Direct curve crossing

Inner shell ionization of electrons to the continuum in ion-atom collisions can occur by two different processes. For low Z (projectile) particles on high Z2 (target) atoms the only available process is Coulomb excitation which is variously treated by plane wave Born approximation (PWBA), the binary encounter approximation (BEA), and the semiclassical approximation (SCA). When Z becomes comparable to 7/1 and the ion velocity v is lower than the velocity of the bound electron in question, v, the electrons adjust adiabati-cally to the approach of the two nuclei and enter molecular orbitals (MO) which in the limit of fused nuclei approach the atomic orbitals of the united atom Z = Z + Z2. This stacking of electrons can lead to a promotion of an innershell electron to the continuum or to a vacant outer orbital by direct curve crossing, rotational coupling, or radial coupling between molecular levels when such channels are available. 

The Kaplan-Meier estimates produce a step function for each group and are plotted over the lifetime of the animals. Planned, accidentally killed, and lost animals are censored. Moribund deaths are considered to be treatment related. A graphical representation of Kaplan-Meier estimates provide excellent interpretation of survival adjusted data except in the cases where the curves cross between two or more groups. When the curves cross and change direction, no meaningful interpretation of the data can be made by any statistical method because proportional odds characteristic is totally lost over time. This would be a rare case where treatment initially produces more tumor or death and then, due to repair or other mechanisms, becomes beneficial. 

Right Histograms of the olg(3II) population vs the iodine-iodine separation at 2 time points. Shown are the first two exits for such points in time that the dissociative population has reached an average intramolecular separation of —4 A (This distance is about the upper limit of the experimental probing window.) [16] The localized nature of the dissociative population is a direct result of the vibrational localization on the parent 6 state, as an exit to the a state occurs only when there is population in the curve-crossing region. 

The initial interlayer anion also plays a strong role in determining the pathway of the reaction. Neither the hexagonal nor rhombohedral forms of LiAl - NO3 exhibit staging the alpha vs. time curves cross at o 0.5, strongly suggesting a direct transformation from host to product. This is illustrated in Fig. 13b,c. 

A number of investigations have previously been carried out to elucidate the spectroscopy and dynamics of CHD. Experimental investigations [2-7] have been paired with quantum chemical calculations [8-11], to refine the orbital symmetry concepts developed by Woodward and Hoffman, and van der Lugt and Osterhoff [12]. However, a direct and unambiguous experimental study regarding the timescales involved in the curve-crossing from the initially excited state to the ground state, for isolated molecules in the gas phase, is not yet available. 

To summarize, Jean shows that coherence can be created in a product as a result of nonadiabatic curve crossing even when none exists in the reactant [24, 25]. In addition, vibrational coherence can be preserved in the product state to a significant extent during energy relaxation within that state. In barrierless processes (e.g., an isomerization reaction) irreversible population transfer from one well to another occurs, and coherent motion can be observed in the product regardless of whether the initially excited state was prepared vibrationally coherent or not [24]. It seems likely that these ideas are crucial in interpreting the ultrafast spectroscopy of rhodopsins [17], where coherent motion in the product is directly observed. Of course there may be many systems in which relaxation and dephasing are much faster in the product than the reactant. In these cases lack of observation of product coherence does not rule out formation of the product in an essentially ballistic manner. 

FIG. 13.47 Small strain tensile creep curves of rigid PVC quenched from 90 °C (i.e. about 10 °C above Tg) to 40 °C and further kept at 40 0.1 °C for a period of 4 years. The different curves were measured for various values of time te elapsed after the quench. The master curve gives the result of a superposition by shifts that were almost horizontal the arrow indicates the shifting direction. The crosses refer to another sample quenched in the same way, but only measured for creep at a te of 1 day. From Struik (1977,1978). Courtesy of the author and of Elsevier Science Publishers. 

In refs. 30-32 this dependence was explained by participation of the proton quantum shift—along with reorganization of classical medium—in the reaction. The dependence of on A was thoroughly analyzed elsewhere [14, 27, 32] in connection with the theory of electrode reactions for which the dependence of the rate constant on the energy difference of initial and final states (equal to the overvoltage) can be measured directly. As follows from relations (12) and (14), the probability of the nonradiative transition is maximal when the final-state potential curve crosses the initial-state term close to its minimum. Such reactions were called activationless . The transition probability is described by... 

Consider first a single equation /(X) = 0, in which f (X) is a function of the single variable X. Our purpose is to find a root of this equation, i.e., the value of X for which the function is zero. A simple function is illustrated inFig. I.l it exliibits a single root at the point where the curve crosses the X-axis. When it is not possible to solve directly for the root, a numerical procedure, such as Newton s method, is employed. 

In discussing the alternative theoretical approaches let us limit ourselves to those which have been applied directly to processes in which we are interested in this article, but first of all let us stress once more the importance of the work of Delos and Thorson (1972). They formulated a unified treatment of the two-state atomic potential curve crossing problem, reducing the two second-order coupled equations to a set of three first-order equations. Their formalism is valid in the diabatic as well as the adiabatic representation and also at distances of closest approach near Rc. Moreover the problem of the residual phase x(l) is solved implicitly. They were able to show that a solution of the three first-order classical trajectory equations is not sensitive to all details of the potentials and the coupling term, but to only one function which therefore can be used readily for modelling assumptions. The resulting equations should be solved numerically. Their method has been applied now to the problem of the elastic scattering of He+ + Ne (Bobbio et at., 1973) but unfortunately not yet to any ionization problem. 

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