Modeling collisional energy transfer in highly excited molecules
Kieran F. Lim and Robert G. Gilbert
Citation: The Journal of Chemical Physics 92, 1819 (1990); doi: 10.1063/1.458064
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Modeling collisional energy transfer in highly·excited molecules
Kieran F. Lima) and Robert G. Gilbert
Department of Theoretical Chemistry. University ofSydney. Sydney. New South Wales. Australia 2006
(Received 3 October 1989; accepted 26 October 1989)
Data from classical trajectory simulations of the collision of a highly excited molecule with a
monatomic bath gas are used to test the validity of the precepts used in the biased-randomwal~ (BRW) ~?del. for collisional energy transfer. This model assumes that energy migration
dunng the ~olllslo~ IS pseudorandom except for the constraint of microscopic reversibility, and
leads to a Simple displaced Gaussian form for the energy-transfer probability distribution. The
~RW ~sumpti~ns are show~ to be of acceptable validity to exact classical trajectory
slm~l.atl~ns. A ~Imple ~alytlcal approximation to the mean-square energy transfer per
colllsl~n IS obtamed which reproduces the trajectory data to within an average of ± 20%, and
also gives acceptable accord with experimental data. The model shows that the magnitude of
the average energy transferred per collision is governed by the time taken to traverse the
overall interaction potential in and out from the appropriate collision diameter, by the internal
energy, and by the average force exerted at the classical turning point of individual reactantatom-bath-gas interactions.
I. INTRODUCTION
Collisional energy transfer between a highly excited species and a bath gas is the primary process in thermal unimolecular and recombination reactions, wherein molecules
gain or lose sufficient energy to form a product. There is a
considerable body of experimental data on this process l - 3 ;
however, it is difficult to use these results to understand the
actual mechanism involved. It is now possible4-JO to carry
out trajectory simulations of the process, and indeed to obtain quite favorable comparison with experiment. 6 ,s However, it is not immediately apparent how such computations
can be used to obtain phenomenological understanding of
the collision process. Moreover, such trajectory calculations
are very computationally demanding (even using very efficient methods 7 ), and cannot be carried out routinely by experimentalists wishing to fit, for example, falloff data. There
is a clear need for approximate models for the energy-transfer process which accurately reflect the physical processes
occurring and which require less computational resources to
evaluate.
Such a model would fulfill several functions:
( 1) It would provide a physically based functional form
for the energy-transfer probability distribution function for
use in the master equation, rather than empirical ones such
as the exponential-down I which are usually used at present.
Having the correct distribution function is important for
certain systems that are sensitive to the actual functional
form, rather than just a single average; one example of such
sensitivity is the pressure dependence of the rate coefficients
in multiple-channel reactions, but indeed a falloff curve for
even a single-channel process usually shows significant sensitivity to the functional form.
(2) It would enable one to predict energy-transfer rates
for use in fitting or predicting falloff curves.
( 3) It would give a better physical understanding of the
process.
a)
Present address: Department of Chemistry, Stanford University, Stanford, California 94305-5080.
J. Chern. Phys. 92 (3), 1 February 1990
( 4) It would enable one to gain information about the
potential function governing the process by fitting potential
parameters to experimental data much more readily than is
possible through cumbersome complete trajectory calculations.
(5) Perhaps the most important result would be that a
successful model would enable one to understand why energy-transfer quantities have the values that they do. For example, it is now well established that the root-mean-square
average energy transferred in collisions between a mona\omic bath gas and an excited reactant is typically 100-500
cm -I; while one can certainly predict this successfully from
trajectory calculations, there has not yet been a satisfactory
explanation of the order of magnitude of this quantity.
A number of approximate models have appeared in the
literature. One class of these involves statistical theories
which assume some redistribution of energy amongst all degrees of freedom of the "supermolecule" or collision complex that is formed when two colliders come together. In
general, most such theories assume that energy redistribution is rapid compared to the lifetime of the collision complex. II A variant ofthis has redistribution only in the kinetic
energy of the complex: the "impulsive ergodic collision theory" (IECT).12 Near room temperatures, such models appear to give an acceptable description of the energy-transfer
process if the bath gas is a large polyatomic. However, for
monatomic and diatomic bath gases, they give results for the
average energy transferred per collision ( (Il.E ) ) that are far
too large when compared with experiment. Moreover, they
are unable to predict the effects of different monatomic bath
gases or isotopic substitution of reactant, both of which are
observed experimentally to give significant changes in (1l.E )
compared with zero or very small effects predicted by these
ergodic theories.
An alternative approach is that of the "biased-randomwalk" (BRW) model. 13,14 This assumes that, during the collision, the dynamics of energy transfer are dominated by
pseudodiffusive energy exchange between the degrees of
freedom of the reactant and those of the bath gas, Prelimi-
0021·9606/90/031819-12$03.00
© 1990 American Institute of Physics
1819
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K. F. Lim and R. G. Gilbert: Energy tran$fer in highly excited molecules
1820
nary trajectory studies 14 seemed to validate the. assumptions
involved in the model, and to indicate that the model gave
acceptable agreement with experimental data for nonpolyatomic bath gases; this overcomes a major failing of other
treatments. The basic assumption of the biased-randomwalk model is that, during a collision, energy exchange between the internal degrees of freedom of the excited molecule and the bath gas is essentially random but for certain
constraints. This implies that the time evolution of the internal energy distribution of an ensemble would be governed by
a Smoluchowski equation. The distribution function for the
probability of energy transfer by collision will then be the
value of the internal energy distribution at the end of a collision, subject to the constraint that the energy distribution
before the collision is a l) function centered on a given initial
energy. The constraints on the randomness of the energy
migration during the collision are microscopic reversibility
and total-energy conservation; these give rise, respectively,
to a "force" term and a boundary condition in the Smoluchowski equation.
The objective of this paper is to investigate this model
further, in the light of recent advances. These advances are of
two kinds: (1) the availability of extensive, and essentially
exact, trajectory data,s.9 and (2) more thorough investigation of model experimental systems 15.16 (which among other
things provide a validation of the classical trajectory methodology employed for these systems). These advances together enable us to put the model on a firmer theoretical
basis. We are also able to derive readily applied expressions
for average energy-transfer quantities that have been properly tested against "exact" trajectory results with the same
assumed interaction potential (completely replacing our
earlier semiempirical BRW formula 13).
Testing a model against trajectory data, as we do here, is
preferable to testing against experiment. This is because simulation of experiment requires both the potential function
and the dynamics; if a model disagrees with experiment, this
may then be due to inadequacies of either one's treatment of
the dynamics, or of an inaccurate potential, or due to failings
in both, and one cannot discriminate between these different
possibilities. Testing against trajectory data, however,
means that any shortcomings must be ascribed to lack of
validity of the treatment of the dynamics.
