Angular shape

Cartesian Gaussian-type orbitals (GTOs) Jfa.i.f( ( characterized by the quantum numbers a, b and c, which detail the angular shape and direction of the orbital, and the exponent a which governs the radial size . 

For two and three dimensions, it provides a erude but useful pieture for eleetronie states on surfaees or in erystals, respeetively. Free motion within a spherieal volume gives rise to eigenfunetions that are used in nuelear physies to deseribe the motions of neutrons and protons in nuelei. In the so-ealled shell model of nuelei, the neutrons and protons fill separate s, p, d, ete orbitals with eaeh type of nueleon foreed to obey the Pauli prineiple. These orbitals are not the same in their radial shapes as the s, p, d, ete orbitals of atoms beeause, in atoms, there is an additional radial potential V(r) = -Ze /r present. However, their angular shapes are the same as in atomie strueture beeause, in both eases, the potential is independent of 0 and (j). This same spherieal box model has been used to deseribe the orbitals of valenee eleetrons in elusters of mono-valent metal atoms sueh as Csn, Cun, Nan and their positive and negative ions. Beeause of the metallie nature of these speeies, their valenee eleetrons are suffieiently deloealized to render this simple model rather effeetive (see T. P. Martin, T. Bergmann, H. Gohlieh, and T. Lange, J. Phys. Chem. 6421 (1991)). 

The factor iji < 1 must be determined experimentally for particles of interest. Typical values are iji = 0.77 for particles of rounded shape iji = 0.66 for particles of angular shape iji = 0.43 for particles of a flaky geometry. 

Using the previous equations and the calculated P/t- ratios, we can determine the angular shapes of the /detection( d) distribution for m J = 0, 1, or 2. The resulting distributions are shown in Fig. 4. Note that since the ratios of V k were used, the absolute scale is arbitrary only the relative shapes and magnitudes are important. We then fit the angular distribution data to a sum of these functions multiplied by the /dissociation term to obtain the relative m,j populations and the / value ... 

These detectivity functions were derived for each magnetic sublevel by setting the population of one sublevel equal to f while setting the others to zero in Eq. (12), and then solving for the angular shape /detection( d) by transforming the multipoles to the probe frame and summing over K. 

Figure 1.34 An approximately tetrahedral arrangement of the electron pairs for a molecule of water that results when the pair of nonbonding electrons are considered to occupy corners. This arrangement accounts for the angular shape of the H20 molecule.

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

Still another aspect of the Li and F valence orbitals is modified by ionic-bond formation. In an isolated ionic or neutral species, each NAO retains the characteristic angular shape of the pure s and p hydrogenic orbitals shown in Fig. 1.1, reflecting the full rotational symmetry of atoms. However, in the presence of another atom or ion this symmetry is broken, and the optimal valence orbitals acquire sp hybrid form (assumed for simplicity to include only valence s and p orbitals), as represented mathematically by... 

Chia and Simmons388 calculated the resonance energies (ER)20 of four mono- and dibenzotetraazapentalenes (Scheme 24). Values are comparable with those of o-condensed aromatic systems (naphthacene, ER = 110 kcal mol-1 chrysene, ER = 116.5 kcal mol-1), and, like these carbocyclic systems, angularly-shaped molecules are more stable than linear ones. HMO calculations of delocalization energies (DE) show that the tetraazapentalene structure 15 is more stable than the tetra-azacyclooctatetraene valence isomer 324 (Scheme 14, Section IV,B,2) whether 324 is planar or tub-shaped. Calculations of electrophilic reactivity (Section IV,C,4,d), electronic spectra (by the PPP method employing all singly excited configurations), and bond orders have been carried out, and they confirm the aromatic nature of these systems. 

The connection between these associated Legendre polynomials and the angular shape of certain real atomic orbitals px, pz, dxz, dyz, dxz, dxi yZ, and dz2 is explicitly made between "quotes " the connection is exact for m 0, but inexact for m O, where the ( -dependence of exp( imq>) gets involved. From the generating function... 

An alternative approach would generate precursor macromolecules with some disorder of the rigid geometry. A possible way of achieving this is the generation of polymers that have structural units which cause an angular shape of the double-stranded chain. Such precursor polymers then have to be converted into the final ladder macromolecules by means of an elimination or rearrangement step [2, 5]. 

Tyle (1993) studied oral perception of grittiness and viscosity of synthetic particles (60 mg) suspended in fruit syrups (5 mL). The particles were of different size distributions and shapes angular-shaped garnet 5.2-33.0, rounded micronized polyethylene 7.2-68.9, and flat mica platelets coated with titanium dioxide 28.1-79.6 p,m. There was no effect on the thickness ratings of the studied particulate dispersions. Particles... 

We thus obtain an immediate explanation for the angular shape of CI2O, which is based on a tetrahedral arrangement of four electron pairs in the valency shell of the oxygen atom and for the shape of SF4, which is based on a trigonal bipyramidal arrangement of five electron pairs. 

Table 5.3 Theoretical angular shapes from calculated electrostatic interaction expanded to R 6 versus experimental results for some H-bonded VdW dimers...

These rules are simple and immediately intuitive, once the electrical characterization of a molecule in terms of its point-like multipoles is accepted. The underlying physical assumption is that the electrostatic interaction is the dominant attractive component of the intermolecular potential determining the angular shape of the dimer, while short-range forces are assumed to provide a repulsive uniform background balancing attraction at the VdW minimum. Monomer size enters the model through rule 7, which corrects for deviation from uniform repulsion when steric interactions occur below the sum of the respective VdW radii.10... 

Provided this situation is avoided, the exact value of the intermolecular distance R is not particularly relevant for the qualitative prediction of the angular shapes of the dimers. 

Figure 5.9 Classification of the first few multipole interactions and stability of the resultant angular geometries of some dimers. Reprinted from Journal of Molecular Structure THEOCHEM, 204, Magnasco, V. et al., A model for the elementary prediction of the angular shape ofVander Waals dimers. 229-246, Copyright (1990), with permission from Elsevier...

Magnasco, V., Costa, C., and Figari, G. (1989b) On the angular shape of van der Waals dimers of small polar molecules. Chem. Phys. Lett., 160, 469 178. 

Compounds of non-transition elements containing odd numbers of electrons are few in number, but they can be included in the present scheme since an odd electron, like an electron pair, occupies an orbital. Thus a 17-electron system has the same angular shape as an 18-electron one, as described later. 

The expected angular shape of (IR2) has been demonstrated in crystalline (C6Hs)2l (ICl2) , the angle between the iodine bonds in the cation being 94° and... 

---