External or Internal Coordinates

Molecular structures may be described and compared in terms of external or internal coordinates. The question of which is to be preferred depends on the type of problem that is to be solved. For example, one problem that is much easier to solve in a Cartesian system is that of finding the principal inertial axes of a molecule indeed, if only internal coordinates are given then, in general, the first step is to convert them to Cartesian ones and then proceed as described in Section 1.2.4. Similarly, the optimal superposition of two or more similar molecules or molecular fragments, i.e. with the condition of least-squared sums of distances between all pairs of corresponding atoms, is best done in a Cartesian system. On the other hand, systematic trends in a collection of molecular structures and correlations among their structural parameters are more readily detectable in internal coordinates. 

With pure advection processes, we refer to continuous phenomena that cause continuous changes in the external and internal coordinates. Continuous changes of the particle s position in real space are quantified by the real-space advection (or free-transport) term ... 

T[[dotb]he nature of the initial attack by the water (eq. 10) is a matter of some controversy (205,206). Stereochemical and kinetic studies of model systems have been reported that support trans addition of external water (207,208) or internal addition of cis-coordinated water (209), depending on the particular model system under study. Other paHadium-cataly2ed oxidations of olefins ia various oxygen donor solvents produce a variety of products including aldehydes (qv), ketones (qv), vinyl acetate, acetals, and vinyl ethers (204). However the product mixtures are complex and very sensitive to conditions. 

There is some similarity between Ferry s treatment of concentrated systems (14), (123) [eq. (4.4)] and Cerf s just mentioned approach. In both cases the normal coordinate transformation is assumed to be possible along the lines given for infinitely dilute solutions of kinetically perfectly flexible chains (Rouse, Zimm). Only afterwards, different external (Ferry) or internal (Cerf) friction factors are ascribed to the various normal modes. 

The 16-electron species invariably must be stabilized by addition of an external ligand or by internal coordination. For example, as will be discussed later, acylation of a [Fe(diene)(CO)3] complex yields a... 

There are various types of classification of attacks. For example, division into passive and active, external and internal attacks, deliberate and unintentional. It should be mentioned that many models of attacks are currently well known one-to-one or one-to-many, i.e., attack proceeds from one point many-to-one and many-to-many, i.e., distributed or coordinated attacks hybrid attacks also named the blended threat [12]. 

Although the most fundamental coordinate system is formed by the cartesian coordinates of each nucleus, the study of F(R) in the absence of external fields may be accomplished only in coordinates (called internal coordinates) which relate to the perimeter or shape of the molecule, disregarding its position or orientation in the space as a whole. Therefore, the discussion and represent-... 

The maintenance of cellular homeostasis implies the dynamic coordination of cellular processes to finely compensate for subtle variations of the external and internal environments (e.g., pH, osmolarity, nutrients, and oxygen supply) but also to monitor and regulate intracellular signalling and compartmentalization. Cellular stress occurs when a threat to homeostasis is detected by highly reactive and efficient protective mechanisms. Depending on the intensity (dose) and duration (exposure) of the stress, these defenses can either cope without any observable change in cellular homeostasis or be overcome, resulting in a detectable shift from basal metabolic or cellular functions. 

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

Another possible scenario is that as a consequence of mass transfer only the number of primary particles changes, whereas their size remains more or less constant. This hypothesis seems to be realistic in the case of negative molar flux, J <0, or, in other words, in the case of shrinking particles. In fact, in this case it is more likely that the external particles will be consumed before the internal ones. The resulting expressions for the continuous rate of change of the two internal coordinates therefore read as... 

If these symbols and models are to have any metrical significance, if they are to portray the relative atomic positions in a reasonably realistic way, they need to be based on numerical data. To define the atomic positions we need numbers - a set of coordinates. These may be external coordinates, referred to some set of coordinate axes, or they may be internal coordinates, defining the various interatomic distances and angles in the molecule of interest. 

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