Expansion, multipole

The multipole expansion may be carried out in several coordinate systems, which may be chosen depending on the symmetry properties of the problem under investigation. Spherical polar and Cartesian coordinates are used most commonly when calculating two-electron integrals. We outline here the derivation for the spherical series. The interested reader may find more detailed discussions, for example, in the books of Eyring, Walter and KimbalP or Morse and Feshbach. A discussion of the multipole expansion in the framework of atomic and molecular interactions and potentials may be found in the article of Williams in this series of reviews (Ref. 34) or the book by Hirschfelder, Curtiss and Bird.  

Before deriving the multipole series, let us start with some nomenclature. We want to evaluate the electron repulsion integral 

respectively. For our task it is handy to express the electronic coordinates by their position relative to the centers A and B as 

Our objective is to find a series expansion of the interelectronic distance ri2, which facifitates the separation into an angular and a radial part and which decouples the coordinates of electrons 1 and 2. We follow here the derivation of Eyring et al.  

With the definitions introduced in the previous paragraph, the interelec-tronic separation can be expressed as 

---

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. 

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. 

The speed of the method comes from two sources. First, all of the macroscopic cells of the same size have exactly the same internal structure, as they are simply formed of tessellated copies of the original cell, thus each has exactly the same multipole expansion. We need compute a new multipole expansion only once for each level of macroscopic agglomeration. Second, the structure of the periodic copies is fixed we can precompute a single transfer... 

Parallelizing this method was not difficult, given that we already had parallel versions of several multipole algorithms to start from. The entire macroscopic assembly, given its precomputed transfer function, is handled by a single processor which has to perform k extra multipole expansions, one for each level of the macroscopic tree. Each processor is already typically performing many hundreds or thousands of such expansions, so the extra work is minimal. 

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

A central multipole expansion therefore provides a way to calculate the electrostatic interaction between two molecules. The multipole moments can be obtained from the wave-function and can therefore be calculated using quantum mechanics (see Section 2.7.3) or can be determined from experiment. One example of the use of a multipole expansion is... 

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

The multipole moments (charge, dipole, quadrupole) of each cell are then calculated mining over the atoms contained within the cell. The interaction between all of the 3 in the cell and another atom outside the cell (or indeed another cell) can then be lated using an appropriate multipole expansion (see Section 4.9.1). 

One recent development in DFT is the advent of linear scaling algorithms. These algorithms replace the Coulomb terms for distant regions of the molecule with multipole expansions. This results in a method with a time complexity of N for sufficiently large molecules. The most common linear scaling techniques are the fast multipole method (FMM) and the continuous fast multipole method (CFMM). 

Another approach to reducing the cost of Coulombic interactions is to treat neighboring interactions explicitly while approximating distant interactions by a multipole expansion. In Eigure la the group of charges, i) at positions = (ri ... 

The multipole expansion gives exactly that expression. If the charge distribution shown has an overall charge Q, an electric dipole pe, an electric quadrupole 0g, and so on, then we write... 

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... 

---