Boundless crystal

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. 

In the theory of electronic structure of crystals, we also use the molecrdar-cluster model being based on physical reasons we choose a molecular fragment of a crystal and somehow try to model the influence of the rest of a crystal on the cluster chosen (for example, by means of the potential of point charges or a field of atomic cores). Prom the point of view of symmetry such a model possesses only the symmetry of point group due to which it becomes impossible to estabhsh a connection of molecular-cluster electronic states with those of a boundless crystal. At the same time, with a reasonable molecular-cluster choice it is possible to describe well enough the local properties of a crystal (for example, the electronic structure of impurity or crystal imperfections). As an advantage of this model it may also be mentioned an opportunity of application to crystals of those methods of the account of electronic correlation that are developed for molecules (see Chap. 5). 

We assume here that there is a finite number of electrons in the system. This is certainly true for molecules, but for crystals it imphes that we are using the model of a finite but boundless crystal (cychc model), i.e. we consider the bulk of a crystal with cyclic boundary conditions imposed on opposite sides. The Hamiltonian He, being an operator acting on the functions depending on the electron coordinates r is invariant under symmetry operations transforming the nuclear equilibrium configuration into itself. Later, we call it the symmetry group of a crystal. 

Although the box matrix has nine elements, only six of them are independent. Thus, in setting up a box matrix for equilibrium simulations from crystal cell parameters, three of the off-diagonal elements can be arbitrarily chosen to be zero. This choice cannot be arbitrary for planar Couette flow. This is because, if the initial values of 21 aud 23 are chosen to be nonzero, and will evolve boundlessly in time. On the other hand, if 21 and f 23 were to be zero at t = 0, then during NPT dynamics they would oscillate around zero, and thus the average rate of change of and hi would be zero. 

In real crystals of macroscopic sizes translation symmetry, strictly speaking, is absent because of the presence of borders. If, however, we consider the so-called bulk properties of a crystal (for example, distribution of electronic density in the volume of the crystal, determining the nature of a chemical bond) the influence of borders can not be taken into account (number of atoms near to the border is small, in comparison with the total number of atoms in a crystal) and we consider a crystal as a boundless system, [13]. 

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See also in sourсe #XX -- [ ]

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