EAM potential

As mentioned earlier, the EAM formulation for the potential energy is

E=12ijjiVij(rij)+iF(ρ¯i)E = \frac{1}{2}\sum_i\sum_{j \atop j\neq i} V_{ij}(r_{ij}) + \sum_i F(\bar{\rho}_i)

where VV is the pair potential, FF is the embedding potential, and ρ¯\bar{\rho} is the host electron density, i.e.,

ρ¯i=jjiρij(rij)\bar{\rho}_i = \sum_{j \atop j \neq i} \rho_{ij}(r_{ij})

where ρij\rho_{ij} is the local electron density contributed by atom jj at site ii.

Let rji\mathbf{r}_{ji} be the vector from atom jj to atom ii with norm rji(=rij)r_{ji} (= r_{ij}), i.e.,

rji=rirj\mathbf{r}_{ji} = \mathbf{r}_i - \mathbf{r}_j

rji=(rixrjx)2+(riyrjy)2+(rizrjz)2r_{ji} = \sqrt{(r_i^x - r_j^x)^2 + (r_i^y - r_j^y)^2 + (r_i^z - r_j^z)^2}

where

rj=rjxex+rjyey+rjzez\mathbf{r}_j = r_j^x\mathbf{e}^x + r_j^y\mathbf{e}^y + r_j^z\mathbf{e}^z

Now, let's prove an important identity,

rjirj=rjirjxex+rjirjyey+rjirjzez=rjixrjiexrjiyrjieyrjizrjiez=rjirji\frac{\partial r_{ji}}{\partial \mathbf{r}_j} = \frac{\partial r_{ji}}{\partial r_j^x} \mathbf{e}^x + \frac{\partial r_{ji}}{\partial r_j^y} \mathbf{e}^y + \frac{\partial r_{ji}}{\partial r_j^z} \mathbf{e}^z = - \frac{r_{ji}^x}{r_{ji}} \mathbf{e}^x - \frac{r_{ji}^y}{r_{ji}} \mathbf{e}^y - \frac{r_{ji}^z}{r_{ji}} \mathbf{e}^z = -\frac{\mathbf{r}_{ji}}{r_{ji}}

which will be used in the force formulation derivation later.

The force on atom kk is

fk=Erk=12ijjiVij(rij)rkiF(ρ¯i)rk\mathbf{f}_k = -\frac{\partial E}{\partial \mathbf{r}_k} = -\frac{1}{2} \frac{\partial \sum_i \sum_{j \atop j \neq i}V_{ij}(r_{ij})}{\partial \mathbf{r}_k}-\frac{\partial \sum_i F(\bar{\rho}_i)}{\partial \mathbf{r}_k}

The first term in the force formulation is non-zero only when kk is either ii or jj, thus it becomes

12[jjkVkj(rkj)rk+ikiVik(rik)rk]=12[jjkVkj(rkj)rkjrkjrkikiVik(rik)rikrikrk]-\frac{1}{2} \left[\frac{\partial \sum_{j \atop j \neq k} V_{kj}(r_{kj})}{\partial \mathbf{r}_k}+\frac{\partial \sum_{i \atop k \neq i}V_{ik}(r_{ik})}{\partial \mathbf{r}_k}\right] = -\frac{1}{2} \left[\frac{\partial \sum_{j \atop j \neq k} V_{kj}(r_{kj})}{\partial r_{kj}}\frac{\partial r_{kj}}{\partial \mathbf{r}_k} - \frac{\partial \sum_{i \atop k \neq i}V_{ik}(r_{ik})}{\partial r_{ik}}\frac{\partial r_{ik}}{\partial \mathbf{r}_k}\right]

With the help of the identity, the term becomes

12[jjkVkj(rkj)rkjrkjrkjikiVik(rik)rikrikrik]\frac{1}{2} \left[\frac{\partial \sum_{j \atop j \neq k} V_{kj}(r_{kj})}{\partial r_{kj}}\frac{\mathbf{r}_{kj}}{r_{kj}} - \frac{\partial \sum_{i \atop k \neq i}V_{ik}(r_{ik})}{\partial r_{ik}}\frac{\mathbf{r}_{ik}}{r_{ik}}\right]

where VkjV_{kj} and VikV_{ik} are the pair potentials for the atomic pairs kjkj and ikik, respectively, while Vkj=VjkV_{kj} = V_{jk} and Vik=VkiV_{ik} = V_{ki}. Since VV is atom type-specific, VkjV_{kj} and VikV_{ik} are likely not the same unless atom ii and jj are of the same type. Thus, if there are two types of atoms in the system, there will be three VV, 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

iF(ρ¯i)rk=iF(ρ¯i)ρ¯iρ¯irk=iF(ρ¯i)ρ¯ijjiρij(rij)rk=iF(ρ¯i)ρ¯ijjiρij(rij)rijrijrk-\sum_i\frac{\partial F(\bar{\rho}_i)}{\partial \mathbf{r}_k} = -\sum_i\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\frac{\partial \bar{\rho}_i}{\partial \mathbf{r}_k} = -\sum_i\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\sum_{j \atop j \neq i}\frac{\partial \rho_{ij}(r_{ij})}{\partial \mathbf{r}_k} = -\sum_i\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\sum_{j \atop j \neq i}\frac{\partial \rho_{ij}(r_{ij})}{\partial r_{ij}}\frac{\partial r_{ij}}{\partial \mathbf{r}_k}

which is non-zero when kk is either ii or jj, i.e., the term becomes

F(ρ¯k)ρ¯kjjkρkj(rkj)rkjrkjrkiikF(ρ¯i)ρ¯iρik(rik)rikrikrk-\frac{\partial F(\bar{\rho}_k)}{\partial \bar{\rho}_k}\sum_{j \atop j \neq k}\frac{\partial \rho_{kj}(r_{kj})}{\partial r_{kj}}\frac{\partial r_{kj}}{\partial \mathbf{r}_k}-\sum_{i \atop i \neq k}\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\frac{\partial \rho_{ik}(r_{ik})}{\partial r_{ik}}\frac{\partial r_{ik}}{\partial \mathbf{r}_k}

