Asymptotic density model

The marching-cube algorithm has been used also by Kolle and Jug (1995) to define the tesserae of isodensity surfaces. The procedure is implemented in the semiempirical SINDOl program (INDO with Slater-type orbitals, Li et al., 1992). To compute AS charges the asymptotic density model ADM (Koster et al., 1993) is used. This is an approximation to the calculation of molecular electrostatic potentials based on the cumulative atomic multipole moment procedure (CAMM, Sokalski et al., 1992). 

ADM = asymptotic density model MESP = molecular electrostatic potential OAO = orthogonalized atomic orbital. 

With this asymptotic density model (ADM) an advantage over the standard multipole expansion can be achieved, so that the MESP is well approximated not only outside the molecule, but also near the atoms. This scheme has been implemented in SINDOl for first- and second-row elements. The evolving expansion has been truncated after the cumulative dipole moments terms. The three-center integrals have been approximated on the NDDO level. In order to achieve good agreement with ab initio based MESPs, the atomic hybrid moments had to be scaled. In the application to. solvation energies the atomic electronic charges had to be scaled too. 

Fig. 6. Comparison of simulated and asymptotical densities of the estimated parameter 62 for the mono-exponential model >/(x 0) = ()t cxpi—x/dn) for two SNR (a) Q /a = 10 and (b) 0] /(j — 100. The exponential curve was sampled at two points x = [10, 80]T and the parameters of the model were obtained by LS estimator from Rician distributed samples. Simulated density results from 105 data vectors, the asymptotical density is a Gaussian distribution centered on the true value 02 = 70 with a variance equal to the CRBs.

In this work a simple analytical atomic density model is obtained from the expression of a modified Thomas-Fermi-Dirac model with quantum corrections near the nucleus as the minimization of a semiexplicit density functional. The use of a simple exponential analytical form for the density outside the near-nucleus region and the resolution of a single-particle Schrodinger equation with an effective potential near the origin allows us to solve easily the problem and obtain an asymptotic expression for the energy of an atom or ion in terms of the nuclear charge Z and the number of electrons N. 

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

Diquark condensation makes the EoS harder, which leads to an increase in the maximum mass of the quark star configuration when compared to the case without diquark condensation. For finite temperatures the masses are smaller than at T = 0. For asymptotically high temperatures and densities the EoS behaves like a relativistic ideal gas, where the relation pressure versus energy density is temperature independent. In contrast to the bag model where this behavior occurs immediately after the deconfinement transition, our model EoS has a temperature dependent P(e) relation also beyond this point. 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Fig. 6 compares the nuclearity effect on the redox potentials [19,31,63] of hydrated Ag+ clusters E°(Ag /Ag )aq together with the effect on ionization potentials IPg (Ag ) of bare silver clusters in the gas phase [67,68]. The asymptotic value of the redox potential is reached at the nuclearity around n = 500 (diameter == 2 nm), which thus represents, for the system, the transition between the mesoscopic and the macroscopic phase of the bulk metal. The density of values available so far is not sufficient to prove the existence of odd-even oscillations as for IPg. However, it is obvious from this figure that the variation of E° and IPg do exhibit opposite trends vs. n, for the solution (Table 5) and the gas phase, respectively. The difference between ionization potentials of bare and solvated clusters decreases with increasing n as which corresponds fairly well to the solvation free energy of the cation deduced from the Born solvation model [45] (for the single atom, the difference of 5 eV represents the solvation energy of the silver cation) [31]. 

By assuming an Arrhenius type temperature relation for both the diffusional jumps and r, we can use the asymptotic behavior of /(to) and T, as a function of temperature to determine the activation energy of motion (an example is given in the next section). We furthermore note that the interpretation of an NMR experiment in terms of diffusional motion requires the assumption of a defined microscopic model of atomic motion (migration) in order to obtain the correct relationships between the ensemble average of the molecular motion of the nuclear magnetic dipoles and both the spectral density and the spin-lattice relaxation time Tt. There are other relaxation times, such as the spin-spin relaxation time T2, which describes the... 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

---