys. 68 (1978) 3028. [S] M. Bruehl and G.C. Schatz, J. Chem. Phys. 89 ( 1988) 770. [9] M.J. Rossi, J.R. Pladziewicz and J.R. Barker, J. Chem. Phys. 78 (1983) 6695. [lO]J.ShiandJ.R.Barker,J.Chem.Phys.88 (1988)6219. [ 111 H. Hippler, L. Lindemann and J. Troe, J. Chem. Phys. 83 [ 121 J. Troe, J. Phys. Chem. 83 (1979) 114.
Press, New York, 1976).
( 1985) 3906.
[ 131 D.C. Tardy andB.S. Rabinovitch,
Chem. Rev. 77 (1977) 369. [ 141 M. Quack and J. Troe, in: Specialist periodical reports, Vol. 2. Gas kinetrcs and energy transfer,
eds. P.G.
Ashmore and R.J.
Donovan (Chem. Sot.. London, 1977) p. 175. [ 151P.J. Robinson and K.A. Holbrook, Unimolecular reactions (Interscience, New York, 1972). [ 161 W. For%, Theory ofunimolecular reactions (Academic Press, New York, 1973). [ 171 D.C. Tardy and B.S. Rabinovitch, J. Chem. Phys. 48 (1968) 1282. [ 181 M.B. Grinchak, A.A. Levitsky, L.S. Polakand S.Y. Umanskii, Chem. Phys. 88 (1984) 365. [ 191 W.L. Hase, QCPE 3 ( 1983) 14. [20] K.F. Lim and R.G. Gilbert, J. Chem. Phys. 84 ( 1986) 6129; R.C. Gilbert, J. Chcm. Phys. SO (1984) 5501. [2l]K.D.King,T.T.NguyenandR.G.Gilbert,Chem. Phys.61 (1981) 223.
381
CHEMICAL
THE CALCULATION
PHYSICS LETTERS
IS November
1988
AND INTERPRETATION
OF AVERAGE COLLISIONAL
ENERGY TRANSFER
PARAMETERS
Andrew R. WHYTE, Kieran F. LIM ‘, Robert G. GILBERT Department of Theoretical Chemistry, University of Sydney, Sydney, NSW 2006, Australia
and William L. HASE Department of Chemistry, Wayne State University, Detroit, MI 48202. USA Received
10 August 1988
It is pointed out that the proper comparison to make between theoretical and experimental results on collisional energy transfer in highly cxcitcd molecules involves the rate of energy transfer (or its moments) rather than the average energy transfer per colhsion. This rate can be computed from trajectory studies without the need to define a “collision” or to omit arbitrary defined elastic trajectories. Illustrative calculations for CH,CH,CI/Ne collisions are presented,
1. Introduction
In this Letter we consider the correlation of theoretical and experimental results in collision-induced energy transfer for highly excited polyatomic molecules in low-density bath gases. The following point, although straightforward, does not appear to have been generally appreciated. Experimental results for such systems are usually reported as the average energy transferred per collision, (A@, or any other equivalent quantity such as (A.!?). To compute such a quantity apparently requires one to define a collision. However, it is well known that (except for hard spheres) no such definition is possible. The dynamical observables of a system can be obtained directly, e.g. from trajectory calculations [ 13, as an appropriate ensemble average over a sufficiently large number and range of initial conditions (impact parameter, relative translational energy, etc.). A number of recent trajectory studies of collisional energy transfer processes [ 2-5 ] have nonetheless relied upon ad hoc definitions of a collision in order to eliminate the contributions to energy transfer parameters from elastic or near-elastic collisions which predominate in the limit of large impact parameters (classically, no trajectory is exactly elastic in collisions in a system with internal degrees of freedom). However, the establishment of criteria for excluding these elastic events is problematic [ 5] and, since it cannot be rigorously defined, is likely to lead to significant errors in the derived energy transfer parameters. Here we note that the observed quantity in such energy transfer experiments is in fact the average rate of energy transfer or some other equivalent quantity such as the root-mean-square (rms) rate of energy transfer. We present an explicit definition of the energy transfer rate distribution and its moments, which arc the quantities appropriate for proper comparisons among the results of different experimental and theoretical studies in this area. The derivations of the equations given here w,ill be presented in a future paper [ 61, Our approach is illustrated in a comparison of calculated and experimental results for the chloroethane/neon system; this example also shows that our method provides a reliable and economical means of determining the maximum ’ Present address: Department
0 009-2614/88/$ ( North-Holland
ofchemistry,
Stanford
University,
03.50 0 Elsevier Science Publishers Physics Publishing Division )
Stanford,
B.V.
