# What is the origin of the many-body expansion?

I'm looking for the original introduction of the many-body expansion (MBE) in the scientific literature. More specifically, I'm interested in a theoretical justification of the rapid convergence of the expansion, especially in the context of molecular physics. The MBE is often introduced in scientific papers without referring to foregoing work that introduces the concept. Instead, one often states something like: "Following the well-known many-body expansion, one writes ...". See for example http://pubs.acs.org/doi/abs/10.1021/ct600253j. (freely available at http://t1.chem.umn.edu/Truhlar/docs/758FAV.pdf)

The many-body expansion is a scheme to decompose the energy of a general system of $N$ particles as follows:

$$V = \sum_{i = 1}^N V_i$$

with

$$V_1 = \sum_{i=1}^N E_i$$

$$V_2 = \sum_{i=1}^N \sum_{j=i+1}^N E_{ij} - E_i - E_j$$

$$V_3 = \sum_{i=1}^N \sum_{j=i+1}^N \sum_{k=j+1}^N \biggl( (E_{ijk} - E_i - E_j - E_k) - (E_{ij} - E_i - E_j)\\ - (E_{jk} - E_j - E_k) - (E_{ki} - E_k - E_i)\biggr)$$

and so on. In these equations, $E_i$ is the energy consisting only of particle $i$, $E_{ij}$ is the energy of a system containing only particles $i$ and $j$, $E_{ijk}$ is the energy of a system with three partilces, $i$, $j$ and $k$, and so on.

-
There is some justification of convergence speed based on perturbative expansion of intermolecular interaction combined with multipole expansion. E.g. the Axilrod–Teller potential in three-body interaction goes like $R^{-9}$, while London potential goes like $R^{-6}$. – user26143 Jul 21 '14 at 9:46

## 1 Answer

I believe it has to do with the Inclusion-Exclusion Principle from combinatorics. (link: http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle)

In doing so, one is making the assumption that the interaction energies between distinct, separate bodies decreases as the number of bodies being taken into account increases.

Of course, this is not always true, but it suffices in most cases in which we are considering relatively weaker interaction energies, e.g. van der Waal's dispersion, between disparate atoms or molecules that are sufficiently distant from each other, whereby quantum theory doesn't really play a part.

-