Only hydrogen has an analytic solution. Even helium requires approximations because the electrons interact. Approximation requires an understanding of the structure of the wavefunction which gets increasingly complex as we add more particles to the system that interact and entangle with one another like electrons in an atom do. Even further, that's for single atoms that aren't interacting strongly with their environment. Here we're talking about an ensemble of nitrogen atoms interacting with each other in a somewhat extreme environment.