Radii of Atoms in Molecules and Crystals

Replacing the covalent radii for Tq (see Sect. 1.4.1) and multiplying the right part of Eq. 1.24 by fc = 0.9 + 0.05n [37] in order to convert the radius of the maximum electron density Tq to the radius of its minimum, r, which can be regarded as the atomic boundary. The results are listed in Table SI.6. The difference between these approaches for calculating radii of free atoms to day allows to present the average values to within 0.1 A (Table 1.10). 

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Radii of Atoms in Molecules and Crystals Table 1.10 Radii of isolated atoms (in A)... 

We underline these results and the implied concepts quoting from a comprehensive review on this subject (Simon 1983). We remember indeed that, ever since it was experimentally possible to determine atomic distances in molecules and crystals, efforts have been made to draw conclusions about the nature of the chemical bonding, and to compare interatomic distances (dimensions) in the compounds with those in the chemical elements. Distances between atoms in an element can be measured with high precision. As such, however, they cannot be simply used in predicting interatomic distances in the compounds. In a rational procedure, reference values (atomic radii) have to be extracted from the individual (interatomic distances) measured values. Various functions have been suggested for this purpose. In the specific case of the metals it has been pointed out that interatomic distances depend primarily on the number of ligands and on the number of valence electrons of the atoms (Pearson 1972). 

As a result of the development of the x-ray method of studying the structure of crystals and the band-spectroscopic method and especially the electron-diffraction method of studying gas molecules, a large amount of information about interatomic distances in molecules and crystals has been collected. It has been found that the values of interatomic distances corresponding to covalent bonds can be correlated in a simple way in terms of a set of values of covalent bond radii of atoms, as described below.1... 

It is, of course, impossible to measure the absolute size of an isolated atom its electron cloud extends to infinity. It is possible to calculate the radius within which (say) 95% of its total electron cloud is confined but most measures of atomic/ionic size are based upon experimental measurements of internuclear distances in molecules and crystals. This means that the measurement is dependent on the nature of the bonding in the species concerned, and is a property of the atom or ion under scrutiny in a particular substance or group of substances. This must always be borne in mind in making use of tabulated radii of atoms or ions. The most important dictum to remember is that radii are significant only insofar as they reproduce experimental internuclear distances when added together. The absolute significance of a radius is highly suspect,... 

The range of atomic radii of the different atoms is not very large, still, the differences in sizes are of importance for the fitting of atoms to molecules, and furthermore for the packing of molecules into liquids and crystals, as will be... 

Batsanov SS (2003) Bond radii of atoms in ionic crystals. Russ J Inorg Chem 48 533-536 Donald KJ, Mulder WH, von Szentpdly L (2004) Influence of polarization and bond-chaige on spectroscopic constants of diatomic molecules. J Phys Chem A108 595-606 Fumi FG, Tosi MP (1964) Ionic sizes and born repulsive parameters in the NaQ-type alkati halides. J Phys Chem Solids 25 31-43... 

Waber JT, Cromer DT (1965) Orbital radii of atoms and ions. J Chem Phys 42 4116 123 Batsanov SS (2011) Calculating atomic charges in molecules and crystals by a new electronegativity equalization method. J Mol Struct 1006 223-226... 

Consider first a few of the possible ways of arranging atoms round a point, ignoring crystal structure for the moment and thinking of groups of atoms in isolation. For this purpose we cannot do better than to recall the structures of a few simple molecules and ions. Fig. 131 is a gallery of simple types. (It should be noted that in these drawings the spheres mark the positions of atomic centres the effective external radii of the atoms are much larger.)... 

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],... 

From the early days of structural chemistry there has been considerable interest in discussing bond lengths in terms of radii assigned to the elements, and it has become customary to do this in terms of three sets of radii, applicable to metallic, ionic, and covalent crystals. Distances between non-bonded atoms have been compared with sums of van der Waals radii , assumed to be close to ionic radii. The earliest covalent radii for non-metals were taken as one-half of the M—M distances in molecules or crystals in which M forms % — N bonds N being the number of the Periodic Group), that is, from molecules such as F2, HO-OH, H2N--NH2, P4, Sg, and the crystalline elements of Group IV with the diamond structure. This accounts... 

Its exponential behavior makes this term the dominant one when short atom-atom distances between the interacting molecules are produced. Consequently, this term prevents molecules from getting closer than some limiting distance this is the physical principle behind the hard-sphere model and Kitaigorodsky s close-packing principles [2]. Moreover, the shortest atom-atom distances that one can find in different intermolecular interactions for the same pair of atoms always fall in a restricted range, a fact that allows one to define atomic radii. They differ in ionic and neutral crystals due to the different electronic structure of ionic and neutral species, as easily shown when comparing the contours at 90% probability in electron density maps for isolated atoms and their ions. 

The atkaU-metal fullerides are intercalation compounds with the metal atoms embedded in the octahedral and tetrahedral gaps of the fuUerene crystal. In the face-centered cubic crystal there are two tetrahedral and one octahedral gap per molecule of Ca>, with radii of 1.12 A and 2.06 A, respectively. Resulting from a different occupation of these interstitial sites, four different stractures are conceivable (Figure 2.40a). 

In Nature, atoms are located at different interatomic distances depending on a kind of the forces between them either by cohesion forces or chemical bonds. The latter prevail at the distances which are smaller or equal to the sum of van der Waals radii of atoms. At such distances atoms form a molecule. By definition, the van der Waals (vdW) radii of a given atom is the half of the shortest distance that is observed in crystals between the nuclei of the same atoms. The vdW radii of atoms are listed in Table 1. At the distances beymid the sum of van der Waals radii of atoms, there exists a specific van der Waals interaction often referred to as the dispersion interaction between atoms, after Johannes Diderik van der Waals who first postulated its existence in his well-known equation of state derived in his PhD thesis in 1837 and which won him the 1910 Nobel Prize in Physics. For the first time van der Waals explained the deviations of gases from the ideal behavior. Let us consider a vessel filled by a gas of atoms. Within this vessel, the pressure exerted by a gas of atoms on its wall is lower compared to that predicted by the ideal gas law since the atoms may collide with the wall and are thus retained by the attraction they undergo from the other atoms in the bulk of the gas that results in the pressure P obeying the equation [94],... 

It has been found that the effective sizes of molecules packed together in liquids and crystals can be described by assigning similar van der Waals radii to each atom in the molecule. Values of these radii are given in Table 6-6. 

In making drawings of atoms or molecules, the van der Waals radii may be used in indicating the volume within which the electrons are largely contained. For ions, the ionic radii (crystal radii) discussed in Section 6-10 may be used. The van der Waals radius of an atom and the ionic radius of its negative ion are essentially the same. For example, the van der Waals radius of chlorine is 180 pm, and the ionic radius of the chloride ion is 181 pm. 

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