Higher Multipolarities

The experimental access to all these other modes has to use methods beyond simple photoabsorption measurements. They have been studied extensively in the case of nuclei by various sorts of scattering experiments (with electrons, protons, ions), which could even provide the full transition formfactors Detailed scattering experiments in the 

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As we have seen from the selection rules for non-relativistic magnetic dipole (M1) radiation, these transitions are permitted only between levels of one and the same configuration. However, this is not so for higher multipolarities (k > 1). Therefore we present here the appropriate formulas to cover the general cases needed in practice. So, it may be of use to have the following expression for the submatrix element of the operator of Mfc-transitions between the levels of two different two-shell configurations ... 

Theoretical formulations of reorganization in the course of electron-transfer processes have undergone a number of advances in recent years. The relative importance of various solvent contributions (including translational as well as orientational response, and inductive and dispersion as well as elecrostatic interactions) can depend strongly on the polarity (i.e., dipolar, higher multipolar, or nonpolar) as well as other molecular features of the solvent [21, 47-49]. Molecular-level perspectives on solvent response are of great utility in helping to parameterize effective cavity models (e.g., in conjunction with conventional [50] or spatially nonlocal [47] dielectric models). Additivity relationships traditionally assumed to pertain to sol-... 

The Ce dispersion coefficients for dipole dipole dispersion between pairs of interacting species, the coefficients for terms involving higher multipolar dispersion, and coefficients for three-body dispersion terms can be and have been evaluated by ab initio techniques [114 119] as well as through relations to experimental optical data based on moments of the dipole oscillator strength [120 122]. These are parameters of the interaction, not properties. However, as noted in Section IVA, values for Ce coefficients of like pairs (e.g., A-A), and possibly for other dispersion coefficients, can be used in simple [Eq. (4)] or in more complete forms [Eq. (2)] as an intrinsic property of a molecule. The basis set and correlation requirements for adequate evaluation show, in part, the same requirements for describing polarizabilities however, there are further needs and other than atom-centered functions are seen as being suited [49 52]. 

In the context of this discussion, surface heterogeneity will be expressed in terms of the adsorptive potential of the material. The adsorptive potential is a measure of the net attraction between a solid surface and an adsorbed probe molecule. For physical adsorption, these forces arise chiefly from London-type dispersion interactions (van der Waals forces) resulting from induced-dipole/induced-dipole and higher multipolar attractions which in turn depend on the size. 

Polar molecules generally interact more strongly with each other than nonpolar ones (per unit volume). The reason for this is that in addition to van der Waals interactions they also interact via dipolar and higher multipolar interactions and, possibly, hydrogen bonds. 

At the same time, of course, the molecules can have static moments of higher multipolarity both in the ground and in the excited state. Generally speaking, these moments are different in the ground and in the excited states. The molecules in these states also have different energies of interaction with the environment so that the frequencies of the intramolecular transitions are somewhat shifted with respect to the transition frequencies in the vacuum. We shall assume below that the only difference between the tensor aij(u>) and the respective tensor for a molecule in the vacuum is determined by this frequency shift. 

At the dipolar level, these approaches are similar. Howevei contributions from quadrupolar and higher moments present in the first method are absent in the second. The calculated contributions from the higher multipolar moments can be significant (0.9 V for the water-hydrocarbon interface ). The difference in results from the two approaches is puzzling in light of the published proofs that the potential drop for a uniform planar interface should depend only on the polarization density and not on higher order terms like the quadrupolar density discussed above. 

All other representations require higher multipolar (/ = 4,6) operators. The are not to be confused with the individual OJ operators in the CEF potential [eq. (1)]. The latter are in general not cubic representations. However, the O can be expressed as linear combinations of the O . The magnetoelastic Hamiltonian has to be invariant under the relevant point group. To first order in the local strains this leads to (Callen and Callen 1963)... 

In these terms it is strongly recommended to redefine the term polarity. Instead of meaning solely dipolarity, it should also include higher multipolar properties,... 

In case of radiation transitions of higher multipolarity (quadrupole with L = 2, octupole with L = 3, and so on) this expression requires a standard substitution... 

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