Single-particle properties density

Spectral functions, described before, determine single-particle properties of the system, such as quasiparticle energy, broadening of the levels (life-time), and density of states. These functions can be modified in nonequilibrium state, but most important kinetic properties, such as distribution function, charge, and current, are determined by lesser Green function... 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

The three main approaches based on the single-particle density are the density functional theory (DFT), quantum fluid dynamics (QFD), and studying the properties of a system through local quantities in 3D space. In this chapter, we present simple discussions on certain conceptual and methodological aspects of the single-particle density for details, the reader may consult the references listed at the end of this chapter. 

The respirable powders of a DPI cannot be characterized adequately by single-particle studies alone bulk properties must also be assessed since they contribute to ease of manufacture and affect system performance. Primary bulk properties include particle size, particle size distribution, bulk density, and surface area. These properties, along with particle electrostatics, shape, surface morphology, etc., affect secondary bulk-powder characteristics such as powder fiow, handling, consolidation, and dispersibility. 

In this section, various issues concerning solid particles are presented. The analysis covers the most important particle properties (surface area, particle shape and size distribution, mechanical strength, and density) as well as the behavior of a single particle in suspension (terminal velocity) and of a number of particles in fluidization state. Finally, the diffusion of molecules in a porous particle (diffusion coefficients) is also discussed. 

The essential information about transport properties in many-particle systems is given by the single-particle density matrix or by the singleparticle Wigner distribution. The equations of motion (1.18) and (1.23) for these important quantities are called kinetic equations. For the further consideration we write the latter equation in the momentum representation ... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

Optical study indicates that at low temperatures the low-energy electronic properties of some organic metal-like conductors (e.g., TTF-TCNQ) are dominated by charge density wave (CDW) effects. Frequency-dependent conductivity of TTF-TCNQ, obtained from the IR reflectance, at 25 K displays a double-peak structure with a low-frequency band near 35 cm-1 and a very intense band near 300 cm-1 [45]. The intense band may be ascribed to single-particle transitions across the gap in a 2kF (Peierls) semiconducting state, while the 35-cm-1 band is assigned to the Frohlich (i.e., CDW) pinned mode. Low-temperature results based on the bolometric technique [72,73] (Fig. 15) confirm the IR reflectance data. Such a con-... 

During the 1970s and 1980s, density-functional theory became an important tool for calculating static electronic and structural properties of solids. The theory represents in principle an exact formulation of the many-electron problem in terms of a single particle moving in the mean field of the other electrons. All the difficulties associated with the solution of the many-electron problem are enclosed in this mean field, for which some approximation must be adopted. In practice, most calculations have been carried out using the local-density approximation (LDA), which has... 

This natural coordinate system is of particular use in the determination of atomic properties. As discussed in Chapter 6, the atomic value of a property F is given by the average over the atomic basin of an effective single-particle density /(r). Thus the value of the property F for atom 2 is... 

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y)... 

The result is multiplied by JV, the total number of electrons, in the definition of an atomic property. The reader is reminded that the mode of integration indicated by N dx [l/ ijy as used in this definition of an atomic average is the same as that employed in the definition of the electronic charge density, p r) (eqns (1.3) and (1.4)). From this point on the subscript T will be dropped from the coordinates of the electron whose coordinates are integrated only over 2 and all single-particle, unlabelled coordinates and operators will refer to this electron. 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The opening section of this chapter stressed the importance of the presence of the surface integral in the hypervirial theorem for an open system, eqn (6.2). Unlike the theorem for a total system, eqn (6.4), in which case the average of the commutator of any observable G with the Hamiltonian H vanishes, the corresponding result for an atom in a molecule is proportional to the flux in the effective single-particle vector current density of the property G through the atomic surface. As a result, the hypervirial theorem plays an important role in determining the properties of an atom in a molecule. It also enables one to relate an atomic property to a sum of bond contributions, as is now demonstrated. 

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