CA 94305, USA.
377
Volume 152, number 43
CHEMICAL
PHYSICS
I8 November
LETTERS
198 8
impact parameter for use in a trajectory study of this type. We note that various comments relating to these issues have been made previously [2-5,7-121, and indeed have been properly taken into account by some workers [ 81, Our intention hcrc is to clarify some apparent areas of confusion as well as to propose some new methods.
2. Theory Consider a system consisting of an excited polyatomic molecule A which may undergo unimolecular reaction as a result of collisions with a bath gas M. The competing processes of collisional activation/deactivation and reaction may be described by the integral master equation:
Q(E) -=Wlj: dt
(1)
[R(E,E')g(E')-R(E',E)g(E)ldE'-k(E)g(E),
0
where g(E) is the population of A molecules at an internal energy level E,R(E,E') is the rate coefficient (which has the dimensions of (energy x concentration x time ) - ’ ) for the transfer by collisions of A molecules from initial energy E' to final energy E, [M] is the concentration of bath gas, and /c(E) is the microscopic rate coefficient for reaction at energy E.For notational simplicity in this paper, we consider only the master equation in terms of E, although extensions to include the total angular momentum are straightforward. Eq. ( 1) describes a range of different types of experiment on collisional energy transfer, including both “direct” [9-l 1 ] (based on observations related to the time evolution of the average energy) and “indirect” measurements [ 13,141 (based on observations of the reaction rate)_ It is often convenient for visualization purposes to rewrite the rate coefficient R (E,E')in terms of an energy transfer probability function P( E,E')together with some reference collision frequency w, or collision number Z, viz.
R(E,E')=w(E)P(E, E’)/[M]=Z(E) If ~1)(or 2) is independent
d&E)
dt=W
P(E,E').
(2)
of energy this leads to the more familiar form of the master equation:
m s[f’(E, E’) g(E’ ) -P(E’, E) g(E) 1do’ -40
g(E) .
(3)
0
The requirement
that the probability
Z=co,,M]=jR(E,Er)dE. 0
distribution
P(E,
E')be normalized
leads to
(4)
Note that eq. (2) defines a total (inelastic) collision rate. Given R(E,E') (or P(E,E')and o) and the microscopic rate coefficients k(E) (obtainable from e.g. RRKM theory [ 15,16]), then solution of eqs. ( 1) or (3) will allow the full dynamical behaviour of the A-M system to be calculated. In fact, a knowledge of the detailed form of R(E,E') (or P(E,E')) itself is not usually necessary - it is by now well-established that, to an excellent approximation, the falloff curve (“indirect” measurements) and the time dependence of the average internal energy (“direct” measurements), resulting from solution of eqs. ( 1) or (3), can be specified completely in terms of the first (or a higher) moment of the distribution in question, or of some related measure [ 171. In what follows, we present expressions for the rate distribution R(E,E')and its moments, based on the assumption that classical mechanics is applicable (i.e. that the system is executing chaotic motion and is in the quasicontinuous region of its internal state distribution), The principal point of this Letter is that each of these moments of R(E,E') is independent of the collision number to within a multiplicative factor which 378
Volume 152, number 4,s
CHEMICAL
PHYSICS LETTERS
I8
November I988
turns out to be just the corresponding moment of P(E, E’ ). Further, the first moment of R(E, E’ ) is measurable in “direct” experiments and is trivially calculable from the results of “indirect” ones; thus it is the quantity of choice for comparisons of the results from different experimental studies of a given system. Similarly, the second moment of R(E, E’ ) can be obtained (indirectly) from experimental studies, and may also be calculated very conveniently from a small number of classical trajectory simulations, rendering it suitable for both theory-experiment and theory-theory comparisons. This will be seen to obviate the problems of defining a collision and elimination of “anomalous elastic” trajectories. Elementary considerations from classical mechanics lead to the following expression for the rate coefficient R(E, E’ ) for transfer of a molecule A from an initial state at energy E’ to a final state at E, if A is in translational equilibrium with a bath gas M at temperature T:
