A test of RRKM theory against numerical simulation for classical chain molecules. III. Heavy mass barrier to intramolecular vibrational relaxation

105

Chemical Physics 108 (1986) 105-114 North-Holland, Amsterdam

A TEST OF RRKM THEORY AGAINST NUMERICAL FOR CLASSICAL CHAIN MOLECULES. III. HEAVY MASS BARRIER TO INTRAMOLECUUR

SIMULATION VIBRATIONAL

RELAXATION

Harold W. SCHRANZ and Sture NORDHOLM Department

of Theoretical Chemistry,

University of Sydney, N.S. W. 2006, Australia

and Ben C. FREASIER Department

of Chemistty,

Faculty of Militav

Studies, University of New South Wales, Duntroon, A.C. T, 2600, Australia

Received 15 July 1985; in final form 9 May 1986

Previously reported simulations of uniform classical chain molecules in one dimension are here extended to consider the effect of non-uniformities in the chains on the dissociation and vibrational relaxation rates. Non-uniformities are introduced by increasing the mass of the central atom and/or by making all bonds except one at the end twice as strong The validity of the RRRM theory is found to be adversely affected by non-uniformities in the chain.

1. Intruduction

In previous papers in this series [1,2] we used the method of numerical simulation to examine the validity of the RRKM model of the internal dynamics. RRKM theory eliminates a detailed treatment of the internal dynamics by assuming microcanonical equilibrium. This requires that the reactant phase space is ergodically covered in a time that is short compared with the reaction time. In this paper we extend our simulations to non-uniform chains. We examine the case when one end-bond is weak and all the others twice as strong. We also consider how a heavy central mass can inhibit the extent of intramolecular energy transfer in a linear chain. Similar simulations have recently been performed by Lopez and Marcus [3] and by Swamy and Hase [4] who considered energy transfer within and between chains tetrahedrally attached to a central atom. Experimentally, the question of heavy mass blocking of intramolecular vibrational relaxation (IVR) is still controversial. Rogers, Rowland and co-workers [5,6] have reported the non-RRKM decomposition of

chemically activated radicals with a tin or a germanium central mass. However, Wrigley and Rabinovitch [7,8] report that, for similar systems containing tin or lead central masses, energy randomization was still found to occur on a subpicosecond timescale. It appears that the rate of energy transfer between chains may not only be related to the mass of the central atom but sensitive to the detailed composition of the chains [4]. While the experiments above cannot be directly addressed in the present model the results we generate are relevant and suggestive. Finally, we have carried out simulations where a heavy central mass occurs in a chain with a single weak bond, the others being twice as strong.

2. Effects of heavy central masses on rate constants

In a previous paper [2] we considered the normal modes for uniform chains. These tended to be global in nature and had motions associated with all the atoms in the molecule. However, once a

0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

106

H. W. Schranz et al. / A test of RRKM theory. III

heavy central mass is introduced, the normal modes, figs. 1 and 2, tend to localize the relaxation to the side chains around the central mass. The uniform chain systems discussed in the previous section were characterized by global normal modes which facilitate relaxation, and hence one might expect very efficient IVR for a chain system. Here we consider the effect of a heavy central mass on IVR and on the rate of dissociation. The simulation method has been described in a preceding paper [l]. The number of trajectories calculated for each condition (N, E, m,) is typically 500. Since we are simulating an isolated molecule the relevant quantity in RRKM theory is k,(E), the specific decay rate coefficient for the energy

(a) m, = 2.

surface of energy E. We estimate this quantity in a manner described in our earlier articles such that anharmonic effects on the density of states are correctly accounted for. The corresponding rate coefficients are labelled RRKM in the figures below. The simulations yield lifetime probability densities which we reinterpret in the form of a time-dependent rate coefficient k( f; E) denoted SIM in the figures below. If we ignore the non-exponential character of the lifetime probability density, we can summarize the simulation in the form of a least-squares fitted rate coefficient denoted LSFS in the figures. It is an average value of k(t; E) which can be compared with the RRKM value of k,(E). In order to provide a basis for an

(a) m,

= 2.

Normal modes

6

o.ooo

o---+

0.99G

o-----+

1.408

o-

O0

O-0

pi%Jpg

O-

-0

(b) mC = 4.

(c)

I

tn,

v,

=

(b) m,

= 4.

(c) m,

= IO.

IO

I

Normal modes

0.996

oo---

I.091

@---

I I 0.000

O--0 -0

I O-

-0

O-

(d) VI, = ioo

Fig. 1. The normal modes of motion (translation and vibration) of a N = 3 chain with a heavy central mass, m,, in mass weighted coordinates [9-111: (a) m, = 2, (b) m, = 4, (c) m, = 10 and (d) m, = 100. Arrows representing amplitudes less than 0.1 are omitted for clarity.

