1.7: Notes to Chapter 1
- Page ID
- 141445
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Since aij = aji for all known physical forces, we may substitute Equation \ref{1.2.9} in Equation \ref{1.2.11} as follows:
\[\begin{aligned}
\sum_{\mathrm{i}} \mathbf{f}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}} & =-\sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \mathrm{na}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{(\mathrm{n}-2)}\left[\left(\mathbf{r}_{\mathrm{i}}-\mathbf{r}_{\mathrm{j}}\right) \cdot \mathbf{r}_{\mathrm{i}}+\left(\mathbf{r}_{\mathrm{j}}-\mathbf{r}_{\mathrm{i}}\right) \cdot \mathbf{r}_{\mathrm{j}}\right] \\[4pt]
& =-\mathrm{n} \sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \mathrm{a}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{(\mathrm{n}-2)}\left[\left(\mathbf{r}_{\mathrm{i}}-\mathbf{r}_{\mathrm{j}}\right) \cdot \left(\mathbf{r}_{\mathrm{i}}-\mathbf{r}_{\mathrm{j}}\right)\right]=-\mathrm{n} \sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \mathrm{a}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{(\mathrm{n}-2)} \mathrm{r}_{\mathrm{ij}}^2
\end{aligned}.\label{N 1.1.1}\]
Thus
\[\sum_\mathrm{i} \mathbf{f}_\mathrm{i} \cdot \mathbf{r}_\mathrm{i}=-\mathrm{n} \sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \mathrm{a}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{\mathrm{n}}=-\mathrm{n} \sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \Phi\left(\mathrm{r}_{\mathrm{ij}}\right).\label{N 1.1.2}\]
Since the second summation is only over j > i, there is no "double-counting" involved, and the double sum is just the total potential energy of the system.
1.2
As in Section 2, let us assume that the force density is derivable from a potential which is a homogeneous function of the distance between the source and field point.5 Then, we can write the potential as
\[\Phi(\mathbf{r})=\int_{\mathrm{v}^{\prime}} \rho(\mathbf{r})\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV}^{\prime} \quad \forall \mathrm{n}<0,\label{N 1.2.1}\]
and the force density is then
\[\boldsymbol{f}(\mathbf{r})=-\rho(\mathbf{r}) \nabla_{\mathrm{r}} \Phi(\mathbf{r})=-\rho(\mathbf{r}) \int_{\mathrm{v}^{\prime}} \rho\left(\mathbf{r}^{\prime}\right) \nabla_{\mathrm{r}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV}^{\prime},\label{N 1.2.2}\]
while the force density at a source point due to all the field points is
\[\boldsymbol{f}\left(\mathbf{r}^{\prime}\right)=-\rho\left(\mathbf{r}^{\prime}\right) \nabla_{\mathrm{r}^{\prime}} \Phi\left(\mathbf{r}^{\prime}\right)=-\rho\left(\mathbf{r}^{\prime}\right) \int_{\mathrm{v}} \rho(\mathbf{r}) \nabla_{\mathrm{r}^{\prime}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV},\label{N 1.2.3}\]
where ∇r and ∇r' denote the gradient operator evaluated at the field point r and the source point r’ respectively. Since the contribution to the force density from any pair of sources and field points will lie along the line joining the two points,
\[\nabla_{\mathrm{r}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}}=-\nabla_{\mathrm{r}^{\prime}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}}=\mathrm{n}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}-2}\left(\mathbf{r}-\mathbf{r}^{\prime}\right).\label{N 1.2.4}\]
Now \(\int_{\mathrm{v}} \boldsymbol{f}(\mathbf{r}) \cdot \mathbf{r}\mathrm{dV}=\int_{\mathrm{v}^{\prime}} \boldsymbol{f}\left(\mathbf{r}^{\prime}\right) \cdot \mathbf{r}^{\prime} \mathrm{dV}^{\prime}\), so multiplying Equation (N 1.2.2) by r and integrating over V produces the same result as multiplying Equation (N 1.2.3) by r' and integrating over V'. Thus, doing this and adding Equation (N 1.2.2) to Equation (N 1.2.3). we get
\[2 \int_{\mathrm{v}} \boldsymbol{f} \cdot \mathbf{r}\mathrm{ d V}=-\int_{\mathrm{v}} \rho(\mathbf{r}) \int_{\mathrm{v}^{\prime}} \rho(\mathbf{r}) \mathbf{r} \cdot \nabla_{\mathrm{r}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV} ^{\prime} \mathrm{dV}=-\int_{\mathrm{v}^{\prime}} \rho\left(\mathbf{r}^{\prime}\right) \int_{\mathrm{v}} \rho(\mathbf{r}) \mathbf{r} \nabla_{\mathrm{r}^{\prime}}\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV} ^{\prime} \mathrm{dV}.\label{N 1.2.5}\]
1.3
It should be noted that the left hand side of 1.5.2 is zero if the system is periodic and the integral is taken over the period.