Field cavity

One-dimensional model Onsanger cavity field Onsanger equation Orbital polarization Ordered phase Ordered state... 

Yh fyT = X)( z) > as cubic and completely disordered lattices (incidentally, the type of arrangements for which in the classical Lorentz cavity-field calculation the contribution of the dipoles inside the small sphere vanishes) these lattice sums vanish for large spherical samples. The sums and... 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

The effective properties in solution with the cavity-field effects taken into account are formulated in terms of /X, i.e. in terms of the m matrix. For example, the IR intensity can be expressed as [8] ... 

In Equation (2.183) new surface charges, qex, have been introduced these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of qex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium more details on this equivalence will be given in Section 2.7.4. 

The OWB model describes the solute as a classical polarizable point dipole located in a spherical or ellipsoidal cavity in an isotropic and homogeneous dielectric medium representing the solvent. In the presence of a macroscopic Maxwell field E, the solute experiences an internal (or local) field E given by a superposition of a cavity field Ec and a reaction field ER. In terms of Fourier components E -n, Ec,n, ER,n of the fields we have... 

The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors FR n, (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors /c,n. 

The corresponding PCM expressions (2.193) and (2.194) show that the same physical effects are considered the static cavity field effects are explicitly represented by the matrices m°, while the static reaction field effects are implicit in the coupled perturbed HF (or KS) equations which determine the derivative of the density matrix. 

For 0 the surface charge density, aR+c, obtained by solving Equation (2.260), contains a contribution which accounts for the cavity field, i.e. the electric field generated in the cavity by the applied field. 

The usefulness or the IEF approach relies on the possibility of expressing physical properties, which depend on the interaction between the molecular charges and the reaction/cavity field, as surface integrals of the general form [8,9,18] ... 

Table 2.13 Explicit form of the functions appearing in Equation (2.263) for the calculation of the dipole induced at the Jth site by the reaction and cavity fields, ix "d, and of the electrostatic free energy, W. This free energy includes a contribution from the reaction field, WR, and one from the cavity field, WE. In the absence of an applied field gc vanishes, and only the reaction field contribution to the induced dipole and to the electrostatic free energy remains...

If Ae 0 the reaction/cavity fields, and then the molecular property /, depend on the molecular orientation. Such a dependence affects the physical observables, which are obtained by averaging over the orientational distribution. Considering in general a tensorial property, we can express the average value as ... 

The total local field FA is equal to the vector sum of the Onsager cavity field FA, and the Onsager reaction field.101 The latter is independent of EA, and results from the dipoles own fields.101 In the medium of dielectric constant n2, FA is ... 

Booth73 used Frolich s108 modification of the Onsager expressions101 for the cavity field in non-associated polar liquids, and corresponding modifications of Kirkwood s equation for associated polar media. Booth s assumptions in deriving Eq. (28) for the Kirkwood case are important to determine the validity of his final expressions. He used the Onsager-Frolich cavity field ratio... 

In the reaction field model (Onsager, 1936), a solute molecule is considered as a polarizable point dipole located in a spherical or ellipsoidal cavity in the solvent. The solvent itself is considered as an isotropic and homogeneous dielectric continuum. The local field E at the location of the solute molecule is represented by (78) as a superposition of a cavity field E and a reaction field (Boettcher, 1973). 

In the case of a static field, the macroscopic relative permittivity e° has to be used in (82) for the cavity field factor, while the optical relative permittivity extrapolated to infinite wavelength e can be applied to estimate the static polarizability a(0 0) in (84). In this way the Onsager-Lorentz factor for a pure dipolar liquid is obtained (87). 

Hie polarizability a(-o) a)) is involved in several linear optical experiments including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor/ " and by the reaction field factor For typical media the magnitude of the product is of the order of 1.3-1.4. In the case of... 

Onsager derived an improved formula by adopting a better model for the calculation of the local field at a molecule. His model consists of a spherical cavity which is excised in the dielectric material and which is just large enough to accommodate one molecule. The molecular dipole is supposed to be a point dipole fj, at the centre of the sphere, radius a. Onsager then said that the local field operating on the dipole at the centre of the cavity could be resolved into two components, a cavity field G and a reaction field R ... 

The cavity field is defined as that present at the centre of an empty cavity in the presence of the applied field E, i.e. 

---