Bloch-type

Let us choose now a set of one-electron wave functions, e.g. of Bloch type. We suppose that cj)(Ri... Rn) is the Slater-Fock determinant of these wave functions ... 

For the case of typical ionic crystals aP 1-10, and the weak coupling limit is applicable. The most important conclusion from this treatment is that the weak coupling limit leads to a perturbed Bloch type wave function characterized by equal probability for finding the electron at any point of the medium. Thus, in the case of the ionic crystals, the current description of the polaron is that of a mobile electron followed by lattice polarization. 

For the N sites in the lattice, we need Bloch-type functions of the type... 

For the system in Fig. la, we assume that excitation induced by the circularly-polarized light produces a Bloch-type state on the donor ... 

Looking at Tables 9 and 10 one can see that the valence and conduction bands of the stacked bases and of the polySP chain are several tenths of an eV in width (values between 0.16 and 0.86 eV) indicating that there is the possibility of a Bloch-type conduction in these systems if free charge carriers are generated in them. On the other hand, the gap in all cases is more than 10 eV. Although one knows that the Hartree-Fock calculation gives too large a gap for conduction, this rules out the possibility of intrinsic semiconduction in DNA. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

At low temperatures, the small-polaron moves by Bloch-type band motion, while at elevated temperatures it moves by thermally activated hopping mechanism. Holstein (1959), Friedman and Holstein (1963), Friedman (1964) performed the theoretical calculations of small-polaron motion and showed that the temperature dependencies of the small-polaron mobility in the two regimes are different. In the high-temperature hopping regime, the electrical conductivity is thermally activated and it increases with increasing temperature. As shown by Naik and Tien (1978), its temperature dependence is characterized by the following equation... 

The (+1) peaks in this case are best approximated by a simple Gaussian/ Lorentzian overlap of two bands present in solution, and not the Bloch-type analysis appropriately used for rapidly exchanging systems. It is concluded that this particular organic mixed-valence complex is localised on the infrared timescale. 

Let us summarise in the effective Hamiltonian language [20] the general fea-tmes of a CP /SO method, in its simple Bloch-type version. In a first step, the scalar relativistic secular equations for states under interest are solved, and extensive Cl calculations define a determinant target space dim providing accurate energies and the corresponding multiconfigurational states of interest Om) m E In a second step, a determinant inter-... 

In order to take into account the correlated results obtained in the first step, a spin-free effective Bloch-type Hamiltonian is defined... 

A Bloch-type effective Hamiltonian technique in a CI /SO method simply amoimts to replace the diagonal energy of the intermediate model space by the fiall correlated one, Em, coming from the target space of the first step calculation, without knowing — at least in principle — the corresponding wave... 

Within the Bloch-type effective Hamiltonian DGCP schemes, Vallet et al proposed a very different method, the so-called EPCISO algorithm [11] (Effective and Polarised CISO) with the aim to enhance the possibilities of the... 

In representative graphs of regular polymers the monomer unit is depicted by a simple graph, and the bonds between adjacent monomer imits are indicated by arcs which are weighted by conjugate complex numbers. The characteristic pdynomial of the representative graph, and the Bloch type crystal orbital description of the polymer, are equivalent. 

Positive muons injected into matter are at first decelerated by Bethe-Bloch type enagy los processes. When their velocity becomes close to that of the valence electrons, it becomes possible for to pick up an electron and form energetic Mu, which, on subsequent collision, will be dissociated. 

Are the rate constants estimated from the Bloch type treatment consistent with expectations based on theory The -1 mixed valence state of complex (Id) is a Robin-Day class II complex, and thus its electron transfer rate constant can be independently estimated from Marcus theory. The semiclassical expression for the rate constant for intramolecular electron transfer, k, in a symmetric mixed valence complex with no net free energy change is given by... 

Yafet (1963) calculated the relaxation of the conduction electrons to the lattice <5 l due to spin-orbit scattering. Set can be separated into (intrinsic) and S eB-X. Hereby x denotes the concentration of the extrinsic scattereis. Has awa (1959) analyzed the situation which is shown in fig. 1 by Bloch-type equations. In this scenario the paramagnetic ions are... 

It is obvious that at every k point there is the same number p of Bloch-type basis states. 

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. 

It is a variational method based on the first-principles approach, t.e. without preliminary knowledge of Bloch-type delocalized functions. The localized functions with the symmetry of Wannier functions and depending on some number of parameters are used. 

M electrons occupying M Bloch-type states can be written as... 

The local properties of the electronic structure of a periodic system are defined by the density matrix p R,R ),see (4.91) the electron position vectors R and R vary within the basic domain of a crystal consisting of N primitive unit cells. For a one-determinant wavefunction the density matrix can be expressed through Bloch-type spin orbitals = ,/ ) ... 

To minimize the essential dependence of results on the basis-set choice and POPAN, it is reasonable to generate Wannier-type atomic functions that are orthogonal and localized on atomic sites. These functions can be generated from Bloch-type functions (after relevant symmetry analysis), see Chap. 3. In the next section it is shown that the results of POPAN with use of Wannier-type atomic functions weakly depend on the basis choice in Bloch function LCAO calculations. 

It is traditional for quantmn theory of molecular systems (molecular quantum chemistry) to describe the properties of a many-atom system on the grounds of interatomic interactions applying the hnear combination of atomic orbitals (LCAO) approximation in the electronic-structure calculations. The basis of the theory of the electronic structure of solids is the periodicity of the crystalline potential and Bloch-type one-electron states, in the majority of cases approximated by a linear combination of plane waves (LCPW). In a quantmn chemistry of solids the LCAO approach is extended to periodic systems and modified in such a way that the periodicity of the potential is correctly taken into account, but the language traditional for chemistry is used when the interatomic interaction is analyzed to explain the properties of the crystalhne sohds. At first, the quantum chemistry of solids was considered simply as the energy-band theory [2] or the theory of the chemical bond in tetrahedral semiconductors [3]. From the beginning of the 1970s the use of powerful computer codes has become a common practice in molecular quantum chemistry to predict many properties of molecules in the first-principles LCAO calculations. In the condensed-matter studies the accurate description of the system at an atomic scale was much less advanced [4]. 

Various interesting properties of 4f-electron systems such as CK and HF compounds have been generally studied using crystalline materials, where there is also periodicity of 4f electrons. The formation of the Fermi surface from 4f-electron bands is essential to many, if not all of these cooperative properties. Therefore, such behaviors are not expected to occur in disordered alloys without translational symmetry where the 4f electrons have non-Bloch-type states. However, interesting behaviors have recently been observed in Ce-based structurally disordered materials. Here, we introduce amorphous CeRu system as a typical example. Crystalline CeRu2 compoimd is one of the famous itinerant 4f-electron materials to show SC (T = 6.3 K, y = 27 mj/mol K ) (Hedo et al., 1998 Matthias et al., 1958). On the other hand, amorphous fl-CeRu also shows FlF-like behavior in the Ce-rich side and SC in the Ru-rich side (Homma et al., 1997). 

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