Some Numerical Results

With the aid of the recurrence procedure described in the previous section, the )) dependence of I2 incorporated in the interactive displacement parameters and Aj as well as in the cross-frequency parameters P21 been investigated. 

Note that if cf) / 0, the two vibrational components are interdependent, which is manifested in an unusual behavior of the distribution I2. This is illustrated in 

When Aj2 and A 2 are small (Ajj 1), Ii is not so simply described graphically, since I2 decreases nearly exponentially as the numbers wi and ri2 increase. In this case, the maximum of I2 lies in the vidnity of wi = mi, W2 = m2. As the angle (() 

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This chapter is intended to provide basic understanding and application of the effect of electric field on the reactivity descriptors. Section 25.2 will focus on the definitions of reactivity descriptors used to understand the chemical reactivity, along with the local hard-soft acid-base (HSAB) semiquantitative model for calculating interaction energy. In Section 25.3, we will discuss specifically the theory behind the effects of external electric field on reactivity descriptors. Some numerical results will be presented in Section 25.4. Along with that in Section 25.5, we would like to discuss the work describing the effect of other perturbation parameters. In Section 25.6, we would present our conclusions and prospects. 

Some numerical results have been obtained for Schottky defects in sodium chloride and for cation vacancies in sodium chloride doped with manganese chloride so that the number of intrinsic defects is negligible. Figure 9 shows the results for the... 

Here we demonstrate some numerical results we replaced the original OGE by the contact interaction with the cutoff around the Fermi surface in the momentum space,... 

The general form of g P-) is quite complicated. We shall present some numerical results of gj, (2) as a function of parameters h, K, and q. Here, it is instructive to present the formal form of the pair correlations (in the limits m —> o<> and A, 0) for the case AT= 1. The first few pair correlations are... 

In Fig. 8.15 we present some numerical results for this model with the parameters... 

For comparison, we present in Fig. 5.3.2 some numerical results for the following non-locally-electro-neutral generalization of the classical Teorell-Meyer-Sievers (TMS) model of membrane transport (see [11], [12] and 3.4 of this text). 

In this section we present some of the results obtained so far, although some numerical results not previously published are now included. They are organized according the sort of strip and the ansatz that has been used. 

Some numerical results for the states with n = 1,2 are also available [26]. Second-order recoil corrections ((Za)4(m3/M2)) are known [27] but their contribution is negligible. 

Cannon JR (1966) Some numerical results for the solution of the heat equation backwards in time. Proceedings Adv Symposium Numerical Solutions of Nonlinear Differential Equations, Madison, Wisconsin, Wiley, New York, pp 21 -54... 

Some numerical results are given in Table 10. We observe rather small effects the contraction of the equilibrium distance is 0.3 % only and the energy per nearest neighbour interaction increases in absolute value by 3.5 %. In three dimensions the effects are somewhat larger. The expressions for fn, dn, fx and d are free of parameters and valid for all Lennard-Jones systems, independently of the particular values of Re and De. 

The expressions for fn, dn, and d do not contain parameters and are valid independently of the particular values of C4, p or a. To give an example, the further neighbour effects are the same in the three differently oriented chains of dipoles shown in Fig. 11. What changes with a of course, is the nature of Re and De. In Table 10 some numerical results are compiled. As expected, further neighbour effects are more important for polar systems since the R-3 potential is further reaching than the R"6 potential (see Fig. 10). 

Application of the Hartree-Fock Method. - Since numerical Hartree-Fock programs dealing with complex numbers are available in many research groups, it seemed natural to apply this scheme also to the scaled Bom-Oppenheimer Hamiltonian (4.15). As a consequence, some numerical results were obtained before the theory was developed, and - as we have emphasized in the Introduction - some features seemed rather astonishing. 

In this section we present some numerical results to examine the efficiency of the new introduced methods. Consider the numerical integration of the Schrodinger equation ... 

