Dipole and electric field gradient

[DrClaude]

I'm developing a classical model of a dipolar ion in an external electric field. It consists of two charges $\delta_+$ and $\delta_-$, located at a fixed distance from each other. For the special case I'm considering, I end up with the potential energy
$$
(\delta_+ + \delta_-) \Phi(\vec{R}) + \sum_{\xi = x,y,z} \left[
\delta_+ (\vec{r}_+')_{\xi} + \delta_- (\vec{r}_-')_{\xi} \right]
\left. \frac{\partial}{\partial \xi} \Phi \right|_{\vec{R}} + \sum_{\xi = x,y,z} \left[
\delta_+ (\vec{r}_+')_{\xi}^2 + \delta_- (\vec{r}_-')_{\xi}^2 \right]
\left. \frac{\partial^2}{\partial \xi^2} \Phi \right|_{\vec{R}}
$$
where $\Phi$ is the electric potential, $\vec{R}$ is the position of the center of mass and $\vec{r}_\pm'$ the positions of the charges $\delta_\pm$ with respect to the center of mass. The notation $(\vec{r})_\xi$ means the $\xi \in \{x,y,z\}$ component of $\vec{r}$.

The first term I can identify as the total charge times the electric potential and the second I can rewrite as
$$
\vec{\mu} \cdot \left. \vec{\nabla}\Phi \right|_{\vec{R}} = \vec{\mu} \cdot \vec{E}(\vec{R})
$$
i.e., the interaction of the dipole with the electric field. But I'm struggling to find a physical interpretation to the third term. I've tried to rewrite it in other terms, but didn't come up with anything interesting. Is this simply because the center of mass is not exactly midway between the two charges?

Any input appreciated.

 

[dEdt]

Well, the formula $$U=\mathbf{\mu}\cdot\nabla \Phi$$ is only appropriate for a point-like dipole, so it just seems like your third term is a correction term that arises when you avoid assuming that the dipole is point-like.

 

[DrClaude]

so it just seems like your third term is a correction term that arises when you avoid assuming that the dipole is point-like.

I had thought about that possibility also, but forgot to write it in my post. Thanks for your reply.