..:ll_:al Fig. 2. As in fig. 1 but for N = 5.

H. W. Schranz et al. /A

test of

interpretation of the deviations from RRRM behaviour we study the internal vibrational relaxation (IVR) rate. The microcanonical average value for the energy of each vibrational mode is obtained by the microcanonical sampling part of the simulation program. The mean square deviation of the actual simulated vibrational energies from the microcanonical average (accounting for anharmonk effects) is then calculated and averaged over all trajectories to y&ld ((AC)‘),. -l-he square root of the quantity is (AC). Its decay is monitored and a least-squares fitted rate coefficient is obtained and referred to as IVR in the figures below. Mathematical expression for these relevant quan-

RRKM theory. III

(0)

0 0

tities can readily be constructed on the basis of the definitions given above and are discussed in our earlier articles [1,2]. Figs. 3-6 show the time dependent rate constant k(t; E) obtained from the ordered set of lifetimes observed in the simulations by the procedure described before [l] for a pentatomic chain, N = 5, with various central masses, m, = 1, 10, 100 and energies, E = 1.25, 1.5. Figs. 3 and 4

relate to a system with identical bonds so that all bonds can break at the same energy, i.e., a multichannel system. As the central mass is increased there is more non-RRICM behaviour. At higher energies, E = 1.5, the heavy central mass encourages more rapid short time dissociations and slows down long time dissociations. At low energies, E = 1.25, more states are trapped by the heavy mass and the major effect is that of slow long time dissociations. One can also observe the failure of the normal mode IVR model to predict the relevant IVR rate constant. In fig. 4 IVR is predicted to be much faster than dissociation and yet one has substantial non-RRKM behaviour for m, = 10, 100. -l-his failure can be qualitatively explained as an effect of the localization of the vibrational relaxation to the side chains surrounding the central mass. Thus, the IVR parameter, which depends on normal modes to predict the rate of attainment of microcanonical equilibrium in a global average sense, will have insufficient sensitivity to slow energy transfer through heavy central masses. The parameter overestimates the rate of IVR because it is an average over the fast initial decay and the slow

N=S

107

E-1.5

12

ibl

N.5

24

,

36

46

60

24

,

36

46

60

46

60

Er.l.5

OL 0

12

. . . . . . . . . . . . . . . . i.y

P t *

1

01 0

12

24

,

36

Fig. 3. A comparison of RRKh& LSFS, and IVR rate constants with the timedependent rate constant k(t; E) as a function of the mass of the central atom, mc, for a uniform chain with N = 5, E=1.5 and &,,(r, - re) =lO: (a) m, =l, (b) mC = 10 and (c) m, = 100.

intermediate to final decay of (AC), the measure of relaxation described in our earlier work. The fast initial decay is a reflection of the fast IVR that occurs in the side chains around the heavy

H. W. Schranz et al. /A

108

;: ..

O

(al

N = 5

E = l.Z!l

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._. . . . .1.y.F ........_.......

test of RRKM

theory. III

In figs. 5 and 6 one sees much the same effect as above for a system which allows only one of the terminal bonds to break at the energies studied. This single channel system is a pentatomic (N = 5) chain with a standard 1,Zbond (between atoms 1

lal

N=5

E-1.5

mc = 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ic;ii’ . . . . . . . . . . . . . .._

0’ d‘

. ............................................................................ids

01

......_.....

0

lb)

N=5

16

32

f

40

64

et

E-1.25

s s *

0‘. , lb) ~15 E = 1.5 nl, = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I’ve

o

. . . . . . . . . . . . . . .._.

z d

-IVR

(cl

I

01 0

N=S

E = 1.25

16

,

32

48

80

64

mc = 100

a s (I

RRKM . . . .. . . .. . . . .. . .. . Y

0 0

12

24

,

.

. . . . . . . . .LSFS . .. . .. . . .. .. . . .. . .. .. . .. . . •l PIU

36

48

o

60

Fig. 4. As for fig. 3 but at E = 1.25.

B

5 I

central mass. The slow decay is a reflection of the lack of complete vibrational relaxation that is caused by the IVR blocking effect of the central mass. The average rate constant obtained pans too much attention to the fast initial decay of (AC) (or the IVR within the side chains).

....................~~ ...... k (cl

E = 1.5

N=5

I

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m, = 100

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.K!Y ....._..._..___...._

. .LSFS . . . . .. .

.. .. . .. . . .. . .. .. . .. . . . . .. .

