Orbital crystal

In many crystals there is sufficient overlap of atomic orbitals of adjacent atoms so that each group of a given quantum state can be treated as a crystal orbital or band. Such crystals will be electrically conducting if they have a partly filled band but if the bands are all either full or empty, the conductivity will be small. Metal oxides constitute an example of this type of crystal if exactly stoichiometric, all bands are either full or empty, and there is little electrical conductivity. If, however, some excess metal is present in an oxide, it will furnish electrons to an empty band formed of the 3s or 3p orbitals of the oxygen ions, thus giving electrical conductivity. An example is ZnO, which ordinarily has excess zinc in it. 

Another question is whether the filled orbitals are of a bonding or antibonding character. This is displayed on a crystal orbital overlap population (COOP) plot as shown in Figure 34.3. Typically, the positive bonding region is plotted to the right of the zero line. 

FIGURE 34.3 Crystal orbital overlap plot for CoNb4Si. 

COOP (crystal orbital overlap population) a plot analogous to population analysis for band-structure calculations... 

Appearance of the metallic structure of CNT is based on the crossing of the highest occupied (HO) and the lowest unoccupied (LU) bands (see, e.g.. Fig. 3), each accompanying pseudo rt-type crystal orbital. Note that pseudo n-type orbital, particularly when all the valence atomic orbitals (AO) are taken into consideration, implies that its main AO component is normal to the cylindrical CNT surface. The band crossing mentioned above is possible when these two... 

By making the approximation of setting matrix B to zero and A to Ao, the resolvent matrix becomes diagonal, every coupling between the crystal orbitals disappears and the polarizability reads ... 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

BAC-MP4 method, description, 344-346 Band theory, crystal orbital method, 149 Be clusters, structures, 24-25 Bifurcation... 

Fig. 9.10 CASTER densities of states for LigZn2Ge3. Nonbonding electron density distribution over selected bunches of crystal orbitals in the valence-band dispersion (a) six orbitals between -2.4 and -0.7 eV, (b) four orbitals ranging from -3.3 to -1.9 eV.

Fig. 10.2 Crystal Orbital Overlap Population (COOP) and Densities of States (DOS) plots for SrCa2ln2Ce (a) COOP plots of the In-In (solid) and In-Ge (dashed) interactions (b) DOS plots of the total DOS (dotted), ln-5py lone pair (dashed), and ln-5px p-states (solid).

The "Crystal Orbital Overlap Population" (COOP) [20] shows (Fig. 4) that all levels arising below the Fermi level are a and Jt bonding and the highest energy levels are ct and n antibonding however the specific COOP curves for each Mo-0 distance (Fig. 5) show a... 

Top DOS contributions of the different bands of a PtX - chain and their superposition to give the total density of states. Bottom COOP contributions of the different bands and their superposition to give the crystal orbital overlap population... 

Schematic sketch of the density of states and the crystal orbital overlap population for metals...

The different shift mechanisms may be understood in more detail by considering the effect of the magnetic field on the populations and energies of the different crystal orbitals (Figure 7a). Transfer of electron density via the 90° interaction arises due to a direct delocalization of spin density due to overlap between the half-filled tzg. oxygen jt, and empty Li 2s atomic orbitals (the delocalization mechanism. Figure 7b).This overlap is responsible for the formation of the tzg (antibonding) molecular orbital in a molecule or the tzg crystal orbital (or band) in a solid. No shift occurs for the 180° interaction from this mechanism as the eg orbitals are empty. 

A second mechanism (the polarization mechanism) arises due to the polarization of the fully occupied (bonding) crystal orbitals formed by the eg. oxygen 2p. and Li 2s atomic orbitals in the presence of a magnetic field. A fully occupied crystal (or molecular) orbital in reality comprises one one-electron orbital occupied by a spin-up electron and a second one-... 

The maximum number of localized states which can be formed is two. This result depends on our assumptions that only one orbital on the foreign atom and only one band of crystal orbitals are in interaction and that the perturbation of the crystal by the foreign atom does not extend beyond the first crystal atom. If we extend the perturbation (i.e., modify the Coulomb integrals) to the first and second crystal atoms, we find a maximum of three localized states. In general, the maximum number of localized states... 

We further proceed, as previously, by building a one-electron wave function from the associated atomic orbitals, Eq. (1.14), but imposing the Bloch condition expressed in Eq. (1.25). This is the well-known TB approximahon (Ashcroft Mermin, 1976). The resulting wave funchon is often called the crystal orbital and has the form ... 

Just as for the cyclic group each irreducible representation is represented once in the collection of molecular (now called crystal) orbitals, and there are now a very large number of them corresponding to the number of atoms in the crystal. For the translation group however, the characters take on a particularly simple form. 

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