Cross multipole terms

The idea of correlating momentary multipoles stands behind the customary modeling of dispersion interaction in the form of a multipole expansion, including dipole-dipole (D-D), dipole-quadrupole (D-Q), quadrupole-quadrupole (Q-Q), and so on, terms. We owe the earliest variational treatments of this problem not only to Slater and Kirkwood [34], but also to Pauling and Beach [35], However, when the distance R decreases and reaches the Van der Waals minimum separation, the assumption that electrons of A and B never cross their trajectories in space becomes too crude. The calculation of the intermonomer electron... 

Very often it is not possible to obtain all basic scattering functions with the same accuracy. Even worse, in resonant scattering we are often left with the cross term I ,(h) only. This is still quite an acceptable situation, if the resolution to a monopole approximation of the structure is required, as A fh) and B fh) may be determined completely from the two remaining functions I (h) and I (h). However a straight-forward method for the evaluation of higher multipoles in the sense of the above calculation then does not exist any more. The analysis of resonant scattering in this case has to refer to models. 

Some of the types of contributing elements combined in Eq. (1) can give rise to potential pieces that are not additive. These would involve products of property or parameter values for more than two molecules, and these are often referred to as cooperative or nonpairwise additive elements. A simple illustration is in the electrical interaction contributions. While the interaction of permanent moments is pairwise additive, involving products of moments of only two different molecules at a time, the polarization energy can have a cooperative part. For some cluster of the molecules A, B, and C, the dipole polarization energy of A will be the polarizability of A, Ka, multiplied by the square of the field experienced at A, F. That field is a sum of contributions from B and C ( F = Fb + Fc) proportional to their multipoles, and its square has a cross term, FbFc, involving a multipole of B times a multipole of C. The net interaction element includes Ka FbFc. thereby giving an overall A B C or three-body term. Mutual or back polarization can be shown to produce contributions up to A-body for a system of N species. 

The interaction of higher multipoles (permanent as well as induced first, the octupole with the corresponding octupole polarizabilities and hyperpolarizabilities, etc.) with the higher derivatives of electric field together with the corresponding cross terms denoted as H-... 

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