Curve-crossing approach

More elaborate and more reliable procedures that can be used for estimates of rate coefficients of free-radical reactions are the bond energy-bond order method (BEBO) of Johnston and Parr [13] and the curve-crossing approach of Pross [14]. 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

These curves have some interesting properties. At any given pH, evidently Fhs + Fs = 1. At the point where the two curves cross, Fhs = Fs = 0.5, and from Eqs. (6-61) and (6-62), at this point [H ] = K, or pH = pK. That this point corresponds to the inflection point can be shown by taking the second derivative d F/dpH and setting this equal to zero one finds pHj n = pf a- In the limit as [H ] becomes much greater than K. Fhs approaches unity and Fs approaches zero... 

The upper, packing, curve of Fig. 9.6 will decrease to zero for large e the lower, chemistry, curve will increase, cross the packing curve, and approach the desired... 

Relative permeability is the reduction of mobility between more than one fluid flowing through a porous media, and is the ratio of the effective permeability of a fluid at a fixed saturation to the intrinsic permeability. Relative permeability varies from zero to 1 and can be represented as a function of saturation (Figure 5.8). Neither water nor oil is effectively mobile until the ST is in the range of 20 to 30% or 5 to 10%, respectively, and, even then, the relative permeability of the lesser component is approximately 2%. Oil accumulation below this range is for all practical purposes immobile (and thus not recoverable). Where the curves cross (i.e., at an Sm of 56% and 1 - Sm of 44%), the relative permeability is the same for both fluids. With increasing saturation, water flows more easily relative to oil. As 1 - SI0 approaches 10%, the oil becomes immobile, allowing only water to flow. 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix... 

An interesting approach to SN2 reactions has been recently advanced by Shaik which leads to verifiable predictions (Shaik, 1981 Pross and Shaik, 1981 Shaik, 1982 Shaik and Pross, 1982). The basic contention of this theory is that the origin for the barrier of SN2 reactions arises from an avoided curve-crossing between two curves containing the reactant and product Heitler-London VB forms. In this treatment, the reacting pair is analysed in terms of a nucleophile being the electron donor (D) and the substrate the electron acceptor (A). For the simple reactions of type (47), the... 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

Specific for ISC and other predissociative curve-crossings is that the response function approach can be afflicted by instabilities which has to be treated with some care. The instabilities encountered for the MCLR eigenvalue equation near curve crossings is a structural problem of the method itself. Partitioning the MCSCF Hessian to orbital and configuration parts on one hand, and excitation and deexcitations on the other, gives the structure... 

The plot of Eq. (16-7) shown in Fig. 16-1 is the conventional form for such plots both ordinate and abscissa are dimensionless. We have plotted values from the model potential (discussed in Appendix D) as points for comparison they are the result of a full calculation giving tables for all the simple metals (listed also in Harrison, 1966a, p. 309). We shall sec that the screening calculation requires that IV, approach — (2/3) . at small q, with fJp the Fermi energy, so both curves approach that limit. We have chosen such that the two curves cross the horizontal axis first at the same point. Corresponding values for r arc listed for all of the simple metals and for some other elements in Table 16-1. We shall see that most properties depend principally, or only, upon the values of the form factor for... 

Figure 4.9 Morse potential energy curves for chloromethane and its ions. The curves are calculated using the activation energy determined from data in Figure 4.8. The high-temperature data is for unimolecular dissociation via the curve crossing on the approach side of the molecule. Only the VEa is negative and dissociation occurs in the Franck Condon transition. The thermal energy dissociation occurs through the thermal activation of the molecule, as is the case for all DEC(l) molecules.

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