Multipolar interaction

An explicit development of Eq. 7.2 in terms of the Cartesian coordinates from Table 7.1 yields 

In terms of running a molecular dynamics (MD) simulation, the quantity of interest is the interaction potential, U. This quantity is defined by the work done on an MTP Qi brought from infinity to a point r in a region populated by the (multipolar) potential, [/ = Qiic (derived from first-order perturbation theory [30, 31]). Thus, the interaction energy between sites (e.g., atoms, molecules) a and b can be written as tab 

For a given interaction between two MTP moments Qf and Q on sites a and b, respectively, the tensor element describing the geometry as is required, where q forms a set of basis 

From Table 7.1, one realizes that placing MTPs up to quadrupoles on a given site will yield nine independent parameters in spherical coordinates (i.e., one for the monopole, three for the dipole, and five for the traceless second-rank tensor). However, the main computational hurdle in MD simulations is the force calculation. Although Eq. 7.6 refers to the pairwise interaction potential, it shows that the associated force (and of course energy) will consist of n x n independent terms, where n is the number of MTP coefficients. As such, the interaction between two MTP sites, described up to quadrupole, will involve 9x9 = 81 terms—to be put in perspective with the single term prescribed by the Coulomb interaction in standard PC force fields. This certainly provides one major reason why MTP force fields have not become routine in the MD community. 

Most equations so far included an infinite collection of terms a distributed MTP expansion without truncation. Formally, the infinite sum in Eq. 7.4 is capable of rep reducing the potential with arbitrary accuracy, given the observation point, r, is located far enough from the molecule (recall that the above-mentioned expansion requires r /r 1—the direct consequence of the convergence properties 

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Let us reconsider the four-level system shown in Fig. 4.1(6), which has two doublets in the spectrum split by 2A and 2e (Fig 4.4(a)). Since diagonal elements of G,kjm are the same in impact and Markovian theories we assume that F , = 0 without any restriction of generality. This is actually the case for any electric multipolar interaction and hence Ac0 = 0. The non-zero elements of the perturbation... 

Sagui C, Pedersen LG, Darden TA (2004) Towards an accurate representation of electrostatics in classical force fields Efficient implementation of multipolar interactions in biomolecular simulations. J Chem Phys 120 73-87... 

For electric multipolar interactions, the energy transfer mechanism can be classified into several types, according to the character of the involved transitions of the donor (D) and acceptor (A) centers. Electric dipole-dipole (d-d) interactions occur when the transitions in D and A are both of electric dipole character. These processes correspond, in general, to the longest range order and the transfer probability varies with l/R, where R is the separation between D and A. Other electric multipolar interactions are only relevant at shorter distances dipole-quadrupole (d-q) interaction varies as l/R, while quadrupole-quadrupole interaction varies as l/R °. 

The dependence on R of the transfer probability due to electric multipolar interactions can be written in a general way as follows ... 

The shape of the I(t) curves of the donor centers carries very useful information about the nature of the interaction process. Assuming that the acceptors A are randomly distributed at various distances from the donor centers D, the Japanese scientists Inokuti and Hirayama (1965) investigated the shape of the donor decay-time curves for the different multipolar interactions and also for the exchange interaction. 

The potential of mean force due to the solvent structure around the reactants and equilibrium electrolyte screening can also be included (Chap. 2). Chapter 9, Sect. 4 details the theory of (dynamic) hydro-dynamic repulsion and its application to dilute electrolyte solutions. Not only can coulomb interactions be considered, but also the multipolar interactions, charge-dipole and charge-induced dipole, but these are reserved until Chap. 6—8, and in Chaps. 6 and 7 the problems of germinate radical or ion pair recombination (of species formed by photolysis or high-energy radiolysis) are considered. 

It was assumed that the quasiresonant transfer of excitation with the rates Wss or Wsr is due to the multipolar interaction between the excited molecules and the decay of the doubly excited molecules 1,3 A is practically instantaneous [263]. 

Nonradiative energy transfer is due to electric or magnetic multipolar interactions or to exchange interactions. ... 

For electric multipolar interactions, the transfer rate IFda is proportional to the probability of the donor and acceptor transitions, to the overlap between the emission and absorption bands of the donor and acceptor, and to an inverse power of the donor-acceptor separation, with = 6, 8, 10 for dipole-dipole, dipole-quadrupole, or quadrupole-quadrupole interaction. For electric dipole-dipole interaction, the transfer rate is proportional to... 

When acceptor ions are present, the decay of donor ions does not exhibit an exponential behavior. In the absence of energy transfer between donor ions and assnming a random distribntion of acceptor ions, for multipolar interactions the decay is given by... 

