Qualitative interpretation

The concept of local properties, that is, properties of atoms or groups of atoms within one molecule, is a useful tool for the interpretation of abstract wave functions in terms of intuitive building blocks. In principle, any molecular property can be partitioned and distributed over a set of subsystems of a molecule, which then provides information for a quaHtative analysis and understanding of bonding. However, such a partitioning is not unique and requires additional ad hoc) assumptions. Still, within the realm of these assumptions, useful qualitative reasoning is possible. One important local concept for open-shell molecules is the decomposition of the total molecular electron spin S,  

Although the spin state of atom (or a group of atoms) A within a molecule is not observable, local spins are employed for the description of spin-spin interactions between magnetic centers, similar to the metal centers of transition-metal clusters, in terms of a Heisenberg spin-coupling model and led to considerable interest in the development of partitioning schemes of the total spin expectation value during the past decade [20, 112-128]. 

If we want to assign local-spin values to (atomic) subsystems, a definition of an atomic basin is required. Clark and Davidson proposed a framework where local molecular spin operators are obtained from projections of the total molecular spin operator [112, 118] upon atomic basins A. The one- and two enter contributions to the total (S ) are then obtained from the expectation value of the local spin operator (S ) and of diatomic products of local spin operators (S Sg), respectively. However, Clark and Davidson s decomposition leads to nonvanishing 

It was, however, demonstrated by Ramos-Cordoba et al. [127] that there actually exists a continuum of different partitioning schemes of (S ) for both single- and multideterminant wave functions that fulfill all physical requirements mentioned and from which the individual decomposition formulas naturally emerge. Their analysis is carried out in three-dimensional space in contrast to the preceding approaches performed in a Hilbert-space formulation. A generalization of the latter to the physical space can be easily accomplished [117,125] and we will employ the physical-space notation as presented in Ref. [127]. 

Given an N-electron wave function, the (S ) expectation value reads in terms of the (spinless) first- and second-order reduced density matrices, /)(rj rJ) and r, rp, respectively. 

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At the micro-scale level, there really is no way to measure concentration fluc tuations. Resort must be made to other qualitative interpretation of results for either a process or a chemical reac tion studv. 

Gauss s theorem, in mathematics, says a change in a volume s density must be accompanied by flow through the boundary. Leak path analysis is a qualitative interpretation of Gauss s theorem. 

Some emphasis has been placed inthis Section on the nature of theel trified interface since it is apparent that adsorption at the interface between the metal and solution is a precursor to the electrochemical reactions that constitute corrosion in aqueous solution. The majority of studies of adsorption have been carried out using a mercury electrode (determination of surface tension us. potential, impedance us. potential, etc.) and this has lead to a grater understanding of the nature of the electrihed interface and of the forces that are responsible for adsorption of anions and cations from solution. Unfortunately, it is more difficult to study adsorption on clean solid metal surfaces (e.g. platinum), and the situation is even more complicated when the surface of the metal is filmed with solid oxide. Nevertheless, information obtained with the mercury electrode can be used to provide a qualitative interpretation of adsorption phenomenon in the corrosion of metals, and in order to emphasise the importance of adsorption phenomena some examples are outlined below. 

K. Janssens and P. van Espen, Implementation of an expert system for the qualitative interpretation of... 

Even though the correct many electron wave function is not available in DFT, we will see in Section 5.3.3 that a related wave function exists, which can often be used for qualitative interpretation. 

Whiteley, J. R., and Davis, J. F., Qualitative interpretation of sensor patterns. IEEE Expert, 8(2) 54-63 (1992a). 

Thus, a suitable refinement of the Debye-Huckel theory must provide a theoretical interpretation of the term CL Originally this term was qualitatively interpreted as a salting-out effect during solvation the ions... 

As the system pressure is decreased at constant temperature, the time between collisions will increase, thereby providing greater opportunity for unimolecular decomposition to occur. Consequently, one expects the reaction rate expression to shift from first-order to second-order at low pressures. Experimental observations of this transition and other evidence support Linde-mann s theory. It provides a satisfactory qualitative interpretation of unimolecular reactions, but it is not completely satisfactory from a... 

