Classical methods definition

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different dynamical footing—for example, quantum mechanics for the electronic DoF and classical mechanics for the... 

From ancient times, humans have pondered what the universe is made of Early philosophers proposed fire, earth, water, and air either individually or in combination as the building blocks of nature. Lavoisier defined an element operationally as a substance that cannot be broken down chemically. Using this definition, the number of elements has increased from around 30 in Lavoisier s time to over 115 today. The initial search for elements involved classical methods such as replacement reactions, electrochemical separation, and chemical analysis. New methods such as spectroscopy greatly advanced the discovery of new elements during the twentieth century. The last half century has been marked by the synthesis of elements by humans. 

Traceability and MU both form parts of the purpose of an analytical method. Validation plays an important role here, in the sense that it confirms the fitness-for-purpose of a particular analytical method [4]. The ISO definition of validation is confirmation by examination and provision of objective evidence that the particular requirements of a specified intended use are fulfilled [7]. Validation is the tool used to demonstrate that a specific analytical method actually measures what it is intended to measure and thus is suitable for its intended purpose [2,11]. In Section 8.2.3, the classical method validation approach is described based on the evaluation of a number of method performance parameters. Summarized, the cri-teria-based validation process consists of precision and bias studies, a check for... 

As for the QM/MM description also for PCM, non-electrostatic (or van der Walls) terms can be added to the Vent operator in this case, besides the dispersion and repulsion terms, a new term has to be considered, namely the energy required to build a cavity of the proper shape and dimension in the continuum dielectric. This further continuum-specific term is generally indicated as cavitation. Generally all the non-electrostatic terms are expressed using empirical expressions and thus their effect is only on the energy and not on the solute wave function. As a matter of fact, dispersion and repulsion effects can be (and have been) described at a PCM-QM level and included in the solute-effective Hamiltonian 7/eff as two new operators modifying the SCRF scheme. Their definition can be found in Ref. [17] while a recent systematic comparison of these contributions determined either using the QM or the classical methods is reported in Ref. [18]... 

The classical method for measuring the solubility parameter of a polymer is to determine the solubility of that polymer in a large number of liquids with known solubility parameters, and then to plot the observed solubility as a function of the solubility parameters of the test liquid. The solubility parameter where the polymer solubility is maximal [40] is by definition the solubility parameter of the polymer, 5pol. 

Definition 5. (See 23.) The corresponding algebraic method is called the classical method. 

The essential aspects have been discussed in the introduction on the use of RMs and CRMs. It should be noted that inorganic CRMs, in particular pure metals, are available on the market from several reliable suppliers. They show usually purity values with associated uncertainties that are negligible compared to the uncertainty of the majority of spectrometric methods in which they serve as calibrants. It is usual to find materials of stated (not by definition certified) purity of 99.999% (five nines in analytical jargon) or better. This would mean that any impurity is below 0.001% as a mass fraction. No relative analytical method has precision performances that go down to such levels. Suppliers of ultra pure metals are numerous. NIST sells such metals as certified RMs (SRMs). The certification of the purity is discussed briefly in Chapter 5. It can be mentioned that the measurements are often based on absolute methods. The ultimate detection of impurities can be made with spark source MS. For pure metals the uncertainty linked to the calculated purity is small. Therefore, compared to the intended use and the uncertainty of classical methods applied by the analyst for the determination of elements, it is totally negligible. 

Methods aimed at the determination of tannin material content in the extract. The classical method of this type still used is the hide-power and derived methods. These methods were devised to determine which percentage of the extract would participate in leather tanning. The main drawback for their use for adhesives in their inability to detect and determine the approximate 3 to 6% of monoflavo-noids and biflavonoids, or phenolic nontannins, present in the extract which do not contribute to tanning capacity but which do definitely react with formaldehyde and contribute to adhesive preparation. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different djmamical footing, e.g. quantum mechanics for the electronic DoF and classical mechanics for the nuclear DoF. As was discussed above, some of the problems with these methods are related to inconsistencies inherent in this mixed quantum-classical ansatz. To avoid these problems, recently a conceptually different way to incorporate quantum mechanical DoF into a semiclassical or quasiclassical theory has been proposed, the so-called mapping approach. " In this formulation, the problem of a classical treatment of discrete DoF such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. In this section we briefly introduce the general concept of the mapping approach and discuss the quasiclassical implementation of this method as well as applications to the three models introduced above. The semiclassical version of the mapping approach is discussed in Sec. 7. 

