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A test of RRKM theory against numerical simulation for classical chain molecules. II. Decomposition and vibrational relaxation in uniform chains
93
Chemical Physics 108 (1986) 93-104 North-Holland. Amsterdam
A TEST OF RRKM THEORY AGAINST NUMERICAL SIMULATION FOR CLASSICAL CHAIN M0LEtXIX.S. II. DECOMPOSITION AND VIBRATIONAL RELAXATION IN UNIFORM
CHAINS
Harold W. SCHRANZ, Sture NORDHOLM Department
of Theoretical Chemistry
University of Sydney, N.S. W. 2006, Australia
and Ben C. FREASIER Department
of Chemistry, Faculty of Military Studies, University of New South Wales, Duntroon, A. C. T 2600, Australia
Received 24 June 1985; in final form 14 March 1986
Deviations from RRKM predictions for the fragmentation of a one-dimensional chain molecule are here studied in detail by numerical simulation and related to the rate of the vibrational relaxation. A measure of the degree of completion of vibrational relaxation is constructed, and its time development followed and compared with the progress of the reaction. The vibrational relaxation is initially rapid but then slows down as the reaction depletes the ensemble of rapidly relaxing trajectories. The observed decay rate starts from the RRKM value and then typically exceeds it at intermediate times crossing over to fall below the RRKM prediction at long times.
1. Introduction
In a preceding paper [l] we presented a method of simulation designed to test the validity of the assumptions of the RRKM theory of unimolecular reactions. Specifically, we wanted to determine whether the decomposition of a molecule microcanonically excited to some energy E above the applicable threshold would decay exponentially with a rate coefficient in accord with the RRKM prediction. The answer will, of course, depend upon the system. We choose to study classical one-dimensional chains for reasons of simplicity. The chains consisted of from three to ten particles of equal mass interacting by identical Morse potentials with neighboring particles in the chain. Initial states were chosen microcanonically at a set of energies above but close to the dissociation energy. Bond breakage was observed and recorded as a lifetime to dissociation from which lifetime probability densities and rate coefficients were constructed. The results obtained for uniform
chain molecules bound by Morse bond potentials led to the following conclusions: (i) anharmonic effects on the relevant densities of states of the chain are substantial, (ii) a proper anharmonic implementation of RRKM theory yields rate coefficients k,(E) in reasonable agreement with simulation, (iii) the deviations between RRKM and simulation diminishes as the energy decreases towards threshold and the number of atoms in the chain increases. The main aim of the present report is to facilitate the interpretation of these results by studying the relative rates of decomposition and vibrational relaxation in greater depth and detail. A time-dependent measure of the vibrational relaxation in an ensemble of trajectories initially in microcanonical equilibrium is constructed. Its observed time dependence is then compared with the corresponding time dependence of the decomposition as observed and as predicted by RRKM theory. The role of the vibrational relaxation rate in de-
0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H. W. Schranr et al. / A test of RRKM
94
termining the validity of the RRKM approximations is thereby illustrated. There is a large body of literature relevant to the general aims of our work [2-41. An extensive, but far from complete, survey is contained in the first report [l]. In our final section of this article we shall discuss our present results in the light of earlier work on the same topic.
theory. II
mate for anharmonic systems at low energies and very inaccurate for anharmonic systems at high energies. Each term in the normal mode hamiltonian corresponds to the hamiltonian for a simple harmonic oscillator with angular frequency, w,. Fig. 1 illustrates the character of the normal modes of the linear chain system. The frequency vi is related to the angular frequency oi as, vi = w/2??.
2. Analysis of intnunokcular
vibrational relaxation
The systems to be studied here are uniform linear chains of N particles with N between 2 and 10. The hamiltonian is
Hz
N
N-l
PZ
iFl G
+
C +(Xi+l -xi),
(2.1)
i-l
with the nearest neighbor interaction type, G(r) =D{exp[-/?(xi+i-xi-r,)]
-lj2.
of Morse
(2.2)
We will use units such that D = 1, and the time unit is 1.47 X lo-l3 s. This information will suffice for our present purposes. A more detailed discussion of the chain models is given in the first article in this series [l]. 2.1. Normal mode analysis It is useful to consider the normal modes of the general N particle linear chain in one dimension. They will be used in order to evaluate a parameter that may reveal the extent of intramolecular vibrational relaxation during the reaction process. Extensive treatments of normal modes as applied to the vibrations of molecules are given in texts by Wilson et al. [S], Herzberg [6], Goldstein [7], and Slater [ 81. It is easily shown that the normal mode hamiltonian can be written in the form, (2.3) This hamiltonian is exact for a system with harmonic bond potentials, but is only approxi-
(2.4)
The length and direction of the arrows in fig. 1 represent the relative amplitudes of the motions of the atoms, in mass weighted coordinates [5-71, that contribute to the particular normal mode. Note that for uniform chain systems, the normal modes tend to be global in nature and have motions associated with all the atoms in the molecule. 2.2. A vibrational relaxation parameter In order to gain some idea of the rate of intramolecular energy transfer in the linear chain molecules studied, a parameter that can measure the extent of intramolecular vibrational relaxation (IVR) will be introduced. It is based on the fact that harmonic systems with more than one bond (N > 2) are non-ergodic [9] and have normal modes with well defined energies. The energy of each normal mode, ci(t), can be identified with an individual term in the normal mode hamiltonian in eq. (2.3), q(t)
= )(if
+ wi’zf),
(2.5)
where the frequencies, wi, and the transformation from cartesian coordinates to normal mode coordinates can be determined by a normal mode analysis [5-71. Anharmonic systems, on the other hand, are associated with a greater degree of ergodicity and considerable energy fluctuation among the normal modes [lo-121. This fluctuation of normal mode energies in time reflects the more chaotic movement of the trajectory in phase space compared to harmonic systems. If, over a long enough period, the time average of the normal mode energies was equal to the microcanonical average of normal
H. W. Schranz et al. /A (a) N = 2.
laxation (IVR) parameter which measures the deviation from equality from eq. (2.6),
Normalmoder
4
95
test of RRKM theory. II
where
ch(t) uwNR(r) =f%(f) 1PI&),
and the microcanonical average of f( t ) over Na( t ) trajectories can be defined as
(c) N = 4.
(2.9) [ j=l where Na(t) is the number of trajectories still in reactant space (i.e., the number of molecules still bound) at time t. We consider several model cases to illustrate the possible behaviour of the IVR parameter defined in (2.7). For a harmonic system the normal modes all have well defined energies,
(d) N = 5.
Fig. 1. The normal modes of motion (translation and vibration) forunifonnchains(mi=l, i-l,...,N):(a) N=2,(b) N=3, (c) N = 4 and (d) N = 5. Arrows representing amplitudes less than 0.1 are omitted for clarity.
G(t)=ci,
i=2 ,..., N,
where the ci are constants. From the formulation in eqs. (2.7)-(2.9), (2.11)
(G)=l, mode energies then one would expect this to be a reflection of ergodic behaviour in the system [13]. Consider a linear chain system with N atoms. It has N normal modes, one external translational mode which shall be ignored, and N - 1 internal vibrational normal modes which according to eq. (2.5) have energies, {
=
Mt))t
=fjddsq(s),
i=2 ,..., N,
(2.6)
where ( )MC is the microcanonical average. The degree to which this equation is satisfied should give some idea of the degree of ergodicity and vibrational relaxation in the chain system. One can formulate an intramolecular vibrational re-
(2.10)
for a harmonic system. For a system which has all internal normal modes approaching microcanonical equilibrium with a rate constant X, L\Ei(t) =&(O)
emX’, i = 2 ,..., N,
(2.12)
then (Z)
= eeX’.
(2.13)
For systems which are only partially ergodic/ non-ergodic one would expect behaviour between these extremes. The normal modes may exhibit different and perhaps non-exponential rates of attainment of microcanonical equilibrium. Perhaps only partial microcanonical equilibrium is atained. In such a case the normal mode parameter (AC) would perhaps follow the behaviour, (z)
= XE emX’+ (I - Xn),
(2.14)
96
H. W. Schranz et al. /A
where xE is a crude measure of the proportion of exponential relaxation with rate constant A. This equation defines exponential relaxation to a nonzero plateau and hence will be termed the plateau-exponential model. The larger this plateau, the more the behaviour corresponds to the harmonic prediction (2.11) and this implies that less microcanonical equilibrium will have been achieved during the reaction process. On the other hand, the lower this plateau, the more the behaviour corresponds to complete microcanonical equilibrium (in normal mode space) which is achieved with a rate constant A, given in eq. (2.13). It will be useful to obtain an average rate constant h for relaxation by fitting the observed behaviour of (E) to eq. (2.13). The IVR rate constant thus obtained is compared to the RRKM and the simulation derived decomposition rate constants in section 3. While a fit to eq. (2.14)
test of RRKM theory. II
would be more realistic, it is the rate of overall microcanonical relaxation that is of interest and hence the former type of fit is more directly relevant. Of course, both types of fits are simplifications of the real behaviour and it is for this reason that direct comparisons of (z) and decomposition are used in the presentation of results in section 4.
