The Multipole Expansion

Multipole moments are useful quantities in that they collectively describe an overall charge distribution. In Chapter 0, I explained how to calculate the electrostatic field (and electrostatic potential) due to a charge distribution, at an arbitrary point in space. 

In principle, we can calculate the potential at the point in space r from 

The multipole expansion gives exactly that expression. If the charge distribution shown has an overall charge Q, an electric dipole pe, an electric quadrupole e, and so on, then we write 

The potential due to a point charge falls off as 1 /r, the potential due to a dipole falls off as 1/r2, and so on, and the expectation in that the series will quickly terminate for chemical problems. 

The equations become a little more compact if we take the point in space as the coordinate origin. The potential due to a dipole is then 

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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 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. 

DP-4 Multipole code, 4 levels of macroscopic expansion, 4 terms in the multipole expansions, low accuracy... 

The convergence sphere of the multipole expansion for a molecule such as butane may be penetrated by r molecule. 

It is conventional to write the multipole expansion as a series in 1/r, and so we need to find alternative expressions for terms higher than the first. From elementary vector calculus we have... 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

In the literature slightly different definitions of the multipole expansion are found, depending on how the pre-factor (2( — l)/2 is distributed between expansion equation and the definition of die coefficients. Cf. (Ward [251], eq. 5.2)... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

The coefficients of the multipole expansion are computed from Eq. (9.8), and after analogous expansions of both the intensity of the perfectly oriented structural entity (i0pt, be), and of the measured intensity (/, ce), Ruland [253] obtains a set of algebraic equations among the expansion coefficients,... 

That is, in the singlet-singlet transition both Coulomb interaction and exchange interaction are involved. However, when the distance between D and A is large, the exchange term can be ignored, and we can use the multipole expansion for e2/rij, that is,... 

The first terms of the power series obtained by the multipole expansion of the Coulomb intermolecular potential account for dipole-dipole interactions prevailing in systems of polar molecules. As an adequate approximation for ensembles of... 

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