Electronic structure Bloch states

The band structure and Bloch functions of metals have been extensively published. In particular, the results are compiled as standard tables. The book Calculated Electronic Properties of Metals by Moruzzi, Janak, and Williams (1978) is still a standard source, and a revised edition is to be published soon. Papaconstantopoulos s Handbook of the Band Structure of Elemental Solids (1986) listed the band structure and related information for 53 elements. In Fig. 4.14, the electronic structure of Pt is reproduced from Papaconstantopoulos s book. Near the Fermi level, the DOS of s and p states are much less than 1%. The d states are listed according to their symmetry properties in the cubic lattice (see Kittel, 1963). Type 2 includes atomic orbitals with basis functions xy, yz, xz], and type e, includes 3z - r-), (x - y ). The DOS from d orbitals comprises 98% of the total DOS at the Fermi level. 

Sometimes the estimation of the electronic structures of polymer chains necessitates the inclusion of long-range interactions and intermolecular interactions in the chemical shift calculations. To do so, it is necessary to use a sophisticated theoretical method which can take account of the characteristics of polymers. In this context, the tight-binding molecular orbital(TB MO) theory from the field of solid state physics is used, in the same sense in which it is employed in the LCAO approximation in molecular quantum chemistry to describe the electronic structures of infinite polymers with a periodical structure -11,36). In a polymer chain with linearly bonded monomer units, the potential energy if an electron varies periodically along the chain. In such a system, the wave function vj/ (k) for electrons at a position r can be obtained from Bloch s theorem as follows(36,37) ... 

Recently use of localised Wannier functions instead of delocalised Bloch states (CP orbitals) in CPMD simulations has proved an efficient and effective approach for study of fluids such as water133 and DMSO/water mixture.134 Use of localised Wannier functions also has the advantage that they allow electrons to be assigned to bonds, making visualisation of bonding and structure of molecules easy and facilitating comparisons with standard chemical bonding models. 

High-resolution transmission electron microscopy can be understood as a general information-transfer process. The incident electron wave, which for HRTEM is ideally a plane wave with its wave vector parallel to a zone axis of the crystal, is diffracted by the crystal and transferred to the exit plane of the specimen. The electron wave at the exit plane contains the structure information of the illuminated specimen area in both the phase and the amplitude.. This exit-plane wave is transferred, however affected by the objective lens, to the recording device. To describe this information transfer in the microscope, it is advantageous to work in Fourier space with the spatial frequency of the electron wave as the relevant variable. For a crystal, the frequency spectrum of the exit-plane wave is dominated by a few discrete values, which are given by the most strongly excited Bloch states, respectively, by the Bragg-diffracted beams. 

We determine the ground-state electronic structure of solids within Density Functional Theory (DFT) and the usual KS variational procedure, all implemented in the computational package gtoff [14]. The results of the all-electron, full-potential calculations are Bloch eigenfunctions p,k(r), expressed as linear combinations of Gaussian Type Orbitals (GTOs), and KS eigenvalues p k-... 

The SIC-LSD still considers the electronic structure of the solid to be built from individual one-electron states, but offers an alternative description to the Bloch picture, namely in terms of periodic arrays of localized atom-centred states (i.e., the Heitler-London picture in terms of the exponentially decaying Warmier orbitals). Nevertheless, there still exist states that will never benefit from the SIC. These states retain their itinerant character of the Bloch form and move in the effective LSD potential. This is the case for the non-f conduction electron states in the lanthanides. In the SIC-LSD method, the eigenvalue problem, Eq. (23), is solved in the space of Bloch states, but a transformation to the Wannier representation is made at every step of the self-consistency process to calculate the localized orbitals and the corresponding charges that give rise to the SIC potentials of the states that are truly localized. TTiese repeated transformations between Bloch and Wannier representations constitute the major difference between the LSD and SIC-LSD methods. 

It has been known since the work of Bloch (1928) that there are universal features in the electronic structures of crystals. The most important of these are energy bands separated by gaps, crystal momentum as a good quantum number, and the form of the wave function they provide the conceptual foundations of much of solid state physics. 

The electronic structure of periodic crystalline solids is usually represented by Bloch orbitals where n and k are quantum numbers of the band and crystal momentum, respectively. The Bloch states are eigenfunctions of the Hamiltonian of the crystal, obeying the same periodicity. Because of the fact that they are usually highly delocalised, it is often difficult to deduce local properties from Bloch orbitals, for instance, bonding between atoms or atomic charges. 

The Wannier functions obtained in Eq. 6.25 are not unique, because it is possible to mix the Bloch states of different band numbers by a unitary matrix The resulting Wannier functions are also a complete representation of the electronic structure, although their localisation features are different ... 

Band theory is basically a one-electron theory. Electron-electron interactions are only included in the form of an average contribution to the effective electron-ion interaction potential. Thus, band theory should be most informative for modeling the electronic structure of liquids for which the MNM transition is of the Bloch-Wilson band-overlap variety. Fig. 2.13 illustrates some typical results for the electronic density of states of mercury in a series of structures with constant interatomic separation. With increasing density, the band-overlap transition is clearly evident as the gap closes between the lower, predominantly s-like band and the upper p-band. These results agree qualitatively with the observed electronic properties of expanded mercury although, as we shall see in chapter 4, the actual MNM transition occurs in a density range for which the band model still predicts a nonvanishing density of states at the Fermi level. 

Hybrid approaches combining ab-initio or DFT and semiempirical approaches have become popular. As an example, we can refer to LEDO (hmited expansion of differential overlap) densities application to the density-functional theory of molecules [262]. This LEDO-DFT method should be well suited to the electronic-structure calculations of large molecules and in the anthors opinion its extension to Bloch states for periodic structures is straightforward. In the next sections we discuss the extension of CNDO and INDO methods to periodic stmctures - models of an infinite crystal and a cyclic cluster. 

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

The second approach (B), proposed for constructing quasiatomic minimal basis orbitals (QUAMBO) in [609] is also closely related to the projection technique (being in fact a projection reverse to the first one) the projection of a given minimal atomic basis is made on the Bloch states obtained in the PW calculations. This method can be regarded as a sort of localization procedure, and it describes the electronic structure of periodic solids in terms of localized quasiatomic minimal basis orbitals (QUAMBO). 

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