3. Comparison of timedependent with RRKM for uniform chains
rate constants
For detailed comparisons of RRKM and simulation it is necessary to consider the behaviour of the time-dependent rate constant k(t; E). Figs. 2-4 show the dependence of the comparison upon molecular size, N, and energy, E. The number of trajectories calculated for each N and E condition
C
N-7
E-
1.5
0
0
60
120
*
160
240
300
0
160
360
,
540
720
900
z d LSFS RRKH
b
N-5
E - 1.5
0
0
12
24
,
36
46
60
Fig. 2. Comparison of Morse RRKM rate constants, least-squares fitted simulation rate constants (LSFS), and IVR rate constants with the time-dependent rate constant k(t; E) (SIM) as a function of chain length N for uniform chains with E = 1.5 and &( r, - rc) = 10: (a) N = 3, (b) N = 5, (c) N = 7 and (d) N = 10.
H. W. Schronz et 01. /A
d
E
N.3
.._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-l
mined from the simulations, tends to reflect the rate of vibrational relaxation compared to the rate of dissociation. When this ratio is small (for small molecules or high energies) the behaviour is very non-RRKM but when this ratio is large one generally finds RRKM behaviour.
1.1
I
test of RRKM theory. ZZ
!vR................
‘I
RRKM LSFS SIM
OL
-
.
-
,
60
80
100
Fig. 3. As for fig. 2 but for a N = 3 uniform /3M(rC-rc)=10and E=l.l.
cllain with
0
20
40
-
has been listed in table 3 of our first report [l]. It is usually 500. In general, small molecules at high energies exhibit rapid dissociations at short times and slow dissociations at long times compares with RRKM predictions. The agreement with RRKM, however, improves for larger N or lower E. The IVR value superimposed is an estimate of the rate constant for vibrational relaxation obtained by the procedure in section 2.2. One assumes all normal modes are approaching microcanonical equilibrium with the same rate constant h and this leads to a simple-exponential form for the relaxation parameter, (AC), ~(2.7). By fitting to the observed behaviour of (AC) this yields a value for A which is labelled as IVR (internal vibrational relaxation) in figs. 2-4. The ratio IVR/LSFS, where LSFS stands for the least-squares fitted decay rate coefficient ob-
2 d
N-r!
E = 1.25
. . . . . . . . . .._...............
_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IVR ... ... ... .... .. .... .
G i’
.
:
RRKM
O?
.
0
16
32
-
-
,
40
-
64
Fig. 4. As for fig. 2 but for a N = 5 uniform &,,M(r, - rc) = 10 and E = 1.25.
80 chain with
4. A comparison of the degree of normal made relaxation with the degree of diisociation for uniform chains In figs. 5-10 we compare the proportion of dissociations, Na/Ns, to the degree of normal mode relaxation, as defined by (AC), as a function of time, t, for various uniform chain systems. These quantities are competing kinetic processes. In a RRKM molecule with a microscopic rate constant k,
NJN,
= eekr.
(4.1)
In similar fashion, if all the normal mode energies relax to their microcanonical values with the same rate constant h, then (as shown in section 2.2), (E)
= e-“.
(4.2)
Thus,> a crude fashion, comparison of Na/Ns and (AC) will yield a comparison of k, the average rate constant for dissociation, and h, the average rate constant for IVR (as represented by the rate at which normal mode energies attain their microcanonical values). In fi&. 5-8 we present a comparison of NJNs and (AC) for different molecular size N and energy E. The general trend is that one gets nonRRKM behaviour if NJzs = (G) and RRKM behaviour for Na/Ns 3 (AC). This agrees with the comparisons of rate constants presented in section 3. Thus, for uniform chain systems, non-RRKM behaviour is expected for small molecules at high energies and RRKM behaviour for large molecules at low energies. Figs. 9 and 10 show the effect of en_ergy on the normal mode relaxation parameter, (AC). At low energies, E = 0.5, the amount of relaxation is low but increases at E = 1.0. However, once reaction begins (E > Eo) less IVR has time to occur and
98
H. W. Schranr et al. / A test of RRKM theory. II
b
E.I.s
N=S
8
0
..--
....
a*----.-.
60
‘tO
20
.
0..
a.0
0
d
N = 10 -.
200
E = 1.5 -.
400
-
,
..-
600
..
.
800
.
.
1000
Fig. 5. Comparison of the proportion of dissociations Na/Ns and the normal mode IVR parameter (z) as a function of chain length N for uniform chains with E = 1.5 and &,( r, - re) = 10: (a) N = 3, (b) N = 5, (c) N = 7 and (d) N = 10.
this is illustrated by the lower relaxation at E = 1.25 and E = 1.5. A longer time comparison for the reactive energies, fig. 10, reveals that the amount of relaxation at E = 1.25 is greater than that for E = 1.5. Examination of NR/Ns reveals that much more reaction occurs on the same timescale at E = 1.5 than at E = 1.25. This suggests
N-3
E-
that dissociation competes with the IVR process. At higher energies, more phase space is close to a dissociation channel and hence irreversible dissociation is more likely than traversing phase space in an ergodic manner. The general shape of the normal model IVR parameter, (z>, plotted versus time t, as in figs.
1.1
.-----..--...-.-..
0,
0
10
20
90
40
50
Fig. 6. As for fig. 5 but for a N = 3 uniform chain with &(rC - re) = 10 and E = 1.1.
Fig. 7. As for fig. 5 but for a N= 5 uniform chain with &(r, - re) = 10 and E = 1.25.
H. W. Schranz et al. /A
01..
.
0
200
-. 400
-. ,
600
-. - 600
test of RRKiU theory II
99
1 1000
0
20
'P
60
60
Fig. 10. As for fig. 9 but for the dissociating trajectories (E =1.25,1.5) at longer times. 0
100
200
t
300
400
500
Fig. 8. As for fig. 5 but as a function of energy E for a N = 10 uniform chain with &(r, - rc) = 4: (a) E =1.25 and (b) E = 1.5.
S-10, roughly corresponds to a plateau-exponential model of the type considered in section 2.2. At low energies, where the initial rapid dissociations are insignificant, it appears that trajectories showing the fastest vibrational relaxation are preferentially eliminated by dissociation. Ibis results in a rapid initial drop in (AC). The remaining trajectories show a smaller degree of vibrational relaxation, take more time to reach a transition state configuration (if at all) and hence tend to be longe~lived. The result is a much slower variation in (AC) at long times. The relative size of the plateau at long times can be considered as a crude measure pf the extent of non-ergodicity at that time.
5. Diiion
0
2
’
I
6
6
10
Fig. 9. The behaviour of the proportion of_dissociations N&N, and the normal mode Im parameter (AC) at various energies for a uniform chain with N = 5 and &(rC - rs) = 10: E = 0.5, 1.0, 1.25, 1.5.
and condusions
It has been found, that for uniform chains, that RRKM effects are greater for large molecules and low internal energies while non-RRKM effects are greater for small molecules and high energies. This behaviour has also been observed for chain systems with masses that are not very different. The trend with internal energy is similar to but more
100
H. W. Schmnz et al. /A
detailed than that observed by Thiele [18] for a model calculation on a linear symmetric triatomic molecule with Morse bond potentials. He based his theoretical description on the gap distribution, g(t; E), which can be related to the lifetime distribution, h(l; E). Note first that any initial reactant state which leads to dissociation must lie on a trajectory which is a closed loop, beginning at the critical surface with the formation of the molecule and then after the time s (the gap) ending with the dissociation. Thus, if g(s; E) is the gap distribution over all such reactive loops, the number of initial states which lead to the dissociation after time t is proportional to the total number of loop trajectories such that s > t, i.e..
