Trajectory studies unimolecular decomposition

Initiated by the pioneering work of Bunker [323,324] classical trajectory simulations have been extensively used to study the decomposition of energized molecules. In a unimolecular classical trajectory study, the motions of atoms for an ensemble of molecules are simulated by solving their classical equations of motion, usually in the form of Hamilton s equations, i.e.,... 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

The first classical trajectory study of unimolecular decomposition and intramolecular motion for realistic anharmonic molecular Hamiltonians was performed by Bunker [12,13]. Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,17,30,M,65,66 and 62] from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3.12.7. Chaotic vibrational motion is not regular as predicted by the normal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9]. For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is formed and the probability of decomposition becomes that of RRKM theory. 

The first step in a unimolecular reaction involves energizing the reactant molecule above its decomposition threshold. An accurate description of the ensuing unimolecular reaction requires an understanding of the state prepared by this energization process. In the first part of this chapter experimental procedures for energizing a reactant molecule are reviewed. This is followed by a description of the vibrational/rotational states prepared for both small and large molecules. For many experimental situations a superposition state is prepared, so that intramolecular vibrational energy redistribution (IVR) may occur (Parmenter, 1982). IVR is first discussed quantum mechanically from both time-dependent and time-independent perspectives. The chapter ends with a discussion of classical trajectory studies of IVR. 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

Classical trajectory studies of unimolecular decomposition have helped define what is meant by RRKM and non-RRKM behavior (Bunker, 1962, 1964 Bunker and Hase, 1973 Hase, 1976, 1981). RRKM theory assumes that the phase space density of a decomposing molecule is uniform. A microcanonical ensemble exists at t = 0 and rapid intramolecular processes maintain its existence during the decomposition [fig. 8.9(a), (b)]. The lifetime distribution, Eq. (8.35a), is then... 

Table 8.5. Classical Trajectory Studies Identifying Intrinsic RRKM and non-RRKM Unimolecular Decomposition.

E. R. Grant and D. L. Bunker, Dynamical effects in unimolecular decomposition A classical trajectory study of the dissociation of C2He, J. Chem. Phys. 68 628 (1978). 

J. Santamaria, D. L. BunRer, and E. R. Grant, Dynamical effects of mode specific excitation in unimolecular decomposition A trajectory study of C2H0, Chem. Phys. Lett. 56 170 (1978). 

For many of the model molecules studied by the trajectory simulations, the decay of P t) was exponential with a decay constant equal to the RRKM rate constant. However, for some models with widely disparate vibrational frequencies and/or masses, decay was either nonexponential or exponential with a decay constant larger than k E) determined from the intercept of P(f). This behavior occurs when some of the molecule s vibrational states are inaccessible or only weakly coupled. Thus, a micro-canonical ensemble is not maintained during the molecule s decomposition. These studies were a harbinger for what is known now regarding inelficient intramolecular vibrational energy redistribution (IVR) in weakly coupled systems such as van der Waals molecules and mode-specific unimolecular dynamics. 

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