Relation between ECI and chemical pair interaction energies?

Alloy Theoretic Automated Toolkit (ATAT)
1

Are pair ECI s are same as pair interaction energy which is commonly used to describe ordering tendencies in alloys?

2

nearest neighbor ECI=(E_AA+E_BB-2E_AB)/4
where E_ij are the energies of a i-j chemical bond.

3

The ECIs we calculate by ATAT are composition independent. In that case ECIs for Ni-Mo system will be one set for Ni4Mo , Ni2Mo and Ni3Mo but these have different pair interaction energies. How to find the pair interaction energies for a particular composition from the composition independent ECIs?

4

The most general answer to your question could be quite involved.
In conventional (concentration-independent) cluster expansion (CE), the concentration-dependence of the pair interactions is incorporated in the multibody interactions.
What energy you assign to pair depends on which environment(s) you average over.
If you consider the environment to be a random solid solution, then you can in principle
compute (E_AA+E_BB-2E_AB)/4 from the CE averaged over all possible random configurations of the environment (everything but those 2 atoms).
If you consider the environment to be an order structure then you compute (E_AA+E_BB-2E_AB)/4 while setting the remaining atoms to be in their ordered configuration.

I hope this helps!