The methods of MM are devoid of this failure since electron correlation effects are implicitly included in the atom-atom potentials . However, these methods are classical by definition and do not describe the quantum effects, which are essential in the overwhelming majority of problems, in fact in all the problems in which the electronic states should be considered explicitly. These considerations lead directly to the idea to separate the molecular system in two (or more) regions in such a way as to find the quantum effects taking place overwhelmingly in only one of them, while the other regions could be considered by classical methods. The QM/MM approaches are based on this idea. 

Besides classical methods, the so-called semi-classical approximation is now widely used. The basic idea of this approximation is the division of the totality of nuclear coordinates into two groups, Q and Q". One is described quantum-mechanically (quantum subsystem Q ) and the other classically (classical subsystem Q"). It is assumed that the classical subsystem moves along a definite trajectory Q"(t) and that the interaction between the two subsystems V(Q, Q") is a time-dependent perturbation causing transitions between states of the quantum subsystem [339, 347]. 

If we consider the ISO and EU definitions instead, the CCp concept can be generalized to address the multivariate sensitivity (which is termed now the capability of discrimination). It can be defined as the analyte concentration that an analytical method is able to discriminate, for preset a and jS errors. Thus, multivariate sensitivity is calculated as the difference between a nominal concentration and the lowest concentration that can be differentiated from it. This means that different multivariate sensitivities exist for different concentration levels, which is advantageous because an analytical method may not be capable of differentiating the same concentration when applied to samples containing different concentration levels, mainly when they are far from the detection limit region. This is a step forward regarding the traditional or classical unique definition. An example is given in Table 5.3. 

This definition of dependability is also called the Kolmogorov s zero-one law. It is one of the laws of large numbers, since according to the definition only two cases exist, either there are dependencies or there aren t. Since we already learned that a complete independency could rarely be achieved, ISO 26262 speaks of a sufficient independency. Failures of common causes or failure dependencies between functions, which can affect through different mechanisms, are often no longer analyz-able with the classical methods. In this case we can often only rely on experience. For functional dependencies we can systematically analyze a lot of things firom the functional chains and their derivation in the different horizontal abstraction levels. A barrier, independent if it is a functional or technical barrier, or whatever technology it is, could be only assessed for its sufficiency or effectiveness, in the specific context and for possible failure effects (Fig. 5.59). 

Classically, aerosols are particles or droplets that range from about 0.15 to 5 p.m ia size and are suspended or dispersed ia a gaseous medium such as air. However, the term aerosol, as used ia this discussion, identifies a large number of products which are pressure-dispensed as a Hquid or semisohd stream, a mist, a fairly dry to wet spray, a powder, or even a foam. This definition of aerosol focuses on the container and the method of dispensiag, rather than on the form of the product. 

Two other methods worth discussing are wet air oxidation and regeneration by steam. Wet oxidation may be defined as a process in which a substance in aqueous solution or suspension is oxidized by oxygen transferred from a gas phase in intimate contact with the liquid phase. The substance may be organic or inorganic in nature. In this broad definition, both the well known oxidation of ferrous salts to ferric salts by exposure of a solution to air at room temperature and the adsorption of oxygen by alkaline pyrogallol in the classical Orsat gas analysis would be considered wet oxidations. 

If, for the purpose of comparison of substrate reactivities, we use the method of competitive reactions we are faced with the problem of whether the reactivities in a certain series of reactants (i.e. selectivities) should be characterized by the ratio of their rates measured separately [relations (12) and (13)], or whether they should be expressed by the rates measured during simultaneous transformation of two compounds which thus compete in adsorption for the free surface of the catalyst [relations (14) and (15)]. How these two definitions of reactivity may differ from one another will be shown later by the example of competitive hydrogenation of alkylphenols (Section IV.E, p. 42). This may also be demonstrated by the classical example of hydrogenation of aromatic hydrocarbons on Raney nickel (48). In this case, the constants obtained by separate measurements of reaction rates for individual compounds lead to the reactivity order which is different from the order found on the basis of factor S, determined by the method of competitive reactions (Table II). Other examples of the change of reactivity, which may even result in the selective reaction of a strongly adsorbed reactant in competitive reactions (49, 50) have already been discussed (see p. 12). 

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