Darwin term

This follows from the Dirac equation and is given by

$$\hat{H}_{\rm Darwin} = \frac{-e\hbar^2}{8m^2c^2} \Delta \Phi({\bf r}),$$ (3.4)

where $\Delta$ is the Laplacian. For the Coulomb potential $\Phi({\bf r})=Ze/4\pi\varepsilon_0 r$ one needs

$$\Delta \frac{1}{r}=-4\pi \delta({\bf r})$$ (3.5)

with the Dirac Delta function $\delta({\bf r})$ in three dimensions.