Point-molecules

The summation is over all pairwise distances between the M selected points (molecules). Power a was set to 1 in all of the experiments reported below however, it could be set to any value between 1 and 6 empirically. 

Molecular Orbital Theory a model that uses wave functions to describe the position of electrons in a molecule, assuming electrons are delocalized within the molecule Molecular Solid a solid that contains molecules at the lattice points Molecule a group of atoms that exist as a unit and are held together by covalent bonds... 

Compounds such as those in Figure 14.4 are known as isomers. Isomers are substances which have the same molecular formula but different structural formulae. The different structures of the compounds shown in Figure 14.4 have different melting and boiling points. Molecule b contains a branched chain and has a lower melting point than molecule a, which has no branched chain. All the alkane molecules with four or more carbon atoms possess isomers. Perhaps now you can see why there are so many different organic compounds ... 

For "up-" and "down-pointing" disorder, T would be the average of the transforms of the (reference) "up-pointing" molecule and the "down-pointing" molecule.) In any case... 

The shape of the metal specimen considered is obviously related to the kind of system to be modelled. For SERS and the others SE phenomena, the presence of curved surfaces, with a curvature radius on the nanometric scale, is fundamental for the enhancement. Thus, spheres, ellipsoids, ensembles of spheres, spheres close to planar metal surfaces and planar metal surfaces with random roughness have been considered. We refer to the review by Metiu [57], which describes most of these analytically solvable models. More recently, the modelling of the electric field acting on the point molecule has moved to more realistic shapes (including fractal metal specimen) [59] which require numerical methods to be tackled. The aim of these approaches is usually to calculate the total electric field around the metal particle, and the molecule does not even appear explicitly in the calculations. Interested readers are referred to some recent reviews on the subject [60] (see also Chapters 2 and 5 of ref. [56]). 

The inversion operation is carried out by joining a point to the inversion center (or center of symmetry) and extending it an equal distance to arrive at an equivalent point. Molecules which possess an inversion center are termed centrosymmetric. Among the eight examples given so far, SF6 (Fig. 6.1.4), cyclohexane (Fig. 6.1.5), trans-N2F2 (Fig. 6.1.6), and BrFJ (Fig. 6.1.8) are centrosymmetric systems. Molecules lacking an inversion center are called non-centrosymmetric. 

Molecules bearing a permanent-dipole moment Think of the interaction between two fairly strong dipoles, /xc ipoie = 2 D = 2 x 10 18 esu cm, slightly larger than the 1.87-D moment of a water molecule.11 Let this molecule be approximately the size of a water molecule, i.e., 3 A across so that the point-molecule approximation would apply at separations much greater than 3 A. In emits of kT the Keesom interaction is... 

The point molecule, while adequate for representing the properties of an ideal gas, is incapable of accounting for molecular collisions. The sim-... 

The same result applies to the calculation of the number of molecules striking a wall. We obtain the result Ng for point molecules and the same result when wo consider hard sphere molecules. The result is increased in the ratio of 1 -b/v when we consider dense gases, where h is the Van der Waals constant. At ordinary conditions, however, the correction h/v is small. 

Simulation was conducted to determine how many clusters of points (molecules) SAGE could cover in comparison with random selection. When the number of sampled points was much smaller than that of clusters that exist in the data set, there was virtually no difference between the rational and random selection. This implies that when the number of sampled compounds was very small compared to that of the natural clusters, rational sampling could not provide any advantages over random sampling. As the number of sampled points increases, SAGE begins to cover more clusters than the corresponding... 

The matrix representation of a molecular set can be associated with a set of finite dimensional vectors representing the molecules. This point of view leads to the concept of Point-Molecules collected as a Molecular Point Cloud. 

Fig. 4. Projection of the point-molecule representation of the nine molecules using the Carbo index obtained from a gravitational-like similarity measure. The studied property is the boiling point...

Up to the gel point all molecules are finite and the weight fraction of solubles, Wg, is thus unity. Beyond the gel point molecules are rapidly incorporated into the network and Wg decreases rapidly. For the homopolymerization of A/ (/ > 2), a randomly chosen A/ molecule will be part of the sol if all / of its arms lead out to finite chains. Thus,... 

Figures 1 to 4 show the projections of the point-molecule representation for the P-diketone compounds described in Table 1, on...

Figures 5 to 8 show projections of the Point-Molecule representation for Cu(II)-P-diketones quelate compounds, on the plane of two principal components for the Cioslowski-like, Coulomb-like, overlap-like and triple density similarity measure matrices, respectively. As before, the figures represent two-dimensional projections of 10-dimensional polyhedrons and the elements of the set are divided in two classes, depending on their respective extraction constant (K ). We can see that the most active compounds (represented by circles) can be shown split from the less active ones (represented by squares).

Figures 13 and 14 are projections of the Point-Molecule representation for retinoid compounds of Table 3, in the plane of two principal components of the similarity matrix. The elements of the set...

---