How to construct a Hamiltonian for an ensemble of atoms interacting with each other?

How to construct a Hamiltonian for an ensemble of atoms interacting with each other?

For example if the one atom hamiltonian can be written as:

$$hat{H}=left(begin{matrix}0&Omega_p(t)&0Omega_p(t)&0&Omega_r(t)&Omega_r(t)&0end{matrix}right)$$

so the many atoms Hamiltonian can be written as

$$hat{mathcal{H}}=hat{H}otimeshat{H}otimes…otimeshat{H}+sum_{j,k}^{N}V_{jk}left|Rrangle_jlangle Rright|otimesleft|Rrangle_klangle Rright|$$
where $Omega_p(t)$ and $Omega_r(t)$ are arbitrary functions, $|Rrangle$ is the Rydberg stat $V$ is the interaction energy between Rydberg states, and $N$ is the number of atoms.

I am using Wolfram Mathematica and every-time I do a simulation for 3 atom, it does not work properly.