Unimolecular classical trajectory studies

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.,... 

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. 

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.

Classical trajectory simulations " are widely used to study the dynamics of unimolecular and bimolecular reactions. In a classical trajectory study, motions of the individual atoms are simulated by solving Hamilton s classical equations of... 

In many classical trajectory studies of unimolecular reactions it is often necessary to accurately sample the microcanonical energy surface of a vibrationally excited molecule e.g., C2Hi. F in equation (4). If the good action-angle variables cannot be found, multidimensional selection procedures must be used since the coordinates q and -p- will not have independent probability distributions. ... 

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. D. McDonald and R. A. Marcus, Classical trajectory study of internal energy distributions in unimolecular processes, J. 

CLASSICAL TRAJECTORY STUDIES OF THE FORMATION AND UNIMOLECULAR DECAY OF RARE GAS CLUSTERS... 

Shalashilin D V and Thompson D L 1996 Intrinsic non-RRK behavior classical trajectory, statistical theory, and diffusional theory studies of a unimolecular reaction J. Chem. Phys. 105 1833—45... 

A power of classical trajectories is that they may be used in a pure simulation mode [326] to investigate how changes in PES properties affect a molecule s unimolecular dynamics. Such a study provides fundamental insight into the relationship between the nature of intramolecular and... 

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. 

Maergoiz, A.I., Nikitin E.E., Tree, J., Ushakov V.G. Classical trajectory and statistical adiabatic channel study of the dynamics of capture and unimolecular bond fission (1996) a) I. Ion-dipole capture, J. Chan. Phys. 105, 6263-6269 b) n. lon-quadrupole capture, ibid. 6270-6276 c) HI. Dipole-dipole capture, ibid. 6277-6284 (1998) d) IV. Valence interactions between atoms and linear rotors, J. Chem. Phys. 108, 5265-5280 e) V. Valence interactions between two linear rotors, ibid. 9987-9998 (2002) g) VI. Properties of transitional modes and specific rate constants H.EJ), J. Chem. Phys. 117, 4201-4213. 

Direct dynamics simulations, in which the methodology of classical trajectory simulations is coupled to electronic structure, have had and will continue to have an enormous impact on the use of computational chemistry to develop [111,112] the theory of unimolecular kinetics. In these simulations the derivatives of the potential, required for numerically integrating the classical trajectory, are obtained directly from electronic stmcture theory without the need for an analytic PES. Direct dynamics is particularly important for studying the unimolecular dynamics of molecules with many degrees of freedom, for which it is difficult to construct an accurate analytic PES. 

A.I.Maergoiz, E.E.Nikitin, J.Troe, and V.G.Ushakov, Classical trajectory and adiabatic channel study of the dynamics of capture and unimolecular bond fission. 

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. 

A number of MD studies on various unimolecular reactions over the years have shown that there can sometimes be large discrepancies (an order of magnitude or more) between reaction rates obtained from molecular dynamics simulations and those predicted by classical RRKM theory. RRKM theory contains certain assumptions about the nature of prereactive and postreactive molecular dynamics it assumes that all prereactive motion is statistical, that all trajectories will eventually react, and that no trajectory will ever recross the transition state to reform reactants. These assumptions are apparently not always valid otherwise, why would there be discrepancies between trajectory studies and RRKM theory Understanding the reasons for the discrepancies may therefore help us learn something new and interesting about reaction dynamics. 

Classical trajectories have been used to investigate vibrational energy transfer of vibrationally excited CS2, CH4, SFg, and SiF4 in collisions with various thermal molecules. The results show evidence for strong collisions (i.e., ones in which large amounts of energy are transferred) but the calculated probabilities ai e much lower than is usually assumed in theoretical treatments of unimolecular reactions in fact, the occurrence of such collisions is sufficiently low that they have little influence on the overall rate of relaxation. This method of successive collisions has also been used to study the relaxation of CS2 by H2, CO, HCl, CS2, and CH4. ... 

Classical trajectories are the only feasible means to explicitly treat all atoms in a dynamical study of a unimolecular reaction. Trajectories have been used extensively to interpret A + BC bimole-cular reactions and a considerable amount of literature exists with respect to these studies. Excitation functions, scattering angles, product energy distributions, and other dynamical properties are usually quantitatively determined by the trajectory calculations. The semiclassical studies of Marcus and Miller have in general confirmed the accuracy of classical trajectories in calculating dynamical properties for bimolecular reactions. However, the trajectories do not describe quantum mechanical effects such as interferences, tunneling, and nonadiabatic electronic transitions. 

Most of the methodology of classical trajectory calculations has been developed for A + BC bimolecular reactions, for which extensive reviews exist.Much of this methodology is transferable to unimolecular studies of polyatomic molecules. However, several modifications pertaining to the selection of initial conditions, the analysis of final results, and the numerical integration of the classical equations of motion are required. These modifictions can be illustrated by considering the reaction given in equation (4). 

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