Coefficient Dependence

The dependence of the DKH Hamiltonians of fifth and higher order on the expansion coefficients in the parametrization of the unitary transformation in Eq. (11.57) clearly shows that the DKH expansion in Eq. (12.2) cannot be unique, in the sense that it is possible to have different expansions of the block-diagonal Hamiltonian in terms of the external potential. At first sight, this seems to be odd because one would expect to obtain a unique block-diagonal and unique Hamiltonian. At infinite order, all infinitely many different expansions with respect to the external potential V formally written in Eq. (11.57) as a single unique one necessarily yield the same diagonal Hamiltonian, i.e., the same spectrum. The individual block-diagonal operators may, however, still be different in accord with what has been said in section 4.3.3. 

Obviously, the different expansions are related to one another. Let us assume two different sequences of unitary transformations. 

However, this formal trick to describe the relation between both expansions does not explain explicitly how the difference between the Hamiltonians (12.49) and (12.50) at a given power n 4 of the expansion parameter V is compensated at some higher power m n. 

The dependence of the even terms on the coefficients does not pose a problem to the infinite-order DKH method without any truncation of the Hamiltonian. For finite-order approximations, however, application of the optimum unitary transformations has been suggested in section 11.4.3. Still, a coefficient dependence remains, but it is, for all meaningful parametriza-tions, orders of magnitude smaller than the effect of the DKH order. Also, the effect on the eigenvalues is diminished with increasing DKH order as it should be in order to converge to the same infinite-order result [645]. 

From a purist point of view, one should use DKH Hamiltonians either only up to fourth order or up to infinite order (within machine precision) in order to avoid ambiguities in the method. However, a discussion of the coefficient dependence on results obtained with fifth or higher order DKH Hamiltonians is not relevant as the numerical effect is tiny (see Fig. 12.2 for an example) and always smaller than any other approximation made (such as the finite size of the one-particle basis set). To have a well-defined model Hamiltonian at hand one may always restrict a calculation to the highest coefficient-free DKH Hamiltonian, which is the fourth-order DKH4 Hamiltonian. 

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A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... 

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

The value of coefficient depends on the composition. As the mole fraction of component A approaches 0, approaches ZJ g the diffusion coefficient of component A in the solvent B at infinite dilution. The coefficient Z g can be estimated by the Wilke and Chang (1955) method ... 

It means that we may determine the cross section of the defect by measuring the variation of the inductance. It is an absolute measurement because the coefficient depends only of... 

Inter-atomic two-centre matrix elements (cp the hopping of electrons from one site to another. They can be described [7] as linear combmations of so-called Slater-Koster elements [9], The coefficients depend only on the orientation of the atoms / and m. in the crystal. For elementary metals described with s, p, and d basis fiinctions there are ten independent Slater-Koster elements. In the traditional fonnulation, the orientation is neglected and the two-centre elements depend only on the distance between the atoms [6]. (In several models [6,... 

Partial molar quantities have per mole units, and for Yj this is understood to mean per mole of component i. The value of this coefficient depends on the overall composition of the mixture. Thus Vj o the same for a water-alcohol mixture that is 10% water as for one that is 90% water. 

The diffusion coefficient depends upon the characteristics of the absorption process. Reducing the thickness of the surface films increases the coefficient and correspondingly speeds up the absorption rate. Therefore, agitation of the Hquid increases diffusion through the Hquid film and a higher gas velocity past the Hquid surface could cause more rapid diffusion through the gas film. 

The heat-transfer coefficient depends on particle size distribution, bed voidage, tube size, etc. Thus a universal correlation to predict heat-transfer coefficients is not available. However, the correlation of Andeen and Ghcksman (22) is adequate for approximate predictions ... 

The rate of heat-transfer q through the jacket or cod heat-transfer areaM is estimated from log mean temperature difference AT by = UAAT The overall heat-transfer coefficient U depends on thermal conductivity of metal, fouling factors, and heat-transfer coefficients on service and process sides. The process side heat-transfer coefficient depends on the mixing system design (17) and can be calculated from the correlations for turbines in Figure 35a. 

Higher virial coefficients are defined analogously. AH virial coefficients depend on temperature and composition only. The pressure series and density series coefficients are related to one another ... 

Under equiUbrium or near-equiUbrium conditions, the distribution of volatile species between gas and water phases can be described in terms of Henry s law. The rate of transfer of a compound across the water-gas phase boundary can be characterized by a mass-transfer coefficient and the activity gradient at the air—water interface. In addition, these substance-specific coefficients depend on the turbulence, interfacial area, and other conditions of the aquatic systems. They may be related to the exchange constant of oxygen as a reference substance for a system-independent parameter reaeration coefficients are often known for individual rivers and lakes. 