The isotherms and heats of adsorption calculated for the system Ar-NaA agree with the experimental data (2) quantitatively. For the system CH4-CaA, we have found in the hterature only the heats of adsorption (3, 8, JO). These experimental data do not agree well with each other. Therefore, it can be said only that for the system CH4-CaA, the calculated values of isosteric heats are not in contradiction with the experimental ones. Some numerical results of our calculations are given also in Ref. 8. 

In this chapter we discuss a novel fast diffusion process that was experimentally discovered by Yasuda, Mori, and co-workers (YM) [7] in nano-sized binary metal clusters, because it is supposed to be a typical manifestation of an anomalous diffusion process peculiar to microclusters. The authors have reported some numerical results of MD simulation on rapid alloying (RA) [8]. Special attention is paid to the diffusion of atoms in cluster [9]. The aim of the present work is to realize RA by elucidating what kind of diffusive motion is relevant to RA. 

Our experience has been that it is usually sufficient to determine first the extremal pairs that are optimal for MP2-R12/A (standard choice of extremal pairs) and to use them afterwards for MP2-R12/B as well as for all CC-R12 calculations. Some numerical results are given in the next section. 

In Tables 1 and 2 some numerical results have been collected. These examples have been chosen among the very large number of charge calculations available in the literature in order to show the great variety of the molecules treated and of the methods employed. 

That electronic correlation is not too sensitive to the form of the one-particle density is a well-known fact. In Section 5.2, we provide some numerical results of this non-variational energy optimization procedure for the beryllium atom and also analyze the effect on the correlation energy of selecting different final densities. 

To complete this account it would have been convenient to report and to analyze some numerical results. Reasons of space forbid us to do it. We have stressed at the maximum the limits established by the Editors in the effort of making this review comprehensible, if not completely self-containing. 

In this section we shall present and discuss some numerical results regarding structures of two series of noncentrosymmetric polyenes NH2(CH=CH)nR (n=l,2), with R=CHO (series I) and with R=N02 (series II). These results have been extracted from a study we have published on the Journal of the American Chemical Society about solvent effects on electronic and vibrational components of linear cUid nonlinear optical properties of Donor-Acceptor polyenes. 

However, it was shown [1] that the adjustment procedure used for the determination of the fundamental constants (Section 2.2) does not have to been carried out again the full, extended adjustment thus does not represent the calculations effectively performed for the new predictions [1]. In fact, optimal predictions for additional energy levels can be obtained in a much simpler way by combining some numerical results from the latest adjustment with covariances between a few theoretical 5-contributions (see Section 2.3.3.2 below). 

In Section 5.1 we describe the independent electron pair approximation (lEPA). We use an approach that leads quickly to the computational formalism but which may give the misleading impression that lEPA is an approximation to DCI. After showing what is involved in performing pair calculations, we will return to the physical basis of the formalism and show that in fact both lEPA and DCI are different approximations to full Cl. In Subsection 5.1.1 we describe a deficiency of the lEPA, not shared by DCI or the perturbation theory of Chapter 6 namely, that the lEPA is not invariant under unitary transformations of degenerate molecular orbitals. In Subsection 5.1.2 we present some numerical results which show that while the lEPA is reasonably accurate for small atoms, it has serious deficiencies when applied to larger molecules. 

Studying moderately thin samples, the molecular time constants can be directly obtained from the experimental data without the need of computer simulations. This situation is termed nearly-free induction decay (NFID) which is related to the case of free induction decay in very thin samples. Some numerical results are depicted in Fig. 1. A specific experimental situation is considered with resonant, weak input pulses of Gaussian shape and duration tp (FWHM) and two values of the normalized propagation length, ai 0.2 and aJl 1. Here a denotes the conventional absorption coefficient at the maximum of the absorption band. The intensity of the transmitted pulse is evaluated from Maxwell-Bloch equations for homogeneous line broadening. 

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