Numerical results-conclusion

It is dangerous to draw too many conclusions from the numerical results of molecular mechanics calculations. In this case, we wanted only to show that the two likely conformations for the E and G-rings of PbTX-1 are roughly equal in energy and that it is plausible that they could both exist in solution at room temperature. It would be very useful to be able to estimate the energy of activation required for the conversions between conformations. Unfortunately, estimation of the energy barrier is beyond the realm of the empirical force field method for all but the simplest cases. 

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. 

If we now consider the numerical results quoted in Table 1 for the optimum exponents, three conclusions follow immediately. Firstly, the 1 s orbital on the heavy atom is unchanged by molecule formation this is to be expected. Second, the sp3 orbitals involved in the X—H bond are all contracted with respect to their free-atom values. Finally, the sp3 orbitals containing Tone pairs of electrons are largely unchanged or expanded slightly on molecule formation. In fact, of course, the optimum separate atoms minimal basis functions do not have the same orbital exponent for the 2 s and 2 p AOs. To facilitate comparisons therefore in Table 1 the optimum n = 2 exponent is given for the atoms when such a constraint is imposed (the qualitative conclusions are, in any event, unchanged by use of these exponents for comparison or a notional exponent of 1/4 (fs + 3 fp) or any reasonable choice). 

We can draw some very important conclusions about the two algorithms from the numerical results obtained in the sample example considered above ... 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

Comparing this value with the order a Zaym result in (3.101) we see that the difference between the exact numerical result and analytic calculation up to order a Zay is about 0.015 kHz for the IS -level in hydrogen, and, taking into account the accuracy of experimental results, one may use analytic results for comparison of the theory and experiment without loss of accuracy. A similar conclusion is valid for other hydrogen levels. 

In conclusion, we believe we are in a rather good shape with our BET, particularly since NCQP [8] has become available. We expect to start shortly to produce numerical results that could be compared with a variety of experimental data. As we stressed above, we can now go beyond the bounds of the BCS theory. Also, the rather general framework of BET, constructed in KT3, now allows us to take into account the non-collective, as well as collective components in a rather systematic way. Further accumulation of data, pertaining to both kinds of levels, is thus highly hoped for,... 

Based on the numerical results obtained for the electrohydrodynamics equations the following conclusions can be drawn ... 

In a theoretical study Verwey (7) came to the conclusion that the negative surface ions of the free surfaces of alkali halide crystals are generally displaced, so that their distances from the next layers of the lattice are increased, and the positive surface ions are displaced toward the inside of the lattice. The numerical results of his calculations for NaCl are that the distance between the sodium ions of the outer layer and the ions of the second layer is 2.66 A. whereas the chloride ions of the outer layer are located at a distance of 2.86 A. from the second layer, the normal distance in the lattice being 2.81 A. The electrical double layer formed by the negative chloride ions, being located in a plane 0.20 A. distant from the plane of the positive sodium ions, is almost compensated for by the effect of the dipoles set up in the negative ions. 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

There are three components in the uncertainty surrounding a given numerical result. One is due to variation in the process by which the test sample was produced, another is due to sampling variability, and the third is due to the measurement process by which the result was obtained. Due attention must be paid to all of these, for maximum reliability in the conclusions based on that result. It should be obvious that good data don t cost so much as they pay ... 

The front page of the report should display a title, your name, the name of any experimental partners, the date on which the report is submitted, and a brief abstract. An abstract is typically 50 to 100 words long the example above contains about 90 words. It should summarize the results of the experiment and state any significant conclusions. Numerical results with confidence limits should be included. 

Fixman gave analytical expressions for high xa and f < 200 mV. Chew and Sen ) offered anal3dlcal equations valid to the first order In (xa)" - As was the case for the electrophoretic mobility, the outcomes of these theories do not differ greatly. Fixman already compared his data with the numerical results of De Lacey and White and came to this conclusion, later elaborations confirmed it We shall now present a relatively simple analytical derivation that should be of relatively wide application. 