II. DERIVATION OF BRW MODEL FROM DYNAMICS OF
COLLISIONAL ENERGY TRANSFER
We first summarize earlier work in this field, in Eqs.
(1 ) - (15). The effect of collisional energy transfer on the
rate coefficient of a unimolecular reaction, k uni , is calculated
using the master equation:
- kunig(E)
= [M] J[R(E,E')g(E ' ) -R(EI,E)g(E)]dE'
- k(E)g(E),
(1)
where geE) is the population of reactant molecules with energy E, [M] is the concentration of bath gas, R (E,E ') is the
distribution function for the rate coefficient of collisional
energy transfer from E to E, and k(E) the microscopic reaction rate coefficient at energy E. Extensions to Eq. (1) to
incorporate angular momentum conservation are straightforward, and will not be given here.
The distribution function R (E,E ') is the concern of the
present paper. Numerical methods have been developed for
the exact calculation of R (E,E ') from trajectories; these
methods also yield the lower moments of R(E,E'), which
are all that are required for the calculation of falloff curves of
single-channel reactions. 7 •s Our present concern is to develop approximate analytical solutions to the dynamical equations for these quantities.
It is conventional, but not esential, to factorize R (E,E ')
into a total collision number for energy transfer, Z (and
hence a total-energy transfer collision frequency
tV = [M]Z) and a probability of energy transfer per collision, P(E,E '):
1
R(E,E') = ZP(E,E ' ) = tVP(E,E')/[M].
(2)
However, it is essential to note that it is not necessary at this
point to attempt to define a "collision," and indeed a rigorous definition of a collision is impossible (an extensive discussion of this in the context of collisional energy transfer
has been given elsewhere7 ). The energy-dependent quantities Z and tV are defined by
Z(E') =
J
R(E,E')dE,
(3)
which involves the normalization condition:
J
P(E,E')dE= 1.
(4)
An important requirement for P(E,E ') or R (E,E ') is
that of microscopic reversibility, viz.,
P(E,EI)f(E ' ) = P(E',E)f(E),
(5)
wheref(E) =p(E)exp( -E/kBn,p(E) being the density of states of the polyatomic.
The first approximations we invoke are the common
ones that the total-energy transfer collision number Z(E ') is
independent of E I, and that Z can be obtained by an appropriate expression such as through the use of collision integral.
The distribution function P(E,E ') is that for the value of
the internal energy of the reactant at infinitely long times
(t..... + 00), given that the energy distribution for t..... - 00 is
a l) function centered on E I. The total Hamiltonian for a
collision between a polyatomic reactant and a collider (bath
gas) can be written in the center-of-mass system as
(6)
where Hi and He are the Hamiltonians of the isolated reactant and collider, respectively, and Hpert is that for the relative degree offreedom (i.e., the kinetic and interaction potential terms for the relative motion). The internal energy of
the reactant, Ei (t), is classically the value of Hi (t) at any
time t. One can from solutions of Hamilton's equations find
the distribution function of Hi (t..... + 00), for a collision
where Hj(t ..... - 00) is E' and the initial relative (translational) energy is E, = Hpert (t ..... - 00). For notationalsimplicity, we assume henceforth that the collider has no inter-
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K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
nal structure, although the generalization of the method is
straightforward. P(E,E') will then be proportional to the
distribution of H; (t --+ + 00), averaged over appropriate distributions ofE, and the impact parameter b, where the range
of b values are confined to those defining a collision in a way
we discuss later.
Solutions of Hamilton's equations show that E; (t) undergoes many rapid, apparently random, changes during a
collision. 13.14,17 This behavior is illustrated in Fig. 1. It arises
because the highly excited reactant has many vibrational
modes, each with randomly distributed phases; although the
collision is too short (a fraction of a picosecond) for randomization to occur (unless the bath gas is a large polyatomic), the collider undergoes many encounters with some or
all of the randomly vibrating atoms of the polyatomic. Each
such encounter causes E; (t) to change phase.
Consider the distribution function B(E;,E',E"b,t) for
the internal energy E; at time t during a collision, for a trajectory with initial energy E', impact parameter b, and translational energy E,. If the evolution of E;(t) were completely
random, B(E,E',E"b,t) would be given by the solution ofa
Smoluchowski equation for diffusion under some external
constraint:
aB
a(zB + aB / aE;)
-=D
,
at
e
aEj
(7)
where De is an "energy diffusion coefficient" and z is a quantity arising from microscopic reversibility. One can then
identify P(E,E') as the value of B(E,E',E"b,t) at some appropriately defined "collision time" te , and ensemble averaged over band E,. Ifit is assumed that De and z are independent of E', E" and b, and can be obtained by an ensemble
average, and that B approaches zero as E; -> ± 00, Eq. (7)
can be simply solved 13 to yield
P(E,E') = (W)-1/2 exp[ - (d + aE)2/4,r], (8)
where AE = E - E', and the value of z is obtained from Eq,
(5):
z=
alnf(E)
aE
,r = Dete'
(10)
The functional form for P(E,E') given by Eq. (8) is a simple
displaced Gaussian; it invokes only the assumption of quasirandom oscillations during collisions and makes use of the
constraint of microscopic reversibility. It will be shown later
that the quasirandom assumption is indeed an excellent approximation for typical systems.
The additional constraint of conservation of energy
comes in by imposing different boundary conditions on the
solution of Eq. (7): B must then have zero flux at the total
energy of the system, rather than vanishing as E; -+ 00. The
resulting solution has been given elsewhere l4 ; it gives a curve
for P(E,E') similar in appearance to that found from Eq.
(8), although now the functional form is much more complex. Quantities such as the average internal energy transferred per collision, (AE), calculated from the two expressions are quite close for given values of the parameter s.
Hence, frequently the refinement of energy conservation
makes little change in the model, and so will not be considered further here. However, it should be noted that an important exception is for transfer of rotational energy in a
collision, when the constraint of energy conservation makes
a major qualitative difference to the form of the collisional
energy-transfer probability distribution; this has been dealt
with in detail elsewhere. 18
We now summarize, in Eqs. (11)-(15), the exact numerical evaluation of R(E,E'). These results will then be
related to the above heuristic assumption about the randomness of the collision dynamics. This is a development from,
but completely replaces, our earlier work in this area. 13.14
It has been shown that the exact classical expression for
the energy-transfer rate distribution is'
E,
(11)
where B(E,E',E"b) = B(E; = E,E',E"b,t--+ 00) and f.l is
the reduced mass of the reactant-bath-gas pair.
