Hush approach

The Levich—Dogonadze—Kuznetsov (LDK) treatment [65] considers that the only source of activation is the polarization electrostatic fluctuations (harmonic oscillations) of the solvent around the reacting ion and uses essentially the same model as the Marcus—Hush approach. However, unlike the latter, it provides a quantum mechanical calculation of both the pre-exponential factor and the activation energy but neglects intramolecular (inner sphere) vibrations (1013—1014 s 1). 

SMOOTHNESS OF THE ELECTRONIC DIPOLE MOMENT AND THE GENERALIZED MULLIKEN-HUSH APPROACH... 

The reorganisation energy consists of internal Ai and solvent As contributions. The former can be calculated in the Born-Hush approach via [107,109,119]... 

The treatment by Balzani et al. in (11) places a different emphasis upon the reorganisational processes compared with that by Marcus and Hush. Bock et al. on the other hand utilised the Marcus-Hush approach and point out that any attempt to predict k must take account of the relative importance... 

We note that, for CT transitions, the expression for /fab is not needed if the Mulliken-Hush approach is used to calculate H h from experimental quantities as discussed in Section 1.3.4. Also, the generalized Mulliken-Hush treatment [32, 33] allows the calculation of //ab from the adiabatic wavefunctions and the complete Hamiltonian the extension of Eq. 56 to include more than two states is then used to obtain Hub-... 

Here we mention as an example that in the coordination-chemistry field optical MMCT transitions between weakly coupled species are usually evaluated using the Hush theory [10,11]. The energy of the MMCT transition is given by = AE + x- Here AE is the difference between the potentials of both redox couples involved in the CT process. The reorganizational energy x is the sum of inner-sphere and outer-sphere contributions. The former depends on structural changes after the MMCT excitation transition, the latter depends on solvent polarity and the distance between the redox centres. However, similar approaches are also known in the solid state field since long [12]. 

In the isoelectronic zirconates this absorption band is not observed [17]. The spectral position of these MMCT bands has been interpreted in terms of the relevant ionization potentials [17], an approach which runs parallel with the Hush theory [10]. The fact that the MMCT transition is at higher energy in the Cr(III)-Ti(IV) pair than in the Fe(II)-Ti(IV) pair is due to the more than 10 eV higher ionization potentials of the trivalent transition-metal ions compared to the divalent transition-metal ions. The fact that the MMCT absorption band is not observed in the zirconates in contradiction to the titanates is due to the higher ionization potential of the Ti(III) species ... 

As with the Marcus-Hush model of outer-sphere electron transfers, the activation free energy, AG, is a quadratic function of the free energy of the reaction, AG°, as depicted by equation (7), where the intrinsic barrier free energy (equation 8) is the sum of two contributions. One involves the solvent reorganization free energy, 2q, as in the Marcus-Hush model of outer-sphere electron transfer. The other, which represents the contribution of bond breaking, is one-fourth of the bond dissociation energy (BDE). This approach is... 

This favorable situation may not be encountered in every case. With radical reductions endowed with high intrinsic barriers, the half-wave potential reflects a combination between radical dimerization and forward electron transfer kinetics, from which the half-wave potential cannot be extracted. One may, however, have recourse to the same strategy as with the direct electrochemical approach (Section 2.6.1), deriving the standard potential from the half-wave potential location and the value of the transfer coefficient (itself obtained from the shape of the polarogram) under the assumption that Marcus-Hush quadratic law is applicable. 

Most of the mixed-valence systems mentioned by Robin and Day and by Hush were in the solid state. The problem of creating discrete chemical systems for which experiments could be carried out either in solution or in the solid state was first attacked experimentally by Creutz and Taube (6). Their approach was to link together the two metal sites through a ligand bridge, which led to dimers and oligomers. 

The general approach is illustrated in detail for the case of aqueous ferrous and ferric ions, and the calculated rate constant and activation parameters are found to be in good agreement with the available experimental data. The formalisms we have employed in studying such complicated condensed phase processes necessarily rely on numerous approximations. Furthermore, some empirical data have been used in characterizing the solvated ions. We emphasize, nevertheless, that (1) none of the parameters were obtained from kinetic data, and (2) this is, as far as we are aware, the first such theoretical determination to be based on fully Ab initio electronic matrix elements, obtained from large scale molecular orbital (MO) calculations. A molecular orbital study of the analogous hexaaquo chromium system has been carried out by Hush, but the calculations were of an approximate, semi-empirical nature, based in part on experi-... 

For typical outer-sphere exchanges at ordinary temperatures, it seems probable that the original assumption of Hush and of Marcus that barrier penetration is a comparatively minor effect is correct. Moreover> it is, in a particular case, quite simple to calculate. The more general questions to which we do not yet have an answer are how adequate is the Golden Rule approach in discussing tunnelling, and, in particular, what would be expected for systems strictly remaining on one surface (electronically adiabatic) A number of fundamental issues involved here have been discussed in a recent series of papers (42-45). 

DR. HUSH For myself, if I gave the impression of any feeling of self-satisfaction, it was certainly in the sense that I think we can say that there is a glimmer of light in the darkness. And in chemistry we must be grateful for the smallest spark. I have stressed also that our present theoretical approaches to rate calculations have been almost entirely confined to outer-sphere processes, and that the task of formulating reliable theories for inner-sphere transfers will be a formidable one. 

Another common approach consists of the comparison between the experimental rate constants and theoretical values calculated by the procedure developed by Marcus (1956), Marcus and Sutin (1985) as well as Hush (1958). This classical procedure is used widely. Premsingh et al. (2004) gave the relevant references and described a detailed procedure to analyze the ion-radical reaction between anilines and chromium (V) complexes of azomethyne derivatives. Lepage et al. (2003) studied transformation of para-substituted thioanisoles to corresponding methylarylsulfoxides... 

The Marcus treatment uses a classical statistical mechanical approach to calculate the activation energy required to surmount the barrier. It assumes a weakly adiabatic electron transfer process and non-equilibrium dielectric polarization of the solvent (continuum) as the source of activation. This model also considers the vibrational contributions of the inner solvation sphere. The Hush treatment considers ion-dipole and ligand field concepts in the treatment of inner coordination sphere contributions to the energy of activation [55, 56]. 

The discussion in the previous section was helpful in identifying the factors at the molecular level which are involved when electron transfer occurs. Two different theoretical approaches have been developed which incorporate these features and attempt to account for electron transfer rate constants quantitatively. The first, by Marcus34 and Hush,35 is classical in nature, and the second is based on quantum mechanics and time dependent perturbation theory. The theoretical aspects of electron transfer in chemical36-38 and biological systems39 have been discussed in a series of reviews. 

For small values of the reorganization energy, the rate constants found a notable difference with those predicted by the BV approach for large applied overpotentials, reaching a plateau rather than an exponential increase. The Marcus-Hush-Chidsey model (MHC) was able to reproduce this behavior accurately in a wide range of temperatures [50]. The rate constant value obeys an exponential dependence on the distance between the electrode surface and the attached redox species [51] ... 

Mulliken-Hush formula for the diabatic electronic coupling. We suggest that the latter approach might be rather useful to thoroughly investigate multistate effects on the electronic diabatic coupling recently pursued by Cave and co-workers [66]. 

To compare the experimental values of AGCF with the predictions of the Marcus-Hush theory, there would seem to be two worth wile approaches. 

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