Critical point Terms Links

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

It is to be stressed that a bond path is not to be understood as representing a ond . The presence of a bond path linking a pair of nuclei implies that the corresponding atoms are bonded to one another. As demonstrated later, the interaction can be characterized and classified in terms of the properties of the charge density at its associated (3, — 1) critical point. The complete description of the interaction, however, requires the evaluation of operators over the associated interatomic surface. These are the topics of later discussions At this point we continue with the identification of the elements of molecular structure with the topological properties of the remaining stable critical points, (3, + 1) and (3, + 3) critical points. 

The value of the charge density at a bond critical point can be used to define a bond order (Bader et al. 1983 Cremer and Kraka 1984). The molecular graphs for ethane, ethylene, and acetylene are shown in Fig. 2.8. In each case the unique pair of trajectories associated with a single (3, — 1) critical point is found to link the carbon nuclei to one another. Multiple bonds do not appear as such in the topology of the charge density. Instead, one finds that the extent of charge accumulation between the nuclei increases with the assumed number of electron pair bonds and this increase is faithfully monitored by the value of p at the bond critical point, a value labelled p, . For carbon-carbon bonds, one can define a bond order n in terms of the values of Ph using a relationship of the form... 

The topology of the electronic charge density ip r)), as pointed out by Bader [55], is an accurate mapping of the chemical concepts of atom, bond and structure. The main topological properties are summarized in terms of their critical points (CP) [55,56]. The nuclear positions behave topologically as local maxima in p(r). A bond critical point (BCP) is found between each pair of nuclei, which are considered to be linked by a chemical bond, with two negative (Ai and ki) and one positive (A3) curvature (denoted as (3,-1) CP). The ellipticity (e) of a bond is defined by means of the two negative curvatures in a BCP as ... 

We noted above that the presence of monomer with a functionality greater than 2 results in branched polymer chains. This in turn produces a three-dimensional network of polymer under certain circumstances. The solubility and mechanical behavior of such materials depend critically on whether the extent of polymerization is above or below the threshold for the formation of this network. The threshold is described as the gel point, since the reaction mixture sets up or gels at this point. We have previously introduced the term thermosetting to describe these cross-linked polymeric materials. Because their mechanical properties are largely unaffected by temperature variations-in contrast to thermoplastic materials which become more fluid on heating-step-growth polymers that exceed the gel point are widely used as engineering materials. 

A critical aspect of integrating sustainability into business is in the area of valuation. How do we know the financial impacts of our decisions How can performing more sustainability help to reduce costs, avoid future costs, increase the company s value proposition, support growth, and so on What are the links between intangibles related to social and environmental performance and a hrm s overall performance What contributes to value creation in the chemical industry How can we reconcile the fact that we may incur costs for performing more sustainably in the short term while the benefits may not be captured until the long term similarly, how do we justify costs associated with our business decisions when the benefits might accrue to others in the value chain We know the value of ecosystem services is both invaluable - that is, our life depends on them - and of no real market value how then should we value ecosystem services in order to protect them for our survival and that of our business institutions This section attempts to offer points of view on these topics. 

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