Density functional equations 206 INDEX

Notice in particular the first term on the right-hand side where a Dirac delta function appears because of integration with respect to z. The second term on the right-hand side has been identified as such based on the assumption that z z since Zj z while integrating the previous equation. We now make use of the symmetry properties of the master density function. In both terms on the right-hand side of the preceding equation, the summands within the inner sums are independent of the index of summation so that we may write... 

We have anployed the parametrized DFTB method of Porezag et al. [33,34]. The approximate DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham [43,44]. In this method, the single particle wave functions l (r) of the Kohn-Sham equations are expanded in a set of atomic-like basis functions < > , with m being a compound index that describes the atom on which the function is centered, the angular dependence of the function, as well as its radial dependence. These functions are obtained from self-consistent density functional calculations on the isolated atoms employing a large set of Slater-type basis functions. The effective Kohn-Sham potential Feff(r) is approximated as a simple superposition of the potentials of the neutral atoms... 

Most analyses of kinetic data have the object of identifying the constants of a rate equation based on the law of mass action and possibly some mass transfer relation.. The law of mass action Is expressed In terms of concentrations of the participants, so ultimately the chemical composition must be known as a function of time. In the laboratory the chemical composition Is determined by some instrument that is suitably calibrated to provide the needed information. Titration, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, magnetic resonance — all of these are used at one time or another to measure chemical composition. In some cases, the calibration to chemical composition is linear with the reading. 

Kurtz and Ward 22) have described a composite function of the refractive index and density which they call the refractivity intercept. The equation is R. Int = n — d/2. The value of the refractivity intercept lies in the fact that for hydrocarbon isomers it is more constant than most other functions. Its chief uses are the rapid checking of physical property data found in the literature and distinguishing between naphthenes, paraffins, and aromatics. 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

The Lorentz-Lorenz equation [2] defines the molar refraction, RD, as a function of the refractive index, density, and molar mass ... 

The Onsager function is dependent on the refractive index of the solvent (n). For supercritical C02 the refractive index is dependent on the fluid density and can be calculated from the Lorentz-Lorenz refraction equation ... 

It is evident from the above equation that for a known size distributional form, (r/c) is a function of the refractive index and density of the particles, the wavelength of the incident light and the parameters describing the particle size distribution. 

In these expressions the index i runs over electrons and a runs over nuclei. The Fermi contact term describes the magnetic interaction between the electron spin and nuclear spin magnetic moments when there is electron spin density at the nucleus. This condition is imposed by the presence of the Dirac delta function S(rai) in the expression. The dipole-dipole coupling term describes the classical interaction between the magnetic dipole moments associated with the electron and nuclear spins. It depends on the relative orientations of the two moments described in equation (7.145) and falls off as the inverse cube of the separations of the two dipoles. The cartesian form of the dipole-dipole interaction to some extent masks the simplicity of this term. Using the results of spherical tensor algebra from the previous chapter, we can bring this into the open as... 

To calculate micelle size and diffusion coefficient, the viscosity and refractive index of the continuous phase must be known (equations 2 to 4). It was assumed that the fluid viscosity and refractive index were equal to those of the pure fluid (xenon or alkane) at the same temperature and pressure. We believe this approximation is valid since most of the dissolved AOT is associated with the micelles, thus the monomeric AOT concentration in the continuous phase is very small. The density of supercritical ethane at various pressures was obtained from interpolated values (2B.). Refractive indices were calculated from density values for ethane, propane and pentane using a semi-empirical Lorentz-Lorenz type relationship (25.) Viscosities of propane and ethane were calculated from the fluid density via an empirical relationship (30). Supercritical xenon densities were interpolated from tabulated values (21.) The Lorentz-Lorenz function (22) was used to calculate the xenon refractive indices. Viscosities of supercritical xenon (22)r liquid pentane, heptane, decane (21) r hexane and octane (22.) were obtained from previously determined values. 

Index of Refraction. The refraction of index of a fluid is usually a function of the thermodynamic state, often only the density. According to the Lorenz-Lorentz equation, the relation between the index of refraction and temperature is given by... 

In a single-component system, the density of a gas can be determined from the ideal gas equation of state. If the pressure in the test section is kept at a constant value and the index of refraction is only a function of density, the temperature distribution in the test section can be evaluated from the equation, given by... 

What physical meaning should one attach to the velocity potential For the flow of an ideal, frictionless fluid, the velocity potential has no physical meaning whatever. To illustrate this, consider the steady flow of a frictionless, constant-density fluid in a horizontal pipe see Fig. 10.3. (Such a frictionless fluid, once started in motion by some external force, would continue moving forever, because there is no force to stop it.) For such a frictionless fluid, the velocity is uniform over the cross section perpendicular to the flow. From Bernoulli s equation we can see that there is no change with distance of pressure, velocity, or elevation, and by straightforward arguments we can show that there is no change of temperature, refractive index, dielectric constant, or any other measurable property. But from Eq. 10.7 we know that, because is constant, there is a steady decrease of (f> in the x direction. Thus the velocity potential for a perfect fluid (f> is not a function of any measurable physical property of the fluid. 

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