1, -1
X
*dt [j 0
J
md4s;
t
E)
(5.1)
where x = j,“dt h (t; E ). For an ergodic surface, x = 1, but if the surface is non-ergodic x represents the fraction of dissociating states. Thiele noted that g(t; E) approached microcanonical (random) behaviour as the internal energy, E, approached E,, from above, while it became more regular (non-random) with increasing internal energy. By estimating the rate of IVR using a simple model based on the behaviour of normal modes, the above trends could be explained in terms of the competition between IVR and dissociation. If IVR is much faster than dissociation, RRKM behaviour is expected and this appears to be the case for long chains or low energies. This is consistent with the experimental deductions that large reactive systems with h,,, > lOi* s-l should behave ergodically [19]. On the other hand, if IVR is on the same timescale or slower than dissociation, non-RRKM behaviour is expected and this is the case for small chains or high energies. This point has also been made, e.g., recently by De Leon and Berne [20] in the context of a model isomerization reaction. They note that, there must be an effective bottleneck to reaction at the transition state, for microcanonical rate constants in general, and
test of RRKM theory. II
the RRKM rate constant in particular, to be valid. This is generally true only for energies close to the barrier maximum, E,, since, at high energies, the transition state is usually much too wide for a sufficient separation in timescales between intramolecular vibrational relaxation and dissociation. In the present work the effort has been focused on dissociation from a given internal energy surface in the hope that the corresponding rate constants would clearly show the extent of nonRRKM behaviour without the complicating effect of averaging over an energy distribution. However, it was found that the extent to which these rate constants are averaged over time also effects their sensitivity to non-RRKM behaviour. When a linear chain system exhibits non-RRKM behaviour, it is more obvious from observing timedependent rate constants than averaged time-independent rate constants. Time-dependent (or timeresolved) rate constants more clearly show the initial state dependence of the rate of dissociation. A similar point has been made by Sellers and State [21] in their simulation of a thermal unimolecular reaction. They noted that the use of the microcanonical rate constant to give an average lifetime for an activated molecule did not provide an adequate description of the entire distribution of lifetimes. In particular, calculated results in the fall-off region were found to be sensitive to details of the short lifetime tail of the distribution. In our simulations we found that non-RRKM behaviour in the time-dependent rate constant takes two forms: (1) Rapid dissociation at short times. This is more prevalent at high internal energies (or small molecules) and less so at low energies (or large molecules). (2) Slow dissociation at long times. This tends to persist to lower energies (or larger molecules) than type (1) above. This behaviour can be understood as follows. Note first that at t = 0 there must be agreement between RRKM theory and simulation since in both cases the reactant molecule is taken to be in microcanonical equilibrium. The deviations from microcanonical equilibrium which arise for l> 0 in the simulation may favour decomposition as
H. W. Schranz et al. /A
clearly illustrated for N = 2 in our first report [l], or they may disfavour decomposition. The latter would clearly be the case at long times for a non-ergodic system where not all states lead to decomposition. The former tends to dominate at short times and high energy. Rapid vibrational relaxation tends to blur these two types of behaviour and produce agreement with the RRKM predictions based on microcanonical equilibrium remaining throughout the decomposition. Similar behaviour has been observed in a series of recent 3D simulations by Hase and Wolf [22-241 on the dissociation of H-C-C. They compared the effect of different initial conditions. If H-C-C is excited randomly they found that the dissociation probability is less than RRKM predictions because the prepared microcanonical ensemble included regular trajectories which are trapped in phase space [22]. This trend is similar to the slow dissociation found at long times in our simulations being more pronounced for smaller chains and higher energies and could be enhanced by addition of a heavy central mass. The fact that randomly prepared H-C-C contains a significant subset of regular trajectories [22] is not too surprising as earlier simulations by Harter et al. [ll] on a 1D chain system showed that C-H stretching and C-C stretching motions do not readily exchange energy. Despite being derived for a 1D chain system this conclusion should also be valid for the 3D system of Hase and Wolf [22], since, the collinear case should allow the most efficient energy transfer possible between bonds ]121. Wolf and Hase also consider the dissociation of chemically activated H-C-C [23,24]. Unlike the microcanonical initial conditions of the present work, their simulation uses chemical activation to provide the initial conditions for the reactant molecule. Thus, by reversibility arguments [24], their simulation, in principle, cannot prepare states that will not dissociate. Thus, states that can dissociate will have been preferentially chosen. Despite this difference in the initial conditions and the obvious difference in molecular models, the same rapid dissociation at short times and slow dissociation at long times have been observed in both cases. The differences in the simulations per-
test of RRKM
theory. II
101
haps only weight these two trends differently. Similar considerations also apply to the observations of Ramaswamy et al. [25] in their model simulation of multiphoton dissociation. They found that while the fraction of surviving trajectories decreased exponentially initially, there was a decrease in the rate constant at long times which they interpreted as due to residual unreacted systems being locked into a regular part of phase space instead of sampling the latter more randomly. Thus, such non-RRKM behaviour can reasonably be suspected to be characteristic of isolated systems with vibrational relaxation that is slow compared with dissociation. Time averaged rate constants tend to agree with RRKM theory since they are an average of the time or initial state dependent non-RRKM behaviour described above. Similar insensitivity of averaged rate constants to non-RRKM behaviour has also been observed by Hase and Wolf [24] in their simulation of H-C-C. Thus, by specifying single time-independent rate constants for each internal energy surface, E, details of the internal dynamics have been obscured. Further averaging of course enters in the average over energy distributions which need to be performed to yield such quantities as the unimolecular rate coefficient, k( w, T), in thermal reactions or the apparent rate constant, wD/S, in chemically activated reactions [26,27]. Thus, another reason that thermal unimolecular reactions seem to obey RRKM theory may well be the averaging over the usually broad distribution of energies required to generate k(w, T) [26,27]. Gn the other hand, chemically activated reactions generally have a much narrower distribution of reacting molecules over energy [26,27], so that oD/S may be more sensitive to nonRRKM behaviour. While even the averaging on each internal energy surface may already obscure non-RRKM behaviour there are a significant number of chemically activated unimolecular reactions that exhibited non-RRKM behaviour [28-321. Generally, non-RRKM behaviour is more frequent for systems which are further state selected on each internal energy surface [33-361. It is useful to consider the simulation results in terms of apparent and intrinsic non-RRKM behaviour as defined by Bunker and Hase [3]. Ap-
102
H. W. Schranr et al. /A
parent non-RRKM behaviour is that which may be seen as a result of the non-random excitation of molecular vibrational states. This is by far the most commonly experimentally observed nonRRKM behaviour, partly perhaps because it is more pronounced and partly because of the averaging inherent in most experiments. Examples include the non-random chemical activation of hexafluorobicyclopropyl [28,29], 2-butanone ions (321 and 3-hexyl radicals [37-401 and laser state selected ally1 isocyanide [33]. It appears that some non-randomly excited systems show “a remembrance of things past” [33]. Intrinsic non-RRKM behaviour is that which can be detected even if the internal vibrational states of a molecule are prepared randomly, i.e. microcanonically. This situation can arise from an internal bottleneck in intramolecular energy transfer. This leads to certain transitions between reactant vibrational states being slower than the rate of dissociation. The simulations performed here have microcanonical initial conditions so that the non-RRKM effects that occur can be interpreted as evidence for intrinsic non-RRKM behaviour in our model system. Simulations which prepare the initial states non-randomly as in the case of the impulsive collisional mechanism treated by Baetzold and Wilson [41] or the simulation of chemically activated H-C-C of Hase and Wolf [23,24], can show non-RRKM effects which can be classified as apparent non-RRKM behaviour. Bunker and Hase [3] have performed a detailed simulation of the isomerization of CH,NC. They concluded that CH,NC was an intrinsic nonRRKM molecule, in contrast to experiment [42-471 which implied that CH,NC was an RRKM molecule. They reached this conclusion because of disagreement between the trajectory and RRKM rate constants, the latter tending to overestimate the former. However, examination of their treatment of simulation results in the light of our work indicates a possible source for a significant part of the discrepancy. While they did not employ as rigorous a microcanonical sampling procedure as in our simulations [48], they did employ methods [3] which should, at least approximately, show a correspondence to such sampling. Our results predict that, for a non-RRKM molecule with slow
