Cross-coupling parameters

Fig. 1. Schematic one-dimensional cross section through the Gibbs free energy surface G(R) of a spin-state transition system along the totally symmetric stretching coordinate. The situation for three characteristic temperatures is shown (B = barrier height, ZPE = zero-point energy, 28 = asymmetry parameter, J = electronic coupling parameter, AG° = Gh — GJ...

In the situation where the transformation involved barrier crossing, e.g., associated with a nonpolar to polar transformation, the computational time was substantially reduced using the X-dynamics formalism, compared with a standard FEP method. This is because X-dynamics searches for alternative lower free energy pathways the coupling parameters (A/ and A2) evolve in the canonical ensemble independently and find a smoother path then when constrained to move as A = A2. Furthermore, a biasing potential in the form... 

A hexagonal representation of proper and improper primary ferroics as proposed by Newnham Cross (1981) is given in Fig. 6.54. The order parameter for proper ferroics appears on the diagonals of the hexagon, while the sides of the hexagon represent improper ferroics. They indicate the cross-coupled origin of ferroic phenomena. An improper primary ferroic in this classification is distinguished from a true secondary... 

Cf. Example 1.8.) This result shows that in the 3x3 model, the spin-orbit coupling vector depends on three factors the coefficient Q + of the in-phase (/d + /d) character of the singlet state, the spin-orbit coupling parameter (heavy atom effect), and the spatial disposition of the orbitals Xa and Xb-actual intersystem crossing rate will also depend on the Franck-Condon-weighted density of states. (Cf. the Fermi golden rule. Section S.2.3.)... 

For alkenes more difficult to reduce than CO2, such as butadiene (63a), electron transfer from C02 to the alkene may be involved. Cross-coupling of CO2 and 63a in MeCN has been carried out in an undivided flow cell at constant current. Using Et4N salts of formate or oxalate as supporting electrolyte, the anode process is formation of CO2 and H" ", which are both consumed in the cathode process [167]. The outcome (up to 63% total yield) was a mixture of isomers of C5, Cg, and Cjo unsaturated carboxylic acids and diacids. The detailed mechanism is not known, but the products may arise from initial addition of C02 to the unreduced butadiene [167], although electron transfer from C02 to 63a or direct reduction of 63a (present in large excess) cannot be ruled out. Based on the observed influence of experimental parameters on the distribution of the C5, Cg, and C]o acid products, the authors suggest that the reactions take place between adsorbed intermediates [167]. 

As an example of the latter strategy, consider again the AlPO-40 sites giving the A-C cross-sections in Fig. 7. With the quadrupole coupling parameters extracted from the cross-section simulations and taking into account the experimental conditions (described in the caption of Fig. 7) it is possible to calculate the efficiency of the 3QMAS experiment for the three Al sites and obtain the correct relative intensities (Fig. 10). 

An insertion of realistic rate parameters shows that the second and the third of the conditions (11) are not critical. For instance, the rate constant Ad for the dissociation of R—Y will normally not exceed Ad = 1 s-1, because the compound will otherwise be very unstable, even at low temperatures. Experimental rate constants for the reaction of transient with persistent radicals are normally larger than 106 M 1 s-1. Hence, a realistic upper limit of the equilibrium constant is K = 10 6 M. Self-termination constants of transient radicals are normally diffusion-con-trolled, that is, 1010 > AtR > 108 M 1 s, and finally, practical aspects set the lower limit of the precursor concentration to [I]0 = 10 3 M. Hence, the second and the third of the conditions (11) read K < 10 6 M [I]o and Ad/AtR < 10 8 M [I]n and are always well obeyed. For these numbers, one has Ac[I]o/AtR = 10 7 M. Obviously, the first condition is not met for the extreme parameters chosen here, but it will be fulfilled for a lower equilibrium constant, a larger initiator concentration, and a larger cross-coupling constant. 

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