The Effective Size of Atoms

According to wave mechanics, the electron density in an atom decreases asymptotically towards zero with increasing distance from the atomic center. An atom therefore has no definite size. When two atoms approach each other, interaction forces between them become more and more effective. 

The effectiveness of these forces differs and, furthermore, they change to a different degree as a function of the interatomic distance. The last-mentioned repulsion force is by far the most effective at short distances, but its range is rather restricted at somewhat bigger distances the other forces dominate. At some definite interatomic distance attractive and repulsive forces are balanced. This equilibrium distance corresponds to the minimum in a graph in which the potential energy is plotted as a function of the atomic distance ( potential curve , cf. Fig. 5.1, p. 42). 

The equilibrium distance that always occurs between atoms conveys the impression of atoms being spheres of a definite size. In fact, in many cases atoms can be treated as if they were more or less hard spheres. 

Inorganic Structural Chemistry, Second Edition Ulrich Muller 2006 John Wiley Sons, Ltd. 

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An approach based on orbital radii of atoms effectively rationalizes the structures of 565 AB solids (Zunger, 1981). The orbital radii derived from hard-core pseudopotentials provide a measure of the effective size of atomic cores as felt by the valence electrons. Linear combinations of orbital radii, which correspond to the Phillips structural indices and have been used as coordinates in constructing structure maps for AB solids. 

On compression of non-hydrogen atoms the energy levels, which in this case are occupied by electrons, respond in the same way. Apart from level crossings, interelectronic interactions now also lead to an internal transfer of energy and splitting of the magnetic sub-levels, such that a single electron eventually reaches the ionization limit on critical compression. The calculated ionization radii obey the same periodic law as the elements and determine the effective size of atoms in chemical interaction. 

When two or more atoms are forced together, they repel each other, and experience van der Waals repulsion as the electrons associated with each atom start to occupy a common space. The effective size of atoms is given by the van der Waals radii (Table 4.3), which are related to how close atoms or groups can come without severe repulsion. Van der Waals repulsion is also called steric hindrance, and the energy of that interaction is steric strain. In the eclipsed conformation of propane, the hydrogen atoms at C-1 and C-3 are not close enough to produce a large van der Waals repulsion. 

Other substituted cyclohexanes are similar- to methylcyclohexane. Two chair confonnations exist in rapid equilibrium, and the one in which the substituent is equatorial is more stable. The relative fflnounts of the two confor-rnations depend on the effective size of the substituent. The size of a substituent, in the context of cyclohexane confor-rnations, is related to the degree of branching at the atom connected to the ring. A single... 

Van der Waals radius (Section 2.17) A measure of the effective size of an atom or a group. The repulsive force between two atoms increases rapidly when they approach each other at distances less than the sum of their van der Waals radii. 

Figure 19-4 contrasts the effective sizes of the halide ions. Each of these dimensions is obtained from the examination of crystal structures of many salts involving the particular halide ion. The effective size found for a given halide ion is called its ionic radius. These radii are larger than the covalent radii but close to the van der Waals radii of neutral atoms. 

Here we see clearly the large concentration of density around the oxygen nucleus, and the very small concentration around each hydrogen nucleus. The outer contour is an arbitrary choice because the density of a hypothetical isolated molecule extends to infinity. However, it has been found that the O.OOlau contour corresponds rather well to the size of the molecule in the gas phase, as measured by its van der Waal s radius, and the corresponding isodensity surface in three dimensions usually encloses more than 98% of the total electron population of the molecule (Bader, 1990). Thus this outer contour shows the shape of the molecule in the chosen plane. In a condensed phase the effective size of a molecule is a little smaller. Contour maps of some period 2 and 3 chlorides are shown in Figure 8. We see that the electron densities of the atoms in the LiCl molecule are only very little distorted from the spherical shape of free ions consistent with the large ionic character of this molecule. In... 

