Solvent Polarization Fluctuation Model

A better expression for A including changes in the inner sphere is  

The total energy change of the three steps which represents the energy required to transfer an electron to an ion in solution with an appropriate change of equilibrium polarization of the medium. A, is given by 

The energy A should be independent of intermediate state and, therefore, of and 8i. Let us calculate A for the equilibrium 

This gives the thermal energy required to shift the energy level from Eq to an arbitrary energy E. The energy distribution function D(E) is given by 

Step (72) represents the electron transfer step from the solid to the unoccupied level of ion in its equilibrium state, o ox The energy change for this step is given by qE - E, where E is the initid 

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For the ET affected by the motions of the high-frequency vibrational modes and the solvent polarization fluctuation, the PESs of the donor and the acceptor can be respectively modeled as ... 

Computer simulations have provided further insight into the model of random fluctuations as a prerequisite for e.t. in polar solvents [60], It has been shown that spontaneous local polarity fluctuations of the magnitude envisaged by the Marcus model are so improbable as to be statistically insignificant and it was necessary to assume that the solvent could adjust continuously in order to follow the position of the electron in the course of e.t., as if e.t. would be slow enough to be the rate-determining kinetic step. To what extent such a modification of the model... 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

In addition to orientational polarization response, , may have translational contributions arising from solvent density fluctuations [226], As it was shown by Matyushov et al. [226, 227], molecular translation of the solvent permanent dipoles is the principal source of temperature dependence for both the solvent reorganization energy and the solvation energy. In fact, the standard dielectric continuum model does not predict the proper temperature dependence of E, in highly polar solvents It predicts an increase in contrast to the experimentally observed decrease in ,. with temperature. A molecular model of a polarizable, dipolar hard-sphere solvent with molecular translations remedies this deficiency of the continuum picture and predicts correct temperature dependence of ,., in excellent agreement with experiment [227a],... 

First of all, the model of the solvent was materially improved. In 1969 Dogonadze and Kuznetsov took into account the spatial correlation of the polarization fluctuations in the medium. In Refs. 43-46 the frequency dispersion of the dielectric constant was also taken into account, t Using this model, Vorotyntsev et performed quantum mechanical calculations of the... 

Thompson and Schenter [98] have proposed a QM/MM method at INDO/S QM level including explicitly the polarizability of the MM subsystem through atomic point dipolar polarizabilities. Several groups have proposed QM/MM techniques in which the MM solvent polarization is explicitly taken into account through a fluctuating charge model [61,63] and... 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

Picosecond Raman measurements have led to the proposal of a dynamic polarization model." In this model, 5i tS undergoes reversible changes in vibrational frequencies that are induced by solvent fluctuations. The mixing of a perturbing state with interconverts carbon-carbon double bonds with single bonds that leads S tS near the vertical geometry to proceed along the pathway for isomerization. 

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). 

Solutions of many proteins, synthetic polypeptides, and nucleic acids show large increases in permittivity c (u>) over that of solvent, normally aqueous, at sufficiently low frequencies f = w/2ir of steady state AC measurements, but with dispersion and absorption processes which may lie anywhere from subaudio to megahertz frequencies. Although our interest here is primarily in counterion fluctuation effects as the origin of polarization of aqueous DNA solutions, we first summarize some relevant results of other models for biopolymers. 

It seems universally accepted that AG° can be accurately determined by electrochemistry, but this requires the use of model compounds for most intramolecular cases, and can only be true for reactions that are slow enough to allow equilibration of solvent fluctuations. Quantitative electrochemical measurements are usually made in polar solvents like acetonitrile containing 0.1 M of a supporting electrolyte. The corrections for solvent changes for reactions run in other solvents are not very accurate, although widely used. 

To end this Section, we would like to go back to what we had just mentioned in Section 4.2 about dynamical effects of stochastic solvent fluctuations on chemical reactions. As already said, solvent fluctuations can be introduced in many effective ways as an example we quote an attempt to model them, via the continuum PCM approach (Bianco et al., 1992) using a Sat2 reaction as test case. In this formulation a fluctuation of a reasonable magnitude is modelled as a time-dependent change in the polarization... 

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