. .. . . . . .

. ..

0

0

16

32

,

48

64

80

Fig. 5. As for fig. 3 but for a single channel chain with N = 5 and E=lS.

H. W. Schranz et al. / A test o/RRKM

theory. III

109

is almost RRKM where there was non-RRKM behaviour in the multichannel case. This is an effect, previously discussed, of dissociation competing with IVR. Since the reaction has been slowed compared to the IVR rate then the behaviour should become more RRKM-like. However, as before for the multichannel case,

I

01

0

16

32

,

40

64

80

F

(01

o

lb)

N = 5

@

MULTI-CHANNEL

E s 1.25

G

o.ooe

2

LSFS

t +

0.004

0.

o.*~

.

0

16

32

,

48

64

80

-1

0

1

2

log [In,1

0.010 ICI

N=5

Ibl

SINGLE-CHANNEL

q=100

E-l.25

..""............"..........................,............................~~~

SIH . . . . . . . . .RRKM . . .. . .. . . .. . .. .. .. . . .. . ..

” . ..-..................................................

LSFS

0

16

32

t

40

64

80

Fig. 6. As for fig. 3 but for a single channel chain with N = 5 and E = 1.25. -1

I

1

0

1

J

2

log [in,1

and 2) but all the other bonds have twice the dissociation energy. This slows down the overall rate of dissociation compared with the analogous multichannel uniform chain. Hence, for the uniform mass case, m, = 1, at E = 1.5 the behaviour

Fig. 7. A comparison of mass effects on the least-squares fitted simulation (LSFS) and RRKM rate constants for (a) a multichannel chain and (b) a single channel chain with N= 5, E = 1.25 and &,.,(r, - re) = 10. The LSFS points have 95% confidence limits.

110

H. W. Schranz et al. /A

test of RRKM

theory. III

3. Correlation

of bond dissociation energy distribution

O.O-1 ,a,

““LTl_TNN”

0

-1

,

,

1

2

log [m,l

0.026

1 i I \

0.020

RRKM

t 0.016 t

12

2 0.010

I

LSFS

i

0.006 lb1

SINGLE-CHANNEL

t

o.ooo1 -1

1

0

logIm,l

I

I

1

2

Fig. 8. As for fig. 7 but at E =1.5.

the single channel model exhibits substantial nonRRKM behaviour as the central mass becomes heavier. The normal mode IVR model again fails to predict such effects. Figs. 7 and 8 and table 1 exhibit the central mass effect described for average rate constants for both the multichannel and single channel models. The trends are qualitatively similar to those described above.

with initial

Here, the blocking effect of a heavy central mass on IVR is explored in more detail for systems with N = 3 and N = 5 atoms. We define E, and E, as the right and left excitation energies in the chain obtained by summing bond and kinetic energies on either side of the central atom. For N = 3 if m, = 1, as in figs. 9 and 10, for energies of E = 1.5 and E = 1.1, respectively, there is no significant correlation between the initial energy distribution, (E, - E,)/E, and the side of the molecule where the bond breaks; left (L), right (R), or whether there is no dissociation (N). The distributions are almost symmetrical. However, on addition of a heavy central mass, m, = 10, as in fig. 9b (E = 1.5) and fig. lob (E = l.l), there is significant correlation. The states initially excited mostly on the left side of the molecule, (E, E,)/E < 0, dissociate mainly from the left (L) side while states initially excited mostly on the right side, (En - E,)/E > 0, dissociate mainly from the right (R) side. Similar trends hold for larger systems. In figs. 11 and 12 the central mass is m, = 10 or m, = 100 for a pentatomic (N = 5) chain. As the mass is increased the correlation also increases. Furthermore, the number of trapped states also increases, especially at lower energies, as evidenced by the increase in non-dissociations (N) at E = 1.25 compared to E = 1.5. Fig. 13 illustrates the mass effect for a single channel N = 5 chain for m, = 1, 10 and 100. The mass effect is similar to that observed in the multichannel case except that many more states get trapped with energy on the side that cannot dissociate [i.e. the right side (R)]. Thus, the overall rate constant will be more decreased in the single channel case than in the multichannel case as the central mass is made heavier. This effect was observed in fig. 7. It should be noted that the above correlations of bond dissociation with initial energy distribution will depend on the ratio of the maximal duration of the simulation, t_, to the halftime of the reaction. If t,, is relatively short the correlation is expected to be high while as t,,,, is in-

