\begin{equation}
g^{Breit}\left(1,2\right)=g^{Gaunt}\left(1,2\right)+g^{gauge}\left(1,2\right)=-\frac{c{\alpha}_{1}\cdot c{\alpha}_{2}}{c^{2}r_{12}}-\frac{\left(c{\alpha}_{1}\cdot{\nabla}_{1}\right)\left(c{\alpha}_{2}\cdot{\nabla}_{2}\right)r_{12}}{2c^{2}}
\end{equation}