Dielectric boundary condition

In the previous chapter, we have seen how Born s simple and successful idea of a dielectric continuum approximation for the description of solvation effects has been developed to a considerable degree of perfection. Almost all workers in this area have been trying to obtain more efficient and more precise methods for the solution of dielectric boundary conditions combined with molecular electrostatics, but the question of the validity of Born s basic assumption has rarely been discussed. This will be done in the following sections, with a surprising result. 

This result simply due to the dielectric boundary conditions imposed on the radiated field of a molecule at the interface. Here 1/Tnr and 1/Tril<1 are the non-radiative and the radiative rates, respectively. The quantity in the brackets accounts for the proximity of the interface at a distance z from the molecule. Algebraic expressions for the weighting functions L zyL and LyfylL are known [114] and depend on z and the refractive index ratio of the two materials that form the interface. The parameter 6t is the angle enclosed between the emission dipole moment and the surface normal. The weighting functions approach unity for z > A. For a molecule adsorbed on a PMMA surface in air, the lifetime should be 2.7 times longer when its emission dipole moment is perpendicular to the interface than when it lies in the surface. Also, the fluorescence quantum yield, defined as = 1 — (t/ rnr), is affected by the location and orientation of the molecule. 

With the image effects due to dielectric boundary conditions properly taken into account, the reorganization energy can be viewed as the difference between the equilibrium solvation free energy of the difference electron... 

The lEF-PCM method creates the solute cavity via a set of overlapping spheres [34-36]. The COSMO [37] model differs from the PCM model in that a scaled conductor boundary condition is used instead of the much more complicated dielectric boundary condition for the calculation of the polarization charges of a molecule in a continuum like with lEF-PCM. In the case of lEF-PCM, frequency calculations at the same level of theory were performed to ensure that all structures are minima on the potential energy surface. 

Felderhof B U 1980 Fluctuation theorems for dielectrics with periodic boundary conditions Physice A 101 275-82... 

Since surface charges depend on the electrostatic potential (Eq. 4.20), Eqs. 4.20-4.22 are solved in an iterative way leading to self-consistent surface charges. At the end of this procedure, surface charges and the electrostatic potential satisfy the boundary condition specified in Eq. 4.21. In practical applications, this self-consistent procedure for calculating reaction field potential is coupled to self-consistent procedure which governs solving the Kohn-Sham equations. A special case for infinite dielectric constant outside the cavity... 

As mentioned above, the PCM is based on representing the electric polarization of the dielectric medium surrounding the solute by a polarization charge density at the solute/solvent boundary. This solvent polarization charge polarizes the solute, and the solute and solvent polarizations are obtained self-consistently by numerical solution of the Poisson equation with boundary conditions on the solute-solvent interface. The free energy of solvation is obtained from the interaction between the polarized solute charge distribution and the self-... 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

Membranes certainly introduce cooperative processes, so that a merely molecular approach will not be enough, particularly with reference to boundary conditions. Whether a cell is large enough, on the other hand, to justify statistical averaging as implied by such terms as phase and dielectric field may involve quite a profound distinction. As a speculation, a cell diameter might be conditioned by the natural mode interval in diffusive systems and phase is not a justifiable term. A related question is whether the thickness of a membrane measured in molecular dimensions can play an important role structurally or whether a membrane behaves merely as an indefinitely thin boundary. 

Since our treatment of the ionic atmosphere around a dipolar molecule makes use of the Onsager model, it becomes necessary to adopt a similar model for the ion. Consequently we are going to assume that the ion is also represented by a spherical cavity in the surrounding dielectric with a point charge at its center. Then the constants by the ordinary boundary conditions become... 

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