Bloch theorem wave functions

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) ... 

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) ... 

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],... 

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. 

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 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. 

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... 

Usual crystal orbitals extend all over the system concerned and are obtained so as to fulfill the Bloch theorem (Bloch, 1928). However, it is known to be rather useful to convert the wave function of the system into the localized function for the purpose of the discussion of the local nature of the system, such as the exciton. One such function is the Wannier function ap derived by the Fourier transformation of the crystal orbital... 

In an infinite crystal the potential is invariant under lattice translations. One may therefore apply Bloch s theorem, and take the wave functions in the form of eigenfunctions of the translational operators, i.e.,... 

Since the symmetry generated by the screw operator S is isomorphic with the one-dimensional translational group, Bloch s theorem is valid also in this case. Therefore, the one-electron wave functions will transform under S as... 

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... 

In modem language, Bloch s theorem can be expressed as follows. For a given wave function tp k, r) which fulfills Schrodinger s equation, there exists a vector k such that translation by a lattice vector T is equivalent to multiplication by a phase factor ... 

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. 

Let us assume that we have to do with N equidistant atoms located on a circle, the nearest-neighbor distance being a. From the Bloch theorem. Eq.(9.12). for the wave function we have... 

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. 

At the age of 23, Felix Bloch published an article Uber die Quantenmechanik der Ekktmnen bi KristaUgateni in Zeitschriji flir PhysUc, 52 (1928) 555 (onty two years after Schrodinger s historic pubhcation) on the translation symmetry of the wave function. This result is known as the Bloch theorem. This was the first appheation of LCAO expansion. A book peared in 1931 by Leon Brillouin entitled Qiiaiitenstatistik (Springer Verlag, Berlin, 1931), in which the author introduced some of the fundamental notions of band theory. The first ah bntio calculations for a potymer were carried out Iqr Jean-Marie Andre in a p per Self-Consisteiit Field Theory for the Electronic Stnicture of Polymers published in ihc Journal of the Chemical ff sics, 50 (1%9) 1536. 

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]

According to Bloch s theorem, the Kohn-Sham wave-functions, can be written as... 

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... 

The effect of inter-site hopping is then introduced into the system. The manifold of basis states are limited to those in which the local correlations have been diagonalized. The wave functions for the composite particles then obey Bloch s theorem, which results in the formation of a dispersion relation consisting of two bands for the quasi-bosons the first band describes spinless quasi-boson excitations, the second band describes the magnetic quasi-bosons. Although these composite particles are bosons in that they commute on different sites, they nevertheless have local occupation numbers which are Fermi-Dirac like. 

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 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.,... 

In the previous section extended wave functions were built out of localized basis functions with the aid of Bloch s theorem. In the simple case where only one localized function is used per unit cell, a single band is thus generated. Since the process by which the extended Bloch functions are formed from the localized ones is just a Fourier transform (12), it may be inverted to recover a localized function from the extended wave functions, according to... 

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