Transition Probabilities and Rates

In the first iteration, the unknown time-dependent wavefunction o( )) in the integral in Eq. (3.84) is approximated by the zeroth-order term, leading to 

We can then identify the first-order correction to the time-dependent wavefunction as 

The second-order correction to the wavefunction is obtained, if we iterate once more on Eq. (3.84), which means that we let the unknown function ko(t, . )) in the integral in Eq. (3.84) be equal to g° (t )) - - 4 g (t, . F)). This gives rise to one term more 

The compact integral expressions for the time-dependent wavefunction in Eq. (3.86) and in particular for the first-order correction in Eq. (3.87) will be employed in the derivation of response functions in the following section. However, for the interpretation of the time-dependent wavefunctions it is useful to expand them in the complete set of unperturbed wavefunctions Eq. (2.14) analogous to the perturbed wavefunctions of time-independent perturbation theory in Eq. (3.23) 

Inserting the expression for the first-order wavefunction Eq. (3.87) we obtain for the first-order coefEcient 

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In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

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. 

The laser-induced X-ray count rate data, normalized to laser power and beam current, were fitted with a Lorentzian using a least-squares technique to obtain the resonance centroid in the laboratory frame. The average resonance width, corrected to the rest frame of the ions, was 8.5 0.4 cm-1, compared to the natural width of 8.0 cm-1. This is consistent with some saturation in the transition probability and also in the detection sensitivity of the proportional counter at increased count rate. The wavenumber of the resonance centroid, in the rest frame of the moving ion, is obtained using the relativistic Doppler formula,... 

In these hybrid simulations, coupling happened through the boundary condition. In particular, the fluid phase provided the concentration to the KMC method to update the adsorption transition probability, and the KMC model computed spatially averaged adsorption and desorption rates, which were supplied to the boundary condition of the continuum model, as depicted in Fig. 7. The models were solved fully coupled. Note that since surface processes relax much faster than gas-phase ones, the QSS assumption is typically fulfilled for the microscopic processes one could solve for the surface evolution using the KMC method alone, i.e., in an uncoupled manner, for a combination of fluid-phase continuum model parameter values to develop a reduced model (see solution strategies on the left of Fig. 4). Note again that the QSS approach does not hold at very short (induction) times where the microscopic model evolves considerably. 

The spectrum of radiation from electronically excited states of atoms appears as lines, when the emission from a hot gas is diffracted and photographed, whereas radiation from these excited states of molecules appears as bands because of emission from different vibrational and rotational energy levels in the electronically excited state. Equation (26) shows that the intensity of radiation from a line or band depends upon the temperature and concentration of the excited state and the transition probability (the rate at which the excited state will go to the lower state). Since the temperature term appears in the exponential, as the temperature rises the exponential term approaches unity, as does the ratio of the concentration of the excited (emitting) state to the ground state (as T approaches oo, Ng = Nj). The concentrations of both the ground and excited states, however, reach a maximum, and then decrease due to the formation of other species. The line or band intensity must also reach a maximum and then decrease as a function of temperature. This relationship can be used to determine the temperature of a system. 

If we compare the rate equation above with the phenomenological one for desorption, we see that the transition probability is nothing but the reaction rate constant. This is not always the case. It is quite common that the transition probability and the reaction rate constant differ by a factor that depends on the number of neighbors of a site. It s very important to know that factor when one tries to calculate rate constants quantum chemically, because one can only calculate transition probabilities using quantum chemical methods, and one needs the derivations we present here to obtain the rate constants. 

In practice the relation between transition probabilities and reaction rates are more difficult. This is because often rate equations are only phenomenological approximations, and the rate constants contain implicitly corrections to make the approximation as good as possible. 

Sometimes we find it advantageous to focus our stochastic description not on the probability but on the random variable itself. This makes it possible to address more directly the source of randomness in the system and its effect on the time evolution of the interesting subsystem. In this case the basic stochastic input is not a set of transition probabilities or rates, but the actual effect of the environment on the interesting subsystem. Obviously this effect is random in nature, reflecting the fact that we do not have a complete microscopic description of the environment. 

D. K. Bondi, J. N. L. Connor, B. C. Garrett, and D. G. Truhlar, Test of variational transition state theory with a large-curvature tunneling approximation against accurate quantal reaction probabilities and rate coefficients for three collinear reactions with large reaction-path curvature Cl + HC1, Cl + DC1, and Cl + MuCl, 7. Chem. Phys. 78 5981 (1983). 

Further developments in the use of kinetic chemical activation for energy transfer studies will undoubtedly be based on more detailed attention to the energy dependence of the transition probabilities and more accurately defined initial product excitation functions. Each extension wall require that the unimolecular decomposition rate constant be calculated or measured as a function of energy. In cases where adequate... 

The systematics of E2 transitions have been discussed by Sunyar, by Temmer and Heydenburg , and by Ford. The last two authors plot the square of the nuclear deformation (which is calculated from the transition probability and is proportional to it) against the neutron number (Fig. 66). The deformation rises very sharply to a flat maximum for N near 90, and then falls slowly as the closed shell at Pb is approached. The rapid rise near N = 90, it should be noted, corresponds to the rapid rate of increase in the nuclear deformation measured by the isotope shift. No corresponding analogue of the sharp rise in the isotope shift just above iV = 82, discussed by Brix and Kopfermann , has yet been detected. 

The statistical adiabatic channel model (SACM) " is one realization of the laiger class of statistical theories of chemical reactions. Its goal is to describe, with feasible computational implementation, average reaction rate constants, cross sections, and transition probabilities and lifetimes at a detailed level, to a substantial extent with state selection , for bimolecular reactive or inelastic collisions with intermediate complex formation (symbolic sets of quantum numbers v, j, E,J. ..)... 

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