Periodic infinite systems

In order to simulate more closely the behavior of an infinite system, periodic boundary conditions are imposed on the solution to the equations of motion. If a molecule labeled i is located at position (xj, yt, z,) at time t, we imagine that there are 26 additional images of i located at (x, L, 0 yj L, 0 L, 0). The particle and its 26 images have the same orientation and velocity. Another molecule j may interact with any i within its interaction range. If molecule i should cross a face of the box, it is reinserted at the opposite face. Constant density is thus maintained. These periodic boundary conditions avoid the strong surface effects that would result from a box with reflecting walls. 

Let us consider an ensemble of N molecules in a fixed volume V with a fixed total energy E. This is a microcanonical ensemble of classical statistical mechanics. Typical values for N used in these simulations of chemical interest is of the order of hundreds to a few thousands. In order to simulate an infinite system, periodic boundary conditions are invariably imposed. Thus a typical MD system would consist of N molecules enclosed in a cubic box with each side equal to length L. MD solves the equations of motion for a molecule i ... 

VIII. SIMULATING INFINITE SYSTEMS, PERIODIC BOUNDARY CONDITIONS... 

The LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

Infinite Lattices Although cyclic behavior is certain to occur under even class c3 rules for finite systems, it is a rare occurrence for truly infinite systems cycles occur only with exceptional initial conditions. For a finite sized initial seed, fox example, the pattern either quickly dies or grows progressively larger with time. Most infinite seeds lead only to complex acyclic patterns. Under the special condition that the initial state is periodic with period m , however, the evolution of the infinite system will be the same as that of the finite CA of size N = m-, in this case, cycles of length << 2 can occur. 

For periodic systems special care must be taken, because it is not possible to define properly a vector potential for a constant magnetic field in an infinite system. In finite systems, however, this problem does not exist, and a given magnetic field B can be described by a vector potential ... 

The fourth term on the right-hand side of eq. (11.3) is the electrostatic interaction (Coulomb s law) between pairs of charged atoms i and j, separated by distance r j. Since electrostatic interactions fall off slowly with r (only as r-1) they are referred to as long-range and, for an infinite system such as a periodic solid, special techniques, such as the Ewald method, are required to sum up all the electrostatic interactions (cf. Section 7.1) (see e.g. Leach, Jensen (Further reading)). The... 

Thus the response to a periodic injection for very general boundary conditions can be found by substituting p = +iu into Eq. (34). The results for the general case would be very complicated so, as an illustration of the form of the periodic response, we will consider only the simplest case a doubly infinite system. For such a system, = H/ = Dii,, and Eq. (34) reduces to... 

Another interesting possibility is die use of plane waves as basis sets in periodic infinite systems (e.g., metals, crystalline solids, or liquids represented using periodic boundary conditions). "While it takes an enormous number of plane waves to properly represent the decidedly aperiodic densities that are possible within the unit cells of interesting chemical systems, the necessary integrals are particularly simple to solve, and dius diis approach sees considerable use in dynamics and solid-state physics (Dovesi et al. 2000). 

Both CIS and TDHF have the correct size dependence and can be applied to large molecules and solids (we will shortly substantiate what is meant by the correct size dependence ) [42-51], It is this property and their relatively low computer cost that render these methods unique significance in the subject area of this book despite their obvious weaknesses as quantitative excited-state theories. They can usually provide an adequate zeroth-order description of excitons in solids [50], Adapting the TDHF or CIS equations (or any methods with correct size dependence, for that matter) to infinitely extended, periodic insulators is rather straightforward. First, we recognize that a canonical HF orbital of a periodic system is characterized by a quantum number k (wave vector), which is proportional to the electron s linear momentum kh. In a one-dimensional extended system, the orbital is... 

In contrast to fhe static methods discussed in the previous section, molecular dynamics (MD) includes thermal energies exphcitly. The method is conceptually simple an ensemble of particles represents fhe system simulated and periodic boundary conditions (PBC) are normally apphed to generate an infinite system. The particles are given positions and velocities, fhe latter being assigned in accordance with a... 

