Positions of the Atoms

In 1960 Oka, in the course of a spectroscopic study of foimaldehyde, proposed a new type of structural parameter which he termed r or the zero-point average structure. r% is derived from moments of inertia h. The /z are calculated from the rotational constants of the ground vibrational state (/.e. from the /,) by applying corrections for some of the vibrational effects, rz (and /z) are the structural parameters (and moments of inertia) corresponding to the mean positions of the nuclei, and thus they do have a precisely defined physical significance. If, for example, the molecule has a planar equilibrium structure then the mean positions of the atoms will also lie in a plane althou of course the separations between the mean positions of the atoms are not the same in the ground vibrational state as in the equilibrium structure. Thus the /z for a planar molecule obey the rigid-body relationship, 

The effective moments of inertia 7 do not correspond to any physically defined geometry of the molecule, and because the vibrational averaging is different for all three principal axes 

In fact the effective moments of inertia /g are those of a fictitious rigid rotor which may not, as in the case of a planar molecule, even have the same symmetry as the molecule itself at equilibrium. Oearly, some of the problons of evaluating structural parameters from /g might be removed by working with h. In order to see how 7z differs from 7g and from / , it is necessary to examine the nature of rz. 

Mean Positions of the Atoms.— The discussion follows that of Kuchitsu and Cyvin. For a pair of atoms i and j, a local coordinate system is defined with origin at atom i, z axis in the direction of the equilibrium position of atom y, and jc and y axes arbitrarily chosen normal to the z axis. Then the equilibrium co-ordinates are simply  

When the right-hand side of equation (32) is expanded as a power series correct to terms of the order Az o /re , the result is 

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Knowing the lattice is usually not sufficient to reconstruct the crystal structure. A knowledge of the vectors (a, b, c) does not specify the positions of the atoms within the unit cell. The positions of the atoms withm the unit cell is given by a set of vectors i., = 1, 2, 3... u where n is the number of atoms in the unit cell. The set of vectors, x., is called the basis. For simple elemental structures, the unit cell may contain only one atom. The lattice sites in this case can be chosen to correspond to the atomic sites, and no basis exists. 

These electronic energies dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.T1 for a diatomic molecule. For nonlinear polyatomic molecules having atoms, the energy surfaces depend on 3N - 6 internal coordinates and thus can be very difficult to visualize. In figure B3.T2, a slice tln-oiigh such a surface is shown as a fimction of two of the 3N - 6 internal coordinates. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

Th e vibrational potential in ay be ex pan ded in a Taylor series abon t the etinilibrmni positions of the atoms. 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

The vibrational potential may be expanded in a Taylor series about the equilibrium positions of the atoms. 

Fig. 9.3. An edge dislocation, (a) viewed from a continuum standpoint (i.e. ignoring the atoms) and (b) showing the positions of the atoms near the dislocation.

In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

Each diffracted beam, which is recorded as a spot on the film, is defined by three properties the amplitude, which we can measure from the intensity of the spot the wavelength, which is set by the x-ray source and the phase, which is lost in x-ray experiments (Figure 18.8). We need to know all three properties for all of the diffracted beams to determine the position of the atoms giving rise to the diffracted beams. How do we find the phases of the diffracted beams This is the so-called phase problem in x-ray crystallography. 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

Analysis of the LEED pattern or of spot profiles does not give any quantitative information about the position of the atoms within the surface unit cell. This type of information is hidden in the energy-dependence of the spot intensities, the so-called LEED 7-Vcurves. 

This section displays positioning of the atoms in the molecule used by the program internally, in Cartesian coordinates. This orientation is chosen for maximum calculation efficiency, and corresponds to placing the center of nuclear charge for the molecule at the origin. Most molecular properties are reported with respect to the standard orientation. Note that this orientation usually does not correspond to the one used in the input molecule specification the latter is printed earlier in the output as the Z-matrix orientation. ... 

By examining the positions of the atoms, we can determine what type of displacements would occur when the C=C bond stretches. The carbons in ethylene must both move in the Z direction, since they are situated on the Z-axis, while those in propene must move primarily in the X direction. 

The principal types of tautomerism can be classified according to the nature and the position of the atoms between which the protons move ... 

The precursor of such atomistic studies is a description of atomic interactions or, generally, knowledge of the dependence of the total energy of the system on the positions of the atoms. In principle, this is available in ab-initio total energy calculations based on the loc density functional theory (see, for example, Pettifor and Cottrell 1992). However, for extended defects, such as dislocations and interfaces, such calculations are only feasible when the number of atoms included into the calculation is well below one hundred. Hence, only very special cases can be treated in this framework and, indeed, the bulk of the dislocation and interfacial... 

The diagram at the left of Figure 7.7 emphasizes that the four electron pairs are oriented tet-rahedrally. At the right, the positions of the atoms are shown. Clearly they are not in a straight line the H20 molecule is bent. 

Solid solutions are more rare. Crystals are stable because of the regularity of the positioning of the atoms. A foreign atom interferes with this regularity and hence with the crystal stability. Therefore, as a crystal forms, it tends to exclude foreign atoms. That is why crystallization provides a good method for purification. 

This type of argument leads us to picture a metal as an array of positive ions located at the crystal lattice sites, immersed in a sea of mobile electrons. The idea of a more or less uniform electron sea emphasizes an important difference between metallic bonding and ordinary covalent bonding. In molecular covalent bonds the electrons are localized in a way that fixes the positions of the atoms quite rigidly. We say that the bonds have directional character— the electrons tend to remain concentrated in certain regions of space. In contrast, the valence electrons in a metal are spread almost uniformly throughout the crystal, so the metallic bond does not exert the directional influence of the ordinary covalent bond. 

In Section 18-6.3 the composition of proteins was given. They are large, amide-linked polymers of amino acids. However, the long chain formula (Figure 18-14, p. 348) does not represent all that is known about the structure of proteins. It shows the covalent structure properly but does not indicate the relative positions of the atoms in space. 

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