Unimolecular dynamics

State specific experiments can now test unimolecular rate theories by probing microcanonical rate coefficients. Moore and coworkers [45] have studied the dissociation of ketene close to the reaction threshold in an attempt to test RRKM theory. 

Using a short tunable laser pulse they excited ketene into its first electronically excited state from which internal conversion occurs to give a highly 

In an elegant four laser experiment Temps [46] has also investigated the variation in k E) for the decomposition of the methoxy radical  

Current developments in RRKM theory by Marcus, Wardlaw [47-49], Smith [50-52] and Klippenstein [53-56], have rigorously extended the theory to include the effects of J dependence, the result leading to micro-canonical rate coefficients which are functions of both energy, E, and the magnitude of the angular momentum, J. 

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Faraday Discuss. Chem.. Soc. 1986 Dynamics of molecular photofragmentation. No 82 Faraday Discuss. Chem. Soc. 1995 Unimolecular dynamics. No 112... 

Faraday Discussions of the Chemical Society 112 Unimolecular Dynamics... 

Another important question deals with the intramolecular and unimolecular dynamics of the X-—RY and XR -Y- complexes. The interaction between the ion and molecule in these complexes is weak, similar to the intermolecular interactions for van der Waals molecules with hydrogen-bonding interactions like the hydrogen fluoride and water dimers.16 There are only small changes in the structure and vibrational frequencies of the RY and RX molecules when they form the ion-dipole complexes. In the complex, the vibrational frequencies of the intramolecular modes of the molecule are much higher than are the vibrational frequencies of the intermolecular modes, which are formed when the ion and molecule associate. This is illustrated in Table 1, where the vibrational frequencies for CH3C1 and the Cr-CHjCl complex are compared. Because of the disparity between the frequencies for the intermolecular and intramolecular modes, intramolecular vibrational energy redistribution (IVR) between these two types of modes may be slow in the ion-dipole complex.16... 

Unimolecular dynamics of smaller clusters has also been studied. The HF dimer provides a particularly interesting system because it involves a highly quantal degenerate rearrangement consisting of a concerted double hydrogen-bond switch (Quack and Suhm 1991 Truhlar 1990). 

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... 

It is not immediately obvious, by simply looking at a molecule s Hamiltonian and/or its PES, whether the unimolecular dynamics will be intrinsic RRKM or not and computer simulations as outlined here are required. Intrinsic non-RRKM dynamics is indicative of mode-specific decomposition, since different regions of phase space are not strongly coupled and a micro-canonical ensemble is not maintained during the fragmentation. The phase space structures, which give rise to intrinsic RRKM or non-RRKM behavior, are discussed in the next section. 

A classical diffusion theory model has been proposed to calculate the rate of IVR between the reaction coordinate and the remaining bath modes of the molecule [345]. Following work by Bunker [324], the unimolecular dynamics will be non-ergodic (intrinsically non-RRKM) if A rrkm fciVR. For such a situation, the unimolecular decomposition will be exponential and occur with a rate constant equal to /sivr- The rate of IVR is modeled by assuming a random force between the bath modes and the reaction coordinate. The model was used to successfully analyze the intrinsic non-RRKM dynamics for Si2He -> 2SiH3 dissociation [345]. 

Direct dynamics is applicable to large molecular systems, but a lower level of electronic structure may be required as well as a blend of direct dynamics and analytic potential energy functions. This latter technique, often called quantum mechanical/molecular mechanical (QM/MM) direct dynamics [377], has been used to simulate SID unimolecular dynamics associated with protonated glycine ions, NH3CH2COOH [(gly-H)+j, colliding with a hydrogenated diamond 111 surface [378]. The potential energy for the system is represented by... 

If an intrinsically-RRKM molecule with many atoms is excited non-randomly, its initial classical non-RRKM dynamics may agree with the quantum dynamics for the reasons described above. But at longer times, after a micro-canonical ensemble is created, the classical unimolecular rate constant is much larger than the quantum value, because of the zero-point energy problem. Thus, the short-time unimolecular dynamics of a large molecule will often agree quite well with experiment if the molecule is excited non-randomly. The following is a brief review of two representative... 

