Atomic radii of atoms

The Angstrom (A) is a unit of length, 1 X lO t m, that provides a convenient scale on which to express the radii of atoms. Radii of atoms are often expressed in nanometers. The radius of a phosphorus atom is 1.10 A. What is the distance expressed in centimeters and nanometers ... 

How do the atomic radii of atoms change across the table ... 

The atomic radii of atoms generally decrease going from left to right across a period, and increase going top to bottom down a group (see graphic on next page). 

Consider the A2X4 molecule depicted here, where A and X are elements. The A—A bond length in this molecule is dj, and the four A—X bond lengths are each d2- (a) In terms of di and d2, how could you define the bonding atomic radii of atoms A and X (b) In terms of d and d2> what would you predict for the X—X bond length of an X2 molecule [Section 7.3]... 

Many other attempts have been made to quantify electronegativity. For example, Allred and Rochow developed a scale based directly on the atomic radii of atoms in molecules. They argued that the electronegativity should be proportional to the electric field at an atom s surface within the molecule. The position of this atomic surface was estimated using covalent radii Rc, and the electric field at this distance then depends on the effective nuclear charge Zeff of the atom. In their scheme, the electronegativity xar is calculated via... 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

The solvent dielectric constant, ionic strength and temperature are chosen to fit the conditions of the experimental studies. The protein dielectric constant is assigned some small value, e.g. 4. The PB calculations are currently carried out with the atomic charges and radii of the PARSE parameter set, developed by Honig and coworkers [17] or that for CHARMM [12]. The PARSE parameter set... 

Non-covalent interactions between molecules often occur at separations where the van der Waals radii of the atoms are just touching and so it is often most useful to examine the electrostatic potential in this region. For this reason, the electrostatic potential is often calculated at the molecular surface (defined in Section 1.5) or the equivalent isodensity surface as shown in Figure 2.18 (colour plate section). Such pictorial representations... 

A yet more realistic cavity shape is that obtained from the van der Waals radii of the atoms of the solute. This is the approach taken in the polarisable continuum method (PCM) [Miertus et al. 1981], which has been implemented in a variety of ab initio and semi-empirical quantu/rt mechanical programs. Due to the non-analytical nature of the cavity shapes in the PCM approach, it is necessary to calculate numerically. The cavity surface is divided... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

Table 4.6 Atom Radii and Effective Ionic Radii of Elements...

In Section 4 the data on bond lengths and strengths have been vastly increased so as to include not only the atomic and effective ionic radii of elements and the covalent radii for atoms, but also the bond lengths between carbon and other elements and between elements other than carbon. All... 

The strain energies of these five-membered heterocycles are relatively small with values of 23.5, 24.8 and S.SkJmoF estimated for tetrahydrofuran, pyrrolidine and tetrahy-drothiophene respectively (74PMH(6)199). The closeness of the values for the two former compounds reflects the almost identical covalent radii of oxygen (0.66 A) and nitrogen (0.70 A) atoms. The sulfur atom with a much larger covalent radius of 1.04 A causes a... 

Figure 4.17 Schematic diagram of bound tyrosine to tyrosyl-tRNA synthetase. Colored regions correspond to van der Waals radii of atoms within a layer of the structure through the tyrosine ring. Red is bound tyrosine green is the end of P strand 2 and the beginning of the following loop region yellow is the loop region 189-192 and brown is part of the a helix in loop region 173-177.

Table 3.1. Van der Waals Radii of Several Atoms and Gronps (A) ...

Chapter 30). Similar plots are obtained for the atomic and ionic radii of the elements and an inverted diagram is obtained, as expected, for the densities of the elements in the solid state (Fig. 2.2). 

The effect of the lanthanide contraction on the metal and ionic radii of hafnium has already been mentioned. That these radii are virtually identical for zirconium and hafnium has the result that the ratio of their densities, like that of their atomic weights, is very close to Zr Hf = 1 2.0. Indeed, the densities, the transition temperatures and the neutron-absorbing abilities are the only common properties of these two elements which differ... 

Most of the present implementations of the CPA on the ab-initio level, both for bulk and surface cases, assume a lattice occupied by atoms with equal radii of Wigner-Seitz (or muffin-tin) spheres. The effect of charge transfer which can seriously influence the alloy energetics is often neglected. Several methods were proposed to account for charge transfer effects in bulk alloys, e.g., the so-called correlated CPA , or the screened-impurity model . The application of these methods to alloy surfaces seems to be rather complicated. 

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

The atomic radii of the main-group elements are shown at the top of Figure 6.13 (p. 153). Notice that, in general, atomic radii—... 

Atomic and ionic radii of the main-group elements. Negative ions are always larger than atoms of the same element, whereas positive ions are always smaller than atoms of the same element. ... 

The radii of cations and anions derived from atoms of the main-group elements are shown at the bottom of Figure 6.13. The trends referred to previously for atomic radii are dearly visible with ionic radius as well. Notice, for example, that ionic radius increases moving down a group in the periodic table. Moreover the radii of both cations (left) and anions (right) decrease from left to right across a period. 

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