  We consider ground states of the $N$ coupled fermionic nonlinear
Schr\"{o}dinger systems with the Coulomb potential $V(x)$ in the
$L^2$-subcritical case. By studying the associated constraint variational
problem, we prove the existence of ground states for the system with any
parameter $\alpha>0$, which represents the attractive strength of the
non-relativistic quantum particles. The limiting behavior of ground states for
the system is also analyzed as $\alpha\to\infty$, where the mass concentrates
at one of the singular points for the Coulomb potential $V(x)$.
