Transition probability completeness

In the case of a strongly exothermic reaction the final term turns into an absorbing wall, and the transition is completed whenever the distance AB reaches a certain value and the A-B bond is broken. The intra- and intermolecular coordinates Q and q are harmonic and have frequencies (Oo and oji, and reduced masses mo and mi. At fixed intermolecular displacement the tunneling probability equals... 

Note that in this case is completely specified by the two transition probabilities... 

The attractive energies 4D(cr/r)6 and ae2/2 r4 have two important effects on the vibrational energy transfer (a) they speed up the approaching collision partners so that the kinetic energy of the relative motion is increased, and (b) they modify the slope of the repulsive part of the interaction potential on which the transition probability depends. By letting m °°, we have completely ignored the second effect while we have over-emphasized the first. Note that Equation 12 is identical to an expression we could obtain when the interaction potential is assumed as U(r) = A [exp (— r/a)] — (ae2/2aA) — D. Similarly, if we assume a modified Morse potential of the form... 

The hard-core limiting forms of U(r) do not lead to physically acceptable results. We conclude that this is caused by a complete neglect of the effect of the attractive forces on the slope of the repulsive part in U(r). If the interaction energy is assumed as the sum of a Morse exponential function and the polarization energy evaluated at r = r°, the resulting transition probabilities appear useful for analyzing ion-molecule collisions. 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

Both these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

The probability of a complete Brownian path is then obtained as the product of such single-time-step transition probabilities. For other types of dynamics, such as Newtonian dynamics, Monte Carlo dynamics or general Langevin dynamics, other appropriate short-time-step transition probabilities need to be used [5, 8]. 

For deterministic dynamics the state zt+At at time t + At is of course completely determined by the state of the system zt a time step At earlier. Therefore, the single-time-step transition probability p(zt -> zt+At) can be written in terms of a delta function... 

Formula (2.2) contains only one-dimensional probability density W(xi, t ) and the conditional probability density. The conditional probability density of Markov process is also called the transition probability density because the present state comprehensively determines the probabilities of next transitions. Characteristic property of Markov process is that the initial one-dimensional probability density and the transition probability density completely determine Markov random process. Therefore, in the following we will often call different temporal characteristics of Markov processes the transition times, implying that these characteristics primarily describe change of the evolution of the Markov process from one state to another one. 

The main distinction between the transition probability and the probability to pass the absorbing boundary is the possibility for a Brownian particle to come back in the considered interval (c, d) after crossing boundary points (see, e.g., Ref. 55). This possibility may lead to a situation where despite the fact that a Brownian particle has already crossed points c or d, at the time t > oo this particle may be located within the interval (c, d). Thus, the set of transition events may be not complete that is, at the time t > oo the probability Q(t,xo) may tend to the constant, smaller than unity lim Q(t, x0) < 1, as in the case... 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

The overall transition probability can be expressed in terms of two types of matrix elements, namely < 0 > and < and will thus depend on whether each of these two is allowed or approximately forbidden. [Note that in a real system transitions are frequently not completely forbidden (see e.g., Jaros 1977)]. This point has, for instance, been emphasized by Grimmeiss et al. (1974) and Morgan (1975), who analyze photoconductivity... 

Just as in the unimolecular cases, the basis for the stochastic approach is to consider the reaction 2A-> B as being a pure death process with a continuous time parameter and transition probabilities for the elementary events that make up the reaction process. Letting the random variable X(t) be the number of A molecules in the system at time t, the stochastic model is then completely defined by the following assumptions ... 

What has been presented here is a semiclassical theory of TJ 1) quantum electrodynamics. Here the electromagnetic field is treated in a purely classical manner, but where the electromagnetic potential has been normalized to include one photon per some unit volume. Here the absorption and emission of a photon is treated in a purely perturbative manner. Further, the field normalization is done so that each unit volume contains the equivalent of n photons and that the energy is computed accordingly. However, this is not a complete theory, for it is known that the transition probability is proportional to n + 1. So the semiclassical theory is only appropriate when the number of photons is comparatively large. 

The picture offered by the Fokker-Planck equation is, of course, in complete agreement with the Langevin equation and the assumptions made about the process. If we can solve the partial differential equation we can determine the probability density or eventually the transition probabilities at any time, and thereby determine any average value of functions of v by simple quadratures. 

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