Atomic orbital dimension

The band-structure code, called BAND, also uses STO basis sets with STO fit functions or numerical atomic orbitals. Periodicity can be included in one, two, or three dimensions. No geometry optimization is available for band-structure calculations. The wave function can be decomposed into Mulliken, DOS, PDOS, and COOP plots. Form factors and charge analysis may also be generated. 

Figure 11.13 Hybridization of 5- and p- atomic orbitals, (a) Linear sp hybrid, from one s- and one p-orbital. (b) sp2 hybrid, from one s- and two p-orbitals, with a plane triangular shape, (c) sp3 hybrid, from one 5-orbital and the three p-orbitals, which has a tetrahedral shape in three dimensions.

It was long ago suggested that the low extinction coefficient absorption band of ketones and aldehydes occurring in the 280-360 m i region of the ultraviolet corresponds to an excitation process in which a nonbonding (hence n ) electron is promoted to the antibonding t (i.e., ir ) orbital. This constitutes the n ir process and is conveniently depicted in three dimensions using an atomic orbital representation ... 

It is known that, in the MO framework, the nondynamical electron correlation is accounted for by means of a so-called CASSCF calculation, which is nothing else than a full Cl in a given space of orbitals and electrons, in which the orbitals and the coefficients of the configurations are optimized simultaneously. If the active space includes all the valence orbitals and electrons, then the totality of the nondynamical correlation of the valence electrons is accounted for. In the VB framework, an equivalent VB calculation, defined with pure AOs or purely localized hybrid atomic orbitals (HAOs), would involve all the covalent and ionic structures that may possibly be generated for the molecule at hand. Note that the resulting covalent—ionic VB wave function would have the same dimension as the valence—CASSCF one (e.g., 1764 VB structures for methane, and 1764 MO SCF configurations in the CASSCF framework). 

As you should recall from general chemistry, a favorable equilibrium constant is not sufficient to ensure that a reaction will occur. In addition, the rate of the reaction must be fast enough that the reaction occurs in a reasonable period of time. The reaction rate depends on a number of factors. First, the reactants, in this case the acid and the base, must collide. In this collision the molecules must be oriented properly so that the orbitals that will form the new bond can begin to overlap. The orientation required for the orbitals of the reactants is called the stereoelectronic requirement of the reaction. (,Stereo means dealing with the three dimensions of space.) In the acid-base reaction, the collision must occur so that the atomic orbital of the base that is occupied by the unshared pair of electrons can begin to overlap with the is orbital of the acidic hydrogen. 

In the alkali metals all the electrons may be accommodated within the limits of one zone. This may be shown by comparing the zone in a crystal with a particular atomic orbital. If the interatomic distances in the crystal lattice are assumed to increase, but with the retention of the original symmetry, the zones become narrower and are finally reduced to atomic dimensions. The first zone in a crystal of sodium corresponds to the s orbital and will contain N energy levels, i.e. it can accommodate 2N electrons. But for these there are only N electrons for which Nj2 levels are sufficient for their accommodation. Therefore one half of the levels of the first zone remain unoccupied. The differences in energy between the individual levels of the zone is not great and excitation of the electrons takes place easily in this fact lies the explanation of metallic properties. 

The C-F bond in C F, (HF),j is clearly unusual. The fluorine atoms bound to the carbon cannot withdraw significant electron density from the graphite (T or it-system since the carbon-atom sheets have practically the same dimensions and stretching frequencies as in graphite itself. The bonding molecular orbital composed of fluorine sp and carbon p atomic orbitals is of appropriate symmetry to overlap with the neighbouring carbon p orbitals, and... 

You may need to draw a visualization of atomic orbitals, usually the s, p and d orbitals. This can be simplified by the use of Cartesian co-ordinates which allow a three-dimensional representation on paper. This is neither easy to replicate or often necessary. A simplified approach is, for example, to replace the spherical s orbital with a circle (Fig. 42.7). Similarly, the three p orbitals can be represented in two dimensions by the use of correct labelling of the axes (Fig. 42.7). Finally, the same approach can be replicated on the five d orbitals (Fig. 42.7). It is worthwhile remembering that the d , d,. and d, -orbitals do not reside on the axes (.x, y or z), but in the plane of their respective axes. In addition, the dy2, .2 orbital occupies the. v- and y-axes and the d-2 orbital occupies the z-axis. 

The particle in a box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the three-dimensional Schrodinger equations. The same methods used for the one-dimensional box can be expanded to three dimensions for atoms. These orbital equations include three... 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Electrons Behave as Waves Standing Waves in One and Two Dimensions Standing Waves in Three Dimensions Atomic Orbitals Mixing Atomic Orbitals into Molecular Orbitals Bonding and Antibonding MOs of Hydrogen... 

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