Successor state

Note that since for each state E , d> assigns a unique successor state Eg, each column of the matrix consists of all zeroes except for a single 1 each row, on the other hand, is under no such constraint and can therefore have any number of I s. 

Heuristically, the dynamics proceed as follows the reaction term makes all active ((7 = 1) and refractory a = 2) sites cycle to their respective successor states. The diffusion term defines the manner in which activity (defined by sites with value (7 = 1) diffuses through the lattice. 

Consider an order W system and a random function 4> which maps each of the H = 2 possible binary states Si to unique successor states Sj = cyclic structure of the corresponding state transition graph... 

When a task is finished, it must transport the products to successor states and tokens to successor places. 

The states in the R-region for which resonance is allowed are called active precursor and successor states (APS and ASS). At equilibrium, there is a symmetry between the interconversion viewed from the side of the successor and the precursor states, and this is the detailed balancing. The transition rate from the ik-state towards the set jn multiplied by the density of states for ik should be equal to the transition rate from the jn-state towards the set ik multiplied by the density of states for jn [47],... 

The use of the symbol E in 5.1 for the environment had a double objective. It stands there for general environments, and it also stands for the enzyme considered as a very specific environment to the chemical interconversion step [102, 172], In the theory discussed above catalysis is produced if the energy levels of the quantum precursor and successor states are shifted below the energy value corresponding to the same species in a reference surrounding medium. Both the catalytic environment E and the substrates S are molded into complementary surface states to form the complex between the active precursor complex Si and the enzyme structure adapted to it E-Si. In enzyme catalyzed reactions the special productive binding has been confussed with the possible mechanisms to attain it lock-key represents a static view while the induced fit concept... 

The reason for this difference is very simple. It is that the density functional approach calculates the rate for the adiabatic channel. For AG x> this will proceed via an activated complex with successor-state formal electron density parameter A = 1 for AGq <-x, it will proceed via an activated... 

In this expression A and Q are distance dispersion resulting from electron-vibrational coupling, and frequency tensor (assumed identical in reactant and product states), respectively (work of formation of precursor and successor states is omitted). If we assume that the frequency tensor is diagonal, then we have simply a sum of independent terms for all inner and outer contributing modes. At sufficiently high temperature, the hyperbolic tangents become unity and we obtain the usual (in this approximation) high-temperature expression ... 

The reorganization energy for the successor state AGFC, is given by the difference in environmental energy between the excited state s and the ground state s [Eq. (3)]. Thus (65), subject to the same conditions as govern Eq. (69),... 

At the transition state, electron transfer takes place rapidly. During this brief moment, the nuclear geometry of the transition state remains fixed (the Franck-Condon principle). Following electron transfer, nuclear relaxation to the equilibrated successor state takes place. The products then separate from the successor state into the bulk of the solution. If the products are charged species, then the work, Wj, involved as the ions separate is w = — wc, and can be estimated from Eq. (28). 

Fig. 13. A hypothetical representation of solvent molecule orientation in the transition state involving electron transfer between two charged reactants. Each solvent molecule possesses a permanent dipole, which in this idealized diagram is depicted by an ellipsoidal shape. Solvent molecules reorganize to the geometry of the transition state. The electronic polarization of the solvent molecules, represented here by the bold arrows, then responds practically instantaneously with electron transfer. Finally, the permanent dipoles of the solvent molecules readjust to the successor state. This model may not be applicable to electron transfer between neutral reetants (see text)...

Fig. 1. Schematic free energy-reaction coordinate profiles for a single-electron electroreduction involving solution reactant O and product R at a given electrode potential E, occurring via three different reaction pathways, PAS, P A S, and P A S". Pathway PAS involves energetically favorable precursor and successor states (P and S) but with a weak-overlap transition state. Pathways P A S and P A"S involve energetically similar precursor and successor states, but with the latter involving strong overlap in the transiton state.

Both these processes can be considered to occur in several distinct stages as follows (i) formation of precursor state where the reacting centers are geometrically positioned for electron transfer, (ii) activation of nuclear reaction coordinates to form the transition state, (iii) electron tunneling, (iv) nuclear deactivation to form a successor state, and (v) dissociation of successor state to form the eventual products. At least for weak-overlap reactions, step (iii) will occur sufficiently rapidly (< 10 16s) so that the nuclear coordinates remain essentially fixed. The "elementary electron-transfer step associated with the unimolecular rate constant kel [eqn. (10)] comprises stages (ii)—(iv). 

