That sounds like really exciting research. I guess the ultimate goal in representing environments would be to construct a function that captures all the statistically relevant information of a particular "type" of atomic neighborhood. It would reduce the degrees of freedom to the bare minimum necessary to recreate a similar environment that correctly reproduces any property of interest (kind of like data compression for materials). Would that be right?
The deal with the ions is that we are studying the transport and storage of lithium in a new type of carbon anode. The anode consists of small crystalline domains distributed throughout an amorphous carbon matrix. In order to capture all of the features of this material, we end up with a system of almost a million atoms. At the end of an equilibration run, the lithium ions can be found at different locations within the carbon matrix. It turns out that the potential energy of the lithium ions (as computed from the reactive potential the simulation was performed with) has a wide range of values. So we can sort these ions into bins of a histogram (this is what I meant by "energy levels"). And because there are so many samples, the radial distribution functions (RDFs) can be computed for all Li-C pairs in each separate energy bin. (The computed RDFs are useful because the results can be compared to the experimental RDFs obtained from neutron scattering.)
However, zooming in on ions of different potential energies reveals very little visual difference in the local environment. Yet we know there is definitely a difference in the structure because of the RDFs, but we cannot get a good understanding of it or provide a good representation of it. So that's when I discovered the 2010 paper by Bartók. As you mentioned, I want to figure out how the entire local atomic neighborhood affects the energy (as opposed to only the radial component), and I also want to create a 3D graphic that provides a good visualization of the differences between the atomic neighborhoods. In that sense, I need something that would compare the atomic neighborhoods without regard to translation, rotation, reflection, or permutation of identical atoms.