test of RRKM
theory. II
vibrational relaxation, one should observe rapid dissociations initially, for a high enough energy, and slow dissociations at long times. Since their simulations were performed at high energies (E = 70,100 and 200 kcal/mol compared with E,, = 37.8 kcal/mol) they did indeed observe both the short-time and long-time behaviour. However, in obtaining rate constants from their trajectory lifetime distributions, they neglected the rapid dissociations at short times, assuming them to be an artifact of their not rigorously microcanonical sampling procedure. While this may be true to some extent it is likely that they have also neglected valid contributions to the averaged microscopic rate constant that they derived from the lifetime distributions. Their estimate would, of course, be lower than the true estimate. This means that they may have overestimated the disagreement between trajectory and RRKM theory predictions of k,(E). Our explanation is also consistent with the fact that the disagreement they observe shows a rough increase with energy E, since the initially rapid dissociation at short times becomes more important as the energy is raised. This is not to say that we believe CH,NC to be a true RRKM molecule as implied by experiment. We suspect that CH,NC is characterized by slow vibrational relaxation, consistent with the results of Bunker and Hase [3] but that the non-state-selected and non-time-resolved experiments that are usually performed on thermal unimolecular reactions, such as CH,NC isomerization, are unlikely to be sensitive to such non-RRKM effects. Real molecules are less likely to exhibit intrinsic non-RRKM behaviour than apparent nonRRKM behaviour. The requirement that a molecule not exhibits intrinsic non-RRKM behaviour is less stringent than the requirement of not exhibiting apparent non-RRKM behaviour. The latter restriction requires that, regardless of the pattern of initial energization, the internal dynamics should convert it to a random one in a negligibly short time compared to the timescale of reaction. Of course, real molecules will conform to this restriction in varying degrees, depending on the localization of the internal excitation within the molecule and the strength of internal coupling. Thus, it is not surprising that systems exist that
H. W. Schranz et al. /A
are state selected but confirm to RRKM theory [49,50] while others do not [51-54,351. A system similar to those studied in this simulation was studied by Hutchinson et al. [12]. They chose a 1D linear chain system containing a single Morse bond coupled to a series of harmonic bonds. They noted that by assuming such a linear configuration the coupling in the molecule is increased. This resulted in the individual bonds being poor local modes [55]; the motions tended to be of a global nature and this behaviour favours a treatment by normal modes [5-71. This is consistent with our results which also indicate globality of normal modes unless masses are very different and, even then, some degree. of globality remains. The assumption of collinearity also reduces the number of degrees of freedom in the molecule [12]. Thus, one would expect ergodic behaviour to be favoured in linear systems. An additional consideration, however, is the topology of linear systems compared with more realistic systems. A very long linear chain may well suffer from slow vibrational relaxation of internal energy travelling from one end of the molecule to the other. This would be accentuated by a bottleneck caused by a heavy central mass, as indicated in our simulation results. Hutchinson et al. [12] further contend that IVR does not depend on subtle details of the exact molecular potential surface. They consider their nearest neighbour potential model to be an adequate description and this consideration would also apply to our model, They observed quasiperiodic, almost harmonic, behaviour at low excitation energies. At higher energies the behaviour was anharmonic in character, with internal energy transfer on the timescale of 0.1 ps. This behaviour is similar to that seen in the present simulation. The simulations of Bunker [9] on N,O and 0, produced results in good agreement with RRKM theory. This finding is in contrast to our predictions of non-RRKM effects for such small systems, e.g, triatomic chains. However, the explanation for this apparent impasse is to note that Bunker’s comparisons were all made on the basis of the high-pressure rate coefficient, k,. RRKM theory provides a clear prescription for k, while the lifetime densities generated from simulation could be translated to the unimolecular rate coef-
test
ofRZiKM ficient theory
103
theory. ZZ
k(w)
via Slater’s
new approach
to rate
PI
k(w) =~~~dE~mdth(t;
E) emo’pr(E). (5.2)
In the high-pressure limit, o + co, we-O’ becomes equivalent to a one-sided delta function centered at t = O,, so that k, = E;dWO; I
E) p,(E).
(5.3)
Thus, Bunkers’ comparisons only sampled the short-time behaviour of the lifetime density. From our preceding discussions it is known that at very short times, systems with slow vibrational relaxation can still exhibit the RRKM lifetime density h(t+O;
E) -k,(E),
and hence the RRKM k(t-+0;
E) -+k,(E).
(5.4) rate constant (5.5)
This has also been noted by Hase [2]. Thus, Bunker’s simulations were insensitive to possible slow vibrational relaxation since they were interpreted at high pressures. The agreement at high pressures has also been confirmed in simulations by Thiele [18]. The numerical simulation undertaken above is a logical first step in the continuing effort to understand the internal dynamics in unimolecular reactions. As with similar studies by Thiele [18], Bunker [9,56], and Hase et al. [3,22,23,57,58], we have demonstrated the ability to extract the lifetime probability density, h(t; E), from simulation. This quantity contains more dynamical information than k,(E) and is of value in more sophisticated theories of unimolecular reactions such as Slater’s new approach to rate theory [8] and the separable unimolecular rate theory (SURT) recently proposed by Nordholm [4,59]. The SURT includes the other statistical rate theories as subsets of a more general approach and allows for the incorporation of detailed dynamical information at various levels of complexity to yield predictions about the reaction process. If this step were taken then comparisons with the simple
104
H. W. Schrant ei al. / A test of RRKM
statistical theories, RRKM and Slater’s harmonic theory, could be made and their successes and limitations would be better appreciated. Furthermore, one would have a sophisticated new tool for the accurate prediction of unimolecular reaction rates.
References [l] H.W. Scbranz, S. Nordholm and B.C. Freasier, Chem. Phys. 108 (1986) 69. [2] W.L. Hase, in: Dynamics of molecular collisions, Part B, ed. W.H. Miller (Plenum Press, New York, 1976) pp. 121-169. [3] D.L. Bunker and W.L. Hase, J. Chem. Phys. 59 (1973) 4621. [4] S. Nordholm, Chem. Phys. 10 (1975) 403. [S] E.B. Wilson Jr., J.C. De&s and P.C. Cross, Molecular vibrations (McGraw-Hill, New York, 1955). [6] G. Herzberg, Infrared and Raman spectra of polyatomic molecules (Van Nostrand, Princeton, 1945). [7] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 1980). [8] N.B. Slater, Theory of unimolecular reactions (Methuen, London, 1959). [9] D.L. Bunker, J. Chem. Phys. 37 (1962) 393. [lo] E. Thiele and D.J. Wilson, J. Chem. Phys. 35 (1961) 1256. [ll] R.J. Harter, E.B. Alterman and D.J. Wilson, J. Chem. Phys. 40 (1964) 2137. [12] J.S. Hutchinson, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 79 (1983) 4247. [13] D.W. Noid, M.L. Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267. (141 K. Huang, Statistical mechanics (Wiley, New York, 1963). [15] J. Ford, FundarnentaJ problems in statistical mechanics, Vol. 3, ed. E.G.D. Cohen (North-Holland, Amsterdam, 1975). (161 R. Baleacu, Equilibrium and non-equilibrium statistical mechanics (Wiley, New York, 1975). [17] J.P. Hansen and I.R. McDonald, Theory of simple liquids (Academic Press, New York, 1976). [18] E. Thiele, J. Chem. Phys. 38 (1963) 1959. [19] I. Oref and B.S. Rabinovitch, Accounts Chem. Res. 12 (1979) 166. [20] N. De Leon and B.J. Beme, J. Chem. Phys. 77 (1982) 283. [21] P.V. Sellers and A.J. State, Chem. Phys. 63 (1981) 237. [22] R.J. Wolf and W.L. Hase, J. Chem. Phys. 72 (1980) 316. [23] W.L. Hase and R.J. Wolf, J. Chem. Phys. 75 (1981) 3809. [24] W.L. Hase, R.J. Duchovic, K.N. Swamy and R.J. Wolf, J. Chem. Phys. 80 (1984) 714. [25] R. Ramaswamy, P. Siders and R.A. Marcus, J. Chem. Phys. 74 (1981) 4418. [26] P.J. Robinson and K.A. Holbrook, Unimolecular reactions (Wiley, New York, 1972). [27] W. Forst, Theory of unimolecular reactions (Academic Press, New York, 1973).
