Square scheme

For simplicity we assume that = (d /dx ) + a(d/dx), where o is a constant. The basic steps in the least squares scheme are ... 

Working equations of the least-squares scheme in Cartesian coordinate systems... 

Retaining all of the terms in the w eight function a least-squares scheme corresponding to a second-order Petrov-Galerkin formulation will be obtained. 

Fig. 5. Dual-pathway square scheme mechanism for electron transfer involving the Cun/I([9]aneS3)tt system. The vertical reactions involve ligand gain or loss while the horizontal reactions represent electron transfer. Reproduced from Ref. (6) by permission of the Royal Society of Chemistry.

Fig. 11. Electrochemical square scheme for coupled spin-state change and electron transfer. Reprinted with permission from Ref. (28). Copyright 2002, American Chemical Society.

Overall, the results are taken in support of the sequential (square-scheme) mechanism rather than the concerted alternative. Detailed analysis of the individual systems enabled the authors to distinguish which of the two possible square-scheme pathways was dominant. 

A reaction scheme frequently encountered in practice, the so-called square scheme mechanism, consists of the association of two EC reaction schemes as shown in Scheme 2.3 (which may as well be viewed as an association of two CE mechanisms). In the general case, the cyclic voltammetric response may be analyzed by adaptation and combination of the treatments given in Sections 2.2.1 and 2.2.2. A case of practical interest is when the follow-up reactions are fast and largely downhill. A and D are then stable reactants, whereas B and C are unstable intermediates. When the starting reactant is A (reduction process), the reaction follows the A-B-D pathway. The reoxidation preferred pathway is D-C-A. It is not the reverse of the forward... 

The mechanism of electrochemical reduction of nitrosobenzene to phenylhydroxylamine in aqueous medium has been examined in the pH range from 0.4 to 13, by polaro-graphic and cyclic voltametry. The two-electron process has been explained in terms of a nine-membered square scheme involving protonations and electron transfer steps565. This process is part of the overall reduction of nitrobenzene to phenylhydroxylamine, shown in reaction 37 (Section VI.B.2). Nitrosobenzene undergoes spontaneous reaction at pH > 13, yielding azoxybenzene471. 

If for an oxidation step, the chemical reaction of B leads to the oxidized form of the second redox couple B (and not the reduced one as in the earlier case) and a second chemical transformation from A leads back to A [reaction (14)], we arrive at a square scheme (Figure 11), which forms the basis for many important redox systems [18, 58]. Again SET steps... 

The square scheme discussed above already includes a further common motif in electroorganic mechanisms reaction As A forms a preequilibrium to both ETs in the scheme. The response of such a system in CV depends particularly on the equilibrium constant K = [A]/[A ] and the rate constants a A and If the... 

When both components of the redox couple adsorb on the electrode surface, the mechanism (2.172)-(2.174) transforms into the following square scheme [128] ... 

The two well-separated waves for the reduction of oxygen on mercury were reported by Heyrovsky [91]. The first wave corresponds to the 2e reduction of O2 to H2O2 in acidic or neutral solutions and to H02 in basic media, species that are reduced to water or OH at lower potentials. Jacq and Bloch have developed the square-scheme concept for the discussion of the mechanism of O2 reduction on Hg and on carbon [28]. 

Fig. 18 Square scheme reduction mechanism for loss of the V=0 group.

Based on CV studies carried out at low temperature (—77°C) in 80% methanol/20% water (w/w), Bernardo etal. [114] demonstrated that Cu / ([14] aneS4) (LBI) showed electrochemical behavior identical to that predicted by the Laviron and Roullier square scheme (Scheme 1). Subsequent CV studies by Robandt etal. [115], involving scan rates up to 80000 Vs , made it possible to... 

Fig. 2 Square schemes showing binding and oxidation by M(bpy), where M is Os or Ru. Reprinted with permission from [19]. Copyright (1999) American Chemical Society...

While the above technique can be used in many cases, it does require uncomplicated voltammetry and that significant Ey2 shifts are observed at guest concentrations >10 times the host concentration. If these conditions are not met, then an alternative strategy is needed. The most powerful is to use CV simulation software to fit the experimental CVs to the square scheme or a more complicated mechanism if necessary. This method allows determination of the thermodynamic parameters and possibly the kinetic parameters as well. 

Figure 2.1 A square scheme illustrating a redox-driven intramolecular motion. Species with an asterisk (Ox and Red ) are metastable and tend to rearrange to their stable topological isomer (Ox and Red).

The pioneering papers by Stoddart and Sauvage have stimulated the design of a variety of movable rotaxanes and catenanes, whose controlled motion is promoted by a redox change. In all cases, the process of the redox-driven intramolecular motion can be described by a square scheme, as illustrated in Fig. 2.1. 

There exist other types of redox-driven intramolecular motions that can be interpreted on the basis of the square scheme of Fig. 2.1 and are promoted by... 

Figure 2.6 A square scheme illustrating the pendular motion of an iron center, driven hy the Fe Fe1" redox couple. As judged from voltammetric experiment carried out at varying potential scan rate, the lifetime for both translocation processes is <10 ms.

The complete process can be interpreted on the basis of a square scheme, as shown in Fig. 2.1. Quickness of the mechanical steps (iv) and (ii) is ensured by the high flexibility of the molecular framework. 

Figure 2.18 A square scheme illustrating the disassembling of the [Cu2(16)2]2 + double helicate complex, following Cu -to-Cu" oxidation, and the consequent assembling of two [Cun(16)]2+ mononuclear complexes, following the Cu"-to-Cu reduction. The process ultimately derives from the geometrical coordinative preferences of the two oxidation states Cu1 prefers a tetrahedral coordination, which can be achieved with the double helicate arrangement Cu11 prefers a square coordination geometry, which is fulfilled by the coordination of a single molecule of 16.

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