Intramolecular Dynamics Diffusion Theory: Nonstatistical Unimolecular

Chapter 6

Downloaded by NORTH CAROLINA STATE UNIV on January 11, 2013 | http://pubs.acs.org Publication Date: June 10, 1997 | doi: 10.1021/bk-1997-0678.ch006

Intramolecular Dynamics Diffusion Theory: Nonstatistical Unimolecular Reaction Rates 1

2

Dmitrii V. Shalashilin and Donald L . Thompson

Department of Chemistry, Oklahoma State University, Stillwater, OK 74078-3071

A method based on diffusion theory for calculating nonstatistical unimolecular reaction rates is described. The method, which we refer to as intramolecular dynamics diffusion theory (IDDT), uses short -time classical trajectory results to obtain the rate of IVR (intramolecular vibrational energy redistribution) between the reaction coordinate and the "bath" modes of the molecule in a diffusion theory formalism (i.e., a classical master equation) to calculate the reaction rate. This approach, which requires much less computer time than the conventional classical trajectory method, accurately predicts the classical rates of unimolecular reactions.

The microcanonical rate coefficient k(E,t) for a chemical reaction can be written in terms of the flux through the critical surface that separates reactant and products:

k ^ ( E , t ) ^ \

;

(1)

if(p,q,t)dT V where the integral in the numerator is over the region of the critical surface S* and that in the denominator is over the configuration space V of the reactant, v is the velocity perpendicular to the critical surface, and f(p,q,t) is a distribution function. (We have explicitly indicated the time dependence in the quantities since that is what is of interest to us here.) The initial microcanonical distribution in equation 1 can be written as ±

l

On leave from the Institute of Chemical Physics, Russian Academy of Sciences, 117334 Moscow, Russian Federation

Corresponding author © 1997 American Chemical Society In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

81

HIGHLY E X C I T E D M O L E C U L E S

82

f (p,qJ E

mc

= o) = f (p,q)

=

S(H(p,q)-E)

reactant

\S(H(p,q)-E)aT V

region (2) product region


The time evolution of an ensemble with microcanonical initial conditions corresponding to this distribution is given by the Liouville equation: • 3f

ÔH Of

. df

ôt

ÔH df

(3)

1

In practice, Hamilton's equations of motion for the ensemble are numerically integrated because it is difficult to solve the Liouville equation. The lifetimes of a microcanonical ensemble of reactants computed by using classical trajectories can be fit by ln-

-k^(E)t.

N(t = 0)

(4) 9

aj

The decay time is the reciprocal of the rate coefficient: if" = l/kf (E). The number of trajectories is usually not large enough to resolve the time dependence ofk^fEj). For an initial microcanonical distribution, equation 1 gives J S(H(j>,q)-E)v dS ±

n

stat

k4' (E,t = 0) = k (E)

=£

S(H(p,q)-E)dT

(5)

for the initial rate; that is, if the initial distribution is microcanonical the initial rate is statistical. If the reaction is not sufficiently fast to disturb the shape of the microcanonical distribution, see equation 2, then the rate will be independent of time and thus remain statistical throughout the decay process. This is the basic assumption of statistical theories. In this case the distribution given by the solution of the Liouville equation for the initial conditions in equation 2 is

f(p,q,0 =

mC

N(t = 0)

f (p,q)

(6)

Substituting equation 6 into equation 1 leads to the time-independent dynamical rate coefficient, which is equal to the statistical rate coefficient: ,at

Ι^"(Ε) = k* (E).

In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

(7)

6. SHALASHILIN & THOMPSON

Intramolecular Dynamics Diffusion Theory

This is the result of the rate of IVR (intramolecular vibrational energy redistribution) being fast compared to that of reaction. If, on the other hand, reactions disturb the microcanonical distribution, the rate coefficient (E t) will be time dependent. Then the trajectory rate coefficient 1f (E) is the time-averaged dynamical rate coefficient , and differs from the / = 0 dynamical rate kf (E,t=0) which is equal to the statistical rate ^ (E,t=Q)\ aj

}

yn

yn

e (E) = (CH ) N . + . N 0 . 3

2

2

3

2

(37)

2

The N - N bond length is the reaction coordinate, thus we use the N - N stretch (which does not correspond to a normal mode) as subsystem 1. It is described by the Hamiltonian for a Morse oscillator: 2

#i ( ^ - * Λ ™ ) = ir—Pi-H

+D{exp[-a(R _ -R° . )]-1} . N

N

(38)