theory. II
[28] J.D. Rynbrandt and B.S. Rabinovitch, J. Chem. Phys. 54 (1971) 2275. [29] J.D. Rynbrandt and B.S. Rabinovitch, J. Phys. Chem. 75 (1971) 2164. [30] P. Rogers, D.C. Montague, J.P. Frank, SC. Tyler and F.S. Rowland, Chem. Phys. Letters 89 (1982) 9. [31] P.J. Rogers, J.I. Selco and F.S. Rowland, Chem. Phys. Letters 97 (1983) 313. [32] D.J. McAdoo and C.E. Hudson, J. Phys. Chem. 87 (1983) 2451. [33] K.V. Reddy and M.J. Berry, Chem. Phys. Letters 66 (1979) 223. [34] E. Thiele, M.F. Goodman and J. Stone, Chem. Phys. Letters 69 (1980) 18. [35] J.B. Hopkins, D.E. Powers and R.E. Smalley, J. Chem. Phys. 74 (1981) 745. [36] J.E. Adams, J. Chem. Phys. 78 (1983) 1275. (371 A.B. Trenwith, B.S. Rabinovitch and F.C. Wolters, J. Chem. Phys. 76 (1982) 1586. [38] A.B. Trenwith and B.S. Rabinovitch, J. Phys. Chem. 86 (1982) 3447. [39] S.P. Wrigley, K.W. Smith and B.S. Rabinovitch, Chem. Phys. 75 (1983) 453. [40] S.P. Wrigley and B.S. Rabinovitch, Chem. Phys. Letters 95 (1983) 363. [41] R.C. Baetzold and D.J. Wilson, J. Chem. Phys. 43 (1965) 4299. [42] F.W. Schneider and B.S. Rabinovitch, J. Am. Chem. Sot. 84 (1962) 4215. [43] B.S. Rabinovitch, Y.N. Lin, S.C. Chan and K.W. Watkins, J. Phys. Chem. 71 (1967) 3715. [44] Y.N. Lin, S.C. Chan and B.S. Rabinovitch, J.Phys. Chem. 72 (1968) 1932. [45] K.M. Maloney and B.S. Rabinovitch, J. Phys. Chem. 73 (1969) 1652. [46] S.C. Chan, J.T. Bryant and B.S. Rabinovitch, J. Phys. Chem. 74 (1970) 2055. [47] S.C. Chan, J.T. Bryant, L.D. Spicer and B.S. Rabinovitch, J. Phys. Chem. 74 (1970) 2058. [48] E.S. Severin, B.C. Freasier, N.D. Hamer, D.L. Jolly and S. Nordholm, Chem. Phys. Letters 57 (1978) 117. [49] J.C. Stephenson, S.E. Biakowski, D.S. King, E. Thiele, J. Stone and M.F. Goodman, J. Chem. Phys. 74 (1981) 3905. [50] H. Reisler, F. Kong, A.M. Renhmd and C. Wittig, J. Chem. Phys. 76 (1982) 997. [51] K.E. Johnson, L. Wharton and D.H. Levy, J. Chem. Phys. 69 (1978) 2719. [52] J.E. Kenny, D.V. Brumbaugb and D.H. Levy, J. Chem. Phys. 71 (1979) 4757. [53] K.V. Reddy and M.J. Berry, Chem. Phys. Letters 66 (1979) 223. [54] E. Thiele, M.F. Goodman and J. Stone, Chem. Phys. Letters 69 (1980) 18. [55] B.R. Henry, Accounts Chem. Res. 10 (1977) 207. [56] D.L. Bunker, J. Chem. Phys. 40 (1964) 1946. 1571 W.L. Hase, Chem. Phys. Letters 67 (1979) 263. [58] R.J. Wolf and W.L. Hase, J. Chem. Phys. 73 (1980) 3779. [59] S. Nordholm, Chem. Phys. 15 (1976) 59.
Chemical Physics 108 (1986) 93-104 North-Holland. Amsterdam
A TEST OF RRKM THEORY AGAINST NUMERICAL SIMULATION FOR CLASSICAL CHAIN M0LEtXIX.S. II. DECOMPOSITION AND VIBRATIONAL RELAXATION IN UNIFORM
CHAINS
Harold W. SCHRANZ, Sture NORDHOLM Department
of Theoretical Chemistry
University of Sydney, N.S. W. 2006, Australia
and Ben C. FREASIER Department
of Chemistry, Faculty of Military Studies, University of New South Wales, Duntroon, A. C. T 2600, Australia
Received 24 June 1985; in final form 14 March 1986
Deviations from RRKM predictions for the fragmentation of a one-dimensional chain molecule are here studied in detail by numerical simulation and related to the rate of the vibrational relaxation. A measure of the degree of completion of vibrational relaxation is constructed, and its time development followed and compared with the progress of the reaction. The vibrational relaxation is initially rapid but then slows down as the reaction depletes the ensemble of rapidly relaxing trajectories. The observed decay rate starts from the RRKM value and then typically exceeds it at intermediate times crossing over to fall below the RRKM prediction at long times.
1. Introduction
In a preceding paper [l] we presented a method of simulation designed to test the validity of the assumptions of the RRKM theory of unimolecular reactions. Specifically, we wanted to determine whether the decomposition of a molecule microcanonically excited to some energy E above the applicable threshold would decay exponentially with a rate coefficient in accord with the RRKM prediction. The answer will, of course, depend upon the system. We choose to study classical one-dimensional chains for reasons of simplicity. The chains consisted of from three to ten particles of equal mass interacting by identical Morse potentials with neighboring particles in the chain. Initial states were chosen microcanonically at a set of energies above but close to the dissociation energy. Bond breakage was observed and recorded as a lifetime to dissociation from which lifetime probability densities and rate coefficients were constructed. The results obtained for uniform
chain molecules bound by Morse bond potentials led to the following conclusions: (i) anharmonic effects on the relevant densities of states of the chain are substantial, (ii) a proper anharmonic implementation of RRKM theory yields rate coefficients k,(E) in reasonable agreement with simulation, (iii) the deviations between RRKM and simulation diminishes as the energy decreases towards threshold and the number of atoms in the chain increases. The main aim of the present report is to facilitate the interpretation of these results by studying the relative rates of decomposition and vibrational relaxation in greater depth and detail. A time-dependent measure of the vibrational relaxation in an ensemble of trajectories initially in microcanonical equilibrium is constructed. Its observed time dependence is then compared with the corresponding time dependence of the decomposition as observed and as predicted by RRKM theory. The role of the vibrational relaxation rate in de-
0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H. W. Schranr et al. / A test of RRKM
94
termining the validity of the RRKM approximations is thereby illustrated. There is a large body of literature relevant to the general aims of our work [2-41. An extensive, but far from complete, survey is contained in the first report [l]. In our final section of this article we shall discuss our present results in the light of earlier work on the same topic.
theory. II
mate for anharmonic systems at low energies and very inaccurate for anharmonic systems at high energies. Each term in the normal mode hamiltonian corresponds to the hamiltonian for a simple harmonic oscillator with angular frequency, w,. Fig. 1 illustrates the character of the normal modes of the linear chain system. The frequency vi is related to the angular frequency oi as, vi = w/2??.
2. Analysis of intnunokcular
vibrational relaxation
The systems to be studied here are uniform linear chains of N particles with N between 2 and 10. The hamiltonian is
Hz
N
N-l
PZ
iFl G
+
C +(Xi+l -xi),
(2.1)
i-l
with the nearest neighbor interaction type, G(r) =D{exp[-/?(xi+i-xi-r,)]
-lj2.
of Morse
(2.2)
We will use units such that D = 1, and the time unit is 1.47 X lo-l3 s. This information will suffice for our present purposes. A more detailed discussion of the chain models is given in the first article in this series [l]. 2.1. Normal mode analysis It is useful to consider the normal modes of the general N particle linear chain in one dimension. They will be used in order to evaluate a parameter that may reveal the extent of intramolecular vibrational relaxation during the reaction process. Extensive treatments of normal modes as applied to the vibrations of molecules are given in texts by Wilson et al. [S], Herzberg [6], Goldstein [7], and Slater [ 81. It is easily shown that the normal mode hamiltonian can be written in the form, (2.3) This hamiltonian is exact for a system with harmonic bond potentials, but is only approxi-
(2.4)
The length and direction of the arrows in fig. 1 represent the relative amplitudes of the motions of the atoms, in mass weighted coordinates [5-71, that contribute to the particular normal mode. Note that for uniform chain systems, the normal modes tend to be global in nature and have motions associated with all the atoms in the molecule. 2.2. A vibrational relaxation parameter In order to gain some idea of the rate of intramolecular energy transfer in the linear chain molecules studied, a parameter that can measure the extent of intramolecular vibrational relaxation (IVR) will be introduced. It is based on the fact that harmonic systems with more than one bond (N > 2) are non-ergodic [9] and have normal modes with well defined energies. The energy of each normal mode, ci(t), can be identified with an individual term in the normal mode hamiltonian in eq. (2.3), q(t)
= )(if
+ wi’zf),
(2.5)
where the frequencies, wi, and the transformation from cartesian coordinates to normal mode coordinates can be determined by a normal mode analysis [5-71. Anharmonic systems, on the other hand, are associated with a greater degree of ergodicity and considerable energy fluctuation among the normal modes [lo-121. This fluctuation of normal mode energies in time reflects the more chaotic movement of the trajectory in phase space compared to harmonic systems. If, over a long enough period, the time average of the normal mode energies was equal to the microcanonical average of normal
H. W. Schranz et al. /A (a) N = 2.
laxation (IVR) parameter which measures the deviation from equality from eq. (2.6),
Normalmoder
4
95
test of RRKM theory. II
where
ch(t) uwNR(r) =f%(f) 1PI&),
and the microcanonical average of f( t ) over Na( t ) trajectories can be defined as
(c) N = 4.
