Solid-state systems unit cell

Apart from the methodological aspects, solid state systems possess many interesting properties that are immaterial for single molecules. In single molecules, point symmetry usually decreases as the size of the molecule increases. Molecules with more than, say, 20 atoms often lack symmetry. Crystalline systems, contrarily, usually maintain high point symmetry, even in the case of large unit cell systems, like zeolites and garnets. 

There is, however, an alternative approach, which lends itself particularly well to performing calculations on solid-state systems. This involves us thinking of electrons as being delocalized, (nearly) free particles. If we think of metallic systems, rather than molecular systems, then the concept is clear. Electrons are free to wander throughout the unit cell, their behavior modified by the ionic lattice. 

We have introduced an important concept here - the unit cell. In crystallography, the unit cell represents the budding block from which the infinite three-dimensional crystal lattice is built. If we are to model solid-state systems we must make use of a similar concept, from which we can build an infinite array of rephcas positioned in accordance with the crystallographic space-group symmetry operations. 

X", which can be either the whole space or specified part of it, e.g., a unit cell for solid state systems. The partitioning into the regions /t, obeys at least the two rules ... 

Computational solid-state physics and chemistry are vibrant areas of research. The all-electron methods for high-accuracy electronic stnicture calculations mentioned in section B3.2.3.2 are in active development, and with PAW, an efficient new all-electron method has recently been introduced. Ever more powerfiil computers enable more detailed predictions on systems of increasing size. At the same time, new, more complex materials require methods that are able to describe their large unit cells and diverse atomic make-up. Here, the new orbital-free DFT method may lead the way. More powerful teclmiques are also necessary for the accurate treatment of surfaces and their interaction with atoms and, possibly complex, molecules. Combined with recent progress in embedding theory, these developments make possible increasingly sophisticated predictions of the quantum structural properties of solids and solid surfaces. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

The three-dimensional symmetry is broken at the surface, but if one describes the system by a slab of 3-5 layers of atoms separated by 3-5 layers of vacuum, the periodicity has been reestablished. Adsorbed species are placed in the unit cell, which can exist of 3x3 or 4x4 metal atoms. The entire construction is repeated in three dimensions. By this trick one can again use the computational methods of solid-state physics. The slab must be thick enough that the energies calculated converge and the vertical distance between the slabs must be large enough to prevent interaction. 

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

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). 

Figure 5, Reaction pathways for (1) C+0 to CO on Pt(l 11) (II) C+N to CN on Pt(111) and (III) C+H to CH on Cu(I II). The large grey circles are surface atoms, black circles are C. The white circles are O, N and H atoms in I, II and III, respectively. The solid lines indicate the surface unit cell. For clarity, the periodic nature of the systems is not shown and only two layers of surface atoms are displayed. Panel (d) represents the transition states (TSs) of each reaction.

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