When evaluating energy-transfer parameters from classical trajectories, the most efficient procedure is to evaluate
an appropriate moment, rather than the full distribution
function R (E,E'). A particularly convenient moment is the
second moment R E ',2:
1.76
1.74
a::
.•-a::
fOO
21rbdb Jo dE, (k T)2
B
Xexp( -E,IkBT)B(E,E',E"b),
~
-•
8kB 1\1I2 fOO
--;;;-J Jo
A
1.78
a::
~
(9)
The quantity s, which has the dimensions of energy, is defined by
R(E,E') = (
...~
1821
t
1.72
R E ',2 =
1.70
0
200
100
300
t (fs)
FIG. 1. Typical trajectory data for variation of internal energy E/ (t) (10"
em - I) with time for a collision for trajectory of highly excited azulene with
Ar. Initial energy E' = 17500 em-I above zero-point energy, E, = kBT
where T = 300 K.
LOO (E -
E')2R(E,E')dE
(12)
or the equivalent quantity (aE 2), the mean-square energy
transferred per collision:
(aE 2) =
L"" (E -
E')2p(E,E')dE.
(13)
Classical expressions for these moments can be immediately
written down: They are the same as Eq. (11) except that
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K. F. Lim and A. G. Gilbert: Energy transfer in highly excited molecules
1822
there is an additional intergration over E and the integrand is
multiplied by (E - E)2. For R E'.2' one has
8k 1\1/2 (<Xl
~-J
RE',2 = (
X
<Xl
Sa°
(<Xl
Jo
dE
Jo
db21rb
R E '.2
dE(E-E')2
E
A
'
e-E,IkBT B(EE' E b)
'(kB n 2
.'
'"
== i<Xl /(b)db.
(14)
The actual computational evaluation of these moments is
through the finite-difference form of Eq. (12), where the
integrand will be evaluated by the usual Monle Carlo selection of initial conditions. If it is assumed that B is identically
zero for all impact parameters b exceeding some value bmax '
then Eq. (11) may be rewritten as
= CkBgIl21rb~ax (b
max
R(E,E')
Jo
1rf.L
2:b db
1rb max
X (<Xl dE
E,
e-ErlkBTj(EE' E b).
'(k B n 2
'
",
(15)
Jo
It has been shown 7 that if initial conditions are chosen randomly from an equilibrium distribution of b, E, and E" this
yields
R E'.2
8k gll2
= ( _B_
lim 1rb ~x
1
LI (AE )2,
N
(16)
j
1rf.L
N-<Xl bmax-<Xl
N j=
where N is the number of trajectories, bmax is the maximum
impact parameter chosen, and AEj is the energy change in
thejth trajectory. We note parenthetically that the choice of
the second moment (the mean-square rate R E '.2 or meansquare energy transferred per collision (AE2) rather than
RE',I or (AE), the first moments) will generally lead to faster covergence in practical computation. This is because the
term to be summed, (AEj ) 2, is always positive, rather than
AEj' which can be either positive or negative.
How is the foregoing method for the exact evaluation of
collisional energy-transfer rates related to the distribution
P(E,E')? One may formally identify the numerical factors
in Eqs. (15) and (16) with a collision number
Z = (8kBT 11rf.L) 1/21rb ~ax' Now, implicit in the use ofa collision probability is the assumption that all trajectories
whose impact parameter exceeds some value d have negligible energy exchange: recall that a collision can only be defined for a hard-sphere interaction. The value of d will be
defined purely for the purpose of developing an approximate
model for R(E,E') through finding P(E,E'). Its value
should be close to (although smaller than) the value of bmax
used in actual computation. Given this arbitrary but "sensible" value of d, one has the hard-sphere collision number
defined by
2
(17)
ZHS = (8k BT 11rf.L)1/21rd .
Thus from Eq. (2), we may write
P(E,E') = (d
Jo
lim
21r~
1rd
db
roo dE,
-
E, 2
(kBn
Jo
A
xexp( - EJkBnB(E,E',E"b).
A value of (AE 2) can then be obtained from a chosen value
of a hard-sphere radius d using the following relation:
(18)
= ZHS (AE2).
(19)
It is essential to note that it is implicit in this expression
that the value of d has been chosen sufficiently large that
negligible energy transfer occurs for impact parameters
b > d. In general, this will not give a value of d that is the
same as a conventional hard-sphere or Lennard-Jones parameter. As will be shown, trajectory studies indicate that d
is somewhat larger.
Leaving the point about the value of d aside for the time
being, we note that Eqs. (16) and (19) give a recipe for
finding the mean-square energy transferred "per collision"
as an appropriate average of the square of the energy transfer
over a large number of trajectories. Next, we see how Eq.
( 16), which is exact, may be related to the heuristic development of the biased-random-walk model. Consider the case of
diffusion in the absence of an external field: i.e., for the case
where z = 0 in Eq. (8) (or, more precisely, where
lu I<IAE I). As is well known from the theory of Brownian
motion, 19 the diffusion coefficient is then given by
D ;0) """
fC (E; (r)E;
(20)
(0) ) dr,
where the superscript zero denotes that we are considering
the case where z = 0 (or rather where lui <IAE 1>, and
where (E; (r)E; (0» is the autocorrelation function of the
time derivative of E; (t):
(E;(r)E;(O» =
lim
~.-
X
lim _1_
- <Xl 'r <Xl tf
-
to
J.:f E;(r+ t)E;(t)dt.
(21)
Following the USual l9 proof of the connection between the
differential and integral expressions for the diffusion coefficient in Brownian motion theory (i.e., in three dimensions,
D=
!f
(v(O)"v(t) )dt
= lim (6t)-I(lr(l) -r(OW»,
'-<Xl
Eq. (10) can be rewritten as l4
That is, for the special case where z = 0, the BRW parameter
s is directly related to the mean-square energy transferred
per collision. Now, from Eqs. (8) and (13), one has directly
(AE2) = 2r + s4z2 = 2(S(0»2.
(23)
Upon rearrangement, one then has
~ = Z-2{[ 1 + 2 (to)z) 2] 1/2 -
n.
(24)
Application of a binomial expansion assuming Ito)z I2 .( 1
gives, to first order,
~=
(to»2[1 - !(S(0)Z)2
+ '''].
(25)
Now, evaluation of the dimensionless quantity (to)Z)2 shows
that it is less than 0.15 for molecules such as azulene (at
E' = 3X lQ4 cm- I ), acetyl chloride (at E' = I.4X lQ4
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K. F. Lim and R. G. Gilbert: Energy transfer in highly excited molecules
em-I), chloro- and bromoethane (atE' = 1.8X let em-I),
etc. Hence to a good approximation one can ignore higherorder terms in Eq. (25) to write
r= (;0»2 = !(6.E2) =te Jortc (£; (-r)£; (0) )dT.