The numerical results, summarized below, show the fourth model to be the most probable one according to the data, and also show that this model does better than the others on the goodness-of-fit test. These results are consistent with those of Stewart, Henson, and Box (1996), who found this model to be the most probable a posteriori of the 18 models considered by Tschernitz et al. (1946). The linearized model forms used by Tscher-nitz et al. yield the same conclusion if one uses the appropriate variable weighting for each linearized model form. 

It follows from the above discussion and numerical results that even a simple convective-diffusive model of concentration behaviour mechanism gives realistic results and yields a satisfactory description of the formation of the gaseous layer under the anode surface. The model may be improved by adding the electrolyte circulation and electromagnetic forces yet we hope that it will not change the main conclusions. The finite volume method proves to be a flexible and sufficiently accurate numerical technique for solving both the equations for the Galvani potential and the reactant concentrations. The marker-and-cell approach makes it possible to outline the electrode surfaces easily. 

The justification of jr-electron theories has been repeatedly questioned during recent years indeed, it has become almost fashionable to emphasize the shortcomings of the cr—it separation and the non-validity of the theories based upon it. These are, in fact, approximations and cannot be expected to lead to unconditionally reliable conclusions. However, the numerical results that have provoked the criticisms in question are not a necessary consequence of the a—it separation and the related approximations. Therefore, we shall begin by restating and clarifying the basic concepts on which the whole question of the a—it separation rests. We shall consider the conditions under which the electrons of a molecule can be classified into a and it electrons and indicate what should be understood be a—7i separation and what are the limitations of this approximation. We shall show that the most important part of th e a—n interaction is usually taken into accound within the a—n separation scheme and, finally, discuss whether the a—it interaction has a significant effect on the theoretical predictions made for the physical properties of unsaturated molecules (ionization potentials, electronic spectra, charge densities and dipole moments etc.). 

Although our numerical results (Fig. 1) show that the HF approximated solution Eq. (10) is rather crude, it nonetheless provides a qualitative guidance to the physical behavior. Simple scaling considerations lead us to the conclusion that ctHF depends on two parameters the dimensionless time variable 7hf t and K [Eq. (9)]. Eq. (10) predicts that at t —> 0 the decay is more rapid than exponential, so the survival probability Iff) = o (t) 2 behaves in the HF approximation as PhfV) 1 — 1 Kc ( 1 I 2) ft/7t, becoming exponential at larger times, PhfV) exp(— 7hf t). The deviation from exponential decay is appreciable only for K < 1. There are two ways to attain K < 1. One is to take a small difference between the impurity atom velocity Vi and the critical... 

It is pointless to show the numerous results obtained for each family of complexes since a survey of the whole of the spectra leads to the same conclusions. A curve representative of results obtained in our laboratory and by other researchers (13—19) using different techniques is given in Fig. 4, 5 and 6. 

Exact Solution for the Complex Susceptibility Using Matrix Continued Fractions Approximate Expressions for the Complex Susceptibility Numerical Results and Comparison with Experimental Data Conclusions... 

From the above numerical results the following conclusion can be obtained ... 

The calculated energy differences give a good correlation with Hammett ct+ values. The p parameter (p = -17) is considerably larger than that observed experimentally for proton exchange (p -8). A physical interpretation of this difference is that the computational results pertain to the gas phase, where substituents are at a maximum because of the absence of any leveling effect owing to solvation. Note that the numerical results parallel the conclusions from qualitative application of resonance... 

How reliable are numerical results of theoretical investigations More and more sophistication in calculation methods improve their reliability — but they may become even better in the future A recent case is the comparison of the dissociation energies of the methane- with the benzenediazonium ion in Glaser s work. In 1992, they were calculated to be practically identical, but in the most recent calculation (1995) the values of //diss were significantly different (176 and 122 kJ mol respectively). A caveat is therefore appropriate for conclusions that reach too far. 

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