Band calculation

Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

The ID electronic energy bands for carbon nanotubes [170, 171, 172, 173, 174] are related to bands calculated for the 2D graphene honeycomb sheet used to form the nanotube. These calculations show that about 1/3 of the nanotubes are metallic and 2/3 are semiconducting, depending on the nanotube diameter di and chiral angle 6. It can be shown that metallic conduction in a (n, m) carbon nanotube is achieved when... 

In addition, for two coaxial armchair tubules, estimates for the translational and rotational energy barriers (of 0.23 meV/atom and 0.52 meV/atom, respectively) vvere obtained, suggesting significant translational and rotational interlayer mobility of ideal tubules at room temperature[16,17]. Of course, constraints associated with the cap structure and with defects on the tubules would be expected to restrict these motions. The detailed band calculations for various interplanar geometries for the two coaxial armchair tubules basically confirm the tight binding results mentioned above[16,17]. 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

Switendick was the first to apply modem electronic band theory to metal hydrides [5]. He compared the measured density of electronic states with theoretical results derived from energy band calculations in binary and pseudo-binary systems. Recently, the band structures of intermetallic hydrides including LaNi5Ht and FeTiH v have been summarized in a review article by Gupta and Schlapbach [6], All exhibit certain common features upon the absorption of hydrogen and formation of a distinct hydride phase. They are ... 

After having described molybdenum trioxide, we intend to specify the best finite clusters allowing to represent each of the (010), (001) and (100) faces in order to study surface properties such as energy and electronic distribution. For this purpose, the evolution of the electronic properties will be studied as a function of the cluster size and referred to the results of an EHT - band calculation [12] all calculations have been made with QCPE programs [13,14] and Hoffmann parameters [15],... 

When the cluster size increases (Fig. 3), the occupied O2.S energy levels are concentrated in two blocks around -15.2 and -312 eV of widths 1.3 and 1.7 eV respectively the Fermi level is slightly removed and stabilized at -14.57 eV, the same value as in the band calculation. 

Christensen, N.E. and Seraphin, B.O. (1971) Relativistic Band Calculation and the Optical Properties of Gold. Physical Review B Condensed Matter, 4,3321-3344. 

Thus, the obtained CP of Equation (23) corresponds to the so-called conventional band calculation CPs. 

Figure 4. Dependence of the ratio u u on the number of 5f electrons for light actinide compounds x free ion values, ° experimental values, form band calculations. The hybridisation between 5f and 3d electrons leads to the reduction of the 5f orbital moments (metallic covalency).

Franco Bassani, Methods of Band Calculations Applicable to III-V Compounds E.O. Kane, The k -p Method... 

The IR spectrum which can be measured in argon at 10 K after irradiation of diazo compound 18 with k = 313 nm is relatively complex. But the absorptions of 19 can be extracted by a subsequent irradiation with k > 570 nm. The signals of 19 decrease in intensity during this secondary irradiation. They fit much better with the bands calculated for T-19 than for S-19. The product formed under these conditions (X > 570 nm) is the ring-opened carbene 16, which in this case can directly be detected and shows an IR spectrum which is in agreement with that of S-16. Intermediate 16 can be transferred photochemically to 2-cyano-2/7-azirene (17) with X > 313 nm, which is the main product in the primary irradiation of diazocompound 18 with this wavelength. 

It is well established that the average lengths of CH bonds are consistently 0.003 to 0.004 A longer than the corresponding CD bonds in the ground vibrational state (see Fig. 12.1, its caption, and Section 12.2.3). It remains only to establish the dipole moment derivative, (9p/9r), at the equilibrium bond length. That is available from theoretical calculation or spectroscopic measurement (via precise measurements of IR intensities of vibration-rotation bands). Calculations based on Equation 12.7 yield predicted dipole moment IE s in reasonable agreement with experiment. 

In white tin, the nearest neighbor interatomic distance is increased to about 3 A. As expected, this lowers the energy of the 5s anti-bonding functions, and in a considerable portion of the Brillouin zone the 5 antibonding functions lie below the 5p like functions in energy hence, there are substantially more than one 5s electron and substantially less than three 5p electrons per atom. Judging from the Mdssbauer data (one could get this presumably from the band calculations with enough effort). 

Detailed electronic energy-band calculations have revealed the existence of appropriate surface states near the Fermi energy, indicative of an electronically driven surface instability. Angle-resolved photoemission studies, however, showed that the Fermi surface is very curved and the nesting is far from perfect. Recently Wang and Weber have calculated the surface phonon dispersion curve of the unreconstructed clean W(100) surface based on the first principles energy-band calculations of Mattheis and Hamann. ... 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

At any transformation level if the minimum F statistic were less than or equal to the critical F value, our work was done and the confidence band calculations began. Otherwise we either accepted a lack of fit (and would note it in published results), segmented the graph to shorter lengths, or sought a non-linear or higher order model. 

This part of the discussion is closed with the raising of a question. Is there an inherent truth that plotting itself should be done with respect to maintaining constant variance across the graph The requirement of constant variance is invoked by statisticians when performing confidence band calculations. Thus, is it also a necessity in plotting the points themselves ... 

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

Figure 6.6 Schematic illustration of a two dimensional energy surface with two local minima separated by a transition state. The dark curves are energy contours with energy equal to the transition state energy. The transition state is the intersection point of the two dark curves. Dashed (solid) curves indicate contours with energies lower (higher) than the transition state energy. The MEP is indicated with a dark line. Filled circles show the location of images used in an elastic band calculation.

Figure 6.8 Similar to Fig. 6.6, but showing a set of images from an elastic band calculation that exhibits comer cutting. ... 

Coloma F, Marquez F, Rochester CH, Anderson JA (2000) Determination of the nature and reactivity of copper sites in Cu-Ti02 catalysts. Phys Chem Chem Phys 2 5320-5327 Umebayashi T, Yamaki T, Itoh H, Asai K (2002) Analysis of electronic structures of 3d transition metal-doped Ti02 based on band calculations. J Phys Chem Solids 63 1909-1920 Yamashita H, Ichihashi Y, Takeuchi M, Kishiguchi S, Anpo M (1999) Characterization of metal ion-implanted titanium... 

Zerner s intermediate neglect of differential overlap (ZINDO)/PM3 calculations of thiazinylium compound 35 were compared to its ultraviolet/visible (UVA is) absorption spectrum (Figure 3) <2000JOC6388>. The authors attribute the observed 453 and 403 nm bands (calculated to be at 456 and 412 nm) to highest occupied molecular orbital (HOMO)-LUMO and HOMO-LUMO + 1 transitions of the 1,2-thiazine sulfonium imide. 

For the non-relativistic case (Schrbdinger equation), T = -V. For relativistic case (Dirac equation), T = c a p + 3mc where m is the rest mass of the electron, c is the velocity of light. We have preferred to write the T operator in a general form, covering both cases, given the importance of the relativistic approach in band calculations for actinide solids - see Chap. F... 

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