111

H. W. Schranz et al. /A test of RRKM theov, III Table 1 The effect of the central mass, m,, on the least-squares fitted simulation rate constant (LSFS) and the RRKM single channel and multichannel N = 5 chains at various energies, E, with dissociation cut-off &( r, - rC) = 10

rate constant for

system

E

m,

LSFS

RRKM

LSFS/RRKM

multichannel

1.25

1 10 109 1 4 10 100 1 10 100 1 10 100

(8.74&0.29)x10-’ (6.13*1.18)x10-3 (5.07 f 1.01) x 10-s (4.18&0.72)X lo-* (4.39*0.57)x10-* (3.43 f 0.65) x lo- ’ (3.56*0.67)x lo-’ (5.15*0.80)X10-’ (2.98 *0.62)x 1O-3 (1.72*0.46)x10-’ (2.33*0.31)x lo-’ (1.05 * 0.20) x 10-s (5.50*1.30)x10-3

1.l6x1O-2 1.03x10-2 9.84x1O-3 3.64x lo-* 3.38X1O-2 3.23x1O-3 3.08 x 1O-2 5.17x10:3 4.80x10-’ 4.64x 10-3 1.95 x 10-2 1.81 x 1O-2 1.76~10-~

0.75 f 0.03 0.60*0.11 0.52 f 0.10 1.15 *0.20 1.30*0.17 1.06*0.20 1.15 kO.22 1.00*0.15 0.62kO.13 0.37 f 0.10 1.19kO.16 0.58kO.11 0.31 f 0.07

1.5

single channel

1.25

1.5

ln

t.Ial

rv=3

er1.5

ll4’l .(a)

d et

L

N

tv=3

E~i.1

mc

E

I

R

L

-1

0

N

1

-1 0 I.%-E,)IE

R

1

-1

0

1

ln .

z

(bl

N=J

E = 1.5

+

I

= 10

In

d .

-1

0

1

-1

0

1

-1

0

--.-------..---

lb1

N=J

E=M

m, = 10

_I-

1

IEI - E,IIE

Fig. 9. Tbe correlation of bond dissociation with initial energy distribution for a multichannel chain with N = 3, E = 1.5 and &(rC - rc) 110: (a) m, -1 and(b) m, -10.

Fig. 10. As for fig. 9 but for a multichannel chain with N = 3, E==l.land~M(rC-rc)==lO:(a) ~-land(b) rn,==lO.

H. W. Schranr et al. /A

test of RRKM

theory. III

ul . E = 1.5

q

= 10

“,

I

(0)

d

N =5

E = 1.25

m, = 1

L

I R

N

t* id a


d d

0 -1

0

1

-1

0

1

-1

0

1

-1

0

lb)

N =5

1

-1

IE.-E,IIE

tE.-$/E

-1

l

0

1

In . lb1

z

N =5

E=

mc = 100

1.5

I

R

1 mc=10

E= 1.25

1 I

L

N

R

7

-1

1

0

-1 0 tE,-E,)IE

1

-1

0

1

Fig. 11. As for fig. 9 but for a multichannel chain with N = 5, E = 1.5 and &(r, - re) = 10: (a) m, = 10 and (b) mc = 100.

r

*

-1

0

(C)

N=5

1

-1

0 1 I.E.-E,lIE

-1

0

1

_ E = 1.25

in, = 10

I

E-1.25

m,

0

1

-1

0

IE,-ELI/E

1

-1

0

100

R

N

L

-1

q

1

In . z

_ (bl

N-5

E = 1.25

mc = 100

I

-1

0

1

-1

0

1

-1

0

1

[E,-E,i’E

Fig. 13. As for fig. 9 but for a single channel chain with N = 5, E = 1.25 and &(rC - rc) = 10: (a) m, = 1, (b) m, = 10 and (c) m, = 100.

Fig. 12. As for fig. 9 but for a multichannel chain with N = 5, E = 1.25 and BM(rC - re) = 10: (a) m, = 10 and (b) m, =lOO.

creased the correlation should decrease. The rate and extent of this decrease in correlation for increasing t,, would be larger for ergodic systems which exhibit rapid vibrational relaxation than for systems which are non-ergodic or vibr+ionally

H. W. Schranz et al. /A

relax slowly compared to the rate of dissociation. Systems with rapid vibrational relaxation throughout all of reactant phase space would show no significant correlation once t, was larger than the time required for vibrational relaxation. Systems which contain non-ergodic regions in reactant phase would exhibit non-dissociating trajectories (N) even as t,, is increased without limit. Systems which relax very slowly would exhibit substantial correlations even at relatively long times but such correlation may decrease with increasing t,, depending on the ratio of the rate of IVR to the rate of dissociation.