1823
0.5
(26)
0
this gives the required approximate relationship between the
BR.W model and exact trajectory calculations. It will be seen
to enable the BRW parameter s to be evaluated with acceptable accuracy from the autocorrelation function. Now, in our
earlier work, 14 we had employed this result (which had been
derived in a quite different, and more approximate, manner)
to evaluate energy-transfer quantities from trajectories,
through calculating and fitting the autocorrelation function
to the results of a large number of simulations. It is now
apparent from Eqs. (26), (16), (19), and (23) that, in fact,
it is far better to evaluate (6.E2) directly from trajectories
using the exact result ofEq. (16), rather than going through
the intermediary of the BRW assumptions}4 However, we
now proceed to develop results using the BRW model that
can be used to find analytical approximations to the autocorrelation function, and hence to find approximate expressions
for the average energy transfer that do not require any trajectory data.
To find this approximate expression, we assume l4 that
the pseudorandom nature of the energy exchange suggests
that the time evolution of E; (t) during a collision obeys a
generalized Langevin equation:
d 2E.
-2-' = - a dt
ft
.
1
0.5
/.:'-
~
0
~
0
1
-
....
oW
~
~ 0.5
8
:0=
to)
c
.2
0
1
5
:;
...
G)
l5
-8
:::J
0.5
as
0
"C
1
CD
II)
:aE
l5
z
0.5
(27)
0
- '"
where a is an external "force" related to the microscopic
reversibility quantity z, X ( t) is a randomly fluctuating force,
and K(t) a memory kernel. As shown previously, 14 it is acceptable to assume that K(t) is exponential:
-0.5
K(t)
=
(A
2
K(T)E;(t- T)dT+X(t),
+ C 2 )exp( -
2At),
(28)
where the quantities A and C are to be determined. Solution
of the generalized Langevin equation then yields a damped,
oscillating autocorrelation function 14:
(£;(t)£;(0»
= (£;)exp( -At)
X [COs( Ct)
+ ~ sine Ct) ],
2A
(30)
2
2 •
A +C
lethe assumptions leading to Eq. (30) are acceptable, then
the evaluation of the mean-square energy transferred per
collision can be accomplished by finding a means of determining the quantities (£~), t e , A, and C from the dynamics
of the system in question.
-2
'2
= (E; )te
20
40
60
80
Time (10-'·s)
FIG. 2. Normalized autocorrelation functions of E, (t) for trajectory simulations of azulene-monatomic bath-gas collisions; from top, bath gases are
He, Ne. Ar, Kr, and Xe. Potential functions (intramolecular valence force
field and intermolecular atom-atom Lennard-Jones) as described elsewhere (Ref. 8). Results from averaging over 60 trajectories with impact
parameter vaiues chosen randomly over the range ~(n (2.2).) 1/2 (where 0
is the Lennard-Iones diameter and n (2.2)· is the reduced collision integral).
initial energy E' = 17 500 em - I, random rotational energy and orientation,
and random E, chosen for a temperature of 300 K.
(29)
where (£;) is the mean-square rate of internal energy
change during a collision. Figure 2 shows that the form of
Eq. (29) is an acceptable representation of this autocorrelation function when the true (£; (t)£;(O» is evaluated from
trajectory data. From Eqs. (26) and (29), one then obtains l4
y-
0
III. TEST OF VALIDITY: COMPARISON WITH
TRAJECTORY DATA
We now test the validity of the assumptions in the BRW
model by comparison with the results of extensive trajectory
calculations8 of collisional energy transfer between highly
excited azulene and monatomic bath gases. These systems
were chosen for study because there are extensive experimental data on the rate of energy transfer for these systems IS.16; however, our objective in the present paper is to
compare the dynamical assumptions made in the BRW
model against exact (trajectory) results with the same potential function. Comparison of the simulation results
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1824
K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
against experiment (which is really a test of the potential
function used, since the classical trajectories are essentially
exact) has been given elsewhere. 8 The only point to be made
here with regard to the comparison oftrajectory results and
experiment is that accord is usually acceptable, which indicates that the potentials are qualitatively and (for all except
the lighter bath gases) quantitatively correct. Hence the
conclusions we draw using the trajectory data are likely to be
of wide applicability.
The first assumption in the BRW is that E; (t) is pseudorandom during a collision event. This randomness can be
tested by examining the appropriate autocorrelation function. Figure 2 shows normalized autocorrelation functions
for azulene-monatomic bath-gas collisions, computed from
trajectories generated using the potential functions given
elsewhere. 8 These show a loss of correlation (significant diminution of the envelope of the autocorrelation function) on
the time scale of a collision evep.t, which as exemplified in
Fig. 1 is typically 10- 12_10- 13 s. Hence the fluctuations can
indeed be regarded as pseudorandom ("pseudo" since they
are still subject to microscopic reversibility and energy conservation). This justifies the fundamental BRW assumption
that the flow of energy between reactant and collider can be
described by a random walk in internal energy space during
a collision.
The second BRW assumption is that IsCO)zI2 <1. This has
been discussed above, and indeed seems to be a reasonable
approximation for typical systems.
The third BRW assumption is that the rate of energy
transfer as a function of impact parameter can be approximated by a step function. This can be tested by examining the
contributions to the integral over impact parameter b in Eq.
( 14) for the overall mean-square rate of energy transfer. Figure 3 shows the appropriate integrand, evaluated from trajectory simulations of excited azulene colliding with He bath
gas. It can be seen that there is indeed a maximum impact
parameter d beyond which energy transfer is insignificant.
However, it is most important to note that the trajectory
data 8 show that the value of d is found to be significantly
greater than that which one might expect from the simple
total scattering rate/o viz., 0'(0(2.2)·)1/2 (where 0' is the
Lennard-Jones diameter and 0 (2.2)· is the reduced collision
integral). Nevertheless, it is clearly semiquantitatively acceptable to assume that all trajectories contributing significantly to energy transfer take place with impact parameters
less than CO'(O(2.2)·) 1/2, where c= 1. Moreover, it confirms
that the total energy-transfer collision rate Z can be calculated from the collision integral with acceptable reliability.
This also suggests that one can reasonably define an average
collision duration te : collisions are effectively "switched on"
at average distances less than 0'( 0 (2.2).) 1/2.
The fourth assumption (not an essential part of the
BRW but used in the present application) is that the fluctuations obey a generalized Langevin equation with a singleexponential kernel. As shown elsewhere,14 autocorrelation
functions such as those shown in Fig. 2 can be fitted with
acceptable accuracy by Eq. (29); this suggests the validity of
the exponential kernel approximation. We note parentheti-
400
-
300
.0
'-'
1004
200
100
0
0
2
3
4
5
6
b(A)
FIG. 3. Integrand I(b) (10- 12 m 3 5- 1 em- 2 A-I) for integral over b in
Eq. ( 14) for mean-square rate of energy transfer. For azulene-He collisions
at T = 300 K, E' = 17 SOO em - I, 180 trajectories in total (data from Ref.
