Atom-Centered Approximations

Substitution of Xq for X in (19.14) gives the ZORA equation in the modified Dirac representation 

The expression in square brackets is the matrix of 2mc — V evaluated over the set of basis functions ((a p)/2mc)x/i, which is essentially the small-component basis set. Likewise, T can be regarded as the integral over (or p) between the small- and large-component basis set. This makes it clear that the denominator in ZORA must be evaluated over the small-component basis functions. 

We have discussed ZORA and lORA at length in chapter 18, and will not pursue these approximations here. 

It is even possible to define a free-particle transformation from which we could continue to develop a matrix Douglas-Kroll-Hess or Barysz-Sadlej-Snijders approximation. But with the matrix formalism we now have available, there is the opportunity to take a different approach, one that centers around the properties of the matrix X. 

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LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Expresses the Molecular Orbitals by linear combinations of atom-centered functions (Atomic Orbitals). 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

FIG. 2. A complex of twenty Friauf polyhedra, with icosahedral symmetry (Samson, Ref. 23). This complex contains 104 atoms, if the central icosahedral position is not occupied. Most of the atoms show approximate icosahedral ligan-cy twenty atoms, at the centers of the Friauf polyhedra, have ligancy 15 or 16. The complex was first identified in Mg32(Al,Zn)<9. In the cubic crystals that form the icosatwins and decatwins these complexes are packed in such a way as to approximate an icosahedral arrangement of twelve complexes about a central one, the structure being similar to that of 0-W. 

The structure of iodine at four different pressures. The outlined face-centered unit cell in the 30-Gpa figure corresponds to that of a (distorted) cubic closest-packing of spheres. At 24.6 GPa four unit cells of the face-centered approximant structure are shown the structure is incommensurately modulated, the atomic positions follow a sine wave with a wave length of 3.89 x c. The amplitude of the wave is exaggerated by a factor of two. Lower left Dependence of the twelve interatomic contact distances on pressure... 

Cieplak, P., Caldwell, J. W., Kollman, P. A., Molecular mechanical models for organic and biological systems going beyond the atom centered two body additive approximation aqueous solution free energies of methanol and IV-methyl acetamide, nucleic acid base, and amide hydrogen bonding and chloroform/water partition coefficients of the nucleic acid bases, J. Comput. Chem. 2001, 22, 1048-1057... 

The central approximation, in terms of reducing overall computation, is to insist that atomic orbitals residing on different atomic centers do not overlap. 

Fig. 10-5.—The atomic arrangement in e molecule of decaborane, Bi0Hu. Note that the 10 boron atoms are approximately at corners of a regular icosahedron. Each of the two remaining corners of the icosahedron may be considered to have been replaced by two bridging hydrogen atoms. The other 12 hydrogen atoms are bonded to boron atoms in such a way that the B—H bonds extend out from the center of the icosahedron.

While the harmonic oscillator is a good approximation to the behavior of a molecule in the lower vibrational energy states, marked deviations occur at higher energies. At the lower energy levels the change in the distance between the atomic centers during the... 

Before beginning a discussion of nuclei and their properties, we need to understand the environment in which most nuclei exist, that is, in the center of atoms. In elementary chemistry, we learn that the atom is the smallest unit a chemical element can be divided into that retains its chemical properties. As we know from our study of chemistry, the radii of atoms are approximately 1-5 x 10-10 m, or 1 -5 A. At the center of each atom we find the nucleus, a small object (r 1-10 x 10-15 m) that contains almost all the mass of the atom (Fig. 1.1). The atomic nucleus contains Z protons, where Z is the atomic number of the element under study, Z being number of protons and is thus the number of positive charges in the nucleus. The chemistry of the element is controlled by Z in that all nuclei with the same Z will have similar chemical behavior. The nucleus also contains N neutrons, where N is the neutron number. Neutrons are uncharged particles with masses approximately equal to the mass of a proton ( 1 u). Each proton has a positive charge equal to that of an electron. The overall charge of a nucleus is +Z electronic charge units. 

The degree of short-range order in an amorphous material can be characterized by a hard sphere model if the basic structure of an amorphous material is approximated by spheres. The density of packing of atoms around a reference atom is described by the number of atom centers per volume that lie in a spherical shell of thickness, dr, and radius about the reference atom. In a hard sphere model, the number, n, of neighboring spheres with centers between r and dr is measured as a function of r. 

In Equation 6.31, A and B indicate atomic centers, and SAB is the Kronecker function, yielding 1 for A = B, and 0 for A 15. The NDDO approximation implies that (with orthonormal AOs) the overlap matrix S l v reduces to the unit matrix, and that all two-electron integrals with charge clouds arising from the overlap of AOs from two different atomic centers are ignored ... 

In the NDDO approximation, the second sum in Equation 6.51 vanishes (zero diatomic overlap between AOs from different atomic centers and orthogonality of different AOs at the same site), and thus the total number of electrons in a closed-shell system with n/2 doubly occupied MOs is... 

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