Principles of the Calculations

Recently, a unique approach for using the correlation fiinction method has been demonstrated to extract morphological variables in crystalline polymers from time-resolved syncluotron SAXS data. The principle of the calculation is based on two alternative expressions of Porod s law using the fonu of interference fiinction [33. 36]. This approach enables a continuous estimate of the Porod constant, corrections for liquid scattering... 

The principle of the calculation is to derive an expression for the number of unreacted groups on sol molecules (r ) by summing over finite species of all sizes and to find the number of unreacted groups on the gel (Tg) as the difference... 

A code has been written to enable the velocities of surface waves in multilayered anisotropic materials, at any orientation and propagation and including piezoelectric effects, to be calculated on a personal computer (Adler et al. 1990). The principle of the calculation is a matrix approach, somewhat along the lines of 10.2 but, because of the additional variables and boundary conditions, and because the wave velocities themselves are being found, it amounts to solving a first-order eight-dimensional vector-matrix equation. A... 

The principles of the calculation by means of the cascade theory is sketched in Fig. 58 and compared with the random polycondensation. Instead of selecting a single monomeric unit as root of a tree, a whole primary chain is placed on the zero-th generation, and the same is done for all the other primary chains from the cross-linked polymer. 

The principle of the calculation of the flow profiles is the same as in the previous section. Here, the increased mass-transfer resistance should be taken into account in... 

For C60, the coupling is in fact presumably in an intermediate coupling regime, so that all these effects should be taken into account and JT distortions could manifest themselves in a variety of ways. However, the previous classical treatment is very useful as a starting basis to get an intuitive idea of the situation and will be introduced now. The principle of the calculation is not fundamentally different than in the previous simple model, although symmetry considerations become more complicated as the number of modes involved increases, as well as their degeneracy. 

Such a model of surface modification has been used by Filippini, Gramaccioli, and Simonetta133 to investigate the naphthalene crystal (001) face. Their calculations, based on a Kitaigorodsky-type method,134 accounted for the reconstruction of the surface and the subsurface layers of the crystal. The principle of the calculation is to minimize the cohesion potential written as a sum of atom-atom potentials.134 The potentials include an attractive part in R"6 (of van der Waals or dispersion forces) which is of the type discussed in Section I.A.2, and a repulsive part which is generally given the form e cR to account for the impenetrability of the electronic clouds. The potentials are determined for each type of pair C , or by adjusting... 

We know that the general principle of the calculation of probabilities consists in this ... 

Hamiltonian (3.26) will be used to explain the principles of the calculation of the shallow acceptor levels in these crystals. For a magnetic field B, derived... 

The principle of the calculation is as follows. Let us consider a two-dimensional lattice, infinite in one dimension and of width n in the other one (this lattice is a strip see Fig. 12.2). Let us associate random variables with the lattice... 

Although these two methods of calculation rarely give identical values of the selectivity ratio, the principle of the calculation is the same, and calculated values are comparable. 

This illustrates the principles of the calculation. In practice, this calculation is easily made using a statistical pocket calculator or using dedicated computer software associated with kits for direct analysis of data. 

The principle of the calculation will first be stated. At the equilibrium temperature we have... 

The essential principles of the calculation are contained in the following considerations. If a molecule with a permanent moment... 

There are a number of ways in which methods of conformational analysis can be applied to solve problems of macromolecular structure and function. We have found that a single simple approach to manipulation and refinement can with suitable modifications be used for a variety of purposes. We shall describe first the principle of the calculation and then list various possible applications and the progress which we have made in each. 

The principle of the calculation is similar to that used for a constant-pressure calorimeter as explained by the paths shown in Fig. 11.11 on page 334. When the combustion reaction in the segment of gas reaches reaction equilibrium, the advancement has changed hy and the temperature has increased from T to T2. Because the reaction is assumed to he adiabatic at constant pressure, A7/(expt) is zero. Therefore, the sum of A//(rxn, T ) and A7/(P) is zero, and we can write... 

The band calculations with potentials generated from local density functional approximations give the most detailed one electron band structures. As was discussed in section 2, one obtains Fermi surface dimensions in good agreement with experiments. We will mention briefly the basic principle of the calculation in order to illustrate that the calculation is surprisingly unsophisticated in terms of many-body correlation effects to be discussed later. 

During the final washing with 50 ml of pure water, we find a loss of 9.556 x 10 g. (The principle of the calculation is the same as in question 1.) For the entirety of... 

Notice in this calculation that the amount of oxygen was determined by difference, a common practice. In a sample containing only C, H, and O, one needs to determine the percentages of only C and H oxygen is assumed to be the unaccounted-for portion. You may also apply this practice in situations involving elements other than oxygen if all but one of the elements is determined, the last one can be determined by difference. Today, most calculations are carried out automatically by the computerized instrumentation. Nevertheless, it is often useful for a chemist to understand the fundamental principles of the calculations. 

The principle of the calculation is illustrated in Fig. 2. The first step is to estimate the charge Aq induced by applying a voltage Vo to the gate in an elemental strip of width dx at a distance x from the source. Below threshold, the charge is zero. Above threshold, it becomes... 

We will give only the results, since the principles of the calculations are the same as for the other models. The enthalpy and the heat capacity expressions are, respectively ... 

These estimates come from an analysis of the wave functions of multielectron atoms. An exact solution of the Schrodinger equation can be obtained for the H atom, but for multielectron atoms, only approximate solutions are possible. The principle of the calculation is to assume each electron in the atom occupies an orbiM much like those of the hydrogen atom. However, the functional form of the orbital is based on another assumption that the electron moves in an effective or average field dictated by all the other electrons. With this assumption, the complicated multielectron Schrodinger equation is converted into a set of simultaneous equations—one for each electron. Each equation contains the unknown effective field and the unknown functional form of the orbital for the electron. The approach to solving such a set of equations is to guess at the functional forms of the orbitals, calculate an average... 

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