8). Arrow marks the value of U(O(2.2)O) 1/2.
cally that the autocorrelation functions obtained from trajectories can be fitted significantly better by a double-exponential kernel in the generalized Langevin equation 14;
however, our objective is to find a simply evaluated model
for energy transfer whose parameters have a straightforward
physical interpretation, and how the additional parameters
appearing in a double-exponential kernel do not lend themselves to immediate interpretation, in the way that those in a
single exponential do, will be shown in the next section.
The trajectory results given here all support the assumptions in the BRW model. While these assumptions are not
highly accurate, it would seem reasonable for the purposes of
a simple approximate model to accept their validity, bearing
in mind that exact trajectory calculations offer a much more
accurate, but much less transparent, alternative. We now
examine how the parameters required in the BRW model
can themselves be evaluated approximately from the potential function for a given system.
IV. EVALUATION OF PARAMETERS
The parameters in the BRW model with a single-exponential kernel are ( 1) the oscillatory frequency in the energy
derivative autocorrelation function, C; (2) the collision duration, te; (3) the decay constant of the autocorrelation
function, A; (4) the mean-square rate of internal energy flow
during the collision, (E
We shall determine simple, approximate, means of evaluating these quantities by comparison with data from trajectories, obtained using methods and potential functions given
elsewhere. 8 The trajectory data for this purpose, given in
Table I, were obtained using only impact parameters in the
range O-u(O(2.2)·) 1/2, rather than the true limit of infinite
bmax (while the trajectories used for comparison with experiment 8 were found by taking the proper limit of bmax -+ ~ ).
This is because we wish to develop approximate descriptions
of the dynamics given the basic BRW approximation that
n.
J. Chern. Phys., Vol. 92, No.3, 1 February 1990
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K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
1825
TABLE I. Parameters for azulene-monatomic bath-gas autocorrelation functions calculated from trajectory
simulation method of Ref. 8.
He
Ne
Ar
Kr
Xe
4.58
72
4.72
129
5.04
244
5.14
305
5.33
347
2.8
13
2.9
22
3.3
36
3.4
43
3.6
50
ma(amu), from Eq. (35a)
mb(amu), from Eq. (35b)
2.5
6.7
4.9
6.7
5.6
6.7
6.0
6.7
6.2
6.7
Ie (l0-13 s), from Eq. (32)
1.1
4.4
11.2
18.1
24.1
0.58
0.56
0.59
0.56
0.58
0.56
Bath gas
Overall potential Va.:
(7
(A)
€(K)
Average local atom-atom potential v,oc:
(7\oc
(A)
€Ioc (K)
E'
= 17500 cm- I
C(lOI~S-I):
from trajectories
from Eq. (33)
(E~)/e (l018 cm- 2 S-I):
from trajectories
0.58
0.56
156
0.58
0.56
509
693
868
755
(E~) (lQl3 cm- 2 S-2):
from trajectories
from Eq. (37)
from Eq. (41)
A (l01~ S-I):
from trajectories
from Eqs. (34a)-(36)
from Eqs. (34b)-(36)
s (cm- I):
from trajectories
fromEqs. (41), (34a)-(36), and (32)
from Eqs. (41) and (43)
(tJ12)1/2 (em-I):
from trajectories
from Eqs. (41), (43), and (23)
1.4
1.2
0.6
0.5
0.3
1.1
0.7
1.1
0.5
1.1
0.3
1.1
0.3
0.3
0.12
0.06
0.04
0.022
0.032
0.028
0.010
0.016
O.ot5
0.007
0.011
0.010
0.007
0.008
0.008
1.1
340
165
250
260
210
290
200
185
240
190
185
190
180
185
150
650
430
450
530
330
405
305
300
280
230
E'
= 30 644 cm- I
C(101~S-I):
from trajectories
from Eq. (33)
(E~)/e (10 18 cm- 2 S-I):
from trajectories
(ED (1Ql3 cm- 2 S-2):
from trajectories
from Eq. (37)
from Eq. (41)
A (101~ S-I):
from trajectories
from Eqs. (34a)-(36)
from Eqs. (34b)-(36)
s (em-I):
from trajectories
from Eqs. (41), (34a)-(36), and (32)
from Eqs. (41) and (43)
(tJ12) 1/2 (em-I):
from trajectories
fromEqs. (41), (43). and (23)
0.57
0.56
208
0.61
0.56
934
0.58
0.56
951
0.58
0.56
1272
0.58
0.56
1140
1.9
3.2
2.3
2.1
3.2
2.1
0.9
3.2
1.8
0.7
3.2
1.6
0.5
3.2
1.5
0.097
0.069
0.036
0.019
0.036
0.Q28
0.010
0.017
O.ot5
0.006
0.012
0.010
0.005
0.009
0.008
350
330
270
310
430
310
230
405
255
205
395
200
175
375
160
695
470
570
575
390
435
335
325
280
250
there is no significant energy transfer for b>0'(O(2.2)*)1I2,
and do not wish to have the tests of our dynamical assumptions, assuming b < 0'( 0 (2.2)*) 1/2, be led astray by the relatively sInall contributions from larger impact parameters in
the simulation results. The data in Table I were calculated
with 60 trajectories for each bath gas and energy, sufficient
to give acceptable convergence.
One of the desired quantities can be immediately evaluated from trajectory data, viz., C (from the oscillation period
in the autocorrelation functions); its values are listed in Ta-
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K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
ble I. It is, however, difficult to determine a precise value of
the decay rate of the kernel A from the autocorrelation function. This is because this decay is quite rapid, and the singleexponential functional form ofEq. (29) gives a form for the
autocorrelation function which, when least-squares fitted to
the simulation data, yields a value of A which has a fairly
wide uncertainty. This is because (i) the trajectory data are
somewhat noisy at later times, and (ii) Eq. (29) is obtained
from the assumption of a simple single-exponential kernel,
and cannot be expected to give an exact representation of the
complicated dynamics of these systems. A means of determining A from trajectory data which is subject to much less
uncertainty will be derived after a discussion of the other
quantities.
The product te (E;) can be obtained directly from the
trajectories through the following resultl 4 • 19 :
(E;)te =
~}i~oo ~
i:
f
[E;{t)]2dt
(31)
obtained from Eq. (21). Trajectory results for this product,
for azulene-monatomic bath-gas systems, are shown in Table I. However, the quantities te and (E Dcannot be evaluated with precision separately directly from the trajectory
data. This is because, even though the "start" and "finish" of
a collision are fairly sharp, as illustrated in Fig. I, one still
cannot uniquely define a collision duration. One sees from
Fig. 1 that one cannot assign te a value more accurately than,
say, 20%. For the same reason, the mean-square energy
change (E t ) can also only be defined by nominating a collision duration, and the same uncertainty applies. These problems arise from the approximations inherent in the BRW
model (or any model based on defining a collision), and
cannot be completely obviated. The lack of precision in the
definition of these quantities is the price that must be paid for
a simple model which gives physical insight and is readily
evaluated; it is emphasized that complete rigor can only be
obtained through full trajectory calculations, wherein the
qualities of insight and ease are lost.
