Free-electron bands corrections

All carbon-carbon bonds in the skeleton have 50% double bond character. This fact was later confirmed by X-ray diffraction studies. A simple free-electron model calculation shows that there is no energy gap between the valence and conduction bands and that the limit of the first UV-visible transition for an infinite chain is zero. Thus a simple free-electron model correctly reproduces the first UV transition with a metallic extrapolation for the infinite system. Conversely, in the polyene series, CH2=CH-(CH=CH) -CH=CH2, he had to disturb the constant potential using a sinusoidal potential in order to cover the experimental trends. The role of the sinusoidal potential is to take into account the structural bond alternation between bond lengths of single- and double-bond character. When applied to the infinite system, in this type of disturbed free-electron model or Hiickel-type theory, a non-zero energy gap is obtained (about 1.90 eV in Kuhn s calculation), as illustrated in Fig. 36.9. 

It is important to recognize that there is no flexibility in the choice of Z. There are no corrections to the integral values, and even though, for example, lead may form compounds as a doubly charged ion (the ten-electron compounds mentioned earlier), the true bands in the metal resemble the free-electron bands for Z = 4, not for Z = 2. On the other hand, the values of kp given in Table 15-1 do not really warrant the precision indicated there they depend upon pressure and temperature, and where real accuracy is required, they should be redetermined for the circumstances at hand, from Eq. (15-4). 

Similar to the failures of the free-electron model of metals (Ashcroft and Mermin, 1985, Chapter 3), the fundamental deficiency of the jellium model consists in its total neglect of the atomic structure of the solids. Furthermore, because the jellium model does not have band structure, it does not support the concept of surface states. Regarding STM, the jellium model predicts the correct surface potential (the image force), and is useful for interpreting the distance dependence of tunneling current. However, it is inapplicable for describing STM images with atomic resolution. 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

V-UV Application First Excited State of Linear Polyenes. The first electronic absorption band of perfect linear aromatic polyenes (CH)X, or perfect polyacetylene shifts to the red (to lower energies) as the molecule becomes longer, and the bond length alternation (BLA) would be zero. This was discussed as the free-electron molecular orbital theory (FEMO) in Section 3.3. If this particle-in-a-box analysis were correct, then as x > oo, the energy-level difference between ground and first excited state would go to zero. This does not happen, however first, because BLA V 0, next, because these linear polyenes do not remain linear, but are distorted from planarity and linearity for x > 6. 

In the so-called semiconductors, such as ZnO, Cu20, etc., the bands are just filled for the perfectly pure substance at low temperatures. Conduction can only occur if the number of electrons is increased (excess conduction or N-type semiconductors, Fig. 28E), which extra electrons find a place in a free band, or if the number of electrons is decreased, whereby a hole is produced in the filled band (defect conduction or P-type semiconductors, Fig. 28F). Such a deficit is displaced in an electric field like an electron with a positive charge. Such a change in the number of electrons, more correctly in the number of electrons per lattice unit, is produced by deviations from the stoichiometric composition. 

Tlie bands have long been interpreted as LCAO d bands, crossed by and hybridized with a free-electron-like band (Saffren, 1960 Hodges and Ehrenreich, 1965 and Mueller, 1967). By including some eleven parameters (pseudopotential matrix elements, interatomic matrix elements, orthogonality corrections, and hybridization parameters), it is possible to reproduce the known bands very accurately. We shall also make an LCAO analysis of the bands but shall take advantage of recent theoretical developments to reduce the number of independent parameters to two for each metal, each of which can be obtained for any metal from the Solid State Table. These two parameters will also provide the basis for understanding a variety of properties of the transition metals. 

The problem of size-consistency is particularly frightful for restricted Cl calculations on solids where the correlation energy falsely arrives at zero, which is the worst possible result [126]. The difficulties of Hartree-Fock theory for the solid state or, more correctly, for solid metals, however, show up much earlier. When Hartree-Fock theory is applied to a gas of noninteracting electrons (free electrons) which comes close to the electronic situation in simple metals, such as the alkali elements, the band energy takes the following analytical form [131] ... 

Figure 6.9a-f illustrates a variety of the accepted band structure representations for nearly-free electron model. The Figure introduces the repeated-zone, extended-zone and reduced-zone images. The original free-electron parabola E = fi k Klme) is shown in Figure 6.9a. To leading order in the weak one-dimension periodic potential this curve remains correct except the value of k near the reciprocal lattice vector g. One can imagine that in this point the Bragg plane reflects the electron wave since the Bragg condition holds. Another free-electron parabola is centered at fe = g, and two parabolas are crossed each other at the...

FIGURE 4. The effect of an applied, static magnetic field, B, on the = 5 levels of a free electron. Transitions between the levels may be induced by electromagnetic radiation of the correct frequency. For an applied field of 3400 G, this corresponds to microwave radiation (X-band) of 9.4 GHz. 

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