Electric moments localized

Examples of p(r) are energy density, charge density, current density (see Section 4.6), difference density (difference between a final density and an initial density), electric moment density, magnetic moment density, local reactivity functions (see Section 4.5.2), force density, etc. Note that, for ensuring the stability of matter, the net force density must vanish everywhere in space. The concept of a PDF has generated many significant developments in interpretative quantum chemistry. 

Our discussion of non-bonded interactions began with the example of two noble gas atoms having no permanent electrical moments. We now turn to a consideration of non-bonded interactions between atoms, bonds, or groups characterized by non-zero local electrical moments. 

However, Eq. (2.21) is not very convenient in the context of intramolecular electrostatic interactions. In a protein, for instance, how can one derive the electrostatic interactions between spatially adjacent amide groups (which have large local electrical moments) In principle, one could attempt to define moment expansions for functional groups that recur with high frequency in molecules, but such an approach poses several difficulties. First, there is no good experimental way in which to measure (or even define) such local moments, making parameterization difficult at best. Furthermore, such an approach would be computationally quite intensive, as evaluation of the moment potentials is tedious. Finally, the convergence of Eq. (2.20) at short distances can be quite slow with respect to the point of truncation in the electrical moments. 

However, serious drawbacks of model 3 are that (i) the proportion r of the rotators should be fitted that is, it is not determined from physical considerations and (ii) the depth of the well, in which a polar particle moves, is considered to be infinite. Both drawbacks were removed in VIG (p. 305, 326, 465) and in Ref. 3, where it was assumed that (a) The potential is zero on the bottom of the well (/(()) = 0 at [ fi < 0 < P], where an angle 0 is a deflection of a dipole from the symmetry axis of a cone, (b) Outside the well the depth of the rectangular well is assumed to be constant (and finite) U(Q) = Uq at [— ti/2 < 0 < ti/2]. Actually, two such wells with oppositely directed symmetry axes were supposed to arise in the circle, so that the resulting dipole moment of a local-order region is equal to zero (as well as the total electric moment in any sample of an isotropic medium). 

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

As was recalled in the first part of this paper, the dipole moment of a given molecule, polymer or monomer, depends on its environment, since there is always some polarization by electric charges localized in the neighboring particles. In other words a reaction field exists even in the absence of external field, so that the internal moment p of a polymer chain in dilute solution may differ considerably from the moment y of the same molecule in vacuo. 

It is well known that transferability is an important property for the investigation of related systems. In cases when one can define quantities for molecular parts, which are additive and transferable, certain similarities of different molecules can be simply recognized. One-electron properties, as electric moments or kinetic energy contributions derivable from transferable/localized molecular orbitals, are especially useful for the above purpose. 

It is known, that in the framework of the independent particle model the one-electron properties of the system can be written as the sum of contributions from the individual orbitals. The transferability of the one-electron properties is implied by the transferability of the orbitals. The first and higher order electric moments determined for localized molecular orbitals (LMOs) in different systems can thus be used in comparative studies. 

The first order electric moments obtained for bond and lone pair electron densities using Edmiston-Ruedenberg localization method are summarized in Table 4 and the geometry of the studied systems are presented in Table 3. The origin of (r) is taken at the corresponding heavy atom nucleus. The values... 

The multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

We decompose the charge distribution of the whole electron system into sum of contributions from localized orbitals. If the localized orbitals do not overlap and their electric moments can be considered as transferable, than it is expected that the Coulomb integrals can be approached by the sum of interaction energies... 

It is interesting to see how these electrical moments, and particularly the dipole moment, can be thought of in terms of contributions from different parts of the molecule. The nuclei are, of course, easily treated but it is helpful to think of the electronic contribution, not in terms of symmetry orbitals like (1.2) but rather as localized orbitals which can be pictured as 0-H bonds and lone pairs. This can be done by performing a unitary transformation upon the wave function specified in terms of components... 

The first two terms in eq. 12 are known as "electro-optical parameters." If the instantaneous molecular dipole moment is expressed in terms of effective electrical charges, localized on the atoms, M = Iaqa a dipole moment change can then be expressed as a function of... 

Brant et al [11] who used the dipole approximation, stressed the importance of the electrostatic term in the calculation of intramolecular interactions in the case of a dipeptide. They have also stressed that when only the nonbonded and torsional interactions are considered, the minimum in energy does not correspond to the experimentally observed conformation. Earlier studies (Rein et al, [15]) on the conformations of simple molecules, such as hydrogen peroxide and methanol, have shown that the monopole and dipole approximations may be insufficient for the electrostatic calculations. Lassetre and Dean [16] considered the interactions in terms of a series of multipoles localized at bond midpoints and included terms up to quadrupole interactions. Tang Au-Chin [17] included interactions up to octopole terms. However, at the time of their work [16, 17], wave functions sufficiently accurate for the calculation of reliable electrical moments were not available. As a result, only rough orders of magnitude were used in their calculations. 

Note that majority electrons that are accelerated by the electric field in one of the cobalt layers contribute to the current, not only in that layer (I = J) but in other layers as well, including the copper layers and the cobalt layers on the other side of the copper. On the other hand, minority electrons that are accelerated by a field in one of the cobalt layers contribute very little to the conductivity in the copper or in the cobalt on the other side of the copper. For anti-parallel alignment of the moments, electrons that are accelerated by the field in one cobalt layer contribute to the current in that layer and in the cobalt, but not in the other cobalt layer. The difference in the lolal current due to both channels between parallel and anti-parallel alignment is almost entirely non-local. It comes from those electrons that are accelerated by the applied electric field in one cobalt layer and propagate across the copper to the other cobalt layer where they contribute to the current. It is clear from Figures 1-4 that this process occurs primarily for majority electrons and for the case of parallel alignment. 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

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