Consider an atom of your choice. I wanted to have a way of organizing all of its electron excited states into some kind of mathematical structure, that was a sufficiently generic framework that the same framework can be applied to any atom.
One idea that came to mind was just to consider an ordered list (by excitation energy, with the ground state being the first item on the list) so for a particular energy level there is a unique excited orbital configuration which describes the orbital cloud around our atom.
This makes perfect sense for a hydrogen atom, but immediately seems suspicious for larger atoms say Lithium onwards. Since in the case of our Lithium atom, for a given energy level, there might be more THAN one type of orbital configuration for a given amount of energy. (Ex: any energy level $K$ might involve taking the outermost electron and raising it $K$ times out, OR, perhaps we only raise it a few times and then raise some of the inner electrons instead. It seems there is a combinatorial explosion here of how the energy level actually gets broken down into various "partitions").
At least that's how i think it works after reading the answer to this
So then my question is, what is a good way to think about the "structure" of the excited states.
My tinkering:
I had an idea that they they form some kind of tree. With the ground state at the root node and each excited state occupying a node in this graph. Two nodes have an edge between them if the corresponding excited states can be reached by the emission or absorption of a single photon, AND there is no state that can be reached by emitting or absorbing a lower energy photon. But the problem is that this way of thinking doesn't let me easily answer "For a given amount of additional energy E, how many different orbital configurations are possible" and based on the physics.se post I linked it seems that there can be many different configurations (since multiple electrons could be excited instead of one electron excited many times).
