Molecular Imaging. Markus Rudin

and Their Interactions

      We begin with a model for reactants that is the simplest and the first one used in the theory.

      M. assumes that each reactant may be treated as a sphere which, if the reactant is an ion, may be surrounded by a concentric spherical shell of saturated dielectric, that is, of dielectric completely oriented in the ion’s electric field, with which is in equilibrium. Outside this spherical shell the solvent medium is supposed to be dielectrically unsaturated, in the 1956 paper [1]. The ion plus the rigid saturated dielectric region is treated as a conducting sphere of radius a,⁽²⁾ and in a more general manner in later papers. Let us now have two reacting ions of radii a₁ and a₂. In the first treatment of M., the radii are supposed to be of a fixed value before, during, and after ET. Each reactant is made up also by all of the surrounding solvent molecules (in theory, considering the long-range nature of the electrostatic interactions). The solution is supposed to be dilute enough that other couples of reactants do not interact with the pair we are considering. Moreover, in this first formulation of the theory no ionic atmosphere is considered.

      Let us now consider the free energy of the system made up of two ions with charges q₁ and q₂ at distance R from each other. There are four distinct contributions to it:

      (i)The free energy of interaction of all atoms within the first sphere with each other and with the central ionic charge

      (ii)The same for the second sphere

      (iii)The free energy of interaction of all molecules outside of the two spheres with each other and with the charges of the spheres

      (iv)The interaction of the two ionic spheres with each other

      If we assume that the atoms within the spheres do not change their average position during the mutual approach of the ions, the first two contributions to the formation of the TS are independent of the interionic distance R and therefore do not contribute to the free energy of its formation from the reactants.

      In order to calculate contributions (iii) and (iv), it is necessary to consider the properties of the dielectric outside the spheres.

       2.3.Electrostatic Characteristics of the Transition State

      State X^∗ is considered as a macroscopic system, made up by x^∗ microstates, in which the dielectric medium surrounding the spheres is a continuum characterized by a definite value of the macroscopic polarization at each point of the system. In order to calculate contribution (iii), we need to determine the polarization function in the volume outside that occupied by the spheres. The electronic polarization for X^∗ will be denoted as and the atomic plus orientation as where r is the position vector with respect to an arbitrarily chosen origin. The total polarization function is P^∗(r), that is:

      For state X, the corresponding polarization functions are P_e(r), P_u(r), and P(r). These functions have the following properties:

      (i)As previously seen, the X^∗ and X states have the same atomic and orientation polarization functions, that is,

      (ii) and P_e(r)⁽³⁾ are always in equilibrium with the respective electric field strengths, that is, they are determined by the fields:

      where α_e is the polarizability associated with the E-type polarization.

      (iii)P_u(r) is unrelated to the electric field strengths in the nonequilibrium states.⁽⁴⁾

The meaning of the macroscopic polarization functions in relation to the microscopic structure of the dielectric media will be now defined following Ref. [11] where it is very clearly explained.

      The classical electrostatics definition of P is

      The macroscopic P(r) is the result of very many microscopic dipoles pointing along the direction of the polarizing electric field. What does it mean to represent discrete molecular dipoles by a continuous macroscopic density function P(r)? The microscopic electric field inside matter varies abruptly from point to point and also so does in time: the field can be, for instance, very great near an electron at a certain instant and it can be completely different an instant later as the atoms move about because of thermal agitation. One is unable to calculate this wildly varying microscopic field which is also in general of no interest. The problem would be similar to that of keeping track of the instantaneous microscopic density at a point in a liquid or gas. What one does is to calculate the macroscopic field inside the matter defined as the average field in a region large enough to contain thousands of atoms, say, so as to allow a smoothing out of the uninteresting microscopic fluctuations and be describable by a P(r). The region must also be small enough to follow the large-scale variations of the field. This is what is meant by macroscopic average field inside matter. So this is the meaning of the E(r) and P(r) fields we are considering.

      We now want to determine the contribution to the electric field caused by the polarization itself.

      We know from electrostatics that the individual microscopic dipole p exerts an electric potential:

      What is now the potential caused by the polarization P(r)? In each tiny volume element dV we have

      so that the potential exerted in r by the polarization P will be

      Observing that [11, p. 15]:

      we finally have:

      The complete potential when dielectrics are present [12, p. 281] is in general the sum of a contribution from volume charges with volume density ρ(r), of surface charges with surface density σ(r) plus the contribution earlier considered of the polarized volume elements dV. On the overall we have [1]:

      Note that the integrals run over the entire