(2.9) [ j=l where Na(t) is the number of trajectories still in reactant space (i.e., the number of molecules still bound) at time t. We consider several model cases to illustrate the possible behaviour of the IVR parameter defined in (2.7). For a harmonic system the normal modes all have well defined energies,
(d) N = 5.
Fig. 1. The normal modes of motion (translation and vibration) forunifonnchains(mi=l, i-l,...,N):(a) N=2,(b) N=3, (c) N = 4 and (d) N = 5. Arrows representing amplitudes less than 0.1 are omitted for clarity.
G(t)=ci,
i=2 ,..., N,
where the ci are constants. From the formulation in eqs. (2.7)-(2.9), (2.11)
(G)=l, mode energies then one would expect this to be a reflection of ergodic behaviour in the system [13]. Consider a linear chain system with N atoms. It has N normal modes, one external translational mode which shall be ignored, and N - 1 internal vibrational normal modes which according to eq. (2.5) have energies, {
=
Mt))t
=fjddsq(s),
i=2 ,..., N,
(2.6)
where ( )MC is the microcanonical average. The degree to which this equation is satisfied should give some idea of the degree of ergodicity and vibrational relaxation in the chain system. One can formulate an intramolecular vibrational re-
(2.10)
for a harmonic system. For a system which has all internal normal modes approaching microcanonical equilibrium with a rate constant X, L\Ei(t) =&(O)
emX’, i = 2 ,..., N,
(2.12)
then (Z)
= eeX’.
(2.13)
For systems which are only partially ergodic/ non-ergodic one would expect behaviour between these extremes. The normal modes may exhibit different and perhaps non-exponential rates of attainment of microcanonical equilibrium. Perhaps only partial microcanonical equilibrium is atained. In such a case the normal mode parameter (AC) would perhaps follow the behaviour, (z)
= XE emX’+ (I - Xn),
(2.14)
96
H. W. Schranz et al. /A
where xE is a crude measure of the proportion of exponential relaxation with rate constant A. This equation defines exponential relaxation to a nonzero plateau and hence will be termed the plateau-exponential model. The larger this plateau, the more the behaviour corresponds to the harmonic prediction (2.11) and this implies that less microcanonical equilibrium will have been achieved during the reaction process. On the other hand, the lower this plateau, the more the behaviour corresponds to complete microcanonical equilibrium (in normal mode space) which is achieved with a rate constant A, given in eq. (2.13). It will be useful to obtain an average rate constant h for relaxation by fitting the observed behaviour of (E) to eq. (2.13). The IVR rate constant thus obtained is compared to the RRKM and the simulation derived decomposition rate constants in section 3. While a fit to eq. (2.14)
test of RRKM theory. II
would be more realistic, it is the rate of overall microcanonical relaxation that is of interest and hence the former type of fit is more directly relevant. Of course, both types of fits are simplifications of the real behaviour and it is for this reason that direct comparisons of (z) and decomposition are used in the presentation of results in section 4.
3. Comparison of timedependent with RRKM for uniform chains
rate constants
For detailed comparisons of RRKM and simulation it is necessary to consider the behaviour of the time-dependent rate constant k(t; E). Figs. 2-4 show the dependence of the comparison upon molecular size, N, and energy, E. The number of trajectories calculated for each N and E condition
C
N-7
E-
1.5
0
0
60
120
*
160
240
300
0
160
360
,
540
720
900
z d LSFS RRKH
b
N-5
E - 1.5
0
0
12
24
,
36
46
60
Fig. 2. Comparison of Morse RRKM rate constants, least-squares fitted simulation rate constants (LSFS), and IVR rate constants with the time-dependent rate constant k(t; E) (SIM) as a function of chain length N for uniform chains with E = 1.5 and &( r, - rc) = 10: (a) N = 3, (b) N = 5, (c) N = 7 and (d) N = 10.
H. W. Schronz et 01. /A
d
E
N.3
.._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-l
mined from the simulations, tends to reflect the rate of vibrational relaxation compared to the rate of dissociation. When this ratio is small (for small molecules or high energies) the behaviour is very non-RRKM but when this ratio is large one generally finds RRKM behaviour.
1.1
I
test of RRKM theory. ZZ
!vR................
‘I
RRKM LSFS SIM
OL
-
.
-
,
60
80
100
Fig. 3. As for fig. 2 but for a N = 3 uniform /3M(rC-rc)=10and E=l.l.
cllain with
0
20
40
-
has been listed in table 3 of our first report [l]. It is usually 500. In general, small molecules at high energies exhibit rapid dissociations at short times and slow dissociations at long times compares with RRKM predictions. The agreement with RRKM, however, improves for larger N or lower E. The IVR value superimposed is an estimate of the rate constant for vibrational relaxation obtained by the procedure in section 2.2. One assumes all normal modes are approaching microcanonical equilibrium with the same rate constant h and this leads to a simple-exponential form for the relaxation parameter, (AC), ~(2.7). By fitting to the observed behaviour of (AC) this yields a value for A which is labelled as IVR (internal vibrational relaxation) in figs. 2-4. The ratio IVR/LSFS, where LSFS stands for the least-squares fitted decay rate coefficient ob-
2 d
N-r!
E = 1.25
. . . . . . . . . .._...............
_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IVR ... ... ... .... .. .... .
G i’
.
:
RRKM
O?
.
0
16
32
-
-
,
40
-
64
Fig. 4. As for fig. 2 but for a N = 5 uniform &,,M(r, - rc) = 10 and E = 1.25.
80 chain with
4. A comparison of the degree of normal made relaxation with the degree of diisociation for uniform chains In figs. 5-10 we compare the proportion of dissociations, Na/Ns, to the degree of normal mode relaxation, as defined by (AC), as a function of time, t, for various uniform chain systems. These quantities are competing kinetic processes. In a RRKM molecule with a microscopic rate constant k,
NJN,
= eekr.
(4.1)
In similar fashion, if all the normal mode energies relax to their microcanonical values with the same rate constant h, then (as shown in section 2.2), (E)
= e-“.
(4.2)
Thus,> a crude fashion, comparison of Na/Ns and (AC) will yield a comparison of k, the average rate constant for dissociation, and h, the average rate constant for IVR (as represented by the rate at which normal mode energies attain their microcanonical values). In fi&. 5-8 we present a comparison of NJNs and (AC) for different molecular size N and energy E. The general trend is that one gets nonRRKM behaviour if NJzs = (G) and RRKM behaviour for Na/Ns 3 (AC). This agrees with the comparisons of rate constants presented in section 3. Thus, for uniform chain systems, non-RRKM behaviour is expected for small molecules at high energies and RRKM behaviour for large molecules at low energies. Figs. 9 and 10 show the effect of en_ergy on the normal mode relaxation parameter, (AC). At low energies, E = 0.5, the amount of relaxation is low but increases at E = 1.0. However, once reaction begins (E > Eo) less IVR has time to occur and
98
H. W. Schranr et al. / A test of RRKM theory. II
b
E.I.s
N=S
8
0
..--
....
a*----.-.
60
‘tO
20
.
0..
a.0
0
d
N = 10 -.
200
E = 1.5 -.
400
-
,
..-
600
..
.
800
.
.
