Periodic boundary conditions three-dimensional crystals

Most of the simulations reviewed were performed in two dimensions, although some simulations of three-dimensional detonations are discussed in Sec. 3.1.2. Equilibrated molecular crystals for AB Model I in both two and three dimensions are shown in Fig. 2. This model has slightly longer bond lengths than either Model II or III, but otherwise the starting crystal structures are similar for all three models. To model an infinite crystal, almost all detonation simulations were carried out with periodic boundary conditions enforced perpendicular to the direction of the shock propagation. 

Fig. 3 One-dimensional loading profiles of benzene across a NaX zeolitic, single crystal membrane. The loading is the spatial average over planes perpendicular to the main diffusion direction (three-dimensional simulations are conducted periodic boundary conditions are employed in the transverse direction and Robin at the membrane interfaces exposed to the high- and low-pressure sides). The inset shows a schematic of the membrane. (View this art in color at www. dekker.com.)...

At this point it is appropriate to define the bovuidary conditions of the simulated system. If we wish to simulate a bulk phase with a limited number of particles, the simulation of an isolated cluster or drop introduces unacceptable disturbances due to boundary effects. The usual method-, to avoid such effects is to use periodic boundary conditions the simiilated particles are put in a cubic (or tricllnic) box that is repeated in a regular three-dimensional lattice (fig. 2). Thus in fact one simulates a crystal. Since the periodicity is an artefact of the... 

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 start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

The vectors k are 3D, 2D, or ID for a crystal, slab, or periodic polymer respectively. Keep in mind that the nomenclature nD refers to the number of cartesian directions in which nuclei have periodic ordering. The electron density is three-dimensional, as is r, whatever the system periodicity. Thus, when we treat an ultra-thin film (UTF) with GTOFF, we are not doing a super-cell calculation on a fictitious crystal consisting of the UTF interspersed by layers of vacuum . GTOFF can do such super-cell calculations but more importantly, it can handle the UTF as a fi ee-standing object periodic in two Cartesian directions and of finite thickness in the third direction (conventionally z), subject to vacuum boundary conditions in z. Note also that a 2D GTOFF calculation does not require inversion s)mimetry with respect to z, hence can treat an even number of nuclear planes as readily as an odd number. 

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