The irregular trajectories in Fig. 15.6 display the type of motion expected by RRKM theory. These trajectories moves chaotically throughout the coordinate space, not restricted to any particular type of motion. RRKM theory requires this type of irregular motion for all of the trajectories so that the intramolecular dynamics is ergodic [1]. In addition, for RRKM behavior the rate of intramolecular relaxation associated with the ergodicity must be sufficiently rapid so that a microcanonical ensemble is maintained as the molecule decomposes [1]. This assures the RRKM rate constant k E) for each time interval f —> f + df. If the ergodic intramolecular relaxation is slower than l/k(E), the unimolecular dynamics will be intrinsically non-RRKM. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

The conclusion one reaches is that quantum RRKM theory is an incomplete model for unimolecular decomposition. It does not describe fluctuations in state-specific resonance rates, which arise from the nature of the couplings between the resonance states and the continuum. It also predicts steps in k E), which appear to be inconsistent with the actual quantum dynamics as determined from computational chemistry. However, for molecules whose classical unimolecular dynamics is ergodic and intrinsically RRKM and/or whose resonance rates are statistical state specific (see Section 15.2.4), the quantum RRKM k E) gives an accurate average rate constant for an energy interval E E + AE [47]. 

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. 

Direct dynamics has made it possible to investigate the unimolecular decomposition of a broad group of molecules for different excitation processes, to compare with experiment and determine fundamental information concerning intramolecular and unimolecular dynamics. Summarized in Table 15.1 are the unimolecular direct dynamics simulations performed by the Hase research group [117-129]. Some degree of non-RRKM behavior is present in each of the reactions. It would not have been possible to determine this level of understanding of the unimolecular dynamics of these reactions without access to direct dynamics. 

The authors research in unimolecular dynamics is funded by the National Science Foundation (USA), Welch Foundation (USA), and Deutsche Forschungsgemeinschaft (Germany). Their understanding of unimolecular rate theory has benefited enormously from work with their students, post-doctorals, and collaborators. 

Bandheads are prominent features of a Loomis-Wood plot. A family of straight and curved lines emanates from the bandhead of the mapped branch on the Loomis-Wood plot. The straight lines actually intersect at the bandhead and correspond to various orders of the mapped branch. The curved lines are generated e.g. when the post-head R-branch overlaps with the pre-head segment of itself or with the P-branch described by the same Fortrat equation. Another prominent feature of Loomis-Wood plots is a horizontal strip free of points, which surrounds the mapped branch. The width of this strip is determined by the widths of lines in the mapped branch (instrumental resolution, Doppler width, nonradiative decay width, or unresolved hyperfine splittings). The Loomis-Wood plot can provide a survey map of J-dependent linewidth variations that encode valuable information about unimolecular dynamics. 

Consequently, much of the research in unimolecular dynamics has been devoted to observing and explaining both rapid intramolecular flow of vibrational energy and the possibility of chaos in the vibrational dynamics. The literature on these two related... 

Considerable experimental effort has been aimed at elucidating the collision-free unimolecular dynamics of excited molecules. Processes of interest include the dynamics of highly excited vibrational states, which have been reached by multiphoton absorption, and the various electronic relaxation processes that can occur in electronically excited states of moderate to large molecules, etc. The idealized collision-free limit is approached either by extrapolating data to the limit of zero pressure or by performing experiments in molecular beams. Alternatively, estimates of expected collisional effects are made by using collision cross-sections that are computed from hard-sphere collision rates. These estimates are then utilized to determine whether the experiments are performed in the collision-free domain. 

If the unimolecular dissociation is not random and not in accord with Eq. (2.2) it is thought that classical bottlenecks restricting intramolecular vibrational-energy redistribution (IVR) may be manifested in the quantum dynamics [16]. Thus, there is considerable interest in identifying the nature of the unimolecular dynamics of excited molecules. In this section Monte... 

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. 

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. 

In the previous section excitation of a single, isolated resonance and its ensuing unimolecular decomposition was considered. However, unimolecular dynamics has also been investigated by exciting a superposition of resonance states, which is initially localized in one part of the molecule, for example, a C—H bond. If this superposition contains all the resonance states in the energy width AE of the excitation process, statistical unimolecular decomposition might be expected after complete IVR for the... 

Real-time experiments (Khundkar and Zewail, 1990 Zewail, 1991) with a subpicosecond resolution have probed the unimolecular dynamics of NO2 NO + O (Ionov et al., 1993a) and H + CO2 HOCO -> HO 4- CO (Scherer et al., 1987, 1990 Ionov et al., 1993b). The NO2 experiment is described and discussed in section 6.2.3.1 (p. 196). The H + CO2 reaction and ensuing formation of HOCO is initiated by photodissociation of HI in the HI—CO2 van der Waals complex (Fig. 8.8). A subpicosecond laser pulse is used to initiate the reaction while a second laser pulse probes the product formation. The reactants are vibrationally and rotationally cold prior to excitation, and the experiments demonstrate that the H + CO2 reaction proceeds... 

As discussed elsewhere in this volume (78), unimolecular dynamical eflFects of this type have been rationalized by Bunker in terms of the angular momentum characteristics of energetic substitution reactions (19). 

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