Band theory Bloch function

It was pointed out in my 1949 paper (5) that resonance of electron-pair bonds among the bond positions gives energy bands similar to those obtained in the usual band theory by formation of Bloch functions of the atomic orbitals. There is no incompatibility between the two descriptions, which may be described as complementary. It is accordingly to be expected that the 0.72 metallic orbital per atom would make itself clearly visible in the band-theory calculations for the metals from Co to Ge, Rh to Sn, and Pt to Pb for example, the decrease in the number of bonding electrons from 4 for gray tin to 2.56 for white tin should result from these calculations. So far as I know, however, no such interpretation of the band-theory calculations has been reported. 

Hence one can expand the Bloch functions using the kp perturbation theory (2)-(4), to find the admixture to the functions wc,k(r) of the Bloch functions Ub(r) of all other bands b ... 

Bloch wave function - A solution of the Schrodinger equation for an electron moving in a spatially periodic potential used in the band theory of solids. 

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 energy-band theory, not only the external potential, but also the single-electron potential, have all the symmetry, translational included, of the crystal lattice. The eigenfunctions of eq. (4), therefore, obey Bloch s theorem and are functions of the wave vector, k, in reciprocal space. They may be constructed from the energy-dependent solutions of the wave equation inside the atomic spheres. 

The band theory applies to the perfectly regular organization of a crystal, leading to delocalized Bloch wave functions, Eq. (6). In a now classical paper, Anderson [7] has shown that disorder may result in a localization of the states. In that case, the one-electron wave function takes an exponential form... 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

The electronic stmcture of bulk vanadium oxides is determined to a major extent by the amount of d electron occupation in the vanadium ions. In the ideal three-dimensional periodic bulk, electrons are described by Bloch states with energy dispersions reflected in band stmctures and corresponding densities of states (DOS). These quantities can be calculated with high accuracy by modem band stmcture and total energy methods based on the density functional theory (DFT) method. 

The theory of induced representations of space groups gives the answer to the question of whether it is possible to generate in the space of states of a given energy band the basis of localized functions The answer to this question allows the symmetry connection between delocalized Bloch-type and localized Wannier-type crystalline orbitals to be obtained. This point is discussed in Sect. 3.3. 

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