Calculating the total body order of MACE

[frgoe003]

I have a question regarding calculating the total body order of a given MACE architecture.

In the docs, it is written that

A two layers mace, with --correlation=3 has a total body order of 13.

Which is further explained in ref¹:

In the MACE models studied in this paper, each layer has a body-order of 4 [corresponding to ν = 3]. Therefore, the node features of the first layer are four-body functions. They are then expanded in the second layer in the second one-particle basis, which is 4 + 1 = 5 body functions, since each application of the one-particle basis adds one to the body-order on account of the central atom. Then, taking the tensor products of these features with themselves three times, we obtain 3 × 4 + 1 = 13-body features (note that the five-body terms of the first layer all share the same central atom).

So the body order of a given layer is

In the end, that means that the maximum total body order can be represented by:

where T represents the number of layers and nu the correlation order. Which holds for the corr=3, T=2 example.

In ref² however, this equation is mentioned:

I'm confused why not the exact formula was used. Am I missing something?

Thanks, Franz

Footnotes

1. Kovács, Dávid Péter, et al. "Evaluation of the MACE force field architecture: From medicinal chemistry to materials science." The Journal of Chemical Physics 159.4 (2023). ↩

2. Batatia, Ilyes, et al. "The design space of E (3)-equivariant atom-centred interatomic potentials." Nature Machine Intelligence 7.1 (2025): 56-67. ↩

[ilyes319]

Hey @frgoe003, the body order after T layers at correlation $\nu$ is indeed $\sum_{t=0}^{T} \nu^{t} = \frac{\nu^{T+1} - 1}{\nu - 1}$. I don't remember how that got in the table tbf