4. Discussion and conclusions An important observation confirmed in this study is that linear chains with heavy central masses tend to show blocking of energy transfer. In such cases the global IVR parameter evaluated in this work is insufficiently sensitive to the internal dynamics and does not show this effect. Instead the IVR parameter seems to be more sensitive to the rapid vibrational relaxation that occurs in the side chains rather than to the slow intramolecular energy transfer that occurs through the central mass. The vibrational relaxation is of highly non-exponential character. There is a need for an IVR parameter that is more sensitive to these details of the internal dynamics. Other means of determining the extent of stochastic and quasiperiodic behaviour have been recently reviewed by Noid et al. [12]. Attempts to distinguish between the two types of classical motion - quasiperiodic, regular or stable and chaotic, stochastic, ergodic or irregular - have included the examination of the behaviour of a trajectory in occupying phase space by calculating the PoincarC surfaces of section [13-151, comparison of the microcanonical phase space and time averages of some function of the dynamical variables, analysis of the frequency spectrum of a dynamical variable or correlation function for a trajectory or set of trajectories [16], and comparison of the motion of two neighbouring points in phase space [17,18] which allowed the calculation of an approximation to the Kohnogorov entropy, the k-entropy or

test of RRKM theory. III

113

the Lyaponov characteristic number [19]. Our technique is based on comparing the microcanonical phase space and time averages of the internal normal mode energies, and we calculate the sum of the squared deviations for each internal normal mode as a measure of the rate and extent of internal energy randomization. The calculation of the Lyaponov characteristic number, as used by Farantos and Murrell [19], suggests itself as a promising alternative. The blocking of intramolecular energy transfer by a heavy central mass has also been seen in a simulation on a similar system by Lopez and Marcus [3]. In an earlier simulation, Harter et al. [20] had also noted such effects. They concluded that the presence of anharmonicities appeared not to permit rapid flow of energy in molecules containing atoms of quite different mass. However, while such blocking has been detected in chemical activation experiments by Rogers et al. [5,6] for molecules with large side chains and heavy central masses, it is important to note that the results of numerical simulations are not general&able to all systems. Real molecules have features that are presently not accounted for accurately by simulations such as the present work and that of Lopez and Marcus [3] and Swamy and Hase [4]. Thus, it should not be surprising that real systems apparently exist which do have a heavy central mass but show no evidence of blocking of intramolecular energy transfer. Examples include the chemically activated unimolecular reactions examined by Rabinovitch et al. [7,8].

References [l] H.W. Schranz, S. Nordholm and B.C. Freasier, Chem. Phys. 108 (1986) 69. [2] H.W. Schranz, S. Nordhohn and B.C. Freasier, Chem. Phys. 108 (1986) 93. [3] V. Lopez and R.A. Marcus, Chem. Phys. Letters 93 (1982) 232. [4] K.N. Swamy and W.L. Hase, J. Chem. Phys. 82 (1985) 123. [5] P. Rogers, D.C. Montague, J.P. Frank, S.C. Tyler and F.S. Rowland, Chem. Phys. Letters 89 (1982) 9. [6] P.J. Rogers, J.I. Selco and F.S. Rowland, Chem. Phys. Letters 97 (1983) 313.

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H. W. Schranz et al. / A test of RRKM theory. III

[7] S.P. Wrigley and B.S. Rabinovitch, Chem. Phys. Letters

[8] [9] [lo] [ll] [12] [13]

95 (1983) 363. S.P. Wrigley and B.S. Rabinovitch, Chem. Phys. Letters 98 (1983) 386. E.B. Wilson Jr., J.C. Decius and P.C. Cross, Molecular vibrations (McGraw-Hill, New York, 1955). G. Heraberg, Infrared and Raman spectra of polyatomic molecules (Van Nostrand, Princeton, 1945). H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 1980). D.W. Noid, M.L. Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267. N. De Leon and B.J. Beme, J. Chem. Phys. 77 (1982) 283.

[14] R.A. Marcus, Ber. Bunsenges. Physik. Chem. 81 (1977) 190. [15] D.W. Noid and M.L. Koszykowski, Chem. Phys. Letters 79 (1981) 485. [16] J.D. McDonald and R.A. Marcus, J. Chem. Phys. 65 (1976) 2180. [17] P. Brumer and J.W. Duff, J. Chem. Phys. 65 (1976) 3566. [18] J.W. Buff and P. Brumer, J. Chem. Phys. 67 (1977) 4898. [19] S.C. Farantos and J.N. MtureII, Chem. Phys. 55 (1981) 205. [20] R.J. Harter, E.B. Alterman and D.J. Wilson, J. Chem. Phys. 40 (1964) 2137.