The foregoing theoretical treatment implicitly assumes that the interaction between the reacting species and the electrode is sufficiently weak and non-specific so that the energetics of the elementary electron-transfer step are determined by the properties of the isolated reactant and the surrounding solvent ("weak-overlap pathway, Sect. 2.2). However, as noted in Sect. 2.2, the occurrence of inner-sphere pathways may not only alter the overall reaction energentics via stabilization of the precursor and successor states, but also via alterations in the shape of the electron-transfer barrier itself ("strong-overlap pathway). 

Although the foregoing electron-transfer theory is preoccupied with describing the electron-transfer step itself, in order to understand the kinetics of overall reactions it is clearly also important to provide satisfactory models for the effective free energy of forming the precursor and successor states from the bulk reactant and product, wv and ws, respectively. As outlined in Sect. 2.2, it is convenient to describe the influence of the precursor and successor state stabilities upon the overall activation barrier using relations such as... 

Taken together, these two terms, comprising the conventional "doublelayer effect, can be thought of as the influence of the surface upon the transition-state stability, presuming that the reactant-surface interactions in the transition state are an approximately weighted mean of those in the adjacent precursor and successor states. (The "appropriate weighting factor is the transfer coefficient aet.) This therefore constitutes the "thermodynamic catalytic influence of the surface, as distinct from the "intrinsic catalytic effect as defined above. The former, but not the latter, is conventionally termed the "double-layer effect, even though both, in fact, involve surface environmental influences upon the transition state stability. 

The intrinsic barrier therefore denotes the portion of the additional free energy possessed by the transition state with respect to the free energies of the adjacent ground (precursor and successor) states that arises only as a consequence of the non-equilibrium properties of the former. The elucidation of intrinsic barriers, at least relative values for a series of structurally related reactions or for different surface environments, is clearly of central fundamental importance in electrochemical kinetics. Although not often perceived in such terms, a major objective is therefore the utilization of strategies that correct, or otherwise allow for, the influence of thermodynamic contributions upon the experimental kinetic parameters. 

In the context of the present discussion, it is worth noting that virtually all the experimental systems that exhibit such "anomalous temperature-dependent transfer coefficients are multistep inner-sphere processes, such as proton and oxygen reduction in aqueous media [84]. It is therefore extremely difficult to extract the theoretically relevant "true transfer coefficient for the electron-transfer step, ocet [eqn. (6)], from the observed value [eqn. (2)] besides a knowledge of the reaction mechanism, this requires information on the potential-dependent work terms for the precursor and successor state [eqn. (7b)]. Therefore the observed behavior may be accountable partly in terms of work terms that have large potential-dependent entropic components. Examinations of temperature-dependent transfer coefficients for one-electron outer-sphere reactions are unfortunately quite limited. However, most systems examined (transition-metal redox couples [2c], some post-transition metal reductions [85], and nitrobenzene reduction in non-aqueous media [86]) yield essentially temperature-independent transfer coefficients, and hence potential-independent AS orr values, within the uncertainty of the double-layer corrections. 

Figure 3 Plots of free energy of zero-order precursor and successor states versus reaction coordinate, for electron-transfer reactions / (A+- B) — r(A- B+). (a) AG° > 0 (b) AG° = 0 (c) 0 > AG° > —X (d) AG° < —X. The upward-pointing arrows in (a), (b), and (c) indicate intervalence charge-transfer transitions the downward-pointing arrow in (d) indicates a possible fluorescent transition from the precursor state / (A+- -B)...

Figure 5 Potential energy curves for an electron-transfer reaction p(A+- -B) — s A- -B+), showing vibrational quantization, assuming the same vibrational frequency v in precursor and successor states. Some of the vibrational wavefunctions are indicated. The dotted arrows refer to electron transfer below the energy of the crossing of the two curves...

Figure 8 Free energy surfaces for the precursor and successor states of intramolecular electron transfer in a model charge-transfer system. " On the plot the dashed fines indicate the Marcus theory, circles are simulations, and solid lines refer to the Q-model. The vertical dashed line marked Xq indicates the hoimdary of the energy gap fluctuation band predicted by the Q-model. (Reprinted with permission from Ref 54, 1989 American Chemical Society)...

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