Born-von Karman periodic boundary conditions

One way of seeing this explicitly is to consider the Schrodinger equation modified for a periodic lattice with Born-von Karman periodic boundary conditions assuming a wavefunction ij/(r) = Eqcq exp(iq r) and a potential U(r) which has the periodicity of the lattice U(r) = 2,GUG exp(/G r), where the Fourier58 coefficients UG are given by UG = JCeiiG(r) exp (—zG r) dr, the Schrodinger equation is rewritten as... 

In the theory of electronic structure two symmetric models of a boundless crystal are used or it is supposed that the crystal fills aU the space (model of an infinite crystal), or the fragment of a crystal of finite size (for example, in the form of a parallelepiped) with the identified opposite sides is considered. In the second case we say, that the crystal is modeled by a cyclic cluster which translations as a whole are equivalent to zero translation (Born-von Karman Periodic Boundary Conditions -PBC). Between these two models of a boundless crystal there exists a connection the infinite crystal can be considered as a limit of the sequence of cychc clusters with increasing volume. In a molecule, the number of electrons is fixed as the number of atoms is fixed. In the cyclic model of a crystal the number of atoms ( and thus the number of electrons) depends on the cyclic-cluster size and becomes infinite in the model of an infinite crystal. It makes changes, in comparison with molecules, to a one-electron density matrix of a crystal that now depends on the sizes of the cyclic cluster chosen (see Chap. 4). As a consequence, in calculations of the electronic structure of crystals it is necessary to investigate convergence of results with an increase of the cyclic cluster that models the crystal. For this purpose, the features of the symmetry of the crystal, connected with the presence of translations also are used. 

Fig.3.3. Definition of a macrocristal and illustration of the Born-von Karman periodic boundary conditions...

In the previous section we introduced k = /cibi + K2I32 + /csbs as a convenient index to label the wavefunctions. Here we will show that this index actually has physical meaning. Consider that the crystal is composed of Nj unit cells in the direction of vector aj (j = 1, 2, 3), where we think of the values of Nj as macro-scopically large. N = A iA 2A 3 is equal to the total number of unit cells in the crystal (of order Avogadro s number, 6.023 x 10 ). We need to specify the proper boundary conditions for the single-particle states within this crystal. Consistent with the idea that we are dealing with an infinite solid, we can choose periodic boundary conditions, also known as the Born-von Karman boundary conditions,... 

Fig.2.3. Born-von Karman or periodic boundary conditions. The linear chain we are studying contains N unit cells with lattice parameter a and is part of an infinite chain. For the displacements we require that u( N) = u( )...

It is convenient to begin the theory of lattice dynamics of the three-dimensional crystal with the assumption of an indefinitely extended crystal. The absence of bounding surfaces results in a perfect lattice periodicity which greatly simplifies the theory. The assumption of an infinitely extended crystal, however, leads to infinite values of several quantities, such as the volume, energy, etc. In Sect.3.3, we shall introduce the Born-von Karman or periodic boundary conditions by which these quantities will be normalized to a finite value. 

We started our discussion in this chapter by assuming an infinitely extended crystal. Now we shall introduce the Born-von Karman or periodic boundary conditions as we did for the linear chain in Chap.2. For this purpose we subdivide the infinitely large crystal into "macrocrystals". Each macrocrystal is a parallelepiped defined by the vectors N a, N a, N a, where 1 2 3 primitive translation vectors and N, N2, are large... 

The one-electron wave functions i/ (r) should correspond to propagating rather than standing waves. The mathematical procedure for to obtaining propagating waves was developed by Born and von Karman [11], They considered a cube of volume V with sides L. The electron waves leave the metal through a face of the cube and reenter it simultaneously through the opposite face. The boundary conditions are hence periodic ... 

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