INDEX theory using

Pure, low temperature organic Hquid viscosities can be estimated by a group contribution method (7) and a method combining aspects of group contribution and coimectivity indexes theories (222). Caution is recommended in the use of these methods because the calculated absolute errors are as high as 100% for individual species in a 150-compound, 10-family test set (223). A new method based on a second-order fit of Benson-type groups with numerous steric correctors is suggested as an alternative. Lower errors are claimed for the same test set. 

The consequences of Mie s theory for absorption (i.e., for tinting strength) are now considered. Calculations from Mie s theory, using the relative refractive index n and the absorption index k, are given in Figure 8 [1.30]. The parameter a on the abscissa can once more be taken as a relative measure of the particle size. The following conclusions may be drawn ... 

For the reasons described above, the droplet size distribution of the same emulsion measured on different laser diffraction instruments can be significantly different, depending on the precise design of the optical system and the mathematical theory used to interpret the diffraction pattern. It should be noted, however, that the most common source of error in particle size analysis is incorrect operation of the instrument by the user. Common sources of user error are introduction of air bubbles into the sample, use of the wrong refractive index, insufficient dilution of emulsion to prevent multiple scattering. and use of an unclean optical system. 

For accurate measurements it is important to enter the correct optical constants of the dispersed and continuous phase into the theory used to calculate the droplet size distribution. In most instruments, it is necessary to enter the refractive index and absorptivity of the component phases at the appropriate wavelength of the laser. Significant errors in the measured droplet size distribution will occur if incorrect optical parameters are used, particularly if the... 

This may be determined theoretically over the whole particle size refractive index domain using Mie theory or, over limited ranges, with modified theories. Other interactions between the particles and the incident radiation, such as state of polarization, light flux at a fixed angle to the direction of the incident beam and angular spectra can be used for particle size determination. 

In this section we discuss index theory, a method that provides global information about the phase portrait. It enables us to answer such questions as Must a closed trajectory always encircle a fixed point If so, what types of fixed points are permitted What types of fixed points can coalesce in bifurcations The method also yields information about the trajectories near higher-order fixed points. Finally, we can sometimes use index arguments to rule out the possibility of closed orbits in certain parts of the phase plane. 

Another series of successfully applied topological descriptors is derived from graph theory using atom connectivity information of a molecule. An example is the connectivity index developed by Randic [21], In the simple form,... 

To arrive at a size-extensive expression for W, and at the same time to ensure that our SS-MRCC theory using an IMS is intruder-free, it is necessary to write explicitly the projector P, and change the label of the dummy index /r on the right side of Eq. (61) appropriately. We thus have... 

On very general grounds, (C,)in,t is taken as a function of the refractive index n of the solvent, Cg as a function of the bulk dielectric constant e and (Ce)orient as a function of eg and n. So far, the various theories use the expressions of Er given by Onsager s theory. As a first approximation, the various coefficients are taken as ... 

The entropy index as defined in equation 18.36 is potentially very useful in the fundamental evaluation of any separation processes, not just in solid-liquid separation. Besides the Russian references ", Ogawa et al. derived the same entropy index (but using mass fractions rather than volumetric ones) from information theory and proposed its use for the evaluation of any separation process. 

Using the correct refractive index values is very important in retrieving an accurate particle size distribution when Mie theory is used in the matrix inversion. It is especially important when the relative refractive index approaches unity in which case even a minor error in the choice of a relative refractive index may lead to a significant variation in the result [54]. Figure 3.37 shows an example of the effect of refractive index in resolving particle size using laser diffraction. Correct results can only be obtained when correct values of the refractive index are used in the matrix inversion. Otherwise, an either... 

The nucleophilicity of 32 amine, amide, carbamate, amidine, and pyridine nucleophiles was calculated at the B3LYP/6-31G(d,p) level of theory using the gas-phase ionization potential based on the nucleophilicity index The correlation coefficient for the calculated pyridine nucleophilicity against Mayr s values was 0.947, and it was 0.987 against the inverse of the electrophilicity scale Mm. The site selectivity predicted for the nucleophiles with more than one nucleophilic centre correlates well with that predicted by the local nucleophilicity index and the philicity index m. ... 

Fig. 2 Left Orbital spaces considered in F12 theory and index conventions used in this chapter. Right Representation of the two-particle function space (geminal space), the grey shaded subspaces are projected out by the strong- orthogonality projector Q 2-...

Equations (10.17) and (10.18) show that both the relative dielectric constant and the refractive index of a substance are measurable properties of matter that quantify the interaction between matter and electric fields of whatever origin. The polarizability is the molecular parameter which is pertinent to this interaction. We shall see in the next section that a also plays an important role in the theory of light scattering. The following example illustrates the use of Eq. (10.17) to evaluate a and considers one aspect of the applicability of this quantity to light scattering. 

Using the original Hc2/r values, recalculate M using the various refractive index gradients. On the basis of self-consistency, estimate the molecular weight of this polymer and select the best value of dn/dc2 in each solvent. Criticize or defend the following proposition Since the extension of the Debye theory to large particles requires that the difference between n for solute and solvent be small, this difference should routinely be minimized for best results. 

The abihty of fillers to improve paper brightness increases with their intrinsic brightness, surface area, and refractive index. According to the Mie theory, this abiUty is maximum at an optimum filler particle size, about 0.25 pm in most cases, where the filler particle size is roughly one-half the wavelength of light used for the observation. 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

Among other things, Redfield s paper led to a heightened awareness of something that was already beginning to be realized, namely the interrelationship between Polya s Theorem (and other enumeration theorems) on the one hand, and the theory of symmetric functions, -functions, and group characters on the other it helped to show the way to the use of cycle index sums in the solution of hitherto intractable problems and in a more nebulous way it provided a refreshing new outlook on combinatorial problems. 

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