J’j
R(E, E’ )= (8,xp)‘/2(kgT)3/2
E,exp(
-E,jksT)
B(E, E’; E,, b) 2xb dbdE,
0 n 7E,.exp(-E,ik~T)B(E,E’;E,,b)~dbdEr. (k H:, Here Z(d) = (8nkgT/p)“2d2 duced mass, and B(E, E’; E,, initial impact parameter b and ( 5) may be identified as P(E,
is a gas kinetic collision number, k, is Boltmann’s constant, p is the A-M reb) is the probability that A will undergo a transition from E’ to E for a given initial relative translational energy E,. From eq. (2) the double integral in eq. E’ )_ We define the nth moment of R( E, E’ ) as
ixI
RI;,,,=
s 0
(E-E’)“R(E,
=)EW)
j [
E’) dE
1VW”&B
(6a)
E, exp( -E,/k,
T) B(E, E’; E,., b) 3
db dE:, dE
(6b)
0 0 0
where ( AEn> is the nth moment of P(E, E’ ) with AE=E-E’. In evaluating Rr.;,, from classical trajectories, there is no need to exclude “elastic” trajectories [ 2-4 ] (which contribute with measure zero to the integrand, as E+E’ ), and there is no need to define a collision, since the integral is evaluated in the limit as d+m. Classical trajectory simulations amount simply to a means of evaluating (by a Monte Carlo technique) the conditional energy transfer probability B(E, E’; E,, b) [61. In a calculation of this type, the triple integral in eq. (6b) is replaced by a sum over N trajectories of (pi)“, where AE,=E-E’ for the ith trajectory, and it is assumed that there exists some value of b, b,,, say, which is sufficiently large that all inelastic events will be included in this sum, the convergence of which will be guaranteed provided that N is also sufficiently large. In previous studies [ 25-7 ] b,,, has been found from trial calculations as the minimum value for which some criterion for ( AE) is fulfilled. Alternatively it has been suggested [ 121 that choosing b&, = u?LJ52(2.2)*(where qLLJis the Lennard-Jones collision radius and s;Z(2,2)*is a reduced collision integral which depends on the Lennard-Jones well depth t,_, ) is sufficiently accurate for most purposes. We note here that the second moment of the rate of energy transfer R k.,2, - or (AE2), is a more convenient parameter than RI:-., or (AE) on which to carry out the computation, due to the smoother and more rapid convergence of the former quantities with N [ 181. Details of the algorithms for these calculations will be given later [ 6 1; for the present the utility of our approach is clearly demonstrated in the following example. Table 1 shows the results of a set of 120 trajectory simulations of collisions of chloroethane with neon bath gas at 975 K. These calculations were performed using a modified version of the MERCURY program [ 191. 379
18November 1988
CHEMICAL PHYSICS LETTERS
Volume 152, number 4,5
Table 1 Convergence of the second moment of R(E, E’) for a set of 120 chloroethane-neon trajectories (initial energy=activation energy for decomposition [Zl], T=975 K, ,u= 15.37 amu b,,,= 6.0 A). N is the number of trajectories having b in the indicated range b(A)
N
b (A)
&. L 104Z(bm,,)
N
0 4 2 7 9 8
0 0.012 0.228 0.511 1.923 2.164
(Am
(cm”s-‘)(cm-‘)’
(cm3s-‘)(cm-‘)2 o-o.5 0.5-1.0 1.0-1s 1.5-2.0 2.0-2.5 2.5-3.0
R A.2 1O”Z(L,,)
cm2>
3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0 5.0-5.5 5.5-6.0
6 9 13 18 20 24
2.183 2.446 2.464 2.466 2.467 2.461
The trajectories have been sorted according to their initial impact parameters into “bins” of width 0.5 8. The tabulated second moments of R (E, E’ ) represent cumulative averages of the (A& ) 2 over these bins, the contribution from the ith bin being weighted according to the associated incremental impact area [ 61. It can be seen that the second moment converges rapidly with increasing b, even for such a small number of trajectories; this convergence is much more rapid than that found for the first moment, where individual (hEi) values have many sign changes. The second moment converges for values of 0 greater than about 4 A; this is close to the Lennard-Jones radius g= 3.85 A.
