The spatial dependence of ##f## is in the argument ##\mathbf{r}##, whilst the time dependence is in the source positions ##\mathbf{r}_a##\begin{align*}
\partial_{\alpha} |\mathbf{r} - \mathbf{r}_a| &= \frac{x_{\alpha} - x_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} = n_{a \alpha}
\end{align*}Then taking the time derivative\begin{align*}
\partial_t \partial_{\alpha} |\mathbf{r} - \mathbf{r}_a| = \partial_t \left( \frac{x_{\alpha} - x_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} \right) &= \frac{-v_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} + (x_{\alpha} - x_{a \alpha}) \partial_t \left( \frac{1}{|\mathbf{r} - \mathbf{r}_a|} \right) \\
&= \frac{-v_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} + \frac{-(x_{\alpha} - x_{a \alpha})}{|\mathbf{r} - \mathbf{r}_a|^2} \left( \frac{(\mathbf{r} - \mathbf{r}_a) \cdot (-\mathbf{v}_a )}{|\mathbf{r} - \mathbf{r}_a|} \right) \\
&= \frac{-v_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} + \frac{n_{a \alpha} (\mathbf{n}_a \cdot \mathbf{v}_a)}{|\mathbf{r} - \mathbf{r}_a|}
\end{align*}Hence if ##f = -(k/2) \sum_a m_a |\mathbf{r} - \mathbf{r}_a |##, then\begin{align*}
\frac{\partial^2 f}{\partial t \partial x^a} &= \frac{k}{2} \sum_a m_a \left[ \frac{v_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} - \frac{n_{a \alpha} (\mathbf{n}_a \cdot \mathbf{v}_a)}{|\mathbf{r} - \mathbf{r}_a|} \right]\end{align*}and consequently\begin{align*}
h_{0 \alpha} &= \frac{8k}{2c^3} \sum_a \frac{m_a v_{a\alpha}}{|\mathbf{r} - \mathbf{r}_a|} - \frac{k}{2c^3} \sum_a \left[ \frac{m_av_{a \alpha}}{|\mathbf{r} - \mathbf{r}_a|} - \frac{m_a n_{a \alpha} (\mathbf{n}_a \cdot \mathbf{v}_a)}{|\mathbf{r} - \mathbf{r}_a|} \right] \\
&= \frac{k}{2c^3} \sum_a \frac{m_a}{|\mathbf{r} - \mathbf{r}_a|} \left[ 7v_{a \alpha} + n_{a \alpha} (\mathbf{n}_a \cdot \mathbf{v}_a ) \right]\end{align*}