Transition probabilities alternating

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

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. 

One explanation is that the radiative transition probabilities of the excimers are similar to those of the monomeric singlet states, but the excimers are formed in much lower yields. This could be the case if the initially excited state (the optically bright state populated by absorption) forms the excimer state in competition with other decay channels. An alternative explanation is that the excimer states are formed in high yield, but have low radiative transition probabilities (i.e. they are relatively dark in emission). 

Many of the observed levels have measured g-factors which are closer to the pure case (c) values than to any alternative pure coupling case. However there is extensive rotational electronic coupling which, in many instances, mixes the case (c) states case (e) is then a better limiting basis, as we shall see in due course. First we investigate the electric dipole transition probabilities for the Zeeman components, so that we can understand the pattern of lines illustrated in figure 10.73. 

An alternative way of estimating the excited state lifetime is to compute the ratio of the MD timestep, 8t and the ensemble and time averaged transition probability... 

Calculation of the transition probability by the Landau method is based on the analytical continuation of classical dynamical variables into the classically forbidden region of the potential [5]. Alternatively, the Landau transition probability can be recovered from the Fourier components of certain classical quantities, related to the transitions in question [10,11]. This allows one to write the Landau VR probability as... 

Figure 4.5 The set of transition probabilities from a specific configuration cto a large number of alternative configurations involving a single reaction step (surface diffusion, adsorption, desorption, surface reaction) during the KMC simulation of the electrodeposition of copper.

An alternative form of the above equation in terms of transition probabilities,... 

Fig. 16. Moves used to equilibrate coil configurations for the self-avoiding walk model of polymer chains on the simple cubic lattice (upper party end rotations, kinkjump motions and crankshaft rotations f 107]. From time to time these local moves alternate with a move (lower pan) where one attempts to replace an A-chain by a B-chain in an identical coil configuration, or vice versa. In the transition probability of this move, the chemical potential difference Ap as well as the energy change SjF enter. From Binder [2S8]...

The exponential operator (- ) is one of various alternatives that can be employed to compute the ground-state properties of the Hamiltonian. If the latter is bounded from above, one may be able to use 11 — , where x should be small enough that 0 = 1 — xE0 is the dominant eigenvalue of 11 — . In this case, there is no time-step error and the same holds for yet another method of inverting the spectrum of the Hamiltonian the Green function Monte Carlo method. There one uses ( — ) 1, where is a constant chosen so that the ground state becomes the dominant eigenstate of this operator. In a Monte Carlo context, matrix elements of the respective operators are proportional to transition probabilities and therefore... 

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