The difficulties in finding t e , (E;), and A are readily
overcome once a suitable means of finding te has been found.
Originally, we had suggested 13 that te could be calculated as
the time required to traverse a distance of one hard-sphere
radius: te = 0'( 11'f.l/8kB n 1/2; however, comparison with extensive trajectory data shows that te is approximately a factor of 4 smaller than those predicted by this simplistic
expression. A better value of te can be found by assuming
that the collision time is that for traversing the overall potential function, starting from an appropriately defined "closest
interaction distance" d. This overall potential is taken as a
simple radial Vav (r), the spherical average of the complete
interaction potential. From the discussion given above, one
may approximated as 0'(0(2.2).) 1/2. This expression for te is
therefore twice the integral of the reciprocal of the velocity v
over this radial distance:
te
('0
= 2 Jd
=
V-I
(2p,)1/2
dr
Jd('0 (E, -
Vav - E,
7b2)-1I2 dr,
(32)
where r 0 is the classical turning point [i.e., where v (r) = 0] .
This expression depends on the impact parameter b, and one
could readily average over all b between b = 0 and b = d.
However, for simplicity we assign b its average value of
(2/3 )d. The translational energy E, can be assigned its average value of 2kB T. The reason for the average translational
energy being 2kB Trather than 1.5kB T (which is the normal
value of the average translational energy) is seen in Eq. (11):
It is due to the additional factor of E, weighting the integrand.
The values of te calculated with this recipe for the Vav
corresponding to the Lennard-Jones simulations of Ref. 8
are given in Table I. It is found that they are acceptable
approximations to those estimated from the trajectories as in
Fig. 1, although it is repeated that this involves a certain
arbitrariness in the average value of dE;ldt at which one
decides the collision stops and starts.
Given the value of te , that of (E;) can be found using
the values of the product (E;)te found using trajectory data
with Eq. (31); these values are shown in Table I. The value
of A can be found from the trajectory data by using the trajectory value of s, of (E;)te , and of C, together with Eq.
(30); the trajectory value of s is calculated from the "exact"
value of the mean-square rate R E '.2' converted to a meansquare energy transfer (aE 2) using Eq. (19), together with
Eq. (23). Values are listed in Table I.
Having used trajectory data to deduce numerical values
for the various parameters (A, C, te , and (E;» in the BRW
model, we next develop means of estimating these from approximate descriptions of the dynamical processes.
We first consider the autocorrelation oscillation frequency C. The data in Table I show that the oscillations in
the autocorrelation functions of Fig. 2 are very regular, and
to an excellent approximation are independent of the bath
gas and of the initial internal energy E'. It is not unreasonable to suggest that these oscillations are "driven" by the
highest vibrational frequency in the reactant, Vh' For azulene, this is the CH stretching frequency, -3000 cm- I . If
one supposes therefore that
(33)
then this predicts C = 0.56 fs - I. This is very close to the
values obtained from the trajectory data in Table I. Thus,
one can reliably assume that the fluctuations in the flow of
energy in and out of the reactant molecule will be governed
by the most rapid motions in the system: the slow-moving
collider mainly "sees" the most rapidly oscillating reactant
local mode. The force due to the atom-atom interaction potential between the collider and that local mode will fluctuate in time with the oscillations of that mode, and so will the
fluctuations in E; (t) and hence in the autocorrelation function.
Next, we turn to the decay parameter A. The preceding
discussion, and other examinations of the trajectory results, 8
suggests that the collision dynamics are of multiple atomatom impulsive interactions, dominated by the repUlsive
wall of the interaction potential. The memory loss which A
represents is therefore brought about by these interactions,
which occur on a much shorter time scale than does the
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K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
collision itself. As a first step, therefore, one might suggest
that A might be estimated from the short-time behavior of a
constant external force F acting on a point particle of mass
m. Denoting the energy of this particle as E, A is then the
short-time limit of E -IdE Idt. We give two alternatives for
calculating m and F. The first, denoted model a, assumes
that E is the average vibrational energy per oscillator, viz.,
E = E = E' Invib; because the average energy of the molecule is so high (corresponding to an extremely high internal
"temperature"), the internal energy is approximately equipartitioned between each oscillator irrespective of the energy
of the oscillator. One then finds
(34a)
A = F(2n vib /m a E ') 1/2,
where ma denotes the value of the average mass m appropriate to this model (see below). The second alternative for F,
denoted model b, assumes that E is the same as the average
translational energy 2kB T:
(34b)
where mb denotes the value of the average mass appropriate
to model b.
Next, we tum to finding the masses ma and mb in these
expressions. Because the dynamics here are governed by individual atom-atom interactions, it seems reasonable to assign the average mass a value which is an appropriate atomatom reduced mass. Model a assumes that the atom-atom
reduced mass if for the reactant-bath-gas reduced mass and
the average mass of an atom in the molecule:
(ma)-I=It-I+m-t,
(35a)
where It is the molecule-<:Ollider overall reduced mass and m
the average mass of an atom in the molecule: i.e., the reactant
molecular weight M divided by the number of atoms. Model
b instead ofIt uses the difference between the reactant molecular weight and m:
(mb)-I = (M - m)-I + m-I.
(35b)
We next tum to how this average force may be estimated
from an approximate dynamic model. In view of the impulsive nature of the energy transfer, Fmight be approximated
as the absolute value of the force acting over a local atomatom interaction, Vloc (r), at the turning point:
(36)
where Vl'!, (r) = Vioc (r) + E(b 2/r'l), the energy E is given
either by E' In vib (model a) or 2kB T (model b), and, as in
Eq. (32), bisgivenitsaveragevalueof(2I3)d loc ' Vloc is an
average local atom-atom potential. From the trajectory simulations used as the test here, this is taken as a LennardJones
function
Vloc = 4E'loc [(Uloclr)12 - (Uloclr)6],
wherein the parameters Uloc and E'loc are the averages of
those for the carbon-bath-gas and hydrogen-bath-gas
atom-atom potentials used in the simulations. 8 Note that the
values for the reduced mass and for E [i.e., the differences
between recipes (a) and (b) for these quantities] come into
the calculation of A both through these recipes and through
the calculation of the classical turning point, at which F is
evaluated.
The values of A calculated with Eqs. (34 )-( 36) for the
1827
two recipes for Fare shown in Table I. Comparison with the
values of A computed from trajectories and from Eqs. (34)(36) shows that those from the trajectories and from the two
recipes (a) and (b) are all quite close. Moreover, the results
ofEqs. (34)-(36) show the correct trends with bath gas. It
is surprising that such a simple picture, wherein the quantities can be evaluated on a hand calculator, can give such
semiquantitative accord with the trajectory data, which require large computational resources. It is apparent that the
physical basis of this simple picture is qualitatively correct,
which is the major aim of this work. There are differences
between the values found from the two recipes (a) and (b),
but these differences are not very large; for simplicity therefore we shall henceforth confine ourselves to recipe (a).
