As mentioned earlier, the EAM formulation for the potential energy is
where is the pair potential, is the embedding potential, and is the host electron density, i.e.,
where is the local electron density contributed by atom at site .
Let be the vector from atom to atom with norm , i.e.,
Now, let's prove an important identity,
which will be used in the force formulation derivation later.
The force on atom is
The first term in the force formulation is non-zero only when is either or , thus it becomes
With the help of the identity, the term becomes
where and are the pair potentials for the atomic pairs and , respectively, while and . Since is atom type-specific, and are likely not the same unless atom and are of the same type. Thus, if there are two types of atoms in the system, there will be three , between type 1 and type 1, between type 2 and type 2, and between type 1 and type 2.
The second term in the force formulation can be written as
which is non-zero when is either or , i.e., the term becomes
Again, with the help of the identify, the term becomes
Note that is the local electron density contributed by atom at site . In general, . This is different from the pair potential , for which generally . Also, generally unless atom and atom are of the same type.
In classical EAM, even when atom and atom are of different type. If there are two types of atoms in the system, there are only two , for the contribution from type 1 atom and for that from type 2 atom, regardless of which type of atomic site it contributes to. This is different from the pair potential , which would have three expressions in this case. Extensions of to distinguish contributions at different types of atomic sites have been proposed, e.g., in the Finnis-Sinclair potential.
Adding the two terms in the force formulation together yields
Since and are just dummy indices, it is safe to replace all with . After that, with , , , and , the force on atom becomes
If there is only type of atoms in the system, , and the force formulation is simplified to
which is Equation 15 of Xu et al. Note that the last two equations hold for both classical EAM and Finnis-Sinclair potentials, because the relation between and is not used during the derivation.