Bloch theorem periodic function

Have the correct form to be a solution of Equation 1.8. As a result, the Bloch theorem affirms that the solution to the Schrodinger equation may be a plane wave multiplied by a periodic function, that is [5,6],... 

The Bloch theorem states that the eigenfunctions of the Hamiltonian (4.71), (4.72) are products of a wave of the form (4.73) and a function that is periodic on the lattice, that is. 

The important term electronic band structure is identical with the course of the energy of an extended wave function as a function of k, and we seek for E ip k,r)), the crystal s equivalent to a molecular orbital diagram. As stated before, the fc-dependent wave function ip k,r) is called a crystal orbital, and there may be many one-electron wave functions per k, just as there may be several molecular orbitals per molecule. Due to the existence of these periodic wave functions, there results stationary states in which the electrons are travelling from atom to atom the Bloch theorem thereby explains why the periodic potential is compatible with the fact that the conduction electrons do not bounce against the ionic cores. 

The periodicity of the nuclei in the system means that the square of the wave function must display the same periodicity. This is inherent in the Bloch theorem (eq. (3.75)), which states that the wave function value at equivalent positions in different cells are related by a complex phase factor involving the lattice vector t and a vector in the reciprocal space. 

The Bloch theorem (6.5) also holds for any vector that does not lie in the first zone. The Bloch function is periodic in the reciprocal space. 

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) . 

This way, the crystalline orbital should be constructed from the atomic ones, (j)(r located at the distance respecting the origin of the reference system of (direct) lattice, so that also the Bloch theorems (3.88) and (3.89) to be respected for including the periodicity of the lattice in crystalline eigen-function. 

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

The simpler PW methods are the most popular in the Kohn-Sham periodic-systems calculations. Plane waves are an orthonormal complete set any function belonging to the class of continuous normalizable functions can be expanded with arbitrary precision in such a basis set. Using the Bloch theorem the single-electron wavefunction can be written as a product of a wave-like part and a cell-periodic part ifih = exp(ifer)Mj(r) (see Chap. 3). Due to its periodicity in a direct lattice Uj(r) can be expanded as a set of plane waves M<(r) = C[Pg.281]

In the periodic systems the basis sets are chosen in such a way that they satisfy the Bloch theorem. Let a finite number of contracted GTFs be attributed to the atom A with coordinate in the reference unit cell. The same GTFs are then formally associated with all translationally equivalent atoms in the crystal occupying positions rA + (In (In is the direct lattice translation vector). For the crystal main region of N primitive unit cells there are N tha Gaussian-type Bloch functions (GTBF)... 

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,... 

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],... 

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 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. 

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... 

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... 

To complete the description, it is also necessary to understand the nature of the k-vectors themselves. For the purposes of the present discussion, we consider a one-dimensional periodic chain along the x-direction of N atoms with a spacing a between the atoms on the chain. One statement of Bloch s theorem, which we set forth in one-dimensional language, is the assertion that the wave function in the n cell f/n and that in the n -b 1) cell, is given by... 

The periodicity of the lattice means that the values of a function (such as the electron density) will be identical at equivalent points on the lattice. Likewise there is a relationship between the wavefunction at a point (x in our ID lattice) and at an equivalent point elsewhere on the lattice (for the ID lattice this would he x + na, where n is an integer). Bloch s theorem provides the link each allowed lattice wavefunction must satisfy the following relationship ... 

The TB solution for the periodic system is obtained first by constructing the Bloch functions i k) relative to each AO in the basis for the representation of the TB Hamiltonian (three in our case), and then constructing the representative matrix H (k) in the basis of the Bloch functions The solution is factorised for each point of reciprocal space, k. Following Bloch s theorem we have ... 

When it comes to crystals, it is clear that the system under study is trans-lationally invariant in all three spatial directions, and Bloch s theorem utilizes the translational s)mimetry to generate the crystal s wave function, composed of crystal orbitals which are also called electronic bands. We therefore imagine an idealized solid-state material whose electronic potential V possesses the periodicity of the lattice, expressed by a lattice vector T, that is... 

Notice that the MOs for benzene as given by Eq. (15-4) are produced from an equation having the following form There is an exponential term that, for each MO, supplies the coefficients for various carbons in the molecule, and there is a basis set of functions located on the various carbons. This basis set has the same periodicity as does the molecule. (In the case of benzene, we have so far taken this to be six identical 2p r AOs.) Wavefunctions for all periodic systems have this same form—an exponential (or equivalent trigonometric) expression times a periodic basis. This is the content of Bloch s theorem, which we prove in the next section. 

Because the ions are arranged with periodicity, a periodic potential is generated. Using Bloch s theorem [38], this periodicity can be used to reduce the infinite number of one-electron wave functions, enabling to simply calculate only for the number of electrons within the imit cell. The wave function then takes the form of the product of a wavehke part and a cell periodic part... 

Bloch s theorem states that wave functions i/ (r) can always be chosen to have the periodicity of the Bravais lattice apart from a single multiplicative factor of exp(ik r), i.e.,... 

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