Band width 1414 theory

Further along the series, we saw that stoichiometric MnO, FeO, CoO, and NiO are insulators. This situation is not easily described by band theory because the d orbitals are now too contracted to overlap much, typical band widths are 1 eV, and the overlap is not sufficient to overcome the localizing influence of interelectronic repulsions. (It is this localization of the d electrons on the atoms that gives rise to the magnetic properties of these compounds that are discussed in Chapter 9.)... 

The volume dependence of the d band width is expected from a simple approximation within resonant scattering theory (Heine (1967)) to behave as... 

These various consequences parallel closely the analogous ones of the fluctuation and frequency modulation theories. There is, however, one important point of difference between the classical and quantum viewpoints which does not seem to have been emphasized previously, namely that transitions from the lowest level of the ground state can occur to several levels of the upper curve. This means that even at very low temperatures, when all the molecules are initially in this lowest energy level, a band of considerable breadth with frequencies rXH + m>(XH Y) will still persist. The temperature independent residual band width is a direct result of the perturbations of the system (in particular the finite change in the distance rxymin) caused by the absorption of a large quantum of radiation of frequency vXH. The same type of explanation may apply to other vibrational bands which remain of finite width at low temperatures the occurrence of such bands have been the cause of considerable discussion [34]. 

The structure of this review is as follows. In Chap. 2 we will survey shortly and nonmathematically the theories in use to explain spectral band width and structure. In subsequent chapters we will deal with a couple of different metal... 

VIII. Factors Governing Line-Widths and Shapes of Bands The theory of line-widths and shapes, other than those resulting from instrumental limitations such as magnetic field inhomogeneities, is complicated and will not be discussed here. The purpose of this short section is simply to describe qualitatively some of the factors involved and to indicate that this field is of importance to the chemist and may well influence his results adversely unless due care is taken. Also the study of changes in the line-widths and shapes may well give information of considerable use to the investigator. 

Notice that the gap vanishes for a homopolar semiconductor, which is true also for the exact bands, and if V were equal to V , it would simply be equal to times the predicted band width, p4K,. Thus, qualitatively, the gap is in very simple correspondence with the polarity of the system. The observed splittings are from 30 percent to 45 percent lower than those predicted in the K,-only theory by Eq. (6-12). The value is not modified as we add additional matrix elements within the Bond Orbital Approximation (Pantelides and Harrison, 1975). P rom Eqs. (6-3) and (6-4) we see that the situation is greatly complicated if the Bond Orbital Approximation is not u.scd (that is, bonding antibonding matrix elements are added), though of course the predicted gaps do go to zero as the polarity goes to zero in any case. 

Knergy bands for the transition metals are constructed, using a minimal basis set of aiomic orbitals. The eleven parameters required are reduced to two, the d-band width H, and its position relative (o the. s-band minimum, using Muflin-Tin Orbital theory. Relations giving W, and all interatomic matrix elements in terms of a d-statc radius r,i and the intcrmicicar distance are listed in the Solid State Table, along with values of r, and E,t for all of the transition elements this makes possible elementary calculations of the bands for any transition metal, at any atomic volume. 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width W, to the resonance width T, and is then reformulated again in terms of iransilion-metal pseudopotential theory, in which the hybridization between the frce-electron states and the d state is treated in perturbation theory, The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the frce-electron effective mass in terms of it. Thus it provides all of the parameters for the L.CAO theory, as well as a means of direct calcidation of many properties, as was possible in the simple metals. ... 

Electronic band structures were calculated for several polymeric chains structurally analogous to polyacetylene (-CH-CH) and carbyne (-CbC). Ihe present calculations use the Extended Huckel molecular orbital theory within the tight binding approximation, and values of the calculated band gaps E and band widths BW were used to assess the potential applic ilitf of these materials as electrical semiconductors. Substitution of F or Cl atoms for H atoms in polyacetylene tended to decrease both the E and BW values (relative to that for polyacetylene). Rotation about rhe backbone bonds in the chains away from the planar conformations led to sharp increases in E and decreases in BW. Substitution of -SiH or -Si(CH,) groups for H in polyacetylene invaribly led to an increase in E and a decrease in BW, as was generally the case for insertion of Y ... 

One of the interesting aspects of the correlation of v and Ava for the fundamental mode is its applicability to a wide variety of H bonded systems and its inapplicability to the m/ramolecularly H bonded compounds, salicylaldehyde and methylsalicylate (2054) (indicated in Fig. 3-13 by squares). The linear relation offers a critical test of theories of the band width, and possibly more experience will show that it is a criterion for distinguishing inter- and intramo tc x x H bonds. 

Band Width. The structure of has been discussed at length in Chapter 3 (Section 3.3.8). As indicated there, one of the currently popular theories is based on the assumption of extreme mechanical anharmonicity of Vi (268, 724, 1). The theory is well described (268), and will not be reproduced here in view of the experimental evidence concerning the anharmonicity presented in Section 3.3.9. Of particular interest are the remarks in that section referring to the absence of odd power terms in the potential function for the IR-active Vg mode of a cyclic carboxylic acid dimer. The implication which can be drawn is that the anharmonicity of this vibration may be quite low. Since an explanation of the extreme band-widths must encompass the carboxylic acids (see Fig. 3-20), anharmonicity does not seem to be a likely one. The weight of evidence suggests that Vg is not unusually anharmonic. 

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