The Solvent Coordinate Basics

A dipole in a cavity in a polarizable solvent will polarize the medium and create an electric field at its own position. The simplest model is that of a dipole ]1 at the center of a spherical cavity of radius a embedded in a dielectric. We already encountered its results in Section 9.5. The way Nee and Zwanzig [16] came to their results is as follows Outside the cavity there is a dielectric with dielectric constant c (o)), and inside the cavity we assume only electronic polarization C or vacuum (fj = 1). The frequency dependence of the outside dielectric constant derives from the fact that the molecules in the solvent can rotate to change the polarization. This rotation is diffusional, so the dipoles need time to adjust to a new situation. This does not have an effect on the solution of the boundary value problem. At the boundary, the usual boundary conditions apply the transverse component of the electric field is continuous, as is the normal component of the displacement field. Using these boundary conditions, it is possible to find the fields inside and outside the cavity. Solving this problem gives the electric and displacement fields inside and outside the cavity. The important field is the field created by the outside polarization inside the cavity, the so-called Onsager reaction field [23] E  

The interaction energy of the dipole with its own reaction field is given by  

The dynamical problem that we have to deal with here is much more complicated. We ignored the frequency dependence in in Eq. (9.14), but the solution is equally valid if we do take it into account. The equation can then be used to investigate what happens when the dipole itself becomes time-dependent, or suddenly changes in magnitude and direction. It is not just the energy due to the reaction field that plays a role in the dynamics we also need to consider the force or torque on the dipole if we want to change its magnitude or direction. In this case, we are not so much 

5) There is some confusion in the literature about whether a factor 1/2 should be included here. This is not the case. The free energy A does have a factor 1/2 because there is a negative contribution from the entropy due to the increased order in the solvent. 

For a dipole at the center of the cavity in a dielectric with frequency-dependent e a ), the reaction field is still given by Eq. (9.14). From this, it can be shown that the force on the dipole is given by 

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