Harmonic band structure

The case of three and four electrons is more complicated, but the two characteristic features of the energy spectra observed for small coz, i.e., the nearly-degenerate multiplet structure of the energy levels of different spin multiplicities and the harmonic band structure of these levels, can be rationalized in a similar way. In the case of three electrons, for example, the internal space can be defined by the two correlated coordinates Zb and zc defined by Equation (11). The potential function becomes a sum of two harmonic-oscillator Hamiltonians for the Zb and zc coordinates plus three Coulomb-type potentials originating from the three electron-electron... 

First of aU, the observed 2vd harmonic band is anomalously sharp in comparison to the one theoretically calculated from the main D-band and does not have such a pronounced internal structure as the latter (Fig. 7.9a). Due to a presence of two maxima (around 1570 cm and around 1592 cm ) in the main G-band, one should expect three peaks in the 2vq harmonic band (two peaks corresponding to a doubled frequency of these two peaks and one peak corresponding to their sum) with a spectral interval of 22 cm. However, this is not the case even with consideration of a possible fine structure for the constituting bands. Relatively... 

Contrary to the above considered harmonic bands, the band of the sum tone Vq+Vrbm is much broader than the low frequency band. We explain this fact by a well-pronounced doublet structure of G-band, which has appearance in the considered sum tone. This is also confirmed by the same spectral range between the observed maxima and by similarity in the intensities of the corresponding constituting components of the doublet bands. More detailed analysis of the structure of all the observed bands is required in the future. 

The structure of vibration bands of the first and the second order in SWCNT Raman spectra has also been studied for ordered and disordered forms of graphite. This was accomplished by decomposition of the complex spectral bands into constituting components. We found proximity of spectral positions in most of spectral components of the nanotubes and graphite and considerable variation of their intensities. This also demonstrates variation of the electronic polarizabilities and can explain anomalous shifts of the harmonic bands 2vq and 2vd for nanotubes in comparison to corresponding bands of a single crystalline graphite. Narrow width of the low frequency mode Vrbm 160 cm leads to reproduction of the G-band structure in the sum harmonic band Vg+Vrbm" 1750 cm while the complex stmcture of the broad Vp band is remarkably reproduced in the Vq+Vg sum tone. The narrow width of SWCNT s 2vd and 2vg harmonics in the Raman spectra may be related to group synchronism effects [72]. 

For atoms with high atomic numbers it is important that relativistic effects are included in the band-structure calculations. In SCFC we therefore solve the Dirac rather than the Schrodinger equation, but leave out the spin-orbit interaction. In doing so we obtain an effective one-electron equation which is essentially the Schrodinger equation with the important mass-velocity and Darwin corrections included. The present technique is based on unpublished work by O.K. Andersen and U.K. Poulsen. KoelUng and Harmon [9.8] have taken a related approach. 

Fig. 22. Results of self-consistent band-structure calculations for LaH (bottom) and LaHj (top). Depletion of conduction electrons at the Fermi level and large charge transfer toward t-site hydrogens is clearly seen in the LaHj curves. After Misemer and Harmon (1982).

This difficulty could be avoided by applying linear response theory, which is widely used in solid-state physics to determine directly the dynamic matrix, polarization, and frequency-dependent dielectric functions, as well as phonon dispersion curves in the harmonic approximation. This method has the great advantage that it requires only a band structure at the equilibrium geometry of the solid (chain), i.e., one does not have to determine a potential hypersurface. However, since this theory has not yet been applied to polymers and involves a rather complicated formalism, we cannot enter into details here but refer the reader to standard solid-state physical works and an application to a simple solid (Si). ... 

The harmonic force constants thus obtained have been used in a Wilson GF procedure modified by periodic boundary conditions to compute phonon dispersion curves. These curves enabled the phonon density of states to be evaluated in a manner similar to that employed in obtaining the electronic density of states from the electronic band structure. 

Leitsmann, R., Schmidt, W. G., Hahn, P. H., Bechstedt, F. (2005). Second-harmonic polarizability including electron-hole attraction from band-structure theory. Physical Review B - Condensed Matter and Materials Physics, 72(19), 195209/1-195209/10. 

The semiconductor band structure of CPs permits electronic excitation or electron removal/addition, e.g. from the valence to the conduction band, leading to most of the properties that are of interest in CPs. Excitation of electrons from the valence band to the conduction band, e.g. by photons, yields typical excited state properties such as photoluminescence and nonlinear optical properties (e.g. third harmonic generation). 

In the band structure methods that use the concept of an atomic sphere (muffin-tin sphere), a wave function in an atomic sphere is represented by the product of a radial function and an angle-dependent function Yf (spherical harmonic). By weighting the total DOS by the square of the contribution of the partial functions with a specific / value to the total wave function of each state, a local (site projected) Hike partial DOS is obtained. The total DOS, g(E), is thus spatially divided according to... 

---