Last, we tum to the mean-squared rate of internal energy change, <k;). A primitive estimate of this quantity can be
made by assuming it is the average vibrational energy per
oscillator, E, divided by the frequency of the fastest vibration, Vh' This can be justified by saying that again it is the
fastest vibration that causes the energy fluctuation, and
again that the frequency for this is that of the fastest mode.
One then has
<k;) =. (E'Vhlnvib )2.
(37)
The values obtained by this very simple expression are
shown in Table I. It can be seen that the results are the same
order of magnitude as those found from the trajectory data.
There are of course some important differences. Equation
(37) suggests that <k;) should be independent ofthe bath
gas. On the other hand, the trajectory results show significant changes with increasing size of the collider. However,
the values lie moderately close to that predicted by Eq. (37).
Indeed, the success of such a simple expression is remarkable.
Equation (37) takes no account of the effect of the bath
gas upon <k;), whereas the trajectory data clearly show a
significant variation with bath gas. This can be taken into
account by supposing that the rapid oscillations are somewhat damped by the presence of the bath-gas atom, which
can be assumed stationary over an oscillation period. This
will diminish the energy change in this period by an amount
which we denote 16. V I. We can treat this through a "local"
atom-atom potential, as in Eq. (36). Let the distance moved
by the oscillator be !u = (2E I k) 112, where-leis an appropriate force constant. !u will be small compared to typical internuclear distances during the collision. The diminution in
energy can then be approximated by !u dV10c (x) I dx, evaluated at an average atom-atom distance x. The force constant
k can be approximated as k = 4rmlight ~, where mlight is
the mass of the lightest atom (which, of course, is involved in
the highest-frequency vibration). The average atom-atom
distance x can be approximated as the mean of an outer value
(xl Uloc = 2/3, as in the average impact parameter given
above) and an inner one (XIUloc = l):Thus, the average x is
taken to be given by XIUloc = 5/6. One thus obtains, for a
Lennard-Jones form for Vloc '
6.V= 4E'loc !u [ - 12(uloc lx) 12 + 6(UIoc lx )6] , (38)
x
where xlUloc = 5/6 and !u and k are as given above. Equa-
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K. F. Lim and R. G. Gilbert: Energy transfer in highly excited molecules
1828
tion (37) can then be modified to give
(E;) "'" [(E -iavi )Vh ]2.
(39)
Clearly, this modification to Eq. (37) is inappropriate if
Ia v Iis too great, when the highly simplistic treatment given
above will break down. For this reason, we pragmatically
restrict Ia v I not to exceed (1/2) E:
(40)
This additional restriction, made to avoid the possibility of
unphysical behavior, only has a minor effect on the results,
and then only for the heaviest bath gases. The values of (E;)
resulting from this modified expression are shown in Table I.
It can be seen that Eq. (40) leads to a significant improvement over the predictions of Eq. (37), inasmuch as it now
semiquantitatively predicts the change in (E;) with bath
gas.
We are now able to assemble all the above results, Eqs.
(30) and (32)-(40), to arrive at an easily calculated expression for s. In Eq. (30), we note that A 2..( C 2, which leads to
further simplification:
(41)
where the expressions for t e , A, C, and (E D are as given in
Eqs. (32), (34)-(36), (33), and (38)-( 40). Wenoteparenthetically that the approximation A 2..( C 2 is generally expected to be valid (unless perhaps the reactant contains no light
atoms).
If one takes the simpler, although less accurate, expression for (E~), Eq. (37), then Eqs. (33)-(37) reduce to a
result that is independent of the highest frequency:
~""'~ teA (E'!1rn vib )2.
(42)
This has the implication that, although the oscillations in the
autocorrelation function of the random energy migration are
dominated by the high-frequency modes, these, in fact, do
not have a large effect on the actual mean-square energy
transfer. However, it is to be noted that this result is only
valid within the approximation of Eq. (37), and a significant
(although not enormous) effect of the highest frequency is
reintroduced through the use of the more accurate equations
(38) and (40).
The values of ~ calculated from Eq. (41) are compared
with those deduced directly from the trajectories in Table I.
It is seen that the agreement is good, considering the ease of
evaluation of the approximate formulas and the primitive
nature of the dynamical assumptions.
While the foregoing crude model is in semiquantitative
accord, it is desirable to have an improved version that can
still be used to carry out improved predictions of <aE 2 )
without undue computational effort. Clearly, there are
many variants on the crude models for A, etc., given above.
For our present purpose, it would seem simplest to correct
the shortcomings semiempirically, as follows. We assume
that the corrections to Eq. (41) due to the dynamics are
comparatively small, and can be expanded in a low-order
series in various dimensionless quantities. The trajectory calculations were carried out for systems with different bath gas
and internal energy, and so it would seem appropriate to
writes = sea + bp, + cE'/nvibkB 1)2, wheresisthevalueof
s calculated from Eq. (41). Least-squares fitting of the three
empirical parameters a, b, and c from the ten independent
data available from the trajectory calculations of Table I
yields, using recipe (a) for m and for E:
s = s( 1.5667 - 0.0053p, - 0.2122E'/n vib k B 1)2. (43)
wherep, is in a.m.u.
The results for (aE 2) 1/2 using these semiempirical corrections are also shown in Table I. Because we have now
actually fitted the constants in Eq. (43) to the trajectory
data, there is a significant improvement in the fit, and the
deviations from the trajectory results are generally within
the uncertainty of typical experimental data. Equation (43)
is based on very limited trajectory data, and no doubt the
availability of more extensive simulation data in the future
will enable it to be improved to take account of other factors
which have been ignored in our simplistic treatment.
The area where both Eq. (41) ~nd its semiempirical correction, Eq. (43), are in significant disagreement with the
trajectory data is for He bath gas, where the (aE 2) 1/2 values
from Eqs. (41) and ( 43) are significantly smaller than those
from the simulations. Coincidentally, those from the present
approximate model are much closer to the experimental values 15 •16 (which for He are -200 and -230 cm- I for
E' = 17 500 and 30 600 cm -1, respectively). However, this
is due to fortuitous cancellation of errors, since the approximate model and the trajectories have the same potential
function, while it has been shown8 that the poor agreement
between the trajectory results and experiment for the lightest
bath gases is due to the interaction potential being significantly softer than the Lennard-Jones 12-6 repulsion assumed here.
