Marcus expressions Applications

Occasionally, the successful application of the Marcus expressions (5.35) and (5.37) to a reaction can support its designation as outer-sphere. The reduction of a series of substituted benzenediazonium salts by Fe(CN)5 and (Me5cp)2pe conforms to the simple Marcus expression and represents supporting evidence for the formulation of these reactions as outer sphere (or non-bonded electron transfer in organic systems)... 

The second term in the square brackets is the Born expression applicable at distances n + Ari, i.e., beyond the first hydration shell of thickness Ar. The first term describes the electrostatic interaction inside this shell, characterized by a relative permittivity e now, approximated by the square of the refractive index of water at the sodium D line. With the relevant de /rfT and de/tfT values for water at 25 °C, the enthalpy of hydrationEq. (2.28)is AHeh+A//ei2 = -69.5z2[(0.35(Ari/ri)+1.005)/(ri+Ari)] kJ mor The entropy is then A5eu + A5ei2 = —4.06z [(1.48(Ari/ri)+1.00)/(ri + Ari)] J mol The thickness of the first hydration shell, Ar, depends on the number of water molecules, hi, in it, the hydration number. According to the model (Marcus 1987) hi = 0.36 zi /(r/nm), that is, it is proportional to the charge number of the ion and inversely proportional to its radius. The volume occupied by hi water molecules is nhid l6, where cfw = 0.276 nm is the diameter of a water molecule. Hence the volume of the first hydration shell is given by ... 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

Another property that characterizes solvents is their softness, in terms of the HSAB concept (Pearson 1963), according to which the interactions of soft solvents are strongest with soft solutes, of hard solvents with hard solutes, but are weaker for hard solvents with soft solutes and vice versa. The applicability of the softness property takes into account that it is superimposed on the more general electron pair donation property discussed above. In fact, it can replace (Marcus 1987) the notion of the family dependence of the P scale, expressed by the , parameter (Kamlet etal. 1985). A few quantitative scales have been... 

Equations (22) and (23), similarly to Eq. (9), are applicable to the electron transfer reactions in the normal Marcus region. In the inverted Marcus region, possible vibrational excitation of the reaction products should again be taken into account. Depending on the values of S and V12, some of the accessible reaction channels may be affected by the solvent molecular dynamics. This problem has been discussed in [89], with the main conclusion that the overall reaction rate may be expressed as follows ... 

This work addressed the issue of enzyme catalysis focusing on the principle of physical organic chemistry and the power of computer simulation approaches. It was shown that when such concepts as reorganization energy and Marcus parabolas are formulated in a consistent microscopic way, they could be used to explore the origin of enzyme catalysis. It was also clarified that phenomenological applications of the Marcus formula or related expressions can lead to problematic conclusions. 

It is important to realize that the only approximations that enter into this rate expression is the use of the Fenni golden-rule, which is compatible with the weak coupling nonadiabatic limit, and the Condon approximation which is known to be successful in applications to electronic spectroscopy. The solvent effect on the electronic process, including the slow dielectric response, must arise from the FC factor that contains contributions from all the surrounding intermolecular and intramolecular nuclear degrees of freedom. In fact, if the nuclear component of the solvent polarization was the only important nuclear motion in the system, then on the classical level of treatment used by Marcus Eqs (16.53) and (16.51) with Ea given by (16.49) should be equivalent. This implies that in this case... 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

In this chapter we have focused on the application of a mixed quantum-classical approach for rationalizing the kinetics of chemical reactions involving more than one electronic state. While previous theoretical frameworks like those of Marcus or Lorquet considered a complete decoupling between the quantum and classical phases of evolution of the molecular system, we have proposed an original path where the quantum-to-classical transition operates in a smooth fashion. As a result we have ended up with a new expression for estimating the probability for the system to hop from one step to the other when decoherence occurs. In the second part of this chapter we have shown how the characteristic decoherence times could be evaluated by atomistic simulations on large molecular systems (from 30 to 40 000 atoms in the... 

The requirements for application of the Marcus equation to reactions other than electron transfer are expressed in precise and succinct form hy J. Jortner, Faraday Discuss. Chem. Soc. 74 (1982) 306, 307. See also the contributions to that Discussion by R.A. Marcus, p. 306 and J.R. Murdoch, pp. 297 seq. 

This expression is applicable to RTELs according to Marcus [360], provided the temperature is sufficient for the viscosity to be below 50 mPa s, because when the viscosity is larger (the fluidity is smaller than 0.02 s mPa ) the linearity of 0(T)... 

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