# How to interpret ECI values?

Hi All,

I asked this question before, but I am still confused about how to interpret ECI values given by ICET. My understanding is that ECI represents atomic interactions in a cluster. So, a positive ECI indicates a repulsion between atoms. For example, the ECIs in this paper. But it seems ICET interprets ECIs in a different way. @magnusrahm replied “an ECI is not a measure of, for example, the A-A, B-B or A-B interaction, but rather a difference thereof (this is equivalent to the Ising model where you have only one interaction parameter)”. But from the ClusterExpansion output table, each sublattice, say A-B, has its own ECI. So it does look like an interaction. Could you explain a bit more of the physical meaning of ECIs ICET?

Also, is there a way to know the ECI for a specific atomic pair? For example, my sublattice A is [‘Li’, ‘Mg’] and sublattice B is [‘Ni’, ‘Mn’]. Is there an ECI for Li-Ni pair?

Thanks

But from the ClusterExpansion output table, each sublattice, say A-B, has its own ECI. So it does look like an interaction.

The A-A and A-B sublattices in icet are not referring to atom-types but sub-lattices.

Also, is there a way to know the ECI for a specific atomic pair?

@rmagnus knows this best, but the problem, as I understand it, is that the interaction for a specific pair (e.g. Li-Mg) is not uniquely defined (and therefore not relevant or interesting)

Maybe in order to understand it better you could try to start from a nearest neighbor model
where A and B denotes atom-type for a binary system, and the energy is

1. E = J_0 + N_AJ_A + N_BJ_B + N_{AA}J_{AA} + N_{BB}J_{BB} + N_{AB}J_{AB}

where N_A is the number of A atoms and N_{AA} is the number of nearest neighbor A-A pairs etc, and J is your interaction parameters (5 unknown parameters).
If you now use some known equations like N = N_A+N_B where N is a constant. The fact that the number of pairs involving A atoms must be N_A f= N_{AA} + N_{AB} (where f is how many nearest neighbors a lattice point have on this lattice), then you will find something like

1. E = \tilde{J} _0 + N_A\tilde{J} _A + N_{AB}\tilde{J}_{AB}

where the \tilde{J} are new parameters which is a (linear) combinations of 5 original parameters. Hence, if we know the three \tilde{J} parameters there is no way of uniquely define the “atomically resolved” five J parameters. But feel free to start from 1. yourself and see how it works, I probably made some misstakes along the way.

If I understand it correctly, the ECI values represent interactions between sub-lattices, which consists of a combination of atomic pairs. In the example I mentioned, sub-lattice A is [‘Li’, ‘Mg’] and sub-lattice B is [‘Ni’, ‘Mn’]. A positive ECI of A-B means a repulsion of sub-lattices A and B, which consists of Li-Ni, Li-Mn, Mg-Ni, and Mg-Mn. Am I understanding it correctly? Looks like ICET calculates ECI differently than the work from Ceder’s and Van der Ven’s group.

I dont think this is correct, ECIs exist also when you only have one sublattices.

I dont think this is a correct interpretation. The two sublattices in my mind can not repel each other. Rather it could be that a pair between sublattice A and sublattice B has a high energy when occupied with e.g. Li-Mn and a lower energy when occupied with Li-Ni.

Im not sure how one interprets the ECI for an orbit for which atoms are located on different sublattices, @magnusrahm any thoughts? I’m guessing it is more difficult than for just a simple binary system, since technically it is a 4-component system where you simply do not consider/allow for Li and Mg on certain sites.

I dont know how they define and compute ECIs but I dont think icet does anything out of the ordinary, and I would be surprised if its very different from others cluster-expansion implementations.

A more practical approach for you might be to just run some tests in order to understand which orbits and ECIs favor what type of ordering of atoms, for example you could try

ecis = [0, 0, 0, 1, 0, 0, ...]
ce = ClusterExpansions(cs, ecis)


and use this CE to investigate the energy of a few different structure, and then you should probably be able to figure out the meaning of the nonzero ECI being positive or negative.

Thank you for your prompt reply. I think I still have many misunderstandings of ECI values. Let me think about it.