N N

2

MN-N

We integrated ensembles of 1000 trajectories for 1.2 fs for total energies of 250, 300, and 400 kcal/mol, and computed the rate of energy flow by monitoring the energy in the reaction coordinate, using equation 38, at fixed intervals of AEj. Additional initial conditions were imposed for the energy of the N - N atom pair to be within the interval Ει to Ej+AE]. Figure 4 shows how an initially narrow distribution spreads as a function of time. The rate of this spreading is described by —(ΔΕ*) . Figure 5 shows dt ' λ

2

the dependence o f — ( Δ £ ) onZ^. a To calculate the integral in equation 35 we fit the results in Figure 5 to the function (39)

2

The dependence of — ( Δ Ε ) on E the energy in the N - N stretch, is shown in dt ' Figure 4 total energies Ε = 250, 300, and 400 kcal/mol. The dependence on the total energy is negligible; the parameters α = 0.07 kcal /s, b = 0.3 kcal /s, and c = 10 kcal/mol provide a reasonable fit for all three energies. The IDDT results (triangles) are compared in Figure 6 with the dynamical (circles), statistical (MCVTST) (squares), and R R K (straight line) results reported earlier.(72) The rates calculated by using IDDT are quite close to the results obtained from a classical trajectory simulation,(72) in fact, they are in quantitative agreement. And, as expected, the dynamical results are lower than the statistical rates. The EDDT method accurately predicts the nonstatistical dynamical effects. It is important to note that the trajectory calculations to determine the I V R rate require very short integration times (on the order of femtoseconds) and thus, even u

λ

2

2


91


92

HIGHLY EXCITED MOLECULES

Figure 3. Dimethylnitramine (DMNA). The active mode, subsystem 1, is the N - N atom pair.



6. SHALASHILIN & THOMPSON Intramolecular Dynamics Diffusion Theory though it is necessary to perform calculations for the range of energies from zero to the dissociation energy, the savings in computer time compared to that required for a standard classical trajectory calculation of the rate coefficient is significant. The same amount of computer time is required to calculate the rates for any total energy, while the cost in computer time for classical trajectory simulations drastically increases as the total energy decreases. The standard classical trajectory may not even be feasible at energies at the lower end of the dynamical regime, while the IDDT method can be easily used to compute the rates there. However, at sufficiently low energies where the reaction is slow and statistical, the IDDT approach, which is based on the assumption that reaction is fast compared to IVR, is not valid. Then, the rate must be calculated by using a statistical theory, e.g., MCVTST. The lower of the rates calculated by M C V T S T and IDDT is to be taken as the actual rate of reaction. We have discussed elsewhere (72) how to use a simple interpolation between the two rates; briefly, one can take the actual rate to be y. IDDT

MCVTST

jf-

JçIDDT

fçMCVTST

(40)

In another study,(73) we applied IDDT to the reaction H Si-SiH -> 2 SiH , 3

3

3

(41)

for which the Si-Si stretch can be easily identified as the reaction coordinate. Due to the difference in masses of Si and H the Si-Si stretch is a good normal mode, thus the flow of energy between it and the other molecular modes which comprise the energy reservoir is slow. It is the kind of situation where it is expected that diffusion theory should provide a good description of the reaction dynamics. However, the accuracy of the IDDT approach for the decomposition of D M N A , equation 27, is less obvious; the N - N stretch is not an isolated mode. One of the purposes of the present study was to see if the IDDT approach can be used for reactions where there is strong coupling of the reaction coordinate to the bath modes. The IDDT is a good approximation to the exact classical mechanics when (75)

-^f(fi")'«l, v a

(42)

where ν is the frequency of dissociating bond. The condition in equation 42 is obviously not a strong one. It is expected to hold for the reaction in equation 41 where the Si-Si bond corresponds to an isolated, normal mode, however, we find that it also is valid for the reaction in equation 37 where the reaction coordinate does not correspond to a normal mode. Also, the condition in equation 41 is violated as the energy approaches the dissociation energy D where v-»0, however, we find that this is not a problem in the integration in of equation 35. A pragmatic conclusion of this study is that one can, in general, simply use the stretching coordinate of the breaking bond as the active mode (i.e., subsystem 1 in our notation) in general. It is not necessary to apply IDDT only in cases where the reaction coordinate is an isolated mode. The success in the present application suggests that the IDDT method may be generally applicable. In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

93


94

HIGHLY EXCITED MOLECULES

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

Ej (kcal/mol)

5

6

8

9

10 11 12 13 14 15 16 17 18 19 20

EÏ (kcal/mol)

Figure 4. The spreading of an initially narrow distribution of energy in the N - N stretch for an ensemble of 1000 trajectories. The total energy for all the trajectories was Ç = 400 kcal/mol. The initial conditions for the trajectories were obtained from a Metropolis Monte Carlo random walk with the additional condition that 10 2

kcal/mol

Intramolecular Dynamics Diffusion Theory: Nonstatistical Unimolecular

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