Complex adiabatic paths

The vibrationally adiabatic path of ISM and the LS potential are combined to reflect the fact that a hydrogen-bonded complex brings the structure of the reactants closer to that of the transition state, as shown in Mechanism (V.I). A hydrogen bond can be regarded as an incipient proton transfer, and the bond order at the precursor complex is no longer n=0, but the bond order of the B- -H bond in that complex, n. .. g. Similarly, for the products, the bond order is not n = 1 but the bond order of the H A bond in the successor complex, (1- h - a)- Thus, for a proton transfer in condensed media, the reaction coordinate n is only defined in the interval [%...b. (1 %...a)]- The precursor and successor complexes are included in the classical reaction path of ISM with a simple transformation of the reaction coordinate [3]... 

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour ... 

The operational path for this case is shown as case (3) in Figure 18.3(b). Achievement of this path, in a practical sense, is very difficult, since it requires a complex pattern of heat transfer for the point-by-point adjustment of T. A literal implementation is thus not feasible. A more practical situation is adiabatic operation of a PFR, considered next, which has its own optimal situation, but one different from case (3). 

In this chapter, the reactor dynamics under adiabatic and isoperibolic conditions is analyzed, while the temperature-controlled case is addressed in Chap. 5. It must be pointed out that these conditions can be easily realized in laboratory investigations, e.g., in reaction calorimetry, but represent mere ideality at the industrial scale. Nevertheless, this classification is useful to recognize the main paths leading to runaway without the burden of a more complex mathematical approach. 

Elaborate mechanistic schemes have been suggested for the principal rearrangements of cyclohexenone, 2,5-cyclohexadienone, and bicyclo-hexenone systems induced by w - tt excitation which are compatible with the experimental data outlined above. In essence, these mechanisms are based on the common concept that the complicated structural changes are initiated in an electronically excited state. For the appreciably complex ketones considered, reaction initiation in a vibrationally excited ground state produced by adiabatic ir n demotion is expected to be readily suppressed in solution by collisional deactivation. It has been pointed out that by this general concept the rearrangements provide a decay path for electronically excited states which allows transfer of minimal amounts of enei to the environment in each step. 

A basic means of modelling approximate reaction paths is the adiabatic mapping or coordinate driving approach [123,149]. The energy of the system is calculated by minimizing the energy at a series of fixed (or restrained, e.g. by harmonic forces) values of a reaction coordinate, which may be the distance between two atoms, for example. More extensive and complex combinations of geometrical variables can be chosen. This approach is only valid if one... 

Only one case of the formulation will be considered here. The three rotations of the molecule as a whole (as well as the three translations of the molecule, which make no contribution to the rate) will be considered to be adiabatic. All the internal degrees of freedom will be taken as active and as vibrations (except for the actual motion along the decomposition path in the activated complex). For this model, one finds... 

---