Modeling Quantum Solids

First-principles models of solid surfaces and adsorption and reaction of atoms and molecules on those surfaces range from ab initio quantum chemistry (HF configuration interaction (Cl), perturbation theory (PT), etc for details see chapter B3.1 ) on small, finite clusters of atoms to HF or DFT on two-dimensionally infinite slabs. In between these... 

But many computations of phase-formation based on the application of pseudo-potential, quantum-mechanical techniques, statistic-thermodynamic theories are carried out now only for comparatively small number of systems, for instance [1-3], A lot of papers dedicated to the phenomenon of isomorphic replacement, arrangement of an adequate model of solids, energy theories of solid solutions, for instance [4-7], But for the majority of actual systems many problems of theoretical and prognostic assessment of phase-formation, solubility and stable phase formation are still unsolved. 

The advent of high-speed computers, availability of sophisticated algorithms, and state-of-the-art computer graphics have made plausible the use of computationally intensive methods such as quantum mechanics, molecular mechanics, and molecular dynamics simulations to determine those physical and structural properties most commonly involved in molecular processes. The power of molecular modeling rests solidly on a variety of well-established scientific disciplines including computer science, theoretical chemistry, biochemistry, and biophysics. Molecular modeling has become an indispensable complementary tool for most experimental scientific research. 

Two specific approximations were developed to solve the problems of surface chemistry the periodic approximation, where quantum-chemical methods employ a periodic structure of the calculated system, and the cluster approximation, where the model of solid phase of finite size is created as a cutoff from the system of solid phase (it produces unsaturated dangling bonds at the border of cluster). The cluster approximation has been widely used for studying the interactions of molecules with all types of solids and their surfaces [24]. This approach is powerful in calculations of systems with deviations from the ideal periodic structure like doping and defects. Clay minerals are typical systems having such properties. 

The Raman spectra of solids have a more or less prominent collision-induced component. Rare-gas solids held together by van der Waals interactions have well-studied CILS spectra [656, 657]. The face-centered, cubic lattice can be grown as single crystals. Werthamer and associates [661-663] have computed the light scattering properties of rare-gas crystals on the basis of the DID model. Helium as a quantum solid has received special attention [654-658] but other rare-gas solids have also been investigated [640]. Molecular dynamics computations have been reported for rare-gas solids [625, 630, 634]. 

However, the model in which the (valence) electrons are completely free and are neither feeling the attraction nor the repulsion is certain not properly describing the nature of the chemical bond. In fact, this limitation was also the main objection brought to Thomas-Fermi model and to the atomic or molecular approximation of the homogeneous electronic gas or helium model in solids. Nevertheless, the lesson is well served because Thomas-Fermi description may be regarded as the inferior extreme in quantum known structures while further exchange-correlation effects may be added in a perturbation manner. 

The Figure 3.10 shows how the first few eigen-levels (energy and stationary wave-function) are displaced in the spectrum of free electronic movement within infinite wall cavity modeling the solid - crystal state. One may easily note the fragrant behavior according which the levels are more and more separated as increasing of the quantum number (or excited... 

FIGURE 3.16 Energetic discretization at the frontier of the first Brillouin zone in the quantum model of quasi-free electrons in crystal after (Further Readings on Quantum Solid 1936-1967 Putz,2006). 

The model of a periodic crystalline potential is considered as a generalization of the effectively null potential model of Figure 3.13, by the alternation of the wells with potential barriers of a finite altitude V, as illustrated in Figure 3.21 (Kronig-Penney model), see (Further Readings on Quantum Solid, 1936-1967),... 

Developing the implication of Fermi quantum statistics to model the semiconductors and junctions, including the transistor phenomenology, in modeling the quantum solids at the conducting level. 

The modeling of a solvent—a liquid phase—is especially challenging. In the gas phase, the molecules can be treated as isolated species that are easily modeled using quantum mechanics (Chapter 14) or molecular mechanics (Chapter 2). Modeling a solid is certainly challenging, but at least in the crystalline state there is periodic order, which in principle, simplifies the problem. Still, accurate computer modeling of solids is a major challenge. 

Fig. 6.2 Bandwidth models of solids. Electrons of atoms in solids inUxact and, when located in the same space, cannot have four identical quantum numbers (Pauli exelusion principle). Therefore, their energy splits into very elose lying levels and energy bands are created. In the diagram, only the band formed by valenee electrons (valence band) and the iimnediately higher band (conductivity band) are shown. In ceramic dielectrics and semiconductors these bands are separated by a band gap. Eg an inadmissible energy range ftn the eleetrons. Note the occupation of bands by electrons is indicated by shadowing. iEp denotes the Fermi level which equals the chemical potential of an electrons and, in a different approach, stands ftn the energy level at whieh the probability of electron occupation equals 50 %...

In 1907, Einstein proposed a simple molecular model for solids that reproduces the Petit-Dulong law. He hypothesized that the vibrations of all N atoms in all three dimensions have the same frequency Vg. Each atom is then a simple oscillator, which, according to quantum mechanics, has energy levels... 

Chapter 2 we worked through the two most commonly used quantum mechanical models r performing calculations on ground-state organic -like molecules, the ab initio and semi-ipirical approaches. We also considered some of the properties that can be calculated ing these techniques. In this chapter we will consider various advanced features of the ab Itio approach and also examine the use of density functional methods. Finally, we will amine the important topic of how quantum mechanics can be used to study the solid state. 

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