Multipole contributions

Now, the quadrupole moment can next be calculated by differentiating the potential to get the electric field due to the dipole moment. The reader can now see that an infinite series can be thus generated. The total electric field is simply the sum of all the individual multipole contributions, given by... 

D. Multipole Contributions in Tip-Enhanced Near-Field Spectroscopy. ..263... 

U. Even In a recent series of papers [M. Bixon and J. Jortner], using a model Hamiltonian quantum treatment, it is shown that all multipole contributions to l mixing are negligible when compared with / mixing by low external fields. Thus the long lifetimes associated with ZEKE states are attributed (in atoms and in molecules) to the external fields alone. 

Fig. 8. Orientational dependence of the average (spin-independent) exchange interaction energy AE = Via) ,

Computations of various dipole-multipole contributions have been done by using a procedure similar to that used for Rayleigh spectra. In fact, the main change consists in replacing the coefficients tp 71 "72 of Table III by these of Table V. 

This representation of the scattering from randomly oriented particles is of considerable importance, as it opens up a general method of separating the various multipole contributions to 1(h). In fact, Svergun have proposed an algorithm which allows one to distinguish between groups of multipoles with the same index 1... 

Tan, M.-L., L. Lucan, and T. Ichiye, Study of multipole contributions to the structure of water around ions in solution using the soft sticky dipole-quadrupole-octupole (SSDQO) model of water. /. Chem. Phys., 2006.124 174505. DOI 10.1063/1.2177240, PMID 16689581. 

An odd-rank tensor can be associated with even- or odd-parity contributions to a matrix element, and these are conventionally described as electric and magnetic multipole contributions, respectively (Stassis and Deckman 1976). The orbital matrix element contains magnetic multipole terms, whereas the spin matrix element contains both magnetic and electric multipole terms. In terms of the notation introduced in sect. 2, A K, /C 1) and B K, A 1) are magnetic multipole terms, and B K, K) is an electric multipole term. The latter term vanishes for 5 = 5, L = L and J = J that is it does not contribute to the elastic form factor of an isolated magnetic ion. 

As seen above, the general treatment of the photon-molecule interaction expands the transition operator as a series with electric and magnetic multipole contributions ... 

The condition for this quantity to be non-zero is that the chromophore of interest must have a non-zero magnetic and electric transition dipole moment along the same molecular diiectioa In the absence of pertuibing external fields, this is only trae for molecules that are chiral. Expressions that include higher order multipole contributions to eqs. (8), (9), and (11) can be found in previous theoretical descriptions of CPU theory (Riehl and Richardson, 1976a, 1986). 

However, the second-order nonlinearity of the centrosymmetric bulk is not strictly vanishing because of the higher-order multipole contributions, and its effect on the surface SHG may not be negligible." Fortunately, in most cases of practical interest, the bulk contribution to SHG is either much smaller or can be subtracted from the surface contribution. 

The potential at the point r is the sum of atomic multipole contributions. A very convenient feature of the resulting expression is that the asymptotical value of the potential is expressed with point multipoles, while at shorter distances an exponentially vanishing correction factor, /(2Cr, 2), appears ... 

The leading multipole contribution to the matrices and is the dipole-induced dipole term. If we assume that the atomic polarizabilities a and ay are related to... 

Consider a perfectly tetrahedral distribution of four point charges —q, —q, +q, and +q, drawn schematically in the following figure. Does this arrangement result in a pure quadrupole field If not, what other multipoles contribute to the electric field ... 

The clear derivation of the dispersion energy according to Eqs. (4.32), (4.34) from multipole contributions suggests that it is the macroscopic multipole modes rather than the molecular dipole oscillators of spheres 1 and 2 which undergo fluctuations. This becomes even more obvious if we substitute the macroscopic dielectric permeabilities i(co) and fi2( ) of materials 1 and 2 for the molecular susceptibilities Xi(co) and X2(co) according to the law of Clausius-Mosotti... 

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