Multipoles

The fourth level is similar to the previous one, except that it allows more than two poles to associate in order to form a chain (serial mounting) or a ladder (parallel mounting) or both. Examples of chains are chemical reactions composed of several steps and reactional intermediates, or conduction processes working by exchange or hopping between several sites. Examples of ladders are multiloops of current (solenoids) in electromagnetism, or magnetic domains in parallel. 

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Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

The perturbation theory described in section Al.5.2,1 fails completely at short range. One reason for the failure is that the multipole expansion breaks down, but this is not a fiindamental limitation because it is feasible to construct a non-expanded , long-range, perturbation theory which does not use the multipole expansion [6], A more profound reason for the failure is that the polarization approximation of zero overlap is no longer valid at short range. 

The details of the second-order energy depend on the fonn of exchange perturbation tiieory used. Most known results are numerical. However, there are some connnon features that can be described qualitatively. The short-range mduction and dispersion energies appear in a non-expanded fonn and the differences between these and their multipole expansion counterparts are called penetration tenns. 

Stone A J 1981 Distributed multipole analysis or how to describe a molecular charge distribution Chem. Phys. Lett. 83 233... 

Stone A J and Alderton M 1985 Distributed multipole analysis—methods and applications Mol. Phys. 56 1047... 

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 ... 

Intermolecular quadrupolar 2 Fluctuation of the electric field gradient, moving multipoles Common for />1 In free Ions In solution [la... 

The polarization interaction arises from the interaction between the ion of charge Ze and the multipole moments it induces in the atom or molecule AB. The dominant polarization interaction is the ion-mduced 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... 

The dispersion interaction arises between the fluctuating multipoles and the moments they induce and can occur even between spherically synuuetric ions and neutrals. Thus,... 

The anisotropy of the product rotational state distribution, or the polarization of the rotational angular momentum, is most conveniently parametrized tluough multipole moments of the distribution [45]. Odd multipoles, such as the dipole, describe the orientation of the angidar momentum /, i.e. which way the tips of the / vectors preferentially point. Even multipoles, such as the quadnipole, describe the aligmnent of /, i.e. the spatial distribution of the / vectors, regarded as a collection of double-headed arrows. Orr-Ewing and Zare [47] have discussed in detail the measurement of orientation and aligmnent in products of chemical reactions and what can be learned about the reaction dynamics from these measurements. 

White C A, Johnson B G, Gill P M W and Head-Gordon M 1994 The fast multipole method Chem. Phys. Lett. 230 8-16 White C A, Johnson B G, Gill P M W and Head-Gordon M 1996 Linear sealing density funetional ealeulations via the eontinuous fast multipole method Chem. Phys. Lett. 253 268-78... 

Basiev T T, Orlovskii Y V and Privis Y S 1996 Fligh order multipole interaotion in nanoseoond Nd-Nd energy transfer J. Luminescence 69 187-202... 

Soholes G D and Andrews D L 1997 Damping and higher multipole effeots in the quantum eleotrodynamioal model for eleotronio energy transfer in the oondensed phase J. Chem. Phys. 107 5374-84... 

Niedermeier, C, Tavan, P. A structure-adapted multipole method for electrostatic interactions in protein dynamics. J. chem. Phys. 101 (1994) 734-748. 

Abstract. Molecular dynamics (MD) simulations of proteins provide descriptions of atomic motions, which allow to relate observable properties of proteins to microscopic processes. Unfortunately, such MD simulations require an enormous amount of computer time and, therefore, are limited to time scales of nanoseconds. We describe first a fast multiple time step structure adapted multipole method (FA-MUSAMM) to speed up the evaluation of the computationally most demanding Coulomb interactions in solvated protein models, secondly an application of this method aiming at a microscopic understanding of single molecule atomic force microscopy experiments, and, thirdly, a new method to predict slow conformational motions at microsecond time scales. 

As an example for an efficient yet quite accurate approximation, in the first part of our contribution we describe a combination of a structure adapted multipole method with a multiple time step scheme (FAMUSAMM — fast multistep structure adapted multipole method) and evaluate its performance. In the second part we present, as a recent application of this method, an MD study of a ligand-receptor unbinding process enforced by single molecule atomic force microscopy. Through comparison of computed unbinding forces with experimental data we evaluate the quality of the simulations. The third part sketches, as a perspective, one way to drastically extend accessible time scales if one restricts oneself to the study of conformational transitions, which arc ubiquitous in proteins and are the elementary steps of many functional conformational motions. 

Task 1 Calculate the first non-vanishing multipole moment of the electrostatic potential of composed objects (i.e., structural units and clusters). 

In general, multiple-time-step methods increase computational efficiency in a way complementary to multipole methods The latter make use of regularities in space, whereas multiple-time-stepping exploits regularities in time. Figure 2 illustrates the general idea ... 

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