Octupole Interaction

James and Keenan [1959] showed that the octupole-octupole interactions are dominant in the multipole expansion of Vc in (7.57) ... 

Fig. 2) Octupole correlation energy for 0+ and 1" states in ZZ°Ra as a function of octupole interaction strength.

In terms of the octupole-octupole interaction, the properties of the low lying 1 state in the even even nuclides and the properties of parity doublets in odd mass nuclide are fairly well understood. Although we understand the El transitions qualitatively, a quantitative treatment of the El rates remains an open problem. 

Higher multipole-multipole interaction terms decrease at higher inverse powers of the intermolecular separation, but become important when the dipole-dipole interaction is symmetry forbidden, e.g., in benzene where the octupole-octupole interaction is dominant [161]. The electron-exchange interaction requires overlap of the electronic wave functions of M d and Ma, and it is therefore of short range (<1.5 nm). Due to an exponential decrease in the overlap of electronic wave functions with intersite distance, the energy transfer rate is expected to decrease more rapidly and, in fact, it can be expressed as (see e.g., Ref. 162)... 

Dipole-Octupole Interaction. Equation (58) yields in explicit form (with m= I, n = 3) ... 

Working out dipole lattice sums, for any wave vector k, became a simpler matter soon after, with computers of the IBM 650 generation, in about 1956-7. Stuart Walmsley (now Reader in Chemistry, University College London) came as my first PhD student after I returned to University College from the University of Sydney in 1956. He had a magic touch with the early computers, and made the first dipole-dipole lattice summations on simple aromatic crystals [220]. Dipole-octupole and octupole-octupole interaction sums were first calculated by Thirunamachandran [221] using the Ferranti Mercury Computer and later extended to quadrupole couplings. These sums are not shape dependent, but were of formidable difficulty at the time. 

The 1 states in the even-even nuclides 220< A < 230 are the lowest lying non-rotational states found in even-even nuclides. These states, and the equivalent states in odd mass nuclides, are discussed in terms of octupole correlations, from both a two-body interaction and an octupole deformed one-body potential point of view. Some discussion of a cluster model treatment of these states is given. 

Strutinsky procedure. Because asymmetric shapes is so shallow, it is worthwhile to deal with octupole correlation effects using a microscopic, two-body interaction treatment of the octupole-octupole residual interaction [CHA80]. The pairing force matrix elements, G. come from a density dependent delta interaction. This set of matrix elements [CHA77] explains many features of the actinides at low and... 

In the region of negligible overlap, the three-body interactions can be calculated, just like the two-body ones, from imaginary frequency polarizabilities. This has been done, for example, by Doran78 for interaction of triplets of noble-gas atoms. He gives the coefficients for dipole-dipole-dipole, dipole-dipole-quadrupole, dipole-quadru-pole-quadrupole, quadrupole-quadrupole-quadrupole, and dipole-dipolc-octupole terms. 

Interaction BH-BH in the linear configuration, a) Electrostatic interaction energy and the first terms of its multipole expansion, Intermolecular distance, R,and quadrupole and octupole moments refer to the centers of mass, b) SCF interaction energy and its components. 

Energies of multipole type. In cases when the external fidd is not uniform, one has to consider, in conformity with the expansion (S2a), interactions between the quadrupole moment and the field gradioit, the octupole and gradient of the field gradirat, and so forth. The potential energy of multipole interaction of ordar 1 being given by equation (54), we can by... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

An electric multipole is specified by its value of l as 2l—pole (1=1 dipole, / = 2quadrupole, l = 3 octupole, etc.). Hence, the electrostatic interaction is between l1— 1 poles, the leading term for two dipolar molecules (l = V = 1) being the dipole-dipole interaction. 

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