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The standard mapping between different N-particle states in Fock space is very abstract and consists of combinations of creation and annihilation operators being formally applied to the vacuum state. The resulting formalism provides us with the correct symmetries in the case of fermions and bosons, but the permanents and Slater determinants that are formally joined by the action of these operators must be individually constructed from product states using symmetrizers and antisymmetrizers. Surely, this abstract approach does not correspond to a direct mapping between different N-particle states in Fock space. In the following, we will examine direct mappings in Fock space, which are generalizations of an old determinant division scheme by Schweins (1825). The suggested new scheme is based on a suitable definition of isolated states. We will show that all the classical results obtained by Schweins may be recovered. Furthermore, the new division scheme may easily be extended to permanents. This leads to direct mappings in Fock space, which correspond to creation and annihilation operators.

The electronic structure of a class of [Zn4O(CO2)6] based metal–organic frameworks (MOFs) is theoretically analyzed by means of density functional perturbation theory. The calculated static dielectric constants vary in a range between 1.33 and 1.54, characterizing the structures as ultralow-kdielectric materials and confirming earlier estimates qualitatively. We also present the results of first-principle calculations of the real and imaginary parts of the dielectric function and give the frequency-dependent dielectric constant up to the near-ultraviolet, which is important for high frequency semiconductor and optical applications of MOFs. The dielectric and electronic properties are governed by the linker molecules, so that the band gap and the dielectric constant can be engineered.

Photonic crystals are not only responsible for the beautiful colors of opal gems, butterflies and peacocks. They are also of great technological interest as the key elements of a large variety of modern photonic devices [1]. We present a domestic simulation package, which has been developed for the optimum design of such devices. It basically consists of a number of MATLAB routines, which should allow for a convenient determination of photonic band structures and related properties. We briefly mention some of the main numerical challenges involved in the development of the most effective code. Then we discuss the main features of photonic band structures obtained for complex dielectric materials, and finally describe some useful numerical features, that will be implemented in future versions of our program package.

Nanosized electronic systems are characterized by ballistic transport properties, which implies that the wave character of the electron plays a dominant role in such systems. We summarize some of the basic theoretical concepts involved, and point out various analogies between scalar electronic waves and electromagnetic vector waves. Those analogies should allow for a direct transfer of key technological concepts from optics or photonics to nanoelectronics. As an example, we discuss the concept of electronic metamaterials, and suggest a possible realization of nanometer sized electronic devices, based on novel low-dimensional carbon and boron nanomaterials.

Photonic crystals are not only responsible for the beautiful colors of opal gems, butterflies and peacocks. They are also of great technological interest as the key elements of a large variety of modern photonic devices [1]. We present a domestic simulation package, which has been developed for the optimum design of such devices. It basically consists of a number of MATLAB routines, which should allow for a convenient determination of photonic band structures and related properties. We briefly mention some of the main numerical challenges involved in the development of the most effective code. Then we discuss the main features of photonic band structures obtained for complex dielectric materials, and finally describe some useful numerical features, that will be implemented in future versions of our program package.

Nanosized electronic systems are characterized by ballistic transport properties, which implies that the wave character of the electron plays a dominant role in such systems. We summarize some of the basic theoretical concepts involved, and point out various analogies between scalar electronic waves and electromagnetic vector waves. Those analogies should allow for a direct transfer of key technological concepts from optics or photonics to nanoelectronics. As an example, we discuss the concept of electronic metamaterials, and suggest a possible realization of nanometer sized electronic devices, based on novel low-dimensional carbon and boron nanomaterials.

We illustrate how ab initio numerical simulation methods can be used to check and improve structure models for i(Al–Pd–Mn). By focusing our study on the optimization of a small approximant, we obtain a number of general structural and compositional rules simple enough to be applicable to the quasicrystalline structure itself.

The rhombohedral unit cells of α-boron crystals and of hypothetical α-boron quasicrystals were investigated by applying ab initioquantum-chemical methods. The atomic decorations of such unit cells are generally based on a suitable arrangement of eight boron icosahedra, and thus the mechanism of their cohesion was simulated by relaxing a B96supercluster composed of icosahedral boron clusters located at the vertices of the rhombohedral unit cells. Also, several alternative structures of B96clusters were proposed and compared to the rhombohedral phases. To this end, total energies of those structures were computed on the basis of all-electron calculations at the Hartree–Fock self-consistent field level of theory using the standard STO3G basis set. Due to the large number of degrees of freedom, the geometries of all isomers were optimized under reasonable symmetry constraints, but using analytical gradient methods. The resulting total energies show that segments of nanotubes and quasiplanar sheets turn out to be more stable than the unit cells of α-boron (quasi) crystals, indicating new kinds of materials.

Recently we have modified a two-stage variant of the cut-and-project method used to generate layered 8-fold (Ben-Abraham, 2004) and 12-fold (Ben-Abraham et al., 2004) quasiperiodic structures in three dimensions, and obtained 8-fold, 10-fold and 12-fold quasiperiodic structures via cut-and-project from various four dimensional lattices (Ben-Abraham and Quandt, 2006). However four dimensions are certainly not the most suitable playground to illustrate the basics of our general two-stage projection method. To this end we cut and project a periodic structure in three dimensions onto two-dimensional physical space, such that a second cut and projection onto a line yields a quasiperiodic structure. This simple example contains all of the basic elements of our two-stage projection method, but now the essential steps may directly be visualized.