Determination of miscibility by additive properties such as average molecular area can give only qualitative interpretations of II/A data. For example, if the film components are ideally miscible, the average molecular area of the binary film at a fixed surface pressure will be the sum of each of the molecular areas of the individual components 1 and 2 in their pure films, and will follow equation (11), where N is the mole fraction. Unfortunately,... 

It was suggested in a previous publication (9) that flocculation at the UCFT can be ascribed to the free volume dissimilarity between the polymer stabilizing the particle and the low molecular weight dispersion medium. Incorporating this idea in a quantitative way into the theory of steric stabilization allowed for a qualitative interpretation of the experimental data. This idea is further extended to include the effect of pressure on the critical flocculation conditions. 

Various theories, ranging from qualitative interpretations to those rooted in irreversible thermodynamics and geochemical kinetics, have been put forward to explain the step rule. A kinetic interpretation of the phenomenon, as proposed by Morse and Casey (1988), may provide the most insight. According to this interpretation, Ostwald s sequence results from the interplay of the differing reactivities of the various phases in the sequence, as represented by Ts and k+ in Equation 26.1, and the thermodynamic drive for their dissolution and precipitation of each phase, represented by the (1 — Q/K) term. 

A disadvantage of tracer methods is the frequent presence of exchange processes which are difficult to account for. This hampered Happel s work with 180 (30) and complicated the interpretation of the work of Conner and Bennett (34), who used pulses of tracers and a qualitative interpretation of their data on CO oxidation over NiO. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

While this equation is certainly not exact, it can be used for qualitative interpretations. In particular, the following limiting cases are of interest ... 

Rank Degree of Hazard Maximum Enthalpy of Decomposition/Reactiona (kcal/g) Possible Qualitative Interpretations of the Classifications... 

There are two types of NOE experiments that can be performed. These are referred to as the steady-state NOE and the transient NOE. The steady-state NOE experiment is exemplified by the classic NOE difference experiment [15]. Steady-state NOE experiments allow one to quantitate relative atomic distances. However, there are many issues that can complicate their measurement, and a qualitative interpretation is more reliable [16]. Spectral artifacts can be observed from imperfect subtraction of spectra. In addition, this experiment is extremely susceptible to inhomogeneity issues and temperature fluctuations. 

Along the discussion developed above, the shapes of time-conversion curves in Figures 2 and 3 are qualitatively interpreted. 

This important equation can be qualitatively interpreted in the following way. When the two components Ox and Red are present in solution at certain concentrations, the working electrode will spontaneously find its equilibrium potential (imposed by the Nernst equation) and there will be no overall current flow. In order for Ox to be reduced or Red oxidized, the system must be moved from equilibrium. This can be achieved by setting a potential different from that for equilibrium. The process of oxidation or reduction will be favoured depending on whether... 

Schuster et al. reported work on monitoring a complex ace-tone-butanol-ethanol (ABE) fermentation system.22 They looked at the qualitative nature of the biomass as well as the solvents present in the liquid phase. A hierarchical cluster analysis was performed on samples from various times of the fermentation. The clusters were then classified using classical markers and analyses. The resultant table, combining qualitative interpretation and quantitative results, shows an interesting mosaic of the system over time. Total solvents, optical density, and butyric acid are given as numeric values in either absorbance units of g/1. 

Irreversible follow-up reactions (most simple case EC mechanism) decrease the concentration of the primary redox product. This is again diagnosed in CV (Figure 9) and also in chronocoulome-try. Timescale variation in CV allows to modulate the importance of the C-step at fast v the chemical reaction will have no influence on the curves, while at slower v all product has reacted and the reverse peak disappears. A governing factor is k/a (k = rate constant of C-step, a = nFv/RT). Thus, for a qualitative interpretation, the peak current ratio in CV is evaluated as a function of V (and E ) in order to calculate k [49]. Also, p and ip depend on k/a [28]. 

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