Expressing interaction between two classical charge distributions in terms of multipole moments

Question

I am interested in expressing the interaction energy between two classical charge distributions in terms of the multipole moments of each of the charge distributions. For example, let's assume each of the charge distributions have a net charge of zero. Then the leading order term in the interaction energy will take the following "dipole-dipole" form:

$$J=\frac{\vec{\mu}_0\cdot\vec{\mu}_1}{|\vec{x}|^3}-3\frac{(\vec{\mu}_0\cdot\vec{x})(\vec{\mu}_1\cdot\vec{x})}{|\vec{x}|^5}$$

where $\vec{\mu}_0$ and $\vec{\mu}_1$ are the dipole moments of each of the charge distributions, and $|\vec{x}|$ is the distance between the "centers" of the charge distributions about which all of the dipole moments are calculated. Does anyone know of a reference where the remaining terms in this series are listed and/or calculated? I would be able to express the interaction between two charge distributions in terms of each of their charges, dipoles, quadrupoles, octupoles, etc. Thanks!

Answer 1

This is one of the topics relevant to those who model intermolecular forces, for computer simulation and/or theory of the liquid state (actually, for gaseous and solid states of matter too). The molecular charge distribution is typically modelled by a multipole expansion (sometimes, by a set of distributed multipoles) and the electrostatic interactions between a pair of molecules are calculated in just the way you describe.

Two of the standard works in the field are The Theory of Intermolecular Forces by Anthony Stone, second edition (Oxford University press, 2013), and Theory of Molecular Fluids by Christopher G Gray and Keith E Gubbins (Clarendon Press 1984). Both texts are very clearly written, go into great detail, and give formulations based on both Cartesian tensors and spherical tensors. The issue of convergence of such expansions is also addressed in both books.

In both cases, they go much further, discussing induction and dispersion interactions and so on, and the way the multipoles may be calculated for real molecules through computational chemistry, but these aspects may not be so relevant to your application.