数式メモ:EAM

E=\sum_i F_i(\rho_i) where \rho_i \equiv \sum_{j\neq i} \phi_{(j\rightarrow i)}(r_{ji}) and r_{ji} \equiv |\vec{r}_i - \vec{r}_j|. Gradient of E w.r.t. \vec{r}_i becomes
\nabla_i E = \nabla_i F_i(\rho_i) + \sum_{j\neq i} \nabla_i F_j(\rho_j)
\nabla_i F_i(\rho_i) = F^'_i(\rho_i) \nabla_i \rho_i = F^'_i(\rho_i) B_i
B_i \equiv \nabla_i \rho_i = \sum_{j\neq i} \phi^'_{j\rightarrow i}(r_{ji}) \nabla_i r_{ji}
\nabla_i r_{ji} = v_{ji} \equiv \vec{r_{ji}}/|r_{ji}|
\sum_{j\neq i} \nabla_i F_j(\rho_j) = \sum_{j\neq i} F^'_j(\rho_j) \nabla_i \rho_j
C_{ij} \equiv \nabla_i \rho_j = \sum_{k\neq j} \nabla_i \phi_{k\rightarrow j}(r_{kj})
(j\neq i) therefore k=l.
= \nabla_i \phi_{i \rightarrow j} (r_{ij}) = \phi^'_{i \rightarrow j} (r_{ij}) v_{ji}
So,
\nabla_i E = \sum_{j\neq i} (F^'_i \phi^'_{j \rightarrow i} + F^'_j \phi^'_{i \rightarrow j}) v_{ji}

Hessian

\nabla_i \nabla_i E = \sum_{j\neq i} \nabla_i F^'_i \phi^'_{j \rightarrow i} v_{ji} + \sum_{j\neq i} \nabla_i F^'_j \phi^'_{i \rightarrow j} v_{jl}
First term:
\sum_{j\neq i} \nabla_i F^'_i \phi^'_{(j \rightarrow i)} v_{ji} = \sum_{j\neq i} (\nabla_i F^'_i) \phi^'_{j \rightarrow i} v_{ji} + F^'_i (\nabla_i \phi^'_{j \rightarrow i}) v_{ji} + F^'_i \phi^'_{j \rightarrow i} X_{ji}
where X_{ji} \equiv I/r_{ji} - (\vec{r_{ji}} \otimes \vec{r_{ji}})/r_{ji}^3 and \nabla_i F^'_i = F^{''}_i (\nabla_i \rho_i) \equiv F^{''}_i \vec{B}_i and \nabla_i \phi^'_j = \phi^{''}_j v_{ji}
Therefore the first term is
\sum_{j\neq i} F^{''}_i \phi^'_{(j \rightarrow i)} (\vec{B}_i \otimes v_{ji}) + F^'_i \phi^{''}_{j \rightarrow i} (v_{ji} \otimes v_{ji}) + F^'_i \phi^'_{j \rightarrow i} X_{ji}

The second term
\sum_{j\neq i} \nabla_i (F^'_j \phi^'_i v_{ji})
=\sum_{j\neq i} (\nabla_i F^'_j) \phi^'_i v_{ji} + F^'_j (\nabla_i \phi^'_i) v_{ji} + F^'_j \phi^'_i X_{ji}
For (j\neq i), we set \nabla_i F^'_j = F^{''}_j \nabla_i \rho_j = F^{''}_j \vec{C}_{ij}
\nabla_i \phi^'_{(i \rightarrow j)} = \phi^{''}_{(i \rightarrow j)} v_{ji}
Therefore the second term is
\sum_{j\neq i} F^{''}_j \phi^'_{(i \rightarrow j)} (\vec{C}_{ij} \otimes v_{ji}) + F^'_j \phi^{''}_{(i \rightarrow j)} ( v_{ji} \otimes v_{ji}) + F^'_j \phi^'{(i \rightarrow j)} X_{ji}

Further reductin lreads to
F^{''}_i (\vec{B}_i \otimes \vec{B}_i)+\sum_{j\neq i}(F^'_i \phi^{''}_{j \rightarrow i}+F^'_j \phi^{''}_{(i \rightarrow j)} +F^{''}_j \phi^'_{(i \rightarrow j)}^2)( v_{ji} \otimes v_{ji})+ f_{ji} X_{ji}

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