Classical approaches numerical limitations

The quantum mechanical approach cannot be used for the calculation of complete lattice energies of organic crystals, because of intrinsic limitations in the treatment of correlation energies. The classical approach is widely applicable, but is entirely parametric and does not adequately represent the implied physics. An intermediate approach, which allows a breakdown of the total intermolecular cohesion energy into recognizable coulombic, polarization, dispersion and repulsion contributions, and is based on numerical integrations over molecular electron densities, is called semi-dassical density sums (SCDS) or more briefly Pixel method. [12-14]... 

The force-field method involves the other part of the Born-Oppenheimer approximation, that is the positioning of the nuclei. The electronic system is not considered explicitly, but its effects are of course taken into account indirectly. This method is often referred to as a classical approach, not because the equations and parameters are derived from classical mechanics, but rather because it is assumed that a set of equations exist which are of the form of the classical equations of motion. The problem from this point of view is one of establishing just which equations are necessary, and determining the numerical values for the constants which appear in the equations. In general there is no limit as to what functions may be chosen or what parameters arc to be used, except that the force-field must duplicate the experimental data. 

If the reader has actually made it up to this point, he or she will have the impression that the whole universe of solid-state materials, i.e., insulators, semiconductors, metals, and intermetallic compounds can nowadays be studied by electronic-structure theory, and predictive conclusions are really in our own hands. Indeed, the numerical limitations of most classical approaches - in particular, the ionic model of everything - have been overcome. While the computational methods of today include very different quantum-chemical methods, their varying levels of accuracy and speed are due to differences in their atomic potentials and the choice of the basis sets that are involved. The latter may either be totally delocalized (plane waves) or localized (atomic-like), adapted to the valence electrons only (pseudopotentials) or to all the electrons. In order to understand structures and compositions of solid-state materials, the results of electronic-structure theory are typically investigated in terms of some quantum-chemical analysis. 

The most sophisticated deterministic dynamic analysis of structures requires that the load should be applied in time domain. This is one of the major challenges in the reliability analysis for seismic loading. The classical random vibration-based approaches were used in the past for this purpose however, they did not provide information acceptable to the deterministic community. The classical random vibration-based approaches have numerous limitations including the loads which are applied in the form of power spectral density functions essentially appropriate for linear structural behavior, the uncertainty in the linear or nonlinear structural behavior which may need to be incorporated in approximate ways, several performance-enhancing features currently introduced in structures which cannot be incorporated in appropriate ways, etc. The most severe weakness is that the seismic loading cannot be applied in time domain. 

The above failure (or stability) criteria do not explicitly contain time as a variable. One may apply them to rate sensitive materials, however, if one recognizes that 0 and r will depend on stress history. The classical approach of Eyring and other rate theories will be discussed in Section IV of this chapter. The limits of the applicability of the classical criteria and of their extension to anisotropic materials are analyzed by Ward [20]. Also in recent years numerous papers (e.g., 21—28)... 

The classical or frequentist approach to probability is the one most taught in university conrses. That may change, however, becanse the Bayesian approach is the more easily nnderstood statistical philosophy, both conceptually as well as numerically. Many scientists have difficnlty in articnlating correctly the meaning of a confidence interval within the classical frequentist framework. The common misinterpretation the probability that a parameter lies between certain limits is exactly the correct one from the Bayesian standpoint. 

As noted above, sensitive and specific GC-MS/MS methods for the determination of 3-OH FAs and Mur have been developed. MS is an alternative to the classical LAL assay for determination of LPS, while no other regulated approach exists for PG assessment. These chemical methods are reproducible and provide quantitative, accurate determination of microbial biocontamination. At the present time mass spectrometric measurement of LPS and PG have matured sufficiently to be used for routine assessment of air quality. Numerous products of medical and environmental origin have been analyzed. However, use for assessment of pharmaceutical products remains limited. 

Two limiting approaches of mechanism building have been outlined in Sect. 2.5.3. In a first approach, a comprehensive mechanism is written a priori, and a minimum set of reactions is selected a posteriori, on the basis of numerical tests. In a second approach, which is more familiar to classical kineticists, rules for choosing reacting species and elementary processes are defined a priori and their consistency is checked a posteriori. In theory and practice, the two approaches converge to a same single mechanism on the obvious condition that the same types of elementary processes be considered for possible inclusion in the reaction mechanism. 

In this case the two systems evidently behave independently. Situations like this are fairly common in chemistry, generally associated with an approach to the classical limit in which the quantum potential becomes negligible and non-local interactions insignificant. Although the basic law therefore refers inseparably to the whole universe, it tends to fragment into numerous independent parts, each constituted of further sub-units that are non-locally connected internally. The key to this fragmentation is the lack (or nature) of chemical interaction between sub-units, which can be treated in the traditional way. 

The necessary derivations with respect to the small displacements can be performed either numerically, or, more recently, also analytically. These analytical methods have developed very rapidly in the past few years, allowing complete ab initio calculation of the spectra (frequencies and intensities) of medium sized molecules, such as furan, pyrrole, and thiophene (Simandiras et al., 1988) however, with this approach the method has reached its present limit. Similar calculations are obviously possible at the semi-empirical level and can be applied to larger systems. Different comparative studies have shown that the precise calculation of infrared and Raman intensities makes it necessary to consider a large number of excited states (Voisin et al., 1992). The complete quantum chemical calculation of a spectrum will therefore remain an exercise which can only be perfomied for relatively small molecule. For larger systems, the classical electro-optical parameters or polar tensors which are calibrated by quantum chemical methods applied to small molecules, will remain an attractive alternative. For intensity calculations the local density method is also increasing their capabilities and yield accurate results with comparatively reduced computer performance (Dobbs and Dixon, 1994). 

In discussing the alternative theoretical approaches let us limit ourselves to those which have been applied directly to processes in which we are interested in this article, but first of all let us stress once more the importance of the work of Delos and Thorson (1972). They formulated a unified treatment of the two-state atomic potential curve crossing problem, reducing the two second-order coupled equations to a set of three first-order equations. Their formalism is valid in the diabatic as well as the adiabatic representation and also at distances of closest approach near Rc. Moreover the problem of the residual phase x(l) is solved implicitly. They were able to show that a solution of the three first-order classical trajectory equations is not sensitive to all details of the potentials and the coupling term, but to only one function which therefore can be used readily for modelling assumptions. The resulting equations should be solved numerically. Their method has been applied now to the problem of the elastic scattering of He+ + Ne (Bobbio et at., 1973) but unfortunately not yet to any ionization problem. 

On the other hand, we must bear in mind that no real planet fits the assumptions of Love s classical problem discussed here, with uniform density and uniform elastic properties immediately prior to application of the tide-raising potential. Real planets will certainly not be uniform. Nevertheless, the limits of parameters within which instabilities may occur are not known, and the possibility of instabilities may be worth investigating in greater detail. Such an investigation is currently underway, using numerical and analytical modeling approaches. 

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