Product phase space calculations

Figure 2. Vibrational energy distribution of the CO product formed via the 351 nm photolysis of W(CO)s. Experimental data are indicated asfl. The lines correspond to results obtained by phase space calculations with an available energy of 40 and 35 Kcal/mole.

FIGURE 3.5 Translational energy distribution of the products of the reaction O ( d) + H2 at collision energy of 8.0 kj mol obtained by the QCT method (—) on BR PES and using statistical phase space calculations (- - ) [31], The line (—) and the shaded area represent the experimental results and respective error bars [10]. 

Phase-space calculations have been made for another four-atom system, where symmetrical angular distributions have been demonstrated at low energies. The system is 02 + and the experimental product ratios OH H20" = 10 1 1) are unexpected since the least endoergic channel yields Quasiequilibrium theory cannot predict this ... 

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

The moving invariant manifolds determine the reactivity or nonreactivity of an individual trajectory under the influence of a specific noise sequence. They thus provide the most detailed microscopic information on the reaction dynamics that one can possibly possess. In practice, though, one is more often interested in macroscopic quantities that are obtained by averaging over the noise. To illustrate that such quantities can easily be derived from the microscopic information encoded in the TS trajectory, we calculate the probability for a trajectory starting at a point (q, v) in the space-fixed phase space to end up on the product side of the... 

Ex 35 Kcal/mole.15 We find that the CO product vibrational distribution calculated using the phase space model with Eav = 35-40 Kcal/mole is in good agreement with our experimental results (Figure 2). Thus, the measured CO vibrational distribution indicates that vibrational energy disposal to the photolysis products is determined at a point on the potential surface where the full reaction exoergicity is available. This suggests that the 351 nm excitation of W(CO)g results in the sequence of events, (2)-(4), where the asterisk denotes vibrational excitation. 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

Shor s algorithm for factoring a product N of two prime numbers. At the heart of the algorithm is a periodic function f Z//V Z//V whose period one must calculate in order to find the two prime factors of N. The phase space for computation is a pair of registers of size L, where 2 < N < 2. In other words, the state space for the quantum computer is... 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

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