Electrical multipoles

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

At distances of a few molecular diameters, the interaction will be dominated by electric multipole interactions for dipolar molecules, such as water, the dominant tenn will be the dipole-dipole interaction ... 

Much of our knowledge of molecules is obtained from experimental studies of the way they interact with electromagnetic radiation, and the recent growth in non-linear spectroscopies and molecular electronics has focused attention on our ability (or otherwise) to predict and rationalize the electric properties of molecules. The idea of an electric multipole is an important one, so let s begin the discussion there. 

Typical properties of the charge distribution are summarized by its various electric multipole moments. The electric dipole moment p. induced in the system by the external field is obviously... 

The transition operator, or electric multipole operator, is a tensor of rank k and it is given the symbol We, thus, have... 

Madelung term and, 1 162-171 permanent electrical multipole term Uq and, 1 176-177... 

Permanent electrical multipole term Uq, lattice energies and, 1 176-177 Pemitrous acid, 22 147 Perovskites, ordered, 35 354, 370-371 Perovskite type fluorides, 20 152-166 Peroxides, see also specific compounds fluorinated, 16 109-168 fluoroaUcyl, spectral properties of, 16 154, 155... 

Awq, in terms of interactions of the lowest-order electrical multipoles is often... 

It is seen that the basic operator (65) and strength matrix (66) have now simple expressions via 6p f) from (62). The operator (65) has exchange-correlation and Coulomb terms. For electric multipole oscillations (dipole plasmon,. ..), the Coulomb term dominates. 

Relative contributions of T-odd densities to a given mode should obviously depend on the character of this mode. Electric multipole excitations (plasmons in atomic clusters, E giant resonances in atomic nuclei) are mainly provided by T-even densities (see e.g. [19]). Instead, T-odd densities and currents might be important for magnetic modes and maybe some exotic (toroidal,. ..) electric modes. 

FIM can also be used to study properties, such as the surface induced dipole moment and the effective polarizability of some surface atoms, kink site atoms and adsorbed atoms etc. The charge distribution of a surface atom is obviously completely different from that of a free atom because of its interaction with the surface and in addition surface atoms are partially shielded by itinerant charges of the surface. The charge distribution of a surface atom can be described by the magnitudes of the electric multipoles of the atom. 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

Both photon-assisted collisions and collision-induced absorption deal with transitions which occur because a dipole moment is induced in a collisional pair. The induction proceeds, for example, via the polarization of B in the electric multipole field of A. A variety of photon-assisted collisions exist for example, the above mentioned LICET or pair absorption process, or the induction of a transition which is forbidden in the isolated atom [427], All of these photon-assisted collision processes are characterized by long-range transition dipoles which vary with separation, R, as R n with n — 3 or 4, depending on the symmetry of the states involved. Collision-induced spectra, on the other hand, frequently arise from quadrupole (n = 4), octopole (n = 5) and hexadecapole (n = 6) induction, as we have seen. At near range, a modification of the inverse power law due to electron exchange is often quite noticeable. The importance of such overlap terms has been demonstrated for the forbidden oxygen —> lD emission induced by collision with rare gases [206] and... 

S. Kielich. The determination of molecular electric multipoles and their polarizabilities by methods of nonlinear intermolecular spectroscopy of scattered laser light. In [406], p. 146. 

Figure 9.3 Weisskopf single-particle estimates of the transition rates for (a) electric multipoles and (b) magnetic multipoles. From Condon and Odishaw, Handbook on Physics, 2nd Edition. Copyright 1967 hy McGraw-Hill Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.

Now, by use of formulas (2.23)-(2.26) we are in a position to present in. /-representation all the operators needed. For example, the non-relativistic operator of electric multipole radiation will have the form... 

Radiative and autoionizing electronic transitions. Generalized expressions for electric multipole (Ek) transition operators... 

Starting with Eq. (4.1) after tedious mathematical transformations [18] we finally find the following two general forms of the relativistic expressions for the electric multipole (Ek) transition probability (in a.u.) ... 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

By evaluating the commutator of Q q with Hnr the generalization of the acceleration form of the transition operator to cover the case of electric multipole transitions of any multipolarity was obtained [77]. Unfortunately, the resultant operator has a much more complex and cumbersome form than Q q and Q q, since it contains both one-electron and two-electron parts. 

Relativistic corrections to various forms of the electric multipole... 

Let us consider the intercombination transitions. Then, we shall retain only the corrections containing the spin operator in the expansion. To find the form of the operator describing the electric multipole intercombination transitions and absorbing the main relativistic corrections, one has to retain in the corresponding expansion the terms containing spin operator S = a and to take into account, for the quantities of order v/c, the first retardation corrections, whereas, for the quantities of order v2/c2 one must neglect the retardation effects. Then the velocity form of the electric dipole transition probability may be written as follows ... 

This operator, too, is a scalar in the space of total angular momentum for an electron. Tensors in this space are, for example, the operators of electric and magnetic multipole transitions (4.12), (4.13), (4.16). So, the operator of electric multipole transition (4.12) in the second-quantization representation is... 

The operator of the hyperfine structure, caused by electric multipole radiation, may be presented in the form... 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

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