Successor complexes

A chemical interconversion requiring an intermediate stationary Hamiltonian means that the direct passage from states of a Hamiltonian Hc(i) to quantum states related to Hc(j) has zero probability. The intermediate stationary Hamiltonian Hc(ij) has no ground electronic state. All its quantum states have a finite lifetime in presence of an electromagnetic field. These levels can be accessed from particular molecular species referred to as active precursor and successor complexes (APC and ASC). All these states are accessible since they all belong to the spectra of the total Hamiltonian, so that as soon as those quantum states in the active precursor (successor) complex that have a non zero electric transition moment matrix element with a quantum state of Hc(ij) these latter states will necessarily be populated. The rate at which they are populated is another problem (see below). 

Let us take a simple example, namely a generic Sn2 reaction mechanism and construct the state functions for the active precursor and successor complexes. To accomplish this task, it is useful to introduce a coordinate set where an interconversion coordinate (%-) can again be defined. This is sketched in Figure 2. The reactant and product channels are labelled as Hc(i) and Hc(j), and the chemical interconversion step can usually be related to a stationary Hamiltonian Hc(ij) whose characterization, at the adiabatic level, corresponds to a saddle point of index one [89, 175]. The stationarity required for the interconversion Hamiltonian Hc(ij) defines a point (geometry) on the configurational space. We assume that the quantum states of the active precursor and successor complexes that have non zero transition matrix elements, if they exist, will be found in the neighborhood of this point. 

To make the ideas sharper consider the case of two quasi degenerate quantum states of the active precursor and successor complexes. The discussion made around equation (57) holds true here too. The activated complex will be the place of a coherent electro-nuclear fluctuation that will go on forever, unless there are quantum states belonging to the relaxation channels of Hc(i) and Hc(j). Note that the mechanisms of excitation to get into the quantum activated complex and those required to relax therefrom are related to the actual rate, while the mechanism of interconversion is closely connected with an... 

This is a (minimal) model including the formation of the complex R1-R2, the active precursor complex APC that interconverts to those states belonging to the active successor complex ASC, as discussed in the previous section. The chemical reaction, in this model, ends up with the formation of the products PI and P2. The kinetic parameters k+ and k- hide the effects of quantum interconversions via the intermediate Hamiltonian Hc(ij). Let us introduce this feature in the kinetic model, so that... 

The reaction channel is open as soon as quantum states of the intermediate Hamiltonian become populated. By hypothesis, such states have two possible different relaxation channels One back to the reactants, the other forward to product via the quantum states of the successor complex. 

The interconversion step with Si and Sf standing for the active precursor and successor complexes reads now ... 

Rates of reductive dissolution of transition metal oxide/hydroxide minerals are controlled by rates of surface chemical reactions under most conditions of environmental and geochemical interest. This paper examines the mechanisms of reductive dissolution through a discussion of relevant elementary reaction processes. Reductive dissolution occurs via (i) surface precursor complex formation between reductant molecules and oxide surface sites, (ii) electron transfer within this surface complex, and (iii) breakdown of the successor complex and release of dissolved metal ions. Surface speciation is an important determinant of rates of individual surface chemical reactions and overall rates of reductive dissolution. 

Similarly, inner-sphere and outer-sphere mechanisms can be postulated for the reductive dissolution of metal oxide surface sites, as shown in Figure 2. Precursor complex formation, electron transfer, and breakdown of the successor complex can still be distinguished. The surface chemical reaction is unique, however, in that participating metal centers are bound within an oxide/hydroxide... 

A forms the surface precursor complex (A S) which then forms the successor complex (B S) after overcoming the activation energy of the transition state denoted (A S), and (3) B desorbs from the surface (4 ... 

Intersection region, but small enough so that It may be neglected In calculating the height of the potential barrier (Hab Eth) Under these conditions the rate constant for the conversion of the precursor to the successor complex Is Independent of the magnitude of the electronic coupling and depends only on the nuclear factor... 

Figure 1. Potential energy plot of the reactants (precursor complex) and products (successor complex) as a function of nuclear configuration Eth is the barrier for the thermal electron transfer, Eop is the energy for the light-induced electron transfer, and 2HAB is equal to the splitting at the intersection of the surfaces, where HAB is the electronic coupling matrix element. Note that HAB << Eth in the classical model. The circles indicate the relative nuclear configurations of the two reactants of charges +2 and +5 in the precursor complex, optically excited precursor complex, activated complex, and successor complex.

Figure 2. Equipotential sections through the potential energy surface for an exchange reaction. The sections define ellipses if the surfaces are parabolic the top left set refer to the initial state (precursor complex) and the bottom right set refer to the final state (successor complex). The dashed line indicates the reaction coordinate. Parameters P and Pa reflect the state of polarization of the solvent, and coordinates d2 and da reflect the inner-shell configurations of the two reactants...

Figure 13. Fitted cubic curves representing solute internal energy with respect to the d f distance, corresponding to the precursor and the successor complexes, e symbol represents an electron inside an electrode. (Reprinted from Ref. 64.)...

Figure 14. Adiabatic free energy curves for the precursor and successor complexes with respect to A j values of the reaction coordinate A . (Reprinted from Ref 64.)...

The product of intramolecular electron transfer within the precursor complex is the successor complex... 

Fig. 5.1 Reaction profiles for inner-sphere redox reactions illustrating three types of behavior (a) prercur-sor complex formation is rate-limiting (b) precursor-to-successor complex is rate-limiting and (c) breakdown of successor complex is rate-limiting. The situation (b) appears to be most commonly encountered.

Of course the Co CNHj) breaks down rapidly in acid into Co + and 5NHJ. Precursor complex formation, intramolecular electron transfer, or successor complex dissociation may severally be rate limiting. The associated reaction profiles are shown in Fig. 5.1. A variety of rate laws can arise from different rate-determining steps. A second-order rate law is common, but the second-order rate constant is probably composite. For example, (Fig. 5.1 (b)) if the observed redox rate constant is less than the substitution rate constant, as it is for many reactions of Cr +, Eu +, Cu+, Fe + and other ions, and if little precursor complex is formed, then = k k2kz ). In addition, the breakdown of the successor complex would have to be rapid k > k 2). This situation may even give rise to negative (= A//° +... 

A ] represent the reorganized precursor and successor complexes involved in the electron transfer step. This scheme predicts that the observed activation energy will switch from a positive to a negative value if the relaxation of [D/A] back to [D/ A] has a larger temperature dependence than the reorganization of [D/A] to [D/A]. In the... 

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