Multipole interactions

The multipole interaction of immobile particles (4.1.44) is an additional way to check up advantages of the superposition approximation [8]. The reason is that the tunnelling recombination (3.1.2) serves better as an example of short-range reaction. Indeed, the distinctive scale ro characterizing distant (non-contact) interaction could be defined as 

For the tunnelling recombination ro is typical of the order of 1 A. In this sense the multipole interaction is a long-range one since substitution of (4.1.44) into (5.1.65) yields ro = oo. Generally speaking, the accuracy of the superposition approximation can turn out to be dependent on the actual recombination law. 

To be short, in Fig. 5.10 results of the computer simulation (full curves) and the superposition approximation (broken curves) are compared for d = 1 and 2 and equal particle concentrations, riA t) = n t) = n(t). Time is given is units aJ jaQ, where 3 is a lattice constant. 

As it took place for the tunnelling recombination, divergence in results is not large. It will be shown in Chapter 6 that the reaction depths studied here are enough to establish appearance of the new asymptotic kinetic laws. The superposition approximation giving a lower bound estimate of the kinetics, reproduces correctly the kinetics at long times. Results of the linear approximation are not plotted since they diverge considerably from the statistical simulations. 

For unequal reactant concentrations a divergence is observed to be small - Fig. 5.11. In physics of phase transition long-range interactions are known 

---

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 ... 

The electrostatic interaction results from the interaction of tire ion with the pennanent multipole moments of the neutral. For cylindrically synnnetric neutrals or linear molecules, the ion-neutral multipole interaction is... 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

It is of special interest for many applications to consider adsorption of fiuids in matrices in the framework of models which include electrostatic forces. These systems are relevant, for example, to colloidal chemistry. On the other hand, electrodes made of specially treated carbon particles and impregnated by electrolyte solutions are very promising devices for practical applications. Only a few attempts have been undertaken to solve models with electrostatic forces, those have been restricted, moreover, to ionic fiuids with Coulomb interactions. We would hke to mention in advance that it is clear, at present, how to obtain the structural properties of ionic fiuids adsorbed in disordered charged matrices. Other systems with higher-order multipole interactions have not been studied so far. Thermodynamics of these systems, and, in particular, peculiarities of phase transitions, is the issue which is practically unsolved, in spite of its great importance. This part of our chapter is based on recent works from our laboratory [37,38]. 

Equation (7) can be applied to interactions of multivalent ions of valence z dissolved in a dielectric by multiplying the right-hand side by (z2/Z>), where D is the dielectric constant of the medium. Equations (5) and (6) do not include ion-multipole and multipole-multipole interactions. A major part of these interactions may be represented by a pseudo dielectric constant which is relatively small and should not vary greatly from salt to salt in a given group.8... 

The result of the electrostatic multipole interaction is to deplete primarily the 5p-shell electrons in the I ion. The number of electrons originally in the 5p closed shell are lost to excited states. Since the 5p excitation is primarily to the 65 shell, the direct effect on the 5 electron density at the nucleus will be much smaller than the 5 density. As a result of the depletion of 5p electrons, however, the nucleus will be less shielded, and the 5 electrons will have a greater probability of being at... 

London-van der Waals forces, which are multipole interactions produced by correlation between fluctuating induced multipole moments in two nearly uncharged polar molecules. These forces also include dispersion forces that arise from the correlation between the movement of electrons in one molecule and those of neighboring molecules. The van der Waals dispersion interaction between two molecules is generally very weak, but when many groups of atoms in a polymeric structure act simultaneously, the van der Waals components are additive. 

Electrical moments are useful because at long distances from a molecule the total electronic distribution can be increasingly well represented as a truncated multipole expansion, and thus molecular interactions can be approximated as multipole-multipole interactions (charge-charge, charge-dipole, dipole-dipole, etc.), which are computationally particularly... 

Let us consider as an illustration the multipole interaction which is observed in many solids and liquids due to the interaction of the electronically excited donors D with acceptors A resulting in an energy transfer D + A -4 D + A -4 D + A + hv [16, 17]. Its probability (per unit time) is... 

As it was shown in Chapter 2, even the linear approximation demonstrates emergence of the distinctive scale factors increasing in time - the correlation lengths . So, for the multipole interaction (4.1.44) there is no scale ro at all, whereas dimensionless parameter where o is defined by... 

To illustrate this point, let us consider recombination of immobile particles with the multipole interaction. In the linear approximation there exists a single correlation length 0> equation (4.1.45). From two lengths Iq and 0> we can combine a new one... 

For the multipole interaction (4.1.44) the dissimilar correlation function could be also presented in a form of product (6.1.4), where Yo(r,t) = exp[—cr(r)f]. Neglecting indirect correlation mechanism, the dissimilar particle function Yo(r, t) — exp[— (r/ o)-m], with o defined by (4.1.45), is stationary in term of variable r)0 = r/ o- Indirect mechanism of the correlation formation, as follows from a solution of equations derived in the superposition approximation, results at long times in Y(r, t) z r) t),... 

In analogy with (6.1.1), let us define the asymptotic exponent a of the multipole interaction as... 

The hyperfine structure (splitting) of energy levels is mainly caused by electric and magnetic multipole interactions between the atomic nucleus and electronic shells. From the known data on hyperfine structure we can determine the electric and magnetic multipole momenta of the nuclei, their spins and other parameters. 

The form of this equation makes explicit the fact that intermolecular forces do depend upon their vibrational states as well as on their electronic states. Due to the antisymmetrization of the global electronic wave function, Vaia2(R R12) contains Coulomb exchange terms and a direct term formed by the Coulomb multipole interactions and the infinite order perturbation electrostatic effects embodied in the reaction field potential [21, 22],... 

EMTP is the electrostatic interaction energy calculated as a sum of multipole-multipole interactions using the overlap multipole expansion of the SCF electron density distributions of the host and guest182). 

Figure 18 Models from which the excitonic coupling between pairs of peptide groups were calculated (a) The direction and location of the transition dipole of the amide I mode (118,123) from which the coupling between two peptide groups is calculated according to a dipole-dipole interaction term [Eqaution (28)] (b) The nuclear displacements, partial charges, and charge flow of the amide I normal mode obtained from a DFT calculation on deuterated N -methylacetamide (all experiments were performed in D2O) (42). With this set of transition charges, the multipole interaction is computed, avoiding the limitations of the dipole approximation.

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)... 

---