Band theory bandwidth

It is possible to characterize f-electron states in the actinides in quite a simple manner and to compare them with the states of other transition metal series. To this, we employ some simple concepts from energy band theory. Firstly, it is possible to express the real bandwidth in a simple elose-packed metal as the product of two parts . One factor depends only upon the angular momentum character of the band and the structure of the solid but not upon its scale. Therefore, since we shall use the fee structure throughout, the scaling factor X is known once and for all. 

For an assemblage of two identical molecules spaced d nm apart, the HOMO and LUMO energies split into four levels, each split by 2t eV apart ("dimer splitting") [26] here t is akin to the Hiickel69 resonance integral (i of Section 3.15 Indeed, chemists will remember Eq. (8.6.10) from the simple Hiickel molecular orbital theory for aromatic 7r-electron systems. As the number of molecules N increases, the energy levels become spaced more closely, until they form a quasi-continuous band of bandwidth W, where... 

In organic materials, the weak intermolecular interactions lead to the formation of narrow conduction or valence bands and to low charge carrier mobilities. In some cases the validity of the band theory can be questioned. For the band theory to be valid, in fact, the bandwidth W must be greater than the uncertainty in the energy of the charge carrier,... 

CO. For that matter, in regards to predicting the type of electrical behavior, one has to be careful not to place excessive credence on actual electronic structure calculations that invoke the independent electron approximation. One-electron band theory predicts metallic behavior in all of the transition metal monoxides, although it is only observed in the case of TiO The other oxides, NiO, CoO, MnO, FeO, and VO, are aU insulating, despite the fact that the Fermi level falls in a partially hUed band. In the insulating phases, the Coulomb interaction energy is over 4 eV whereas the bandwidths have been found to be approximately 3 eV, that is, U > W. 

Narrow band electrons (bandwidth < kT) have induced, paramagnetic spin contributions to p. that are temperature dependent, and in the narrowband limit, this temperature dependence approaches that for localized electrons. 11 is not, however, interpretable on the basis of localized-electron theory. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

A possible method for predicting absorption bandwidths of chromogenic molecules or FBAs using PPP-MO theory (section 1.5) has been devised. It is based on the empirical linear relationship stated by the Pestemer rule. Thus theoretical Stokes shifts are computed by the PPP-MO method and related to bandwidths. The requisite MO parameters for various typical absorption bands have been developed for use in these calculations. Reasonable correlation between calculated and experimental half-bandwidth data was found, suggesting that this approach has practical potential in predicting colour tone and brightness intensity [ 19]. 

The large increase of the high-frequency bandwidth is but one challenge for the nuclear dynamics theories of hydrogen bonding, which are the subject of this chapter. Other challenges are the band asymmetry, the puzzling appearance... 

A formalism similar to that presented for actinide metals has been developed for the ground state properties of binary compounds by Andersen et al. leading to a general form of equation of state (see Chap. F). However, this analysis of bonding contributions must draw from detailed results of band calculations more heavily than for the metals case (where the explanation of the qualitative behaviour of ground state properties vs. atomic number needed only the hypothesis of a constant 5f-bandwidth and its volume dependence as predicted by the general theory). In fact, the bond is more complicated ... 

In the Mott-Hubbard theory on the other hand, it is shown that there exists an instability in the narrow-band electronic structure (Peierls instabihty ) and if the bandwidth decreases below a critical value, a sudden transition (Mott transition) takes place toward a complete localized situation. In this approach, it is assumed, in fact, that band magnetism does not exist and one has to deal only with 2 classes of materials... 

Plate height is the constant of proportionality between the variance, cr2, of the band and the distance it has traveled, x. The name came from the theory of distillation in which separation could be performed in discrete stages called plates. Plate height is also called the height equivalent to a theoretical plate. Plate height is approximately the length of column required for one equilibration of solute between mobile and stationary phases. We explore this concept further in Box 23-2. The smaller the plate height, the narrower the bandwidth. 

Since frequencies for EPR spectroscopy are -100 times higher than those for NMR spectroscopy, correlation times (Chapter 3) must be less than 10-9 s if sharp spectra are to be obtained. Sharp bands may sometimes be obtained for solutions, but samples are often frozen to eliminate molecular motion spectra are taken at very low temperatures. For spin labels in lipid bilayers, both the bandwidth and shape are sensitively dependent upon molecular motion, which may be either random or restricted. Computer simulations are often used to match observed band shapes under varying conditions with those predicted by theories of motional broadening of lines. Among the many spin-labeled compounds that have been incorporated into lipid bilayers are the following ... 

However, the available theories have still been restricted to selected parameter orderings. In particular, it has been assumed in theories of exciton transport that the exciton bandwidth is narrower than the phonon bandwidth, and this assumption has been carried over to theories of carrier transport. In fact, carrier bandwidths may well be much larger than phonon bandwidths at low temperatures, becoming smaller than phonon bandwidths as the temperature is raised, owing to polaron band narrowing... 

This effect is also found for the bandwidth of the 0(2p) bands for the alkaline earth metal oxides which, at the LDA level of theory, decrease down the group from a calculated value of 4.44eV (MgO) to 1.83eV (BaO). This is partly due to the increase in lattice parameter, which spaces the O " ions more widely in BaO than in MgO. However, in addition, it is found that the outermost valence electrons for the metal ions interact more strongly with the 0(2p) states in BaO than in MgO, giving more localization of the electron density at O " and so a smaller anion in BaO [50]. 

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