1000
Fig. 5. Comparison of the proportion of dissociations Na/Ns and the normal mode IVR parameter (z) as a function of chain length N for uniform chains with E = 1.5 and &,( r, - re) = 10: (a) N = 3, (b) N = 5, (c) N = 7 and (d) N = 10.
this is illustrated by the lower relaxation at E = 1.25 and E = 1.5. A longer time comparison for the reactive energies, fig. 10, reveals that the amount of relaxation at E = 1.25 is greater than that for E = 1.5. Examination of NR/Ns reveals that much more reaction occurs on the same timescale at E = 1.5 than at E = 1.25. This suggests
N-3
E-
that dissociation competes with the IVR process. At higher energies, more phase space is close to a dissociation channel and hence irreversible dissociation is more likely than traversing phase space in an ergodic manner. The general shape of the normal model IVR parameter, (z>, plotted versus time t, as in figs.
1.1
.-----..--...-.-..
0,
0
10
20
90
40
50
Fig. 6. As for fig. 5 but for a N = 3 uniform chain with &(rC - re) = 10 and E = 1.1.
Fig. 7. As for fig. 5 but for a N= 5 uniform chain with &(r, - re) = 10 and E = 1.25.
H. W. Schranz et al. /A
01..
.
0
200
-. 400
-. ,
600
-. - 600
test of RRKiU theory II
99
1 1000
0
20
'P
60
60
Fig. 10. As for fig. 9 but for the dissociating trajectories (E =1.25,1.5) at longer times. 0
100
200
t
300
400
500
Fig. 8. As for fig. 5 but as a function of energy E for a N = 10 uniform chain with &(r, - rc) = 4: (a) E =1.25 and (b) E = 1.5.
S-10, roughly corresponds to a plateau-exponential model of the type considered in section 2.2. At low energies, where the initial rapid dissociations are insignificant, it appears that trajectories showing the fastest vibrational relaxation are preferentially eliminated by dissociation. Ibis results in a rapid initial drop in (AC). The remaining trajectories show a smaller degree of vibrational relaxation, take more time to reach a transition state configuration (if at all) and hence tend to be longe~lived. The result is a much slower variation in (AC) at long times. The relative size of the plateau at long times can be considered as a crude measure pf the extent of non-ergodicity at that time.
5. Diiion
0
2
’
I
6
6
10
Fig. 9. The behaviour of the proportion of_dissociations N&N, and the normal mode Im parameter (AC) at various energies for a uniform chain with N = 5 and &(rC - rs) = 10: E = 0.5, 1.0, 1.25, 1.5.
and condusions
It has been found, that for uniform chains, that RRKM effects are greater for large molecules and low internal energies while non-RRKM effects are greater for small molecules and high energies. This behaviour has also been observed for chain systems with masses that are not very different. The trend with internal energy is similar to but more
100
H. W. Schmnz et al. /A
detailed than that observed by Thiele [18] for a model calculation on a linear symmetric triatomic molecule with Morse bond potentials. He based his theoretical description on the gap distribution, g(t; E), which can be related to the lifetime distribution, h(l; E). Note first that any initial reactant state which leads to dissociation must lie on a trajectory which is a closed loop, beginning at the critical surface with the formation of the molecule and then after the time s (the gap) ending with the dissociation. Thus, if g(s; E) is the gap distribution over all such reactive loops, the number of initial states which lead to the dissociation after time t is proportional to the total number of loop trajectories such that s > t, i.e..
1, -1
X
*dt [j 0
J
md4s;
t
E)
(5.1)
where x = j,“dt h (t; E ). For an ergodic surface, x = 1, but if the surface is non-ergodic x represents the fraction of dissociating states. Thiele noted that g(t; E) approached microcanonical (random) behaviour as the internal energy, E, approached E,, from above, while it became more regular (non-random) with increasing internal energy. By estimating the rate of IVR using a simple model based on the behaviour of normal modes, the above trends could be explained in terms of the competition between IVR and dissociation. If IVR is much faster than dissociation, RRKM behaviour is expected and this appears to be the case for long chains or low energies. This is consistent with the experimental deductions that large reactive systems with h,,, > lOi* s-l should behave ergodically [19]. On the other hand, if IVR is on the same timescale or slower than dissociation, non-RRKM behaviour is expected and this is the case for small chains or high energies. This point has also been made, e.g., recently by De Leon and Berne [20] in the context of a model isomerization reaction. They note that, there must be an effective bottleneck to reaction at the transition state, for microcanonical rate constants in general, and
test of RRKM theory. II
the RRKM rate constant in particular, to be valid. This is generally true only for energies close to the barrier maximum, E,, since, at high energies, the transition state is usually much too wide for a sufficient separation in timescales between intramolecular vibrational relaxation and dissociation. In the present work the effort has been focused on dissociation from a given internal energy surface in the hope that the corresponding rate constants would clearly show the extent of nonRRKM behaviour without the complicating effect of averaging over an energy distribution. However, it was found that the extent to which these rate constants are averaged over time also effects their sensitivity to non-RRKM behaviour. When a linear chain system exhibits non-RRKM behaviour, it is more obvious from observing timedependent rate constants than averaged time-independent rate constants. Time-dependent (or timeresolved) rate constants more clearly show the initial state dependence of the rate of dissociation. A similar point has been made by Sellers and State [21] in their simulation of a thermal unimolecular reaction. They noted that the use of the microcanonical rate constant to give an average lifetime for an activated molecule did not provide an adequate description of the entire distribution of lifetimes. In particular, calculated results in the fall-off region were found to be sensitive to details of the short lifetime tail of the distribution. In our simulations we found that non-RRKM behaviour in the time-dependent rate constant takes two forms: (1) Rapid dissociation at short times. This is more prevalent at high internal energies (or small molecules) and less so at low energies (or large molecules). (2) Slow dissociation at long times. This tends to persist to lower energies (or larger molecules) than type (1) above. This behaviour can be understood as follows. Note first that at t = 0 there must be agreement between RRKM theory and simulation since in both cases the reactant molecule is taken to be in microcanonical equilibrium. The deviations from microcanonical equilibrium which arise for l> 0 in the simulation may favour decomposition as
H. W. Schranz et al. /A
clearly illustrated for N = 2 in our first report [l], or they may disfavour decomposition. The latter would clearly be the case at long times for a non-ergodic system where not all states lead to decomposition. The former tends to dominate at short times and high energy. Rapid vibrational relaxation tends to blur these two types of behaviour and produce agreement with the RRKM predictions based on microcanonical equilibrium remaining throughout the decomposition. Similar behaviour has been observed in a series of recent 3D simulations by Hase and Wolf [22-241 on the dissociation of H-C-C. They compared the effect of different initial conditions. If H-C-C is excited randomly they found that the dissociation probability is less than RRKM predictions because the prepared microcanonical ensemble included regular trajectories which are trapped in phase space [22]. This trend is similar to the slow dissociation found at long times in our simulations being more pronounced for smaller chains and higher energies and could be enhanced by addition of a heavy central mass. The fact that randomly prepared H-C-C contains a significant subset of regular trajectories [22] is not too surprising as earlier simulations by Harter et al. [ll] on a 1D chain system showed that C-H stretching and C-C stretching motions do not readily exchange energy. Despite being derived for a 1D chain system this conclusion should also be valid for the 3D system of Hase and Wolf [22], since, the collinear case should allow the most efficient energy transfer possible between bonds ]121. Wolf and Hase also consider the dissociation of chemically activated H-C-C [23,24]. Unlike the microcanonical initial conditions of the present work, their simulation uses chemical activation to provide the initial conditions for the reactant molecule. Thus, by reversibility arguments [24], their simulation, in principle, cannot prepare states that will not dissociate. Thus, states that can dissociate will have been preferentially chosen. Despite this difference in the initial conditions and the obvious difference in molecular models, the same rapid dissociation at short times and slow dissociation at long times have been observed in both cases. The differences in the simulations per-
test of RRKM
theory. II
101
