Aperiodic systems

M. A. Dulea, Physical Properties of One-Dimensional Deterministic Aperiodic Systems. Linkoping Studies in Science and Technology, No. 269, Linkdping (1992). 

Liegener, C.-M., A. Sutjianto, and J. Ladik. 1990. The Treatment of Electron Correlation in Aperiodic Systems. HI. Application to Polypeptides. Chem. Phys. 145, 385-388. Mavri, J., F. Avbelj, and D. Hadzi. 1989. Conformation of N-Acetyl-L-Pro-D-Ala-N -Methyl Tripeptide Empirical, Semi-Empirical MO and Ab Initio MO Calculations. J. Mol. Struct. (Theochem) 187, 307-315. 

Surely, only irredundant information related to irreplaceable elements of the system is valuable. The redundant (replicated or duplicated) infor mation also is of vital importance to minimize the probability of destruct ing the valuable information as it is transferred. In this viewpoint, any aperiodic system contains a much larger number of irreplaceable units and, therefore, a greater quantity of irredundant information than a peri odic system of the same gross composition with the interchangeable elements (see Figure 6.1). 

One of the interesting implications of this section is that the walks, and hence moments, may be generated without any recourse to the translational synunetry of the solid, or the point group of the molecule. In the context of extended arrays therefore, the moments method may be used in the study of aperiodic systems such as are found " in amorphous materials and in surface phenomena. In this article we shall exclude such areas fom discussion, and will concentrate on structural problems in molecules and crystalline solids. 

Then the values aj (diagonal elements of H) can be determined from the positions of the bands of the components (the middle point or weighted middle points of the bands) and the off-diagonal elements / , from the widths of the bands. The method is very fast, but from obvious mathematical reasons (5 diagonal or 7 diagonal matrices even in the first neighbors interactions case) it cannot be applied for 2D or 3D aperiodic systems. 

Bohr was well aware of this limitation of the adiabatic principle but continued to use it even for many-electron atoms in the hope that it might stiU remain vahd for such aperiodic systems. He repeatedly acknowledged this point in his writings ... 

Crystal with single point defect (aperiodic system)... 

If a single point defect is introduced into an otherwise perfect crystal, the translational symmetry is, in principle, lost completely (see Fig. 10.1), and equations should be solved for all atoms of the solid, i.e. for the aperiodic system [294]. Obviously, the application of some kind of simphfied model is desirable. In many cases the electronic states of defects in a crystal are locahzed in some relatively small region. The degree of this localization depends on the nature of the defect and of the host crystal. The local... 

The recent trends in the theory of the point defects in soUds and of the crystalline surfaces, formally aperiodic systems, are connected with the widespread application of periodic boundary conditions (PBC) and computer codes developed for the perfect crystals, see [568,678,679] and references therein. This allows one to use the same numerical schemes both for the host and defective crystal to study changes induced... 

In this chapter we have seen that in modern quantum chemistry of sohds the first-principles periodic calculations are successfully extended to formally aperiodic systems - defective crystals and solid solutions. In the next chapter we consider the application of periodic models in the calculations of crystalline surfaces and adsorption. 

Matsuda, H., Ogita, N. Coarse-grained quantities in aperiodic systems. (II) Frequency spectra of aperiodic chains. Progr. Theor. Phys. 38, 81 (1967). 

The following constitutes a skeletal review of the basic ideas which are common to the entire field of band structure, i.e., independent of the details of the techniques used or the substances studied (with the exception of aperiodic systems). The material will be quite familiar to many readers and is therefore covered in a cursory fashion. The standard textbooks of elementary solid-state theory contain fuller descriptions of many of the matters which are reviewed. 

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