Dexter, following the classic work by Forster, considered energy transfer between a donor (or a sensitizer) S and an acceptor (or activator) A in a solid. This process occurs if the energy difference between the ground and excited states of S and A are equal (resonance condition) and if a suitable interaction between both systems exists. The interaction may be either an exchange interaction (if we have wave function overlap) or an electric or magnetic multipolar interaction. In practice the resonance condition can be tested by considering the spectral overlap of the S emission and the A absorption spectra. The Dexter result looks as follows ... 

These imperfections have occasioned to review the spherical DFT approach with respect to a more correct description for fluids which consists of non-spherical particles. The paper applies a statistic thermodynamic approach [7, 8] which uses density functional formulation to describe the adsorption of nitrogen molecules in the spatial inhomogeneous field of an adsorbens. It considers all anisotropic interactions using asymmetric potentials in dependence both on particular distances and on the relative orientations of the interacting particles. The adsorbens consists of slit-like or cylinder pores whose widths can range from few particle diameters up to macropores. The molecular DFT approach includes anisotropic overlap, dispersion and multipolar interactions via asymmetric potentials which depend on distances and current orientations of the interacting sites. The molecules adjust in a spatially inhomogeneous external field their localization and additionally their orientations. The approach uses orientation distributions to take the latter into account. 

Note that the simplified expressions obtained here would be complicated by the inclusion of resonant V-V transfer, which could also give a large many-body contribution to dephasing if multipolar interactions are involved. Also a very common source of motional narrowing arises from vibration-rotation coupling to which the present discussion is not adapted. 

Paulini, R., Muller, K. and Diederich, F. (2005) Orthogonal multipolar interactions in structure chemistry and biology. Angew. Chem. Int. Ed., 44, 1788-1805. 

Schweizer, E., Hoffmann-Roeder, A., Olsen, J. A., el al. (2006) Multipolar interactions in the D pocket of thrombin large differences between tricyclic imide and lactam inhibitors. Org. Biomol. Chem., 4, 2364-2375. 

Dipole interactions are usually weaker than electrostatic monopole interactions but can dominate the intermolecular interactions within a supramolecular assembly. Diederich and coworkers have recently drawn attention onto dipole interactions, and multipolar interactions in general, in such systems based on a statistical analysis of structures [180]. 

In the condensed phase the field at the molecule will be different from the applied macroscopic field due to induced dipole-dipole (and higher order multipolar) interactions with the surrounding molecules, as briefly mentioned in Section 2.1... 

Molecules composed of atoms of different electronegativities have usually an asymmetric distribution of electrons, which produces electronic dipoles. These dipoles within a cell or in aqueous medium can be attracted by a close-by ion, establishing so-called charge-dipole interactions. A permanent dipole can also interact with another permanent dipole leading to a dipole-dipole interaction. In a recent review, Diederich et al. has listed these orthogonal multipolar interactions among which C—X C=0 (X=halogen),... 

The high electronic density associated with the dinuclear centre was suggested to produce electric multipolar interactions and this combined with the sterically rigid core leads to the high transition temperatures in the mesogens. 

In general, to incorporate the matrix elements of Eq. (47) into the rate equation (38), it is necessary to sum, over all molecules in the system, the tensor product entailed in the former—and to this end it proves useful to isolate the one part of the above radiation tensor that is molecule-specific. This simply reflects the fact that the tensor is a field quantity, sensitive to the position of the molecule at which it is evaluated, as follows from the phase factors in (19) and (20). The tensor representing the radiation field for the interaction at molecule E, may, in fact, be written in the following general form, irrespective of the order or nature of the multipolar interactions involved ... 

The evidence available suggests that the two approaches are about equally accurate, although the approach based on site-site correlation functions is more readily generalized to the treatment of multipolar interactions as well as to the effect of the attractive forces upon the structure and free energy at moderate and low density. In addition to the efforts made at extending the WCA theory to interaction site fluids, the Barker-Henderson theory has also been extended to these systems by Lombardero, Abascal, Lago and their co-workers. ... 

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. 

The microscopic behaviour between the ions in dilute systems results from multipolar interaction. On the other hand in the rate equations which are used for measurement of macroscopic data such as quantum efficiencies of fluorescence, the multipole questions are absent. The macroscopic treatment of energy transfer was performed recently independently by Fong and Diestler (5) and Grant ( >) who conclude that the concentration... 

The coupling of adjacent ions in such a case can arise via exchange interaction if their wave-functions overlap, via super exchange interactions involving intervening ions, or via various electric or magnetic multipolar interactions. 

Blasse and Bril (15) came to the conclusion that the exchange interaction is active if the S emission band overlaps the 4/- 4/absorption bands of 4, and by electric multipole interaction if the S emission band overlaps allowed absorption bands of A. Their assumption was based mainly on the fact that the exchange interaction depends on the overlap integral only while the multipolar interaction depends on the absorption cross-section in addition to the overlap. 

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