Density formalism

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Combining the angular and radial functions discussed above leads to a valence-density formalism in which the density of each of the atoms is described as (Hansen and Coppens 1978)... 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

The aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

The atom-centered multipole expansion used in the density formalisms described in chapter 3 implicitly assigns each density fragment to the nucleus at which it is centered. Since the shape of the density functions is fitted to the observed density in the least-squares minimalization, the partitioning is more flexible than that based on preconceived spherical atoms. 

The applications of the many-particle densities will be demonstrated on a full scale in further Chapters. It should be only said here that the many-particle density formalism being combined with the shortened Kirkwood superposition approximation, equation (2.3.64), results in the well-known equations of the standard kinetics for both neutral [83] and charged particles [100] giving just another way of their derivation. On the other hand, the use of the full-scale (complete) Kirkwood s approximation, equation (2.3.62), permits us to take into account the many-particle (cooperative) effects [81, 91, 99-102] we are studying in this book. 

In this Section we consider several approaches which differ from the many-point density formalism discussed above. Szabo et al. [45] have introduced a novel method based on the density expansion for the survival probability, u>(t). Consider a system containing walkers (particles A) and N traps (quenchers B) in volume V in d-dimensional space. We assume that the particles have a finite size but the traps can be idealized as points and hence are ignorant of each other. When the concentration of the walkers is sufficiently low so that excluded volume interactions between them are negligible, one might focus on a single walker. 

Since the many-point density formalism in its practical applications assumes macroscopically homogeneous system, we will restrict ourselves to a particular class of microscopically self-organized autowave processes. Without investigating in Chapter 8 all possible kinds of autowave processes, we are aimed to answer a principal question - whether these two models under question could be attributed to the basic models useful for the study of autowave processes. 

The considerable progress made in the studies of simple bimolecular reactions (which has led to such fundamental conclusions) was achieved by a more rigorous mathematical treatment of the problem, avoiding the use of the simplest approximations which linearize the kinetic equations. We focus main attention on the many-point density formalism developed in [26, 28, 49] since in our opinion it seems at present to be the only general approach permitting treatment of all the above-mentioned problems, whereas other theoretical methods so far developed, e.g., those of secondary quantization [19, 29-32], and of multiple scattering [72, 73], as well based on... 

This formalism is the quantum-mechanical analog of the Klimontovich phase-space density formalism, and is particularly effective in the theory of long-range correlation and fluctuation in the kinetic theory of plasmas. 

In order to validate the reduced model (uncoupled population of Izhikevich neurons) we chose to perform comparisons with a direct simulation model. In this last model the internal state of each neuron is computed at each time step (with a forth-order Runge-Kutta method) using the equations of the Izhikevich model so we have complete access to individual information as opposed to the population density formalism where only the states distribution can be computed. The simulation parameters were 50 000 Izhikevich neurons in tonic spiking mode (a = 0.02, b = 0.2, c = —65, d = 6 as provided in [29]), a gaussian form of p(v, u, t) at t =0 and a constant input current of / =60 qA was applied to all neurons. The firing rate and the mean membrane potential (M M P) were computed at each time for the two methods during 15 ms. 

The PCM-TDDFT excitation energies obtained from Eq. (7-10) reflect the variations of the solute-solvent interaction in the excited states in terms of the effects of the corresponding transition densities. To overcome this limitation (see the Introduction) the PCM-TDDFT scheme may exploits the relaxed density formalism (Section 7.1.1.4) to compute, for each specific electronic state, the variation of the solute solvent-interaction in terms of the changes of the electronic density. 

Yamashita, J., and S. Asano (1983a). Cohesive properties of alkali halides and simple oxides in the local-density formalism. J. Phys. Soc. Jpn. 52, 3506-13. 

Conventional ab initio techniques are based on a wavefunction formalism, as opposed to the density functional theory techniques, which are based on a density formalism. We will shortly describe the most common features of ab initio techniques starting with the lowest level of theory, the Hartree-Fock (HF) method. 

The treatment in terms of induced current is in the mainstream of modem development of the time-dependent density functional theory (TDDFT). Moreover, the current density formalism has been proposed [4] as a variant of TDDFT. The evolution of current density presents properly the response of electrons on an external field. In general words, such a strong basis is promising for a theoretical treatment of many aspects of ion interactions with atoms, molecules and solids. 

Structural Non-directed, giving structures of high co-ordination Spatially directed and numerically limited, giving structures of low co-ordination and low density Non-directed, giving structures of very high co-ordination and high density Formally analogous to metallic bond... 

DFT [95] was generalized to include relativistic effects by MacDonald and Vosko [96] and by Ramana and Rajagopal.[97] By omitting small diamagnetic effects one-electron equations were obtained that contained a scalar effective potential as well as an effective magnetic field caused by the polarization. In cases where the paramagnetic currents cannot be neglected, it is necessary to resort to a current-density formalism.[98,99]... 

Density-functional formalism may be extended to a spin-density formalism,... 

The centroid density formally defines a classical-like effective potential that is [1, 3, 21-23]... 

Fourier Transform and Discrete Variational Method Approach to the Self-Consistent Solution of the Electronic Band Structure Problem within the Local Density Formalism. 

Consider cell operation at the oxygen-limiting current density. Formally, operation at the limiting current is equivalent to infinite voltage loss E. The expression under the last logarithm on the left side of (4.207) should... 

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