In slab calculations, a finite number of layers mimicks the semi-infinite system, with a two-dimensional (2D) translational periodicity. A minimal thickness dmin is required, so that the layers in the slab centre display bulk characteristics. Practically speaking, dm-,n should be at least equal to twice the damping length of surface relaxation effects, which depend upon the surface orientation. In plane wave codes, the slab is periodically repeated... 

Fig. 5 Iso-density map of an excess electron state on TiO2(110). Titanium and oxygen atoms are marked by small and large circles, respectively, and the semi-infinite system is represented by a slab with vacuum above and below and lateral periodic boundary conditions (from Ref. 157).

When we solve the problem numerically, the number of surface elements, and consequently, the size of the dielectric boundary surfaces must be finite. This is in accordance with the practice in a simulation, where the simulation cell is also finite. To approximate an infinite system in a simulation, periodic boundary conditions are applied in the x and y directions. The closest image convention is used not only for the ionic distances but any distances between... 

An infinite system which is periodic in p directions (p = 1, 2, 3 for polymers, surfaces, and crystals, respectively) is described by a reference unit cell and p basis vectors, ai,. ap. These vectors point to the directions of the periodicity... 

The relatively small system size feasible in molecular simulation can have a large effect on the results [21,23]. To minimize such finite size effects, an infinite system is mimicked by appl3dng periodic boundary conditions. These conditions effectively create an infinite number of copies of the system in each direction. The cubic box is simplest to implement, but more complex tilings of space such as the truncated octahedron, or the dodecahedron results in smaller system sizes because they require fewer solvent molecules, and hence are more efficient [21,22]. 

In tune with the above introductory remarks, we have arranged this review in the following way Section II deals with the oriented gas model that employs simple local field factors to relate the microscopic to the macroscopic nonlinear optical responses. The supermolecule and cluster methods are presented in Section III as a means of incorporating the various types of specific interactions between the entities forming the crystals. The field-induced and permanent mutual (hyper)polarization of the different entities then account for the differences between the macroscopic and local fields as well as for part of the effects of the surroundings. Other methods for their inclusion into the nonlinear susceptibility calculations are reviewed in Section IV. In Section V, the specifics of successive generations of crystal orbital approaches for determining the nonlinear responses of periodic infinite systems are presented. Finally,... 

Anticipating such scenarios, we use in many applications periodic boundary conditions as a trick to represent infinite systems by taking the periodic box dimensions to infinity at the end of the calculation. We will see several examples below. 

These results of Bruns have been supplemented by Poincare s investigations 1 these lead to the following conditions Apart from special cases, it is not possible to represent strictly the motion of the perturbed system by means of convergent /-fold Fourier series in the time and magnitudes Jk constant in time, which could serve for the fixation of the quantum states. For this reason it has hitherto been impossible to carry out the long-sought-for proof of the stability of the planetary system, i.e. to prove that the distances of the planets from one another and from the sun remain always within definite finite limits, even in the course of infinitely long periods of time. 

It is the right-hand side that is to be identified with (2.37) and the left-hand side with (2.30c). The latter equation can be thought of as the equation relevant to a sample of material of macroscopic radius R embedded in an infinite system of the same material (i.e., an infinite system of dielectric constant e). Moreover, on the macroscale determined by the length unit R, the system external to the sphere can be regarded dielectrically simply as a continuum of dielectric constant e. Thus (2.37) and (2.30c) are the relevant equations for the same macroscopic spherical sample embedded in continua of dielectric constant 1 and e, respectively. These results can be generalized to a sample embedded in a continuum of arbitrary dielectric constant e, as discussed by de Leeuw, Perram, and Smith, who use the generalization to illuminate the status of Ewald summation in systems with periodic boundary conditions. We review their work in Section III.C. 

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