Density theoretical framework

In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

The electronic properties of single-walled carbon nanotubes have been studied theoretically using different methods[4-12. It is found that if n — wr is a multiple of 3, the nanotube will be metallic otherwise, it wiU exhibit a semiconducting behavior. Calculations on a 2D array of identical armchair nanotubes with parallel tube axes within the local density approximation framework indicate that a crystal with a hexagonal packing of the tubes is most stable, and that intertubule interactions render the system semiconducting with a zero energy gap[35]. 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Relaxation measurements provide a wealth of information both on the extent of the interaction between the resonating nuclei and the unpaired electrons, and on the time dependence of the parameters associated with the interaction. Whereas the dipolar coupling depends on the electron-nucleus distance, and therefore contains structural information, the contact contribution is related to the unpaired spin density on the various resonating nuclei and therefore to the topology (through chemical bonds) and the overall electronic structure of the molecule. The time-dependent phenomena associated with electron-nucleus interactions are related to the molecular system, and to the lifetimes of different chemical situations, for the resonating nucleus. Obtaining either structural or dynamic information, however, is only possible if an in-depth analysis of a series of experimental results provides sufficient data to characterize the system within the theoretical framework discussed in this chapter. 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

There are several problems in the physics of quantum systems whose importance is attested to by the time and effort that have been expended in search of their solutions. A class of such problems involves the treatment of interparticle correlations with the electron gas in an atom, a molecule (cluster) or a solid having attracted significant attention by quantum chemists and solid-state physicists. This has led to the development of a large number of theoretical frameworks with associated computational procedures for the study of this problem. Among others, one can mention the local-density approximation (LDA) to density functional theory (DFT) [1, 2, 3, 4, 5], the various forms of the Hartree-Fock (HF) approximation, 2, 6, 7], the so-called GW approximation, 9, 10], and methods based on the direct study of two-particle quantities[ll, 12, 13], such as two-particle reduced density matrices[14, 15, 16, 17, 18], and the closely related theory of geminals[17, 18, 19, 20], and configuration interactions (Cl s)[21]. These methods, and many of their generalizations and improvements[22, 23, 24] have been discussed in a number of review articles and textbooks[2, 3, 25, 26]. 

Solvophobic theory provides a theoretical framework to evaluate hydrophobic effects. To place a polypeptide or protein into a solvent, a cavity of the same molecular dimensions must first be created. The amount of energy or work required to create this cavity is related to the cohesive energy density or the surface tension of the solvent. The fusion of cavities reduces the total surface area in the combined cavity, and thus the free energy of the... 

So far, theoretically predicted schwarzites do not display this universality, although a number of predictions give an area per C atom close to that found in graphite and C60 (Fig. 2.22). These theoretical frameworks are the result of complex numerical quantum mechanical calculations. The apparent conservation of surface density, irrespective of the curvatures of the surface, is clearly not a direct consequence of standard physics. It will be very interesting to compare the surface densities of actual schwarzites (although they have yet to be prepared in the laboratory) with those of fullerenes and graphite. Given the usefulness of this principle in the study of tetrahedral frameworks, our bet is that they too will lie on the dotted line in Fig. 2.22. 

Density functional theory is a very fruitful theoretical tool to understand and develop reactivity concepts. Several quantities have been defined to construct a theoretical framework of the intrinsic response of chemical species, using either intuitive or formal procedures. However, this framework is far from a final and complete form, and no systematic procedure is known to generate the appropriate quantities to describe a specific problem. This work shows that some of the most successful reactivity parameters appear from a procedure where energy stabilization takes a relevant place. 

The chemical concept of localization is based on the classical assumption that molecular electron densities are confined to finite regions of space. Although various quantum chemical approaches have been proposed to implement this concept within a theoretical framework, this is essentially a classical idea that disregards some aspects of the Heisenberg uncertainty relation. Nevertheless, the interpretation of molecular properties and chemical reactions strongly relies on the inherent assumption that both complete molecules and various, chemically identifiable, molecular pieces can be assigned to various regions of space, at least in some approximate sense. The explanation of the apparent success of this essentially classical assumption within a quantum chemical framework is a nontrivial problem. 

Eqs. 31-32 provide a better theoretical framework to reach the exact ground-state electron density po and energy E0 because p%SCED + p SCED = p0, whereas Kohn-Sham equations cannot lead to po (pES po). 

Baev et al. review a theoretical framework which can be useful for simulations, design and characterization of multi-photon absorption-based materials which are useful for optical applications. This methodology involves quantum chemistry techniques, for the computation of electronic properties and cross-sections, as well as classical Maxwell s theory in order to study the interaction of electromagnetic fields with matter and the related properties. The authors note that their dynamical method, which is based on the density matrix formalism, can be useful for both fundamental and applied problems of non-linear optics (e.g. self-focusing, white light generation etc). 

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