Band structure hydrogen chain

Figure 1 The band structure of a chain of hydrogen atoms spaced 3, 2, and 1 A apart. The energy of an isolated H atom is -13.6 eV.

To get a feeling for this quantity, let s think about what a COOP curve for a hydrogen chain looks like. The simple band structure and DOS were given earlier, 26 they are repeated with the COOP curve in 35. 

Hydrocarbon Based Polymers. The substitution of one hydrogen atom in the -CH2-CH2- unit by some short carbon chains induces subtle modifications in the electronic structure (molecular orbitals) of the polymers. Though these modifications cannot be easily evidenced on the XPS carbon Is core level spectra, it appears that the XPS valence band structures are much more sensitive to these substitutions and that they become unique and readable fingerprints of the polymers (1, 22). We will not speak here of the Cls shake-up data that were revealed useful to distinguish between saturated and unsaturated bonds (this field with various applications was recently reviewed (23)). 

The same trend is observed for the N—H.. X system, as shown in Fig. 5. The NH stretching frequency is high, and the bands are narrow (5 cm- ) for the weak N—H.. n hydrogen bonds encountered in crystalline pyrrole (16). There is a broad vNH band (v1 — 270 cm-1) at much lower frequency, 2720 cm-1, in the Raman spectrum (26) of the triazole crystal in which there are infinite chains of molecules linked by relatively short (N.. N = 2.82 A) hydrogen bonds (27). As far as the band structure is concerned, the pyrrole doublet results from correlation field splitting while the submaxima of the NH stretching band of triazole arise from combination-overtones. 

Fig. 2.9 Semiempirical (extended Huckel theory) band structure E ip k)) for a onedimensional chain of hydrogen atoms with H-H distances of 3 A and 2 A (in grey) and 1 A (in black). The shape of the extended wave function has been iconized for three different k points, r (left), k = n/2a (middle), and X (right).

Figure 2.17 now shows the band structure of the one-dimensional hydrogen chain and its DOS side-by-side for all three atomic spacings. As mentioned before, a steep/flat band is characterized by a low/high EXDS. If the band becomes more atomic-like due to a wide interatomic spacing, the DOS turns into a spike for that atomic level. 

Fig. 2.19 Semiempirical (extended Huckel theory) band structure, density-of-states, and crystal orbital overlap population for a one-dimensional chain of hydrogen atoms spaced at 2.0 A. Due to a Gaussian smoothing, DOS and COOP plots appear slightly...

Polyacetylene already has quite a complex band structure, but as usual, the bands close to the Fermi level (valence bands) are the most important in chemistry and physics. All these bands are of the jv type i.e., their COs are antisymmetric with respect to the plane of the polymer. Fig. 9.17 shows how the valence bands are formed. We can see that the principle is identical to that for the chain of the hydrogen atoms the more nodes there are, the higher the energy is. The highest energy corresponds to the band edge. 

Figure 15.1 Infinite linear chain of regularly spaced hydrogen atoms (a) schematic representation (b) Bloch orbitals at the T and X points (c) Band structure.

Figure 15.2 Uniform hydrogen chain (a) schematic representation of the occupation of the different levels (b) and (c) are two representations of the band structure. Shown in (d) and (e) are two different schematic band structures for a three-band, four-electrons per repeat...

Figure 15.3 (a) Dimerized hydrogen chain (b) schematic band structure (c) crystal... 

Figure 15.4 Linear chain of hydrogen atoms (a) band structure and (b) density of states according to a Huckel approach (c) DOS for a typical ID band.

Figure 15.5 Zigzag chain of hydrogen atoms (a) schematic band structure (b) and (c) schematic COOP curves for the 1,2 and 1,3 interactions.

The EHMO band structure appears in Fig. 15-18b. There are 10 valence AO basis functions per unit cell (two carbons and two hydrogens) so we get 10 lines in the band diagram. Note that, since the chain is no longer linear, the 2p and 2p bands are no longer degenerate. One of these is still perpendicular to the molecular plane, hence is still of 7T symmetry, but the other now lies in the plane and has a symmetry. 

The next step in the investigations of the ab initio band structure of polyglycine, namely its treatment as a two-dimensional periodic system (perio city along the polypeptide chain and perpendicularly through the hydrogen bonds), is in progress. 

In the actual calculation the complicated side chains R were substituted by an H atom. For the first step (calculation of the HF band structures) a 6-3IG basis set (double C + polarization function on both the carbon and hydrogen atoms) was applied. The valence and conduction bands obtained in this way were then corrected using the generalized electronic polaron model [quasi-particle (QP) band structures see Section 5.3]. The lowest sin et-exciton enei es (at K = 0) were then calculated using the QP one-electron levels and performing the three steps described in the previous section after equation (8.22). Table 8.1 shows the results obtained in this way for both PTS and TCDU. ... 

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