３体項から生じる力の計算

$$ \boldsymbol{F}_i =  – \varepsilon \sum_{a}\sum_{b \neq a} \sum_{\stackrel{c > b}{c\neq a}} \frac{\partial h(r_{ab},r_{ac}, \theta_{bac})}{\partial \boldsymbol{r}_i } $$

\(a = i\)の場合、\(b = i\)の場合、\(c = i\)の場合に分けて計算する。

$$ \begin{eqnarray}
\boldsymbol{F}_i &=&  – \varepsilon \sum_{b \neq i} \sum_{c > b} \frac{\partial h(r_{ib},r_{ic}, \theta_{bic})}{\partial \boldsymbol{r}_i }  – \varepsilon \sum_{a \neq i}\sum_{c > i} \frac{\partial h(r_{ai},r_{ac}, \theta_{iac})}{\partial \boldsymbol{r}_i }\\
&&   – \varepsilon \sum_{a\neq i}\sum_{b < i} \frac{\partial h(r_{ab},r_{ai}, \theta_{bai})}{\partial \boldsymbol{r}_i }
\end{eqnarray} $$

第３項目の\(b\)についての和を、\(c\)についての和に書きかえると

$$ \begin{eqnarray}
\boldsymbol{F}_i &=&  – \varepsilon \sum_{b \neq i} \sum_{c > b} \frac{\partial h(r_{ib},r_{ic}, \theta_{bic})}{\partial \boldsymbol{r}_i }  – \varepsilon \sum_{a \neq i}\sum_{c > i} \frac{\partial h(r_{ai},r_{ac}, \theta_{iac})}{\partial \boldsymbol{r}_i }\\
&&   – \varepsilon \sum_{a\neq i}\sum_{c < i} \frac{\partial h(r_{ac},r_{ai}, \theta_{cai})}{\partial \boldsymbol{r}_i }
\end{eqnarray} $$

３体項の対称性から\(h(r_{ac},r_{ai}, \theta_{cai}) = h(r_{ai},r_{ac}, \theta_{iac})\)となるので、第２項と３項が次のようにまとめられる。

$$\boldsymbol{F}_i =  – \varepsilon \sum_{b \neq i} \sum_{c > b} \frac{\partial h(r_{ib},r_{ic}, \theta_{bic})}{\partial \boldsymbol{r}_i }  – \varepsilon \sum_{a \neq i}\sum_{c \neq i} \frac{\partial h(r_{ai},r_{ac}, \theta_{iac})}{\partial \boldsymbol{r}_i } $$

和の添え字を書き換え、以下\(\sum\)の中身の計算を進める。

$$\begin{eqnarray}
\boldsymbol{F}_i &=&  – \varepsilon \sum_{j \neq i} \sum_{k > j} \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial \boldsymbol{r}_i }  – \varepsilon \sum_{j \neq i}\sum_{k \neq i} \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial \boldsymbol{r}_i }\\
&=& – \varepsilon \sum_{j \neq i} \sum_{k > j} \left ( \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial r_{ij}}\frac{\partial r_{ij}}{\partial \boldsymbol{r}_i} + \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial r_{ik}}\frac{\partial r_{ik}}{\partial \boldsymbol{r}_i} \right .\\
&& \left . + \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial \cos\theta_{jik}}\frac{\partial \cos\theta_{jik}}{\partial \boldsymbol{r}_i} \right ) \\
&& – \varepsilon \sum_{j \neq i}\sum_{k \neq i} \left ( \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial r_{ji}}\frac{\partial r_{ji}}{\partial \boldsymbol{r}_i} + \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial \cos\theta_{ijk}}\frac{\partial \cos\theta_{ijk}}{\partial \boldsymbol{r}_i} \right )
\end{eqnarray} $$

ここで、余弦定理より

$$\begin{eqnarray}
\frac{\partial \cos\theta_{jik}}{\partial \boldsymbol{r}_i} & = & \frac{\partial}{\partial \boldsymbol{r}_i}\left ( \frac{r_{ij}^2 + r_{ik}^2 – r_{jk}^2}{2r_{ij}r_{ik}} \right )\\
&= & \frac{\partial}{\partial r_{ij}}\left ( \frac{r_{ij}^2 + r_{ik}^2 – r_{jk}^2}{2r_{ij}r_{ik}}\right ) \frac{\partial r_{ij}}{\partial \boldsymbol{r}_i} + \frac{\partial}{\partial r_{ij}}\left ( \frac{r_{ij}^2 + r_{ik}^2 – r_{jk}^2}{2r_{ij}r_{ik}}\right ) \frac{\partial r_{ik}}{\partial \boldsymbol{r}_i}\\
& = & \frac{r_{ij} – r_{ik}\cos\theta_{jik}}{r_{ij}r_{ik}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}} + \frac{r_{ik} – r_{ij}\cos\theta_{jik}}{r_{ij}r_{ik}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_k}{r_{ik}}
\end{eqnarray} $$

$$\begin{eqnarray}
\frac{\partial \cos\theta_{ijk}}{\partial \boldsymbol{r}_i} & = & \frac{\partial}{\partial \boldsymbol{r}_i}\left ( \frac{r_{ji}^2 + r_{jk}^2 – r_{ik}^2}{2r_{ji}r_{jk}} \right )\\
&= & \frac{\partial}{\partial r_{ji}}\left ( \frac{r_{ji}^2 + r_{jk}^2 – r_{ik}^2}{2r_{ji}r_{jk}} \right ) \frac{\partial r_{ji}}{\partial \boldsymbol{r}_i} + \frac{\partial}{\partial r_{ik}}\left ( \frac{r_{ji}^2 + r_{jk}^2 – r_{ik}^2}{2r_{ji}r_{jk}} \right ) \frac{\partial r_{ik}}{\partial \boldsymbol{r}_i}\\
& = & \frac{r_{ji} – r_{jk}\cos\theta_{ijk}}{r_{ji}r_{jk}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}} – \frac{r_{ik}}{r_{ji}r_{jk}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_k}{r_{ik}}
\end{eqnarray} $$

となるので、

$$\begin{eqnarray}
\boldsymbol{F}_i & = – \varepsilon \sum_{j \neq i} \sum_{k > j} & \left ( \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial r_{ij}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}} + \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial r_{ik}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_k}{r_{ik}} + \right . \\
& & \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial \cos\theta_{jik}}\frac{r_{ij} – r_{ik}\cos\theta_{jik}}{r_{ij}r_{ik}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}} \\
& & \left . + \frac{\partial h(r_{ij},r_{ik}, \theta_{jik})}{\partial \cos\theta_{jik}}\frac{r_{ik} – r_{ij}\cos\theta_{jik}}{r_{ij}r_{ik}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_k}{r_{ik}} \right ) \\
& – \varepsilon \sum_{j \neq i}\sum_{k \neq i} & \left ( \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial r_{ji}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}}\right . \\
& & + \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial \cos\theta_{ijk}}\frac{r_{ji} – r_{jk}\cos\theta_{ijk}}{r_{ji}r_{jk}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_j}{r_{ij}} \\
& & \left . – \frac{\partial h(r_{ji},r_{jk}, \theta_{ijk})}{\partial \cos\theta_{ijk}}\frac{r_{ik}}{r_{ji}r_{jk}}\frac{\boldsymbol{r}_i – \boldsymbol{r}_k}{r_{ik}} \right )
\end{eqnarray} $$

以上で導出完了。