Density, local, arbitrariness

The velocity and density have the same values at the end of the closed cycle as at the beginning. Thus, the kinetic energy is the same at the beginning and the end, and the first term in (5.36) vanishes. The work assumption (5.34) is expected to hold for any arbitrary region of the body and must therefore hold locally at every material particle, so that (5.36) reduces to... 

Here v is the space- and time-dependent velocity field, p is the density of the fluid, p is the local pressure, v is the kinematic viscosity, and / is some arbitrary body-force acting on each small element of the fluid (gravitation, for example). 

In chapter 2, Profs. Contreras, Perez and Aizman present the density functional (DF) theory in the framework of the reaction field (RF) approach to solvent effects. In spite of the fact that the electrostatic potentials for cations and anions display quite a different functional dependence with the radial variable, they show that it is possible in both cases to build up an unified procedure consistent with the Bom model of ion solvation. The proposed procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy. Especially interesting is the introduction of local indices in the solvation energy expression, the effect of the polarizable medium is directly expressed in terms of the natural reactivity indices of DF theory. The paper provides the theoretical basis for the treatment of chemical reactivity in solution. 

Thus, for any p(r) e there exists a unique wavefunction generated by means of local-scaling transformation from the arbitrary generating wavefunction The set of all the wavefunctions thus generated, yielding densities p(f) in J g, is called an orbit and is denoted by... 

From a theoretical point of view XES furthermore provides a very strong basis for the evaluation of methods for population analysis, i.e., the decomposition of the molecular orbitals into atomic contributions [10]. Many different schemes subdividing the charge density into contributions assigned to respective atoms have been proposed, but the lack of means to directly measure the atomic populations in different orbitals has made all techniques somewhat arbitrary and a matter of taste. Due to the strongly localized character of the intermediate core-excited state, however, in combination with the direct dependence of the XES transition probability on the amount of local -population (assuming a Is core hole), XES provides a very sensitive tool to directly measure this atomic population. In all, we have an atom-specific tool, which can be used to address important questions regarding... 

We point out that the results of locally electro-neutral studies should be extrapolated upon the nonreduced systems with a certain caution even for e very small. This is so because, to the best of the author s knowledge, no asymptotic procedure for the singularly perturbed one-dimensional system (4.1.1), (4.1.2) has been developed so far that would be uniformly valid for the entire range of the operational parameters (e.g., for arbitrary voltages and fixed charge densities). 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

As we are dealing with spherical harmonics, and as we are trying to model the aspherical atomic electron density, the orientation of the local atom centered coordinate system is, in principle, arbitrary, appropriate linear combinations always giving the same result. However, in practice it is helpful to choose a local coordinate system such that the multipoles are oriented in rational directions, and thus the most important multipole populations will lie in directions that would be expected to represent chemical bonds or lone pairs [2,20], e.g. for an sp2 hybridized atom, defining one bond as the x direction, the trigonal plane as the xy plane, and z perpendicular defines three lobes of the 33+... 

Because a local-scaling transformation establishes a one to one correspondence between a transformed wavefunction ip and the density p, there also arises (by construction) a one to one correspondence between the functional of the transformed wavefunction, E[ip], and that of the one-particle density (and of the initial arbitrary wavefunction g), [p,ig. This property guarantees [3] that the minimization of the energy with respect to the one-particle density is variational [20]. 

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

The essence of the local density approximation is the assumption that the exchange-correlation energy of a non-uniform electron density can be approximated as a sum of contributions from small volume elements each characterized by uniform electron density (exc(p)(r)). Since the exchange-correlation energy of uniform electron gas is available with arbitrary accuracy, the LDA exchange-correlation functional takes the following simple form ... 

Thus equation (A53) contains the density of the harmonic oscillator for an arbitrary number of particles N. Of course, it is only for this very elementary potential field that the local density assumption (A51) becomes exact. For a... 

In the pseudopotential method, core states are omitted from explicit consideration, a plane-wave basis is used, and no shape approximations are made to the potentials. This method works well for complex solids of arbitrary structure (i.e., not necessarily close-packed) so long as an adequate division exists between localized core states and delocalized valence states and the properties to be studied do not depend upon the details of the core electron densities. For materials such as ZnO, and presumably other transition-metal oxides, the 3d orbitals are difficult to accommodate since they are neither completely localized nor delocalized. For example, Chelikowsky (1977) obtained accurate results for the O 2s and O 2p part of the ZnO band structure but treated the Zn 3d orbitals as a core, thus ignoring the Zn 3d participation at the top of the valence region found in MS-SCF-Aa cluster calculations (Tossell, 1977) and, subsequently, in energy-dependent photoemission experiments (Disziulis et al., 1988). 

The latter free energy can be represented as a surface integral over the solvent accessible surface of the molecule on the basis of a local free energy surface density (FESD) p. This surface density function is represented in terms of a three-dimensional scalar field which is comprised of a sum of atomic increment functions to describe lipophilicity in the molecular environment.The empirical model parameters are obtained by a least squares procedure with experimental log P values as reference data. It is found that the procedure works not only for the prediction of unknown partition coefficients but also for the localization and quantification of the contribution of arbitrary fragments to this quantity. In addition, the... 

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