Because the coefficient depends on c, the equations are more compBcated. A finite difference method can he written in terms of the fluxes at the midpoints, -t- 1/2. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

For purely physical absorption, the mass-transfer coefficients depend on trie hydrodynamics and the physical properties of the phases. Many correlations exist for example, that of Dwivedi and Upadhyay (Ind. Eng. Chem. Proc. De.s. izDev.,... 

The centerline temperature differential in Zone 3 of the diffuser jet is proportional to the value of the K, coefficient, which, along with the fC, coefficient, depends upon jet and diffuser types and supply conditions. The theoretical value of the K2 coefficient, according to Shepelev, is 2.49. Experimental data reported by Grimitlyn show Ki to be 2.0. 

The value of this coefficient depends on the relative height of the flow along the floor h /H (calculated in the free conditions), where... 

The minor loss coefficient depends on the Reynolds number because this dependency is weak, it is normally ignored. The minor loss coefficients for different resistances are given in textbooks. 

Note that if sticking is controled by site-exclusion only, i.e., if S 6,T) = 5 o(P)(l — 0), this rate is that of a first-order reaction at low coverage. This simple picture breaks down when either the sticking coefficient depends dilferently on the coverage, as it does for instance for precursor-mediated adsorption, or when lateral interactions become important. It then does not make much physical sense to talk about the order of the desorption process. 

In general the drag coefficient depends on the shape of the structure. In the case of a human body a value of 1 is considered to be accurate enough. 

The above technique has the practical inconvenience of requiring as many different sets of Tchebyschev coefficients as the unit cell non equivalent sublattices. Furthermore, for non cubic systems, these coefficients depend on the lattice distortion ratios. Namely, for tetragonal lattices different sets of coefficients are required for each value of c/a. This situation has made difficult the implementation of KKR and KKR-CPA calculations for complex lattice structures as, for example, curates. 

At flow speeds well below the speed of sound, the lift coefficient depends only on the shape and orientation (angle of attack) of the body ... 

The drag coefficient for an antomohile body is typically estimated from wind-tunnel tests. In the wind tunnel, the drag force acting on a stationaiy model of the vehicle, or the vehicle itself, is measured as a stream of air is blown over it at the simulated vehicle speed. Drag coefficient depends primarily on the shape of the body, but in an actual vehicle is also influenced by other factors not always simulated in a test model. 

In general terms, the pyroelectric coefficient of a free sample consists of three components. The first, called the real coefficient, depends on the derivative of spontaneous polarization with respect to the temperature. The second is derived from the temperature dilatation and can be calculated based on mechanical parameters. The third coefficient is related to the piezoelectric effect and results from the temperature gradient that exists along the polar axis of the ciystal. 

Attention is directed to the fact that the molar absorption coefficient depends... 

Friction coefficients will vary for a particular material from the value just as motion starts to the value it attains in motion. The coefficient depends on the surface of the material, whether rough or smooth, as well as the composition of the material. Frequently the surface of a particular plastics will exhibit significantly different friction characteristics from that of a cut surface of the same smoothness. These variations and others that are reviewed make it necessary to do careful testing for an application which relies on the friction characteristics of plastics. Once the friction characteristics are defined, however, they are stable for a particular material fabricated in a stated manner. 

Wall-to-bed heat-transfer coefficients were also measured by Viswanathan et al. (V6). The bed diameter was 2 in. and the media used were air, water, and quartz particles of 0.649- and 0.928-mm mean diameter. All experiments were carried out with constant bed height, whereas the amount of solid particles as well as the gas and liquid flow rates were varied. The results are presented in that paper as plots of heat-transfer coefficient versus the ratio between mass flow rate of gas and mass flow rate of liquid. The heat-transfer coefficient increased sharply to a maximum value, which was reached for relatively low gas-liquid ratios, and further increase of the ratio led to a reduction of the heat-transfer coefficient. It was also observed that the maximum value of the heat-transfer coefficient depends on the amount of solid particles in the column. Thus, for 0.928-mm particles, the maximum value of the heat-transfer coefficient obtained in experiments with 750-gm solids was approximately 40% higher than those obtained in experiments with 250- and 1250-gm solids. 

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

A number of techniques are employed to determine activities or activity coefficients, depending upon the nature and type of the system. Often, procedures are used that are very specific for the system being studied. We will describe several commonly encountered examples. 

It was shown that these adhesion-coefficients depend on the ratio of the logarithms of the ratios of the moduli of phases and those of the radii of mesophase and inclusion. Thus, for a certain average size of inclusions, the thinner the mesophase, the better the adhesion quality of the composite. 

With increasing water content, the dielectric constant of the medium and the degree of endgroup ionization will increase.30 This is likely to influence the end-group activity coefficients, depending on whether the polycondenzation reaction involves the condensation of predominantly neutral or ionized species. 

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