Multipolar moments

Then, if one knows how to integrate the charge density into the atomic basin, one can get charges and multipolar moments of the distribution. 

This technique has been applied in the analysis of the electronic structure of stoichiometric and non-stoichiometric (111) surfaces of BaFj and CaF2 crystals [85-88] or in the case of AI2O3 crystals by Guo, Ellis and Lam [89-91]. Vg t could also represent an applied static electric field as used for evaluation of static polarizabilities as discussed below. In the latter case, we could evaluate the multipolar moments separately, thus obtaining the static polarizability, induced by a field of the form [92]... 

Figure 4 Contours of the charge distributions corresponding to the pure multipolar moments Qoo, charge Qio> dipole Q20, quadrupole Q30, octopole. The solid contours are positive and dashed negative (buildup of electron) charge density.

The excess collision-induced pair polarizability of molecules A and B results as a self-consistent solution of a set of equations for local fields polarizing molecules A and B and the multipolar moments induced in molecules A and B. In a first approximation, the long range field-induced change in the pair polarizability, related to the linear multipolar polarizabilities of molecules, is [27] ... 

The author became interested in collision-induced state-to-state processes as a result of experiments in which narrow-line, circularly polarised laser radiation was used to excite fluorescence from diatomic molecules in collision cells. The laser excites known multipolar moments of nij distribution among specific j states of an excited vibronic state. Wavelength and polarisation analysis of the emission allowed the population and orientation multipoles of nij among states accessed by elastic and inelastic collisions to be determined from measured intensity and degree of circular polarisation. An example is shown in Fig. 1, from which it is evident that the initial asymmetric nij distribution is maintained through elastic and inelastic collisions [13] even when substantial amounts of vibrational and rotational energy... 

The second term is a point charge distribution of multipolar order v and unit multipolar moment [71], which means that both /(v,/<,R-fD,r) and the com-... 

At the dipolar level, these approaches are similar. Howevei contributions from quadrupolar and higher moments present in the first method are absent in the second. The calculated contributions from the higher multipolar moments can be significant (0.9 V for the water-hydrocarbon interface ). The difference in results from the two approaches is puzzling in light of the published proofs that the potential drop for a uniform planar interface should depend only on the polarization density and not on higher order terms like the quadrupolar density discussed above. 

An electrostatic model of crystal-field interactions can be developed in which the crystal-field components A n defined by eq. (32) are expanded in a multipolar series (Hutchings and Ray, 1%3). It is assumed that the lattice consists of a series of points, one for each ion in a lattice, that have multipolar moments associated with them that are defined by... 

Using eqs. (42) and (43), it is possible to obtain an explicit form of contributions of ionic multipolar moments into expressions (57), (58) (Malkin 1987). The contributions of exchange charges to electron-phonon interaction parameters are... 

Addendum 1 Direct determination of multipolar moments Direct determination of multipolar moments is the key of the multipolar electrostatic model. Garcia and Faucher (1986) related the electronic multipole magnitude to the X-ray factor structure ... 

To evaluate this expression for distributions expressed in terms of their multipolar density functions, the potential <1> and its derivatives must be expressed in terms of the multipole moments. The expression for charge distribution has been given in chapter 8 [Eq. (8.54)]. Since the potential and its derivatives are additive, a sum over the contributions of the atom-centered multipoles is again used. The resulting equation contains all pairwise interactions between the moments of the distributions A and B, and is listed in appendix J. 

For some of the more common molecules, the low-order molecular multipole moments are known [166, 378] (Landolt-Bornstein 1974). Collision-induced absorption of molecular gases arises mainly from multipolar induction. Studies of collision-induced absorption in the molecular gases provides, therefore, useful information on multipole moments [38]. 

An important induced dipole component of pairs involving molecules is multipolar induction. Specifically, the lowest-order multipole consistent with the symmetry of H2 is the electric quadrupole. Each H2 molecule may be thought of as being surrounded by an electric field of quadrupolar symmetry that rotates with the molecule.-In that field, a collisional partner X is polarized, thus giving rise to an induced dipole moment which in turn is capable of emitting and absorbing light. For like pairs, molecule 1 will induce a dipole in molecule 2 and 2 will induce one in 1. In... 

Fig. 3.27. Left Spectral moments 71 of the rototranslational bands of several molecular pairs, as function of temperature. Various measurements ( , o, etc.) are compared with theoretical data based either on the fundamental theory (H2-H2, H2-He) or on refined multipolar induction models after [58]. Right Same as at left, except the spectral moment 70 is shown.

Fig. 17. (a) Orientation of molecular dipole moments in 3-methyl-4-nitropyridine. Y-oxide fih molecular dipole moment from direct integration methods fi2, from multipolar model and Hj from semi-empirical calculation, (b) Electrostatic potential around the molecule in the plane of ring atoms. Contours at 0.2kcal/mol (reproduced with permission from Hamazaoui et al. [79]). 

As emphasized, the Born—Kirkwood—Onsager (BKO) approach includes only the solute s monopole and dipole interaction with the continuum. That is, the full classical multipolar expansion of the total solute charge distribution is truncated at the dipole term. This simplification of the electronic distribution fails most visibly for neutral molecules whose dipole moments vanish as a result of symmetry. A distributed monopole or distributed dipole model is more... 

Linear Multipolar Electric Polarizabilities. Just as equation (40) served for defining the 2 -pole permanent electric moment, one can generalize the definition of the induced dipole (68) by introducing that of an induced 2 -pole moment. Assuming the inducing factor to be a uniform extamal electric field E, one has ... 

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