The Coulomb potential in SMASH includes the electric and magnetic field. For simplicity we assume magnetostatics such that the fields can be directly calculated as

\[ \vec{E}(\vec{r})=-\vec{\nabla} \phi(\vec{r}) = -\vec{\nabla}\int\frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|} dV' =\int\frac{\rho(\vec{r}')(\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3}dV' \]

and

\[ \vec{B}(\vec{r}) = \vec{\nabla}\times\vec{A}(\vec{r}) = \vec{\nabla}\times\int \frac{\vec{j}(\vec{r}')}{|\vec{r}-\vec{r}'|}dV' = \int\vec{j}(\vec{r}')\times\frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^3}dV'. \]

These integrals are solved numerically on the SMASH lattice, where the discretised equations read

\[ \vec{E}(\vec{r}_j) = \sum_{i\neq j} \frac{\rho(\vec{r}_i)(\vec{r}_j-\vec{r}_i)}{|\vec{r}_j-\vec{r}_i|^3}\Delta V \]

and

\[ \vec{B}(\vec{r}_j)=\sum_{i\neq j} \vec{j}(\vec{r}_i)\times \frac{\vec{r}_j-\vec{r}_i}{|\vec{r}_j-\vec{r}_i|^3} \Delta V \]

with the lattice cell volume \( \Delta V \). For efficiency the integration volume is cut at \( R_\mathrm{cut} \), which is taken from the configuration. Note that in the final eqations the summand for \( i=j \) drops out because the contribution from that cell to the integral vanishes if one assumes the current and density to be constant in the cell.