Bloch theorem generalized

For future reference we cite here without proof a useful identity that involves the harmonic oscillator Hamiltonian H = p /2m + (1 /2)ma> q and an operator of the general formH = explaip + a2q with constant parameters ai and that is, the exponential of a linear combination of the momentum and coordinate operators. The identity, known as the Bloch theorem, states that the thermal average A )t (under the hannonic oscillator Hamiltonian) is related to the thennal average ((aip + Q 2 )2)t according to... 

An ion channel is modeled for simplicity as a rigid, regular double helix of sources. Although there is no particular requirement to represent the channel as a double helix, a single helix will reflect minima for the ion-source interaction that are also helical. The virtue of the double helix, therefore, is an axial symmetry that makes the subsequent analysis reasonably simple. It is in fact possible to carry out a similar analysis for the general single helix system using an extension of the Bloch theorem that has been explored by Mintmire et al. [13] for polymer systems. 

The spinors further commute with the Kohn-Sham Hamiltonian and obey a commutative multiplication law, thereby making them an Abelian group isomorphic to the usual translation group [133]. But this means that they have the same irreducible representation, which is the Bloch theorem. So, we therefore have the generalized Bloch theorem ... 

We now move on to consider a two-dimensional square lattice in the (x, y) plane, where the inter-lattice spacing is still a. The Bloch theorem is now written in the following more general form ... 

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). 

That is of the essence that the resulting pseudoeigenfunction V pseudo may be well represented by plane waves. The most general solution has to satisfy the Bloch theorem (Section 5.2) and boundary conditions. Each electronic wave function in a periodic crystal lattice can be written as the product of a cell-periodic part and a wave-like part. 

The general solution of Eq. (12.6) is given by the Floquet-Bloch theorem, as a sum of products of a spatially periodic amplitudes A( ) and B( ) with oscillating exponential functions... 

Thus, the periodicity of the density of probabilities (3.80) was lowered at the level of eigen-function such as Eq. (3.88), regaining the celebrated Bloch theorem of Eq. (3.34), here in a generalized form the eigen-function of an electron in a periodic potential can be written as a product of a function carrying the potential periodicity and a basic exponential factor exp(/lx) . 

As such there can be immediately verified how the eigen-fimctions of type (3.101) satisfy the Bloch theorem under the general form ... 

Employing the superposition principles with periodic constrains to formulate the wave function general forms specific to solid state -the Bloch theorem and orbitals ... 

The model above describes a free electron in a one dimensional crystal at the BZ boundaries. In the more general case, the solution of the time independent SchrSdinger equation (5) in a three dimensional crystal can be obtained with the help of the Bloch theorem, which states that if the potential energy V(r) is periodic, the solutions ti k(r) of... 

Here, R = n- ais the position of the nth atom measured along the circumference of the ring and k is a quantum number, controlling the period with which the complex phase factor e " oscillates along the ring, k is called the wave vector of state (r). The wave function as specified in Eq. (5.2) is a special expansion (namely, in terms of atomic orbitals) of the electronic wave function in a crystalline solid. More generally, wave functions in a periodic solid obey the Bloch theorem, stating that at two equivalent points r and r + R the wave function (called Bloch junction) differs only by a phase factor e ", that is,... 

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. 

If we know the translational symmetry of an extended solid (Bloch s theorem) and also have a trustworthy strategy on how to deal with the nuclear potential and the electron-electron interactions (by, say, a semiempirical method or by DFT), we are ready to explicitly calculate the band structure of any real material. Although we have done this before for idealized systems (see sketches in Section 2.6), let us now attack the problem once again, but in more general terms. For real materials, one needs to solve SchrBdinger s equation using the true potential v r), namely. 

Q 15-5 General Form of One-Electron Orbitals in Periodic Potentials—Bloch s Theorem... 

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