Molecular Imaging. Markus Rudin

rel="nofollow" href="#fb3_img_img_a64d718f-e047-5fa4-be67-778a4e198857.png" alt="image"/>

      When such work terms occur, the ΔU^∘ in Eq. (3.40) is replaced by that is, the ΔU⁰ at separation distance R. Thus Eq. (3.42) is replaced by:

      where ΔU⁰ is the ΔU at infinite separation and w^p is the work to bring the products together to the separation distance R.

      The earlier equations refer to the contribution given by λ_i to the total λ = λ_i +λ_o. The equations found here in a very simple way considering a cross section of the PES where only simple harmonic forces are involved are the same as those found using the nonequilibrium dielectric polarization theory for the complicated atomic motions and interactions of the solvent, that is,

This “method of the intersecting parabolas” is used, for instance, in Ref. [35] in a simple description of Marcus theory.

      The above calculation of ΔU^* is classical and indeed classical mechanics is commonly used nowadays to treat reactive collisions but for some problems a quantum mechanical treatment is needed as, for example, for treating a proton vibration in the equation:

A quantum treatment is given in Refs. [36, 37].

      The results of Eqs. (3.45–3.47a) may be extended to all vibrations of the reactant(s) in reactions 3.38 or 3.44. The PECs of Fig. 3 are replaced by PESs plotted as a function of all the q’s in the system rather than just one. The coordinate q in the figure represents then in this case some path in the N-dimensional q-space and the R and P curves are profiles of the actual PESs plotted along that path. λ_i becomes a sum of terms of the type in Eq. (3.43), summed over all vibrations, that is,

k_i is related to the force constants of a bond, and in the oxidized and reduced form for Eqs. (3.38) or (3.44). A more sophisticated treatment of λ_i is given in Ref. [18], leading to k_i being a certain average of and More on this in Chapter 6.

      “There remain the solvent fluctuations outside of the inner coordination shell of the reactant in Eq. (3.38a) or reactants in Eq. (3.38b). Here, the potential energy functions do not depend on the solvent coordinates (orientations, translations) in the simple quadratic fashion in Eqs. (3.39) and (3.40) of Fig. 1. The treatment of the solvent coordinates is correspondingly more complicated. However, one feature is immediately clear: Just as a thermal fluctuation of vibrational coordinates was needed to reach the intersection region in Fig. 1, a suitable thermal fluctuation of solvent orientation coordinates or reactant’s vibrations also permits the system to reach the N − 1 dimensional hypersurface (the intersection region). A statistical mechanical treatment of the free energy associated with these fluctuations is given in Ref. [18]. Dielectric continuum theory also permits an estimate of the latter to be made.” [13, p. 166]

      In the very instructive review of Ref. [13], M. gives a simple and elegant derivation of the free energy change needed to reach the intersection region by the fluctuations of solvent dielectric polarization.

      I shall report it now almost verbatim with explanatory notes.

       3.11.*The Nonequilibrium Polarization Expression by a Two Steps Charging Process—A Simplified Derivation

Let us consider the homogeneous reaction system in Eq. (3.44) and the modification for the electrode case in Eq. (3.38a). The charges of the reactants are denoted by e_i and their radii by a_i for reactants 1 and 2 (i = 1, 2). A superscript p to the e_i denotes the charges of the products. Let D_s and D_op be the static and optical dielectric constants of the solvent. The separation distance of the centers of the reactants is denoted by R. Marcus describes here in a simplified way his famous method [38] of calculating the nonequilibrium dielectric polarization of the medium. He produced the state of nonequilibrium polarization P = P_u +P_e by a two stages reversible charging process. “Since each step is reversible, the free energy of formation of this non-equilibrium system, i.e. the free energy of this polarization fluctuation, can be calculated in a relatively straightforward manner.”

      The two-step charging process at a given separation distance R is the following:

      (i)The charge of each reactant i is changed from e_i to where is so chosen as to produce the desired orientational–vibrational dielectric polarization P_u

      (ii)The charge of each particle i is changed back from to e_i holding the earlier orientational–vibrational dielectric polarization P_u fixed. After step (ii), the desired electronic polarization P_e is also obtained because P_e is now in equilibrium with the electrostatic potential, that is, it is determined by ψ(r) which is, on its turn, determined by the ionic charges and by P_u (see Eq. 2.11b). Note then that, once back to the initial charge e_i this charge is not alone in determining P_e