Again, with the help of the identify, the term becomes

F(ρ¯k)ρ¯kjjkρkj(rkj)rkjrkjrkjiikF(ρ¯i)ρ¯iρik(rik)rikrikrik\frac{\partial F(\bar{\rho}_k)}{\partial \bar{\rho}_k}\sum_{j \atop j \neq k}\frac{\partial \rho_{kj}(r_{kj})}{\partial r_{kj}}\frac{\mathbf{r}_{kj}}{r_{kj}}-\sum_{i \atop i \neq k}\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\frac{\partial \rho_{ik}(r_{ik})}{\partial r_{ik}}\frac{\mathbf{r}_{ik}}{r_{ik}}

Note that ρkj\rho_{kj} is the local electron density contributed by atom jj at site kk. In general, ρkjρjk\rho_{kj} \neq \rho_{jk}. This is different from the pair potential VV, for which generally Vkj=VjkV_{kj} = V_{jk}. Also, generally ρkjρij\rho_{kj} \neq \rho_{ij} unless atom kk and atom ii are of the same type.

In classical EAM, ρkj=ρij\rho_{kj} = \rho_{ij} even when atom kk and atom ii are of different type. If there are two types of atoms in the system, there are only two ρ\rho, 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 VV, which would have three expressions in this case. Extensions of ρ\rho 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

fk=12[jjkVkj(rkj)rkjrkjrkjikiVik(rik)rikrikrik]+F(ρ¯k)ρ¯kjjkρkj(rkj)rkjrkjrkjiikF(ρ¯i)ρ¯iρik(rik)rikrikrik\mathbf{f}_k = \frac{1}{2} \left[\frac{\partial \sum_{j \atop j \neq k} V_{kj}(r_{kj})}{\partial r_{kj}}\frac{\mathbf{r}_{kj}}{r_{kj}}-\frac{\partial \sum_{i \atop k \neq i}V_{ik}(r_{ik})}{\partial r_{ik}}\frac{\mathbf{r}_{ik}}{r_{ik}}\right] + \frac{\partial F(\bar{\rho}_k)}{\partial \bar{\rho}_k}\sum_{j \atop j \neq k}\frac{\partial \rho_{kj}(r_{kj})}{\partial r_{kj}}\frac{\mathbf{r}_{kj}}{r_{kj}}-\sum_{i \atop i \neq k}\frac{\partial F(\bar{\rho}_i)}{\partial \bar{\rho}_i}\frac{\partial \rho_{ik}(r_{ik})}{\partial r_{ik}}\frac{\mathbf{r}_{ik}}{r_{ik}}

Since ii and jj are just dummy indices, it is safe to replace all ii with jj. After that, with rjk=rkj\mathbf{r}_{jk} = -\mathbf{r}_{kj}, rjk=rkjr_{jk} = r_{kj}, Vjk=VkjV_{jk} = V_{kj}, and ρjkρkj\rho_{jk} \neq \rho_{kj}, the force on atom kk becomes

fk=jjk[Vkj(rkj)rkj+F(ρ¯k)ρ¯kρkj(rkj)rkj+F(ρ¯j)ρ¯jρjk(rkj)rkj]rkjrkj\mathbf{f}_k = \sum_{j \atop j \neq k}\left[\frac{\partial V_{kj}(r_{kj})}{\partial r_{kj}}+\frac{\partial F(\bar{\rho}_k)}{\partial \bar{\rho}_k}\frac{\partial \rho_{kj}(r_{kj})}{\partial r_{kj}}+\frac{\partial F(\bar{\rho}_j)}{\partial \bar{\rho}_j}\frac{\partial \rho_{jk}(r_{kj})}{\partial r_{kj}}\right]\frac{\mathbf{r}_{kj}}{r_{kj}}

If there is only type of atoms in the system, ρjk=ρkj\rho_{jk} = \rho_{kj}, and the force formulation is simplified to

fk=jjk[Vkj(rkj)rkj+(F(ρ¯k)ρ¯k+F(ρ¯j)ρ¯j)ρkj(rkj)rkj]rkjrkj\mathbf{f}_k = \sum_{j \atop j \neq k}\left[\frac{\partial V_{kj}(r_{kj})}{\partial r_{kj}}+\left(\frac{\partial F(\bar{\rho}_k)}{\partial \bar{\rho}_k}+\frac{\partial F(\bar{\rho}_j)}{\partial \bar{\rho}_j}\right)\frac{\partial \rho_{kj}(r_{kj})}{\partial r_{kj}}\right]\frac{\mathbf{r}_{kj}}{r_{kj}}

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 ρkj\rho_{kj} and ρij\rho_{ij} is not used during the derivation.

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