Dimensionality, moment densities, molecules

According to Eq. (25), the equilibrium radius R of the island is the product of two factors one of them is the equilibrium radius in the absence of other islands (C = 0), and the other one increases exponentially with the area fraction of the islands. R is determined by the line tension (2) and the difference Ap between the two-dimensional dipole moment densities of the LC islands and the LE molecules. Both X and Ap depend on the molecular surface area difference (jAd — Ac) between the LC and LE phases. Since Ac is assumed to be constant and is considered to... 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

The descriptors reflecting structural characteristics may e.g. be element of the actual composition and 3-dimensional configuration of the molecule, whereas descriptors reflecting the electronic characteristics may e.g. be charge densities, dipole moment etc. The descriptors reflecting the hydrophobic characteristics are related to the distribution of the compound between a biological, hydrophobic phase, and an aqueous phase. The fourth type of characteristics, Dx, accounts for possible underlying characteristics that may be known or unknown, such as environmental or experimental parameters as, e.g., temperature, salt content etc. The data may often be associated with a certain amount of systematic and non-quantifiable variability in combination with uncertainties. These unknown variations are expressed as "noise". Thus, the parameter, e, account for possible noise in the system, i.e., the variation in the property that cannot be explained by the model. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series ... 

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