Intexmolecular forces can be repulsive as well as attractive in nature. Tuo molecules ultimately reach a miniimm distance between them as they approach one another, and this distance is the sum of the van der Waals radii of the interacting groape. Hence, the van der Waals radius is considered a measure of the effective size of an atom in noncovalent interactions. Van der Waals radius is correlated with another measure of steric repulsion, Es.)... 

The effective size of an atom or group varies with the phenomenon studied. Atoms are better pictured as somewhat compressible than as hard balls. 

Calculations have been performed where this bond distance has been varied over several angstroms in order to find the best fit with experiment. 8.9 These BtudieB have also shown that there iB a sensitivity of the polar angle distribution to the effective size of the adsorbed atom. Thus, it is important to know more about the scattering potential parameters if this distance is to be determined accurately. It appears, however, that the type of adsorption site may be determined in a reasonably straightforward manner. 

Meta-diisopropylbenzene is reacted with propylene over the acid form of the molecular sieves SAPO-5, mordenite, offretite, beta, hexagonal and cubic faujasite (EMT and FAU), L, SAPO-37, and an amorphous silica-alumina at temperatures around 463 K in a flow-type fixed-bed reactor. A small amount of cracking is observed. However, the main reactions of meta-diisopropylbenzene are isomerization and alkylation. It is proposed that this alkylation can be used as a new test reaction to characterize the effective size of the voids in larger pore (12 T-atom rings or above) molecular sieves by measuring the weight ratio of 1,3,5- to 1,2,4-triisopropylbenzene formed. In most cases, this ratio increases with die increasing effective void size of the molecular sieves in the order SAPO-5 < mordenite < offretite < beta < EMT FAU < L < SAPO-37 < amorphous silica-alumina. 

This approximate form of Gss(z R1 R2) shows a general property of van der Waals interactions when formulated in the approximation (small differences in dielectric response, neglect of retardation) used here. The interaction is independent of length scale. If we were to change all the sizes and separations by any common factor, both the numerator RfR and the denominator z6 would change by the same factor to the sixth power. In reality, because retardation screening effectively cuts off interactions at distances of the order of nanometers, it makes sense to think of this inverse-sixth-power interaction only for particles that are the angstrom size of atoms or small molecules. 

From the discussion of Chapter I, it follows that metallic conduction is to be associated with partially filled bands of collective-electron states. Since the s-p bands of an ionic compound are either full or empty, metallic conduction implies partially filled d bands, and collective d electrons imply Rtt < Rc(n,d). From the requirement Rtt < Rc(n4) it is apparent that the metallic conduction in ionic compounds must be restricted either to transition element compounds in which the anions are relatively small or to compounds with a cation/anion ratio > 1. Also Rc(n,d) decreases, for a given n, with increasing atomic number, that is with increasing nuclear charge, and the presence of eQ electrons increases the effective size of an octahedral cation (627) (see Fig. 66) and similarly UQ electrons the size of a tetrahedral cation. It follows that If the cation/anion ratio < 1, MO d electrons are more probable in ionic compounds with octahedral-site cations if the cations contain three or less d electrons MO d electrons are more probable in ionic compounds with tetrahedral-site cations if the cations contain two or less d electrons. 

It is of interest to note the relationship of the effective size of H on the distension of a BCC metal and the symmetry of the hydride. If one assumes occupancy by H of a typical group of octahedral holes, arbitrarily (V2, V2> 0) and (0, 0, 1/2), then the development of a tetragonal structure as a function of the radius of a hard spherical H atom may be calculated by simple geometry with the... 

The Dutch physicist J.D. van der Waals found that in order to explain some of the properties of gases it was necessary to assume that molecules have a well defined size, so that two molecules undergo strong repulsion when, as they approach, they reach certain distance from one another. [...] It has been found that the effective sizes of molecules packed together in liquids and crystals can be described by assigning Van der Waals radii to each atom in the molecule. The Van der Waals radius defines the region that includes the major part of the electron distribution function for unshared [electron] pairs. Cf. Fig. l.A [2],... 

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