Electric dipole moment definition

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

At the molecular level, electric quadrupoles can lead to useful structural information. Thus, whilst the absence of a permanent electric dipole in CO2 simply means that the molecule is linear, the fact that the electric quadrupole moment is negative shows that our simple chemical intuition of 0 C" 0 is correct. The definition of quadrupole moment is only independent of the coordinate origin when the charges sum to zero and when the electric dipole moment is zero. 

Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

An important consequence of the presence of the metal surface is the so-called infrared selection rule. If the metal is a good conductor the electric field parallel to the surface is screened out and hence it is only the p-component (normal to the surface) of the external field that is able to excite vibrational modes. In other words, it is only possible to excite a vibrational mode that has a nonvanishing component of its dynamical dipole moment normal to the surface. This has the important implication that one can obtain information by infrared spectroscopy about the orientation of a molecule and definitely decide if a mode has its dynamical dipole moment parallel with the surface (and hence is undetectable in the infrared spectra) or not. This strong polarization dependence must also be considered if one wishes to use Eq. (1) as an independent way of determining ft. It is necessary to put a polarizer in the incident beam and use optically passive components (which means polycrystalline windows and mirror optics) to avoid serious errors. With these precautions we have obtained pretty good agreement for the value of n determined from Eq. (1) and by independent means as will be discussed in section 3.2. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

Electric field in 10 an, energies in eV, dipole moment matrix elements in Debye, charges Aq in e for the definitions of various quantities, see Table 1 Adapted from [41]... 

In resolving the apparent paradox of how an antenna with no charge (in free space) can have an electric-dipole moment, one can go back to definitions. In Ref. 14 the fields from a current distribution are evaluated by expanding... 

P is the macroscopic polarization. It consists of a lattice polarization b21 w originating from the electric dipole moment arising from the mutual displacement of the two sublattices, and of a second term b22 P originating from the pure electron polarization. According to definition, P and E are connected by... 

Electrochemistry deals with charged particles that have both electrical and chemical properties. Since electrochemical interfaces are usually referred as electrified interfaces, it is clear that potential differences, charge densities, dipole moments, and electric currents occur at these interfaces. The electrical properties of systems containing charged species are very important for understanding how they behave at interfaces. Therefore, it is important to have a precise definition of the electrostatic potential of a phase [1-6]. Note that what really matters in electrochemical systems is not the value of the potential but its difference at a given interface, although it is illustrative to discuss its main properties. 

Table 8. Increase in electric dipole moments upon formation of hydrogen-bonded complexes (for the definition of Ap see Eq. (7))...

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Assume a perturbing electric field of frequency co in the v-direction with perturbation matrix elements (V + W) = 2DV in (16), and consider the perturbation of the magnetic dipole moment s u-component. With the help of the definitions... 

The conventional description of molecules, which is obviously much more intuitive and straightforward than its quantum-mechanical counterpart, is often adequate. Nevertheless, the manifestations of quantum effects are easily detectable experimentally. For example, species such as HfeCD, HD, or CH D, which are clearly nonpolar by the conventional definition, do possess temperature-dependent % and observable microwave spectra, and do deflect in inhomogeneous electric fields [11,16]. In fact, if one insists upon the conventional approach, these observations can be consistently accounted for by assuming the presence of small (of the order of 0.01 [D]) permanent dipole moments in these molecules. However, a rigorous quantum-mechanical treatment of such cases is clearly preferable. 

By insertine the mean dipole moment (2S2a) into the polarization fonnula (2S1) and using the definitions (246) and (2S0), one obtains the change in molar polarization of a gas due to the square of the electric field strength ... 

The most important electric properties are dipole moment and polarizability derivatives. The theory of dipole moment derivatives has been worked out by Bratoz (1958) and by Gerratt and Mills (1968). Both papers use the obvious definition of the dipole moment derivative the change of the dipole moment with nuclear coordinates. Komornicki and Mclver (1979) have pointed out, in an influential paper, that the alternative definition, namely the derivative of the geometrical force with respect to the electric field, is more useful, as there are only three field components versus 3N nuclear coordinates. Similarly, the polarizability derivatives can be defined as the second derivatives of the forces with respect to the field components. Komornicki and Mclver (1979) suggest numerical differentiation with respect to the field... 

Several important relationships are relevant to the properties of a dielectric. The vector P defined for a dielectric is the net dipole moment per unit volume. When it is combined with the electrical field, one obtains the definition of the electric displacement D. Thus,... 

Finally starting from the usual definitions of atomic electric dipole and quadrupole and magnetic dipole moments fAkel, qfcel, and jnkm... 

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