haps only weight these two trends differently. Similar considerations also apply to the observations of Ramaswamy et al. [25] in their model simulation of multiphoton dissociation. They found that while the fraction of surviving trajectories decreased exponentially initially, there was a decrease in the rate constant at long times which they interpreted as due to residual unreacted systems being locked into a regular part of phase space instead of sampling the latter more randomly. Thus, such non-RRKM behaviour can reasonably be suspected to be characteristic of isolated systems with vibrational relaxation that is slow compared with dissociation. Time averaged rate constants tend to agree with RRKM theory since they are an average of the time or initial state dependent non-RRKM behaviour described above. Similar insensitivity of averaged rate constants to non-RRKM behaviour has also been observed by Hase and Wolf [24] in their simulation of H-C-C. Thus, by specifying single time-independent rate constants for each internal energy surface, E, details of the internal dynamics have been obscured. Further averaging of course enters in the average over energy distributions which need to be performed to yield such quantities as the unimolecular rate coefficient, k( w, T), in thermal reactions or the apparent rate constant, wD/S, in chemically activated reactions [26,27]. Thus, another reason that thermal unimolecular reactions seem to obey RRKM theory may well be the averaging over the usually broad distribution of energies required to generate k(w, T) [26,27]. Gn the other hand, chemically activated reactions generally have a much narrower distribution of reacting molecules over energy [26,27], so that oD/S may be more sensitive to nonRRKM behaviour. While even the averaging on each internal energy surface may already obscure non-RRKM behaviour there are a significant number of chemically activated unimolecular reactions that exhibited non-RRKM behaviour [28-321. Generally, non-RRKM behaviour is more frequent for systems which are further state selected on each internal energy surface [33-361. It is useful to consider the simulation results in terms of apparent and intrinsic non-RRKM behaviour as defined by Bunker and Hase [3]. Ap-
102
H. W. Schranr et al. /A
parent non-RRKM behaviour is that which may be seen as a result of the non-random excitation of molecular vibrational states. This is by far the most commonly experimentally observed nonRRKM behaviour, partly perhaps because it is more pronounced and partly because of the averaging inherent in most experiments. Examples include the non-random chemical activation of hexafluorobicyclopropyl [28,29], 2-butanone ions (321 and 3-hexyl radicals [37-401 and laser state selected ally1 isocyanide [33]. It appears that some non-randomly excited systems show “a remembrance of things past” [33]. Intrinsic non-RRKM behaviour is that which can be detected even if the internal vibrational states of a molecule are prepared randomly, i.e. microcanonically. This situation can arise from an internal bottleneck in intramolecular energy transfer. This leads to certain transitions between reactant vibrational states being slower than the rate of dissociation. The simulations performed here have microcanonical initial conditions so that the non-RRKM effects that occur can be interpreted as evidence for intrinsic non-RRKM behaviour in our model system. Simulations which prepare the initial states non-randomly as in the case of the impulsive collisional mechanism treated by Baetzold and Wilson [41] or the simulation of chemically activated H-C-C of Hase and Wolf [23,24], can show non-RRKM effects which can be classified as apparent non-RRKM behaviour. Bunker and Hase [3] have performed a detailed simulation of the isomerization of CH,NC. They concluded that CH,NC was an intrinsic nonRRKM molecule, in contrast to experiment [42-471 which implied that CH,NC was an RRKM molecule. They reached this conclusion because of disagreement between the trajectory and RRKM rate constants, the latter tending to overestimate the former. However, examination of their treatment of simulation results in the light of our work indicates a possible source for a significant part of the discrepancy. While they did not employ as rigorous a microcanonical sampling procedure as in our simulations [48], they did employ methods [3] which should, at least approximately, show a correspondence to such sampling. Our results predict that, for a non-RRKM molecule with slow
test of RRKM
theory. II
vibrational relaxation, one should observe rapid dissociations initially, for a high enough energy, and slow dissociations at long times. Since their simulations were performed at high energies (E = 70,100 and 200 kcal/mol compared with E,, = 37.8 kcal/mol) they did indeed observe both the short-time and long-time behaviour. However, in obtaining rate constants from their trajectory lifetime distributions, they neglected the rapid dissociations at short times, assuming them to be an artifact of their not rigorously microcanonical sampling procedure. While this may be true to some extent it is likely that they have also neglected valid contributions to the averaged microscopic rate constant that they derived from the lifetime distributions. Their estimate would, of course, be lower than the true estimate. This means that they may have overestimated the disagreement between trajectory and RRKM theory predictions of k,(E). Our explanation is also consistent with the fact that the disagreement they observe shows a rough increase with energy E, since the initially rapid dissociation at short times becomes more important as the energy is raised. This is not to say that we believe CH,NC to be a true RRKM molecule as implied by experiment. We suspect that CH,NC is characterized by slow vibrational relaxation, consistent with the results of Bunker and Hase [3] but that the non-state-selected and non-time-resolved experiments that are usually performed on thermal unimolecular reactions, such as CH,NC isomerization, are unlikely to be sensitive to such non-RRKM effects. Real molecules are less likely to exhibit intrinsic non-RRKM behaviour than apparent nonRRKM behaviour. The requirement that a molecule not exhibits intrinsic non-RRKM behaviour is less stringent than the requirement of not exhibiting apparent non-RRKM behaviour. The latter restriction requires that, regardless of the pattern of initial energization, the internal dynamics should convert it to a random one in a negligibly short time compared to the timescale of reaction. Of course, real molecules will conform to this restriction in varying degrees, depending on the localization of the internal excitation within the molecule and the strength of internal coupling. Thus, it is not surprising that systems exist that
H. W. Schranz et al. /A
are state selected but confirm to RRKM theory [49,50] while others do not [51-54,351. A system similar to those studied in this simulation was studied by Hutchinson et al. [12]. They chose a 1D linear chain system containing a single Morse bond coupled to a series of harmonic bonds. They noted that by assuming such a linear configuration the coupling in the molecule is increased. This resulted in the individual bonds being poor local modes [55]; the motions tended to be of a global nature and this behaviour favours a treatment by normal modes [5-71. This is consistent with our results which also indicate globality of normal modes unless masses are very different and, even then, some degree. of globality remains. The assumption of collinearity also reduces the number of degrees of freedom in the molecule [12]. Thus, one would expect ergodic behaviour to be favoured in linear systems. An additional consideration, however, is the topology of linear systems compared with more realistic systems. A very long linear chain may well suffer from slow vibrational relaxation of internal energy travelling from one end of the molecule to the other. This would be accentuated by a bottleneck caused by a heavy central mass, as indicated in our simulation results. Hutchinson et al. [12] further contend that IVR does not depend on subtle details of the exact molecular potential surface. They consider their nearest neighbour potential model to be an adequate description and this consideration would also apply to our model, They observed quasiperiodic, almost harmonic, behaviour at low excitation energies. At higher energies the behaviour was anharmonic in character, with internal energy transfer on the timescale of 0.1 ps. This behaviour is similar to that seen in the present simulation. The simulations of Bunker [9] on N,O and 0, produced results in good agreement with RRKM theory. This finding is in contrast to our predictions of non-RRKM effects for such small systems, e.g, triatomic chains. However, the explanation for this apparent impasse is to note that Bunker’s comparisons were all made on the basis of the high-pressure rate coefficient, k,. RRKM theory provides a clear prescription for k, while the lifetime densities generated from simulation could be translated to the unimolecular rate coef-
test
ofRZiKM ficient theory
103
theory. ZZ
k(w)
via Slater’s
new approach
to rate
PI
k(w) =~~~dE~mdth(t;
E) emo’pr(E). (5.2)
In the high-pressure limit, o + co, we-O’ becomes equivalent to a one-sided delta function centered at t = O,, so that k, = E;dWO; I
E) p,(E).
(5.3)
Thus, Bunkers’ comparisons only sampled the short-time behaviour of the lifetime density. From our preceding discussions it is known that at very short times, systems with slow vibrational relaxation can still exhibit the RRKM lifetime density h(t+O;
E) -k,(E),
and hence the RRKM k(t-+0;
E) -+k,(E).
(5.4) rate constant (5.5)
This has also been noted by Hase [2]. Thus, Bunker’s simulations were insensitive to possible slow vibrational relaxation since they were interpreted at high pressures. The agreement at high pressures has also been confirmed in simulations by Thiele [18]. The numerical simulation undertaken above is a logical first step in the continuing effort to understand the internal dynamics in unimolecular reactions. As with similar studies by Thiele [18], Bunker [9,56], and Hase et al. [3,22,23,57,58], we have demonstrated the ability to extract the lifetime probability density, h(t; E), from simulation. This quantity contains more dynamical information than k,(E) and is of value in more sophisticated theories of unimolecular reactions such as Slater’s new approach to rate theory [8] and the separable unimolecular rate theory (SURT) recently proposed by Nordholm [4,59]. The SURT includes the other statistical rate theories as subsets of a more general approach and allows for the incorporation of detailed dynamical information at various levels of complexity to yield predictions about the reaction process. If this step were taken then comparisons with the simple
104
H. W. Schrant ei al. / A test of RRKM
statistical theories, RRKM and Slater’s harmonic theory, could be made and their successes and limitations would be better appreciated. Furthermore, one would have a sophisticated new tool for the accurate prediction of unimolecular reaction rates.
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