3. Discussion and conclusion It is a relatively trivial matter to calculate average energy transfer parameters directly from the results of classical trajectory simulations via eq. (6b). However, we wish to stress that the significant quantity in a comparison of the results of two or more such studies is the product RE,, = Z( b,,, ) (Al?“) rather than (AI?‘) in isolation. Experimental results are reported as a value of RE,,, (for n = 1 or 2) divided by an “experimental” collision frequency w (or equivalently a collision number ZeXP) computed from e.g. the Lennard-Jones form [ 121. The relation between different measures of the energy transferred per collision ( ( AE) , ( A&_), (Al?> ‘I*) requires an assumed form of the distribution R (E, E’ ) or P(E, E’ ), although the interconversion is not strongly sensitive to the chosen functional form as long as it is physically realistic. For this purpose, we employ the displaced Gaussian form suggested by the biased random walk model [20], which is based on physically reasonable requirements of microscopic reversibility, energy conservation and randomness of the internal energy over a series of collisions. We emphasize that the choice of a Lennard-Jones, or any other, prescription for Z,,, in interpreting experimental data is solely a matter of convenience: further, given that the Lennard-Jones parameters for any system are likely to be uncertain and that there exist several different methods of calculating the Lennard-Jones collision frequency [ 9,111, this prescription is itself a potential source of uncertainty. It is therefore vital that any report of experimental (dE) (or (A&,,,, }) values includes details of the method used to calculate Z. We now compare the calculated and experimental values of the energy transfer parameters for the chloroethane/Ne system. This can be done directly through the second moment of the energy transfer rate, RF,.,. Alternatively (and admittedly, more conveniently for purposes of visualization) one can “renormalize” to the value of ( AE’), through the factor Zexpor Z( b,,,). The results (“renormalized” using a hard-sphere collision frequency calculated with dz4.2 A) are (AI?*) ‘/2= 621 cm-’ (experimental [ 2 11) and 620 cm ’ (calculated). This agreement is so good as to be fortuitous, particularly since the present calculations yield the total energy transfer rather than energy transfer in the active degrees of freedom (vibration plus active external rotor) as observed in the experiment. Nevertheless, the results show that the potential parameters employed in the calculation are quite acceptable. It is to be noted that the present method, wherein care has been taken to 380
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PHYSICS LETTERS
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1988
ensure convergence in all quantities used in the trajectory calculations (bmax+cc and N-a), represents an accurate classical evaluation of the second moment of the energy transfer rate. When expressed in finite difference terms, the double integral in eq. (5) is equivalent to the histogram distributions used by other workers in previous trajectory calculations of collisional energy transfer parameters [ 2-5 1. In these studies, in which attempts have been made to lit either P(E, E’ ) or the differential energy transfer cross section with some analytical function, the contributions of elastic events to derived energy transfer parameters have been eliminated by excluding the central (A,!?,% 0) peak in the fitting procedure. However, this “exclusion” approach is unnecessary (and liable to introduce arbitrary errors) for the derivation of average energy transfer parameters for comparison with experimental results, since the classification of trajectories as “anomalous” or otherwise [ 21 and the values of the resultant parameters depend on the arbitrary form assumed for the fitting function [ 3 1. Further, this procedure also alters the effective collision number (albeit in a well-defined manner) and, more importantly, may depend critically on the choice of bin width - use of too large an energy grain may result in the elimination of significant information if the true distribution is sharply peaked about AE= 0. The use of eq. (6) obviates the need for any such ad hoc definition of a collision and the concomitant arbitrary rejection of “anomalous” events. This approach is also consistent with the result noted above that fitting of falloff data is insensitive to the functional form of P(E, E’ ). The technique discussed here, which consists of comparing the energy transfer rate (or equivalent measure) obviates the apparent problem of comparing theory and experiment.
Acknowledgement The support of the Australian Research Grants Award (KFL) are gratefully acknowledged.
Scheme, and of a Commonwealth
Postgraduate
Research
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