Frontier band

Fig. 2-4. Lattice potential energy and electron energy bands in crystals IB s inner band FB s frontier band.

Electron occupation in the frontier bands of metal crystals varies with different metals as shown in Fig. 2-7. For metallic iron the frontier bands consist of hybridized 4s-3d-4p orbitals, in which 4s and 3d are partially occupied by electrons but 4p is vacant for electrons. Figure 2-8 shows the electron state density curve of metallic iron, where the 3d and 4s bands are partially filled with electrons. Electrons in metals occupy the energy states in a frontier band successively fix>m the lower band edge level to the Fermi level, leaving the higher levels vacant. 

Fig. 2-8. State density distribution curve of 3d and 4s frontier bands partially occupied by electrons in metallic iron [From Fiyita, 1996.]...

As Figure 2 illustrates, the distinction between semiconductors and insulators is rather arbitrary and resides with the magnitude of the energy band-gap [E between the filled and vacant bands. Semiconductors typically have E in the 1-4 eV range (Table 1). The vacant frontier band is termed a conduction band, CB (Figure 2). We shall see later that E has an important bearing on the optical response of a semiconductor. 

We normally define the energy level of electrons in a solid in terms of the Fermi level, eF, which is essentially equivalent to the electrochemical potential of electrons in the solid. In the case of metals, the Fermi level is equal to the highest occupied level of electrons in the partially filled frontier band. In the case of semiconductors of covalent and ionic solids, by contrast, the Fermi level is situated within the band gap where no electron levels are available except for localized ones. A semiconductor is either n-type or p-type, depending on its impurities and lattice defects. For n-type semiconductors, the Fermi level is located close to the conduction band edge, while it is located close to the valence band edge for p-type semiconductors. For examples, a zinc oxide containing indium as donor impurities is an n-type semiconductor, and a nickel oxide containing nickel ion vacancies, which accept electrons, makes a p-type semiconductor. In semiconductors, impurities and lattice defects that donate electrons introduce freely mobile electrons in the conduction band, and those that accept electrons leave mobile holes (electron vacancies) in the valence band. Both the conduction band electrons and the valence band holes contribute to electronic conduction in semiconductors. 

Unlike in molecular systems, semiconductor energy levels are so dense that they form, instead of discrete molecular orbital energy levels, broad energy bands. Consider a solid composed of N atoms. Its frontier band will have IN energy eigenstates, each with an occupancy of two electrons of paired spin. Thus, a solid having atoms with odd number of valence... 

Figure 6 a) Calculated band structure of PPYV and b) pattern of the frontier band of PPYV at fc=l. i) conduction band, ii) valence band and iii) /t>band. 

If the perturbations thus caused are relatively slight, the accepted perturbation theory can be used to interpret observed spectral changes (3,10,39). The spectral effect is calculated as the difference of the long-wavelength band positions for the perturbed and the initial dyes. In a general form, the band maximum shift, AX, can be derived from equation 4 analogous to the weU-known Hammett equation. Here p is a characteristic of an unperturbed molecule, eg, the electron density or bond order change on excitation or the difference between the frontier level and the level of the substitution. The other parameter. O, is an estimate of the perturbation. 

Electron Level Position. One essential condition of spectral sensitization by electron transfer is that the LUMO of the dye be positioned above the bottom of the conduction band, eg, > —3.23 eV in AgBr or > —4.25 eV in ZnO (108). To provide the desired frontier level position respectively to the valence and conduction bands of the semiconductor, it is necessary to use a polymethine with suitable electron-donor abiHty (Pq. Increasing the parameter (Pq leads to the frontier level shift up, and vice versa. Chain lengthening is known to be accompanied by a decrease of LUMO energy and hence by a decrease of sensitization properties. As a result, it is necessary to use dyes with high electron-donor abiHty for sensitization in the near-ir. The desired value of (Pq can be provided by end groups with the needed topological index Oq or suitable substituents (112). 

Molecules have some occupied and some unoccupied orbitals. There occur diverse interactions (Scheme 1) when molecules undergo reactions. According to the frontier orbital theory (Sect 3 in Chapter Elements of a Chemical Orbital Theory by Inagaki in this volume), the HOMO d) of an electron donor (D) and the LUMO (fl ) of an electron acceptor (A) play a predominant role in the chemical reactions (delocalization band in Scheme 2). The electron configuration D A where one electron transfers from dio a significantly mixes into the ground configuration DA where... 

With the power of the donors and acceptors, changes occur in the important frontier orbital interactions (Scheme 2) and in the mechanism of chemical reactions. The continuous change forms a mechanistic spectrum composed of the delocalization band to pseudoexcitation band to the electron transfer band. 

According to the calculations at high levels of theory, the [4+2] cycloaddition reactions of dienes with the singlet ( A oxygen follow stepwise pathways [37, 38], These results, which were unexpected from the Woodward-Hoffmann rule and the frontier orbital theory, suggest that the [4+2] cycloadditions of the singlet ( A oxygen could be the reactions in the pseudoexcitation band. 

The reactions in this band are controlled by the frontier orbital interactions (Sect... 

Chemical bonds are defined by their frontier orbitals. That is, by the highest molecular orbital that is occupied by electrons (HOMO), and the lowest unoccupied molecular orbital (LUMO). These are analogous with the top of the valence band and the bottom of the conduction band in electron band theory. However, since kinks are localized and non-periodic, band theory is not appropriate for this discussion. 

In addition to shape selectivity, which is primarily a steric directing effect, orbital confinement,5 a quantum6 electronic7 effect, can also dramatically influence reactivity in nanospace.8 10 This concept, first introduced by Corma and coworkers,8 points out that when molecular orbitals are confined and not allowed to extend over all space that their energies increase. The HOMO is more sensitive than the LUMO to size restrictions resulting in a decrease of the frontier molecular orbital band gap. This effect can be experimentally demonstrated in systems where the size of the... 

Fig. 2-12. Electron energy band formation of silicon crystals from atomic frontier orbitals number of silicon atoms in crystal r = distance between atoms rg = stable atom-atom distance in crystals, sp B8 = bonding band (valence band) of sp hybrid orbitals sp ABB = antibonding band (conduction band) of sp hybrid orbitals.

Most metal oxides are ionic crystals and belong to either the class of semiconductors or insulators, in which the valence band mainly comprises the frontier orbitals of oxide ions and the conduction band contains the frontier orbitals of metal ions. In forming an ionic metal oxide ciTstal from metal ions and oxide ions, as shown in Fig. 2-21, the crystalline field shifts the frontier electron level of metal ions to higher energies to form an antibonding band (the conduction... 

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