Bloch s theorem

A1.3.4 ELECTRONIC STATES IN PERIODIC POTENTIALS BLOCH S THEOREM... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

One of the first models to describe electronic states in a periodic potential was the Kronig-Penney model [1]. This model is commonly used to illustrate the fundamental features of Bloch s theorem and solutions of the Schrodinger... 

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. 

Bloch s theorem states that in a periodic solid each electronic wave function can be expressed as the product of a wave-like component (with wave vector k) and a cell-periodic component/ (r) ... 

When an external electric field is applied along the periodicity axis of the polymer, the potential becomes non periodic (Fig. 2), Bloch s theorem is no longer applicable and the monoelectronic wavefunctions can not be represented under the form of crystalline orbitals. In the simple case of the free electron in a one-dimensional box with an external electric field, the solutions of the Schrddinger equation are given as combinations of the first- and second-species Airy functions and do not show any periodicity [12-16],... 

In this section, we consider how to model a bulk (i.e., infinite) substitution-ally disordered binary alloy (DBA), in light of its intrinsic randomness. The fact that the DBA lacks periodicity means that the key tool of Bloch s theorem is inapplicable, so specialized methods (Ehrenreich and Schwartz 1976, Faulkner 1982, Yonezawa 1982, Turek et al 1996) must be used. 

Our lengthy discussion of k space began with Bloch s theorem, which tells us that solutions of the Schrodinger equation for a supercell have the form... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

These fundamental ideas of band theory can be extended to three dimensions. In particular, Bloch s theorem takes the form... 

The phase factor automatically guarantees that ( ) satisfies Bloch s theorem, eqn (5.30), since... 

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) ... 

This statement is known as Bloch s theorem, which in three dimensions reads... 

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. 

The wave function has the same amplitude at equivalent positions in each unit cell. Thus, the full electronic structure problem is reduced to a consideration of just the number of electrons in the unit cell (or half that number if the electronic orbitals are assumed to be doubly occupied) and applying boundary conditions to the cell as dictated by Bloch s theorem (Eq. 4.14). Each unit cell face has a partner face that is found by translating the face over a lattice vector R. The solutions to the Schrodinger equation on both faces are equal up to the phase factor exp(zfe R), determining the solutions inside the cell completely. 

The disorder of the atomic structure is the main feature which distinguishes amorphous from crystalline materials. It is of particular significance in semiconductors, because the periodicity of the atomic structure is central to the theory of crystalline semiconductors. Bloch s theorem is a direct consequence of the periodicity and describes the electrons and holes by wavefunctions which are extended in space with quantum states defined by the momentum. The theory of lattice vibrations has a similar basis in the lattice symmetry. The absence of an ordered atomic structure in amorphous semiconductors necessitates a different theoretical approach. The description of these materials is developed instead from the chemical bonding between the atom, with emphasis on the short range bonding interactions rather than the long range order. 

This is the parabolic curve shown in Fig. 4.4(a). A linear atomic lattice will provide a periodic rather than a constant potential, i.e. V(x) = V[x + a), where a is the repeat distance of the array. Bloch s theorem, also known as Floquet s theorem, states that possible solutions of the Schrodinger equation with a periodic potential are ... 

For any molecule, including polymers, the LCAO approximation and Bloch s theorem can be used to describe the delocalized crystalline orbitals as a periodic combination of functions centered... 

Firstly, note that it is the electron density that is observed experimentally in X-ray diffrachon (XRD) experiments and so is used to define the unit cells of crystalline solids. In a periodic system, this means that the electron density has to have the same repeat distance as the lathee in all directions. Bloch s theorem points out that the restriction this imposes on the underlying wavefunchons is actually less rigorous since the relationship between electron density, p(r), at an arbitrary point, r, and the one-electron wavefunctions obtained from calculations (HF or DFT) is ... 

Both approaches treat the solid as an infinite three-dimensional array of unit cells. This enables Bloch s theorem to be applied so that the electronic wave function of the solid can be written... 

LDA methods have been employed to investigate solids in two types of approaches. If the solid has translational symmetry, as in a pure crystal, Bloch s theorem applies, which states that the one-electron wave function n at point (r + ft), where ft is a Bravais lattice vector, is equal to the wave function at point r times a phase factor ... 

The proof of Bloch s theorem can be found in any text of solid state physics and will not be reproduced here. In the course of that proof it is shown that the eigenfunctions T/f k (r) can be written in the form... 

Because of the presence of the regularity associated with a crystal with periodicity, we may invoke Bloch s theorem which asserts that the wave function in one cell of the crystal differs from that in another by a phase factor. In particular, using the notation from earlier in this section, the total wave function for a periodic solid with one atom per unit cell is... 

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