  In perovskite-type compounds, the interplay of cooperative Jahn-Teller
effect, electronic correlations and orbital degree of freedom leads to
intriguing properties. $\mathrm{NaCrF_{3}}$ is a newly synthesized Jahn-Teller
active fluoroperovskite where the $\mathrm{CrF_{6}^{4-}}$ octahedrons are
considerably distorted. Based on the first-principles calculation, we analyze
its electronic structure and magnetic properties. Our numerical results show
that the $\mathrm{Cr^{2+}}$ ions adopt the high-spin
$t_{2g\uparrow}^{3}e_{g\uparrow}^{1}$ configuration with $G$-type orbital
ordering. We also estimate the magnetic exchange couplings and find that the
in-plane and interplanar nearest-neighbor interactions are ferromagnetic and
antiferromagnetic, respectively. The ground state of this material is $A$