Deformation dipole model

Figure 8 also shows values of f that have been calculated by two other methods. In the first, Jaswal (19) has used lattice-vibration eigen-frequencies and eigenvectors which have been calculated in the first Brillouin zone using the deformation-dipole model for the lattice. This... 

Computations of minimum-energy configurations for some off-centre systems were first carried out on the basis of polarizable rigid-ion models, mainly devoted to KChLi" " [95,167-169]. Van Winsum et al. [170] computed potential wells using a polarizable point-ion model and a simple shell model. Catlow et al. used a shell model with newly derived interionic potentials [171-174]. Hess used a deformation-dipole model with single-ion parameters [175]. At the best of our knowledge, only very limited ab initio calculations (mainly Hartree-Fock or pair potential) have been performed on these systems [176,177]. 

It should also be mentioned that other lattice dynamical model exist which, in many respects, are equivalent to the shell model an important one is the deformation dipole model put forward by HARDY [4.47]. 

A splitting of the dipole resonance into three peaks has been observed in some sodium clusters [64], This observation is interpreted as corresponding to collective vibrations of the valence electrons in the directions of the principal axes of a triaxially deformed cluster, and has motivated an extension of the deformed jellium model to fully triaxial shapes. Lauritsch et al. [65] have applied this model to Nai2 and Nai4. 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The outer angle brackets < > in F t) and C (t) imply an average over the different rods to which the fluorophore is bound. It has been assumed that the motions in the different factors in Eq. (4.24) are statistically independent. Equation (4.24) is expected to be rather generally valid for deformable macromolecules with mean local cylindrical symmetry. Relaxation of the FPA by rotation of the rods around their symmetry axes is contained in C (r). Likewise, relaxation of the FPA by rotation, or end-over-end tumbling, of the rods about their transverse axes is contained in F (t). Motion of the transition dipole with respect to the frame of the rod in which it is attached is contained in / ( ) Further progress requires the evaluation, or estimation, of / (/), F (t), and C (t) for particular models. 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

Most of the potential energy surfaces reviewed so far have been based on effective pair potentials. It is assumed that the parameterization is such as to account for nonadditive interactions, but in a nonexplicit way. A simple example is the use of a charge distribution with a dipole moment of 2.ID in the ST2 model. However, it is well known that there are significant non-pairwise additive interactions in liquid water and several attempts have been made to include them explicitly in simulations. Nonadditivity can arise in several ways. We have already discussed induced dipole interactions, which are a consequence of the permanent diple moment and polarizability of the molecules. A second type of nonadditive interaction arises from the deformation of the molecules in a condensed phase. Some contributions from such terms are implicitly included in calculations based on flexible molecule potentials. Other contributions arises from electron correlation, exchange, and similar effects. A good example is the Axilrod-Teller three-body dispersion interaction ... 

Simulations of solvation dynamics following charge transfer at the water liquid/vapor interface[53,80] have shown that the solvent relaxation rate is quite close to that in bulk water, even though one might expect (based on the reduced interfacial dielectric constant and simple continuum model arguments) to have a significantly slower relaxation rate. The reason for this behavior is that the interface is deformed and the ion is able to keep its first solvation shell nearly intact. Since a major part of the solvation dynamics is due to the reorientation of first shell solvent dipoles, the rate relative to the bulk is not altered by much. 

Energetic explanations of these properties have been given by the band model (Sato and Toth, 1965) and by the spatial correlation model (Schubert, 1964) the latter model explains more facts In Cu3Auan Al-VEC aCu3Au =aA1(l 1 1) may be assumed. If N( is increased by substituting some Zn for Cu, the number of electron places per cell N(p must also be increased. Since the above commensurability between VEC and atomic correlation (AC) is energetically favorable, it is profitable to loosen commensurability only in one direction by inserting further electron planes perpendicularly to that direction. As by this insertion the crystal will be deformed, the above-mentioned direction must be of minimum elastic modulus it is a3. By the incommensurability between AC and VEC a wave-like distribution of electric dipole vectors arises at the atoms, which will be different at the minority component of the struc-... 

In several cases, the polarizability distribution can be found by chemical intuition. For instance, in the case of naphthalene, which is made up of two identical fragments, the polarizability can be decomposed into two equivalent parts. Also, group or atom contributions can be deduced from a variety of schemes such as Stone s approach [74], the theory of atoms in molecules [75], the localization of molecular orbitals into chemical functions [76], atom/ bond additivity [77], the use of the acceleration gauge for the electric dipole operator [78], quantum mechanically determined induction energies [79], or calculated molecular quadrupole polarizabilities and their derivatives with respect to molecular deformations [80]. Several of these models consider charge... 

The usual practice in modeling the valence deformation of H-atoms is to terminate the expansion at the dipolar level (/max = U with a bond directed dipole) and to fix the radial exponents. In addition, H-atoms bonding to the same type of atoms are often constrained to be identical, regardless of their involvement in non-bonded interactions. Model studies on radial functions, obtained by projection of theoretical densities onto nucleus-centered spherical harmonics, show that (a) second neighbors have significant effects on H-atoms,... 

Kirkwood31 has also considered, and treated by statistical mechanics, the orientation polarization but not the deformation effects in a polar liquid. He considers a sphere in vacuocontaining a set of nondeformable molecules characterized by an internal moment p (and not fi). The total potential energy U is divided in two parts Ulf due to London-Van der Waals and dipole forces, is practically independent of the field U2 is due to electrostatic interactions of the dipoles with the external field. In Kirkwood s model, one finds ... 

These difficulties have been avoided by Frdhlich20 whose Reasoning is very similar to Kirkwood s but who has chosen his model in such a manner that he, need consider no boundary effect. He has treated the deformation polarization as a macroscopic phenomenon. Molecules are replaced by a set of nondeformable point dipoles, having a moment p and placed in a continuous medium of dielectric constant n2(n — refractive index), accounting for deformation effects. The moment of a spherical molecule is given by... 

When the model is refined to include polarization effects, i.e. to allow for that fact the electric field set up by each ion in the monomer deforms the electron cloud in the other, calculated molecular dipole moments are in significantly better agreement with the observed values. Polarization effects do not, however, lead to significant improvement of the fit between calculated and experimental dissociation energies. 

Figure 7.5 Dipole photoabsorption strength for the cluster Na2i in the diffuse jellium model after averaging over the thermal ensemble of and shape fluctuations at T = 480 K. The experimental strength from [79] is also shown for comparison (upper panel). The potential energy surfaces for P2 and deformation compared with the thermal energy atT = 480 K are displayed in the lower panel...

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