Qualitative aspects

Chemical and electrochemical reactions in condensed phases are generally quite complex processes only outer-sphere electron-transfer reactions are sufficiently simple that we have reached a fair understanding of them in terms of microscopic concepts. In this chapter we give a simple derivation of a semiclassical theory of outer-sphere electron-transfer reactions, which was first systematically developed by Marcus [1] and Hush [2] in a series of papers. A more advanced treatment will be presented in Chapter 19. 

A natural question is In which temporal order do the reorganization processes and the proper electron transfer take place The answer is given by the Frank-Condon principle, which in this context states First the heavy particles of the inner and outer sphere must assume a suitable intermediate configuration, then the electron is exchanged isoenergetically, and finally the system relaxes to its new equilibrium 

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This venerable book was written in 1935, shortly after the birth of modern quantum mechanics. Nevertheless, it remains one of the best sources for students seeking to gain an understanding of quantum-mechanical principles that are relevant in chemistry and chemical physics. Equally outstanding jobs are done in dealing with both quantitative and qualitative aspects of the subject. More accessible to most chemists than Landau and Lifschitz. 

Molecular modeling is an indispensable tool in the determination of macromolecular structures from NMR data and in the interpretation of the data. Thus, state-of-the-art molecular dynamics simulations can reproduce relaxation data well [9,96] and supply a model of the motion in atomic detail. Qualitative aspects of correlated backbone motions can be understood from NMR structure ensembles [63]. Additional data, in particular residual dipolar couplings, improve the precision and accuracy of NMR structures qualitatively [12]. 

This example illustrates the qualitative aspects of SPEAR, using the chlorine tanker loading case study as a basis. 

Equilibrium (continued) calculations, 192 constant, 151, table, 154 crystallization and, 144 dynamic nature of, 144, 165 effect of catalyst, 148 effect of concentration, 148 of energy, 167 of randomness, 166 of temperature, 67. 148, 167 factors determining, 155, 158 law of chemical, 152, 173 liquid-gas, 66 qualitative aspects of, 142 quantitative aspects of, 151 recognizing, 143 slate of, 142, 147 sugars, 425 thermal, 56... 

In the physics and chemistry of many-particle systems comprising atoms, molecules, solid state, and nuclei, quantum mechanics has given very important contributions to the theory of both a qualitative and quantitative nature. The Hartree-Fock scheme has usually been considered as a rather sophisticated approach, but, if one seriously studies the typical errors listed in Tables I and II and Eqs. 11.83 and 11.84 it becomes clear that the qualitative aspects... 

In these developments there were two distinct stages the first one, which dealt with the nearly linear oscillations, found its perfect algorithm in the theories of Poincare (both topological and analytical), which constitute the major part of this review here the discoveries were most striking as well as systematic. Numerous phenomena that had remained as riddles for many years, sometimes even for centuries, were systematically explained. We indicate in Part I of this chapter the qualitative aspect of this progress and in Part II, the quantitative one. 

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

Verification of this structure is provided by the comparison with calculated intensity curves. In Fig. 6 curve A represents the model described above and curve B a similar model with the C=C bond distance equal to 1.38 A., as given by the original table of covalent radii. Each of these curves reproduces closely the qualitative aspect of the photographs curve A also shows quantitative agreement, whereas curve B shows a systematic difference of about 3%. The quantitative comparison of measured ring diameters and r values for the maxima and minima of curve A is shown in Table VIII. 

Woods SP, Conover E, Rippeth ID, Carey CL, Gonzalez R, Marcotte TD, Heaton RK, Grant I (2004) Qualitative aspects of verbal fluency in HIV-associated dementia a deficit in rule-guided lexical-semantic search processes Neuropsychologia 42(6) 801-809... 

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