To test the present approximate BRW model further,
we give two more comparisons: with trajectory data from
simulations of the methylperoxy/Ar system,9 and with experimental data on deuterated and undeuterated iso-propyl
bromide and tert-butyl bromide with various monatomic
colliders. 21
For the trajectories on CH30 2/ Ar, the potential parameters used in the simulations9 can be used to deduce the values of the various parameters required for the present BRW
model. These are E = 202 K, 0' = 3.7 A, E' = 104 cm - I,
I
Vh = 3000 cm- , ma = 6.6 amu, Eloc = 54 K, and
(TIDe = 3.15 A. One then obtains, using Eq. (43), etc.,
s = 290 and 350 cm - I at temperatures of 440 and 582 K,
respectively. These compare with the trajectory values of
185 and 250 cm- I , respectively. The accord is considered
acceptable.
It is interesting to carry out a sample comparison with
experiment, although there is of course the problem that the
potential parameters are unknown, and therefore no definite
conclusions as to the correctness of either the model or the
assumed potential can be made. We choose the systems
C3H7Br, C3D7Br, C4H 9Br, and C4D9Br with various monatomic bath gases because the experiments21 were all done
using the same technique (pressure-dependent very-Iowpressure pyrolysis) and equipment, and so relative values
and trends should be able to be meaningfully compared. For
these reactants, there is the question as to whether the 'local"
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K. F. Urn and R. G. Gilbert: Energy transfer in highly excited molecules
1829
TABLE II. Potential parameters and experimental (Ref. 21) and BRW values for s for iso-propyJ bromide and
ten-butyl bromide-inert-gas collisions.
s (em-I)
E
(T
System
(K)
(A)
C 3H,Br/Ne
114
279
114
279
222
275
222
275
4.3
~H,Br/Xe
~D,Br/Ne
C 3D,Br/Xe
C.HJJr/Ar
C.HJJr/Kr
C.D,Br/Ar
C.D,Br/Kr
S.O
4.3
S.O
4.S
4.6
4.S
4.6
P.
(amu)
T
E'
(K)
(10" em-I)
7.1
10.1
7.4
10.8
7.3
8.1
7.9
8.8
870
870
870
870
720
720
720
720
1.9
1.9
2.0S
2.05
1.7
1.7
1.9
1.9
atom-atom potential Vloc ' etc., should be that found by averaging over all atoms or only the outer ones. Both alternatives
were used, and the differences were found to be very minor.
The various potential parameters (found using only the outer atoms, viz., H, D, and Br), and the experimental and
BRW values of s are shown in Table II. The values of E and 0'
were those chosen to interpret the original experimental
data, while those for E loc and O'loc were chosen as the average
of the atoms, wherein each was treated as the corresponding
inert gas. The comparison with experiment is comforting,
inasmuch as model and experimental absolute values are
quite close. However, the trends with bath gas for a given
reactant tend in the wrong direction when one of the bath
gases is Ne, but this shortcoming could be ascribed to either
the incorrectness of the assumed Lennard-Jones potential
[as has been shown for other systems involving Ne (Ref. 8) ]
or to the inadequencies of the approximate dynamical treatment given here. The trend of decreasing average energy
transfer going from iso-propyl to tert-butyl bromide is successfully reproduced, which is good because the same type of
potential was of course used for each, and predicting the
wrong trend would have indicated significant inadequacies.
The effects of deuteration of reactant predicted theoretically
and experimentally are both small: less than 10%, which is
well within experimental error. However, again the BRW
predicts the wrong direction. This is not regarded as a major
shortcoming, since it could be remedied (for example) by
slightly different ways of choosing the value of E' (the experimental method, being of the "indirect" class, actually
has contributions from a wide range of internal energies) or
of finding the average O'loc' E loc ' and rna'
v. CONCLUSIONS
By comparison with results of trajectory simulations,
we have validated the physical approximations in the "biased-random-walk" model for collisional energy transfer.
Th~assumptions are that the migration ofintemal energy
between the molecule and the bath gas is pseudorandom,
subject to the constraints of microscopic reversibility and
energy conservation; these assumptions are expected to be
valid for all reactant molecules except those containing oilly
a few (say, four or less) atoms. The mean-square energy
transfer per collision can then be obtained from a "diffusion
~r)
3.1
3.4
3.1
3.4
3.0
3.1
3.0
3.1
tioc
(K)
34
87
34
87
58
73
S8
73
Sq. (41) Expt.
470
370
SIO
390
300
310
320
330
390
430
360
460
195
235
170
210
coefficient in internal energy space" and an average collision
duration. The former can be obtained by the integral of the
autocorrelation function of the internal energy transfer rate
during the collision. It was shown that this is mathematically
equivalent to an approximate evaluation directly from trajectories.
While this formalism should not be used for evaluation
of (fl.E 2) from trajectories (much more reliable means for
this are now available7•8 ), it can be used to find a crude but
readily evaluated analytic approximation to (I1E2) from the
properties of the colliding species. This is given in Eqs. (40),
(43), (34 )-(36), and (32). These simple expressions reproduce trajectory data to within an average of about 20%. It is,
however, pertinent to add here that if the bath gas is a large
polyatomic, then ergodic collision theories (which grossly
overestimate (fl.E 2) for nonpolyatomic bath gases) can then
be used with acceptable accuracy.
A major result of the present work is that it enables one
to understand readily the magnitudes of experimentally observed average energy transfer values. The model shows that
energy migration occurs through a large number of impulsive interactions between the bath gas and the reactant
atoms. From the very simplest treatment given in Eq. (42),
it can be seen that the energy transferred per collision is
governed by (1) the collision time t c ' which is that to traverse the overall interaction potential in and out from the
appropriate collision diameter; (2) the average internal energy per oscillator, E '/n vib ; and (3) a decay rate A which is
governed by the average force exerted at the classical turning
point of individual reactant atom-bath-gas interactions.
This therefore fulfills a major goal of the present work,
which is to explain the magnitude of the average energy
transfer for monatomic colliders.
The method given here has a number of shortcomings,
but also some important advantages. If an accurate value of
the mean-square energy-transfer rate is required, then one
must use a full classical trajectory calculation. However, the
present semi-empirical model gives results of an accuracy
sufficient for many applications such as a first estimate of the
energy transferred per collision for falloff calculations. In
addition, the mechanistic precepts of the model lead to an
increased phenomenological understanding of the energytransfer process.
J. Chern. Phys., Vol. 92, No.3, 1 February 1990
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1830
K. F. Lim and R. G. Gilbert: Energy transfer in highly excited molecules
ACKNOWLEDGMENTS
Insightful comments from Gregory Russell, and stimulating discussions with Dr. Mark Sceats, are gratefully acknowledged, as is the financial support of the Australian
Research Council and (for K.F.L.) a Commonwealth Postgraduate Research Award.
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J. Chern. Phys., Vol. 92, No.3, 1 February 1990
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