Brillouin-Bloch bands

Sung and Lazar [85] provided confirmation of Mason s [79, 80] controversial hypothesis of hydrocarbon binding to proteins, which is based on the the so-called Brillouin—Bloch bands in proteins (primarily the delocalised bond in peptide moieties) the energy difference between the highest occupied electron band and the first unoccupied (so-called conduction) band is approximately 3 eV. Mason observed that, for carcinogenicity in polycyclic aromatic hydrocar-... 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

First, the irreducible part of the Brillouin zone now varies from k = 0 to k = Tr/d = tt/2d. Indeed, doubling the parameter of the unit cell in real space halves the size of the Brillouin zone (or the reciprocal-space unit cell). Second, recall that orbital interactions are additive and that the final MO diagram (or band structure) is just the result of the sum of all the orbital interactions. Within each individual H2 unit the interactions simply correspond to the bonding (a) and antibonding (a ) MOs of each individual H2 unit. There are three types of interactions involving the MOs of different H2 units interactions between all the a orbitals interactions between all the a orbitals and interactions between the a and the a orbitals. Since all the an orbitals are equivalent by translational symmetry, their interaction is described by the Bloch function ... 

Fig. 14 Band structure of a fully oxygen defective (1 x 1) MgO(lOO) surface along the three symmetry lines J-F-M of the 2D Brillouin Zone, as obtained through the FP-LMTO calculation (Full Potential- Linear MufiSn-Tin Orbital method). The dashed horizontal line represents the Fermi level, black dots (st indicate the energy positions of the filled (empty) Bloch states at F calculated in a (2v x 2- /2) supercell. The dashed line in the gap of the projected bulk bandstructure gives the dispersion of the F, centre band. The dashed-dotted line is used for the surface conduction band of lowest energy (from Ref. 69).

Fig. 5.12. (a) Variation of electron energy with wave number for a two-dimensional metal on the basis of the Bloch theory. ka, kb and kc are values of the first forbidden wave number in different directions in the first Brillouin zone. The band of forbidden energy in (b) indicates that the energy levels of the first and second zones do not overlap, (c) As (a) but for the case in which there is overlap between the energy levels of the first and second zones, as shown in ([Pg.99]

At the age of 23, Felix Bloch published an article called t/ter die Quantenmechanik derElektronen in Kristcdlgittern" in Zeitschriftflir Physik, 52,555 (1928) (only two years after Schrbdinger s historic publication) on the translational symmetry of the wave function. This was also the first application of LCAO expansion. In 1931. Leon Brillouin published a book entitled Quantenstatistik (Springer Verlag, Berlin), in which the author introduced some of the fundamental notions of band theory. The first ab initio calculations for a poljmer were made by Jean-Marie Andre in a paper Self-consistentfield theory for the electronic structure of polymers f publidied in the Journal cf Chemical Physics, 50, 1536 (1969). 

In (6.5) the subscript n indicates the band index and fe is a continuous wave vector that is confined to the first Brillouin zone of the reciprocal lattice. The index n appears in the Bloch theorem because for a given k there are many solutions to the Schrodinger equation. Because the eigenvalue problem is set in a fixed finite volume, we generally expect to find an infinite family of solutions with discretely-spaced eigenvalues which we label with the band index n. The wave vector k can always be confined to the first Brillouin zone. The vector k takes on values within the Brillouin zone corresponding to the crystal lattice, and particular directions like r,A,A,Z (see Figures 4.13-4.15). 

We have developed our model for the simplest possible system, one p orbital per unit cell, and then argued qualitatively about the system with two AOs per unit cell. In fact, there is a rigorous, analytical solution for the generalized problem of a unit cell with any number of MOs in an infinite polymer. This produces the Bloch equations that are the direct analogues of HMO theory as applied to infinite systems. To create computed band structures of the sort shown throughout this chapter, we simply solve the Bloch equations at several points throughout the Brillouin zone and then interpolate between these points to produce the smooth curves shown in band structures. 

The dispersion relation (Eq. (5.34)) yields two bands, which for an infinite ( bulk ) crystal are displayed by the thick lines in the left part of Eigure 5.12. As can be seen, an energy gap of width 2Vg opens up at the Brillouin zone boundary k = f Note that for an infinite crystal the wave vector k has to be real, since otherwise the Bloch wave function F(z) = e" Ut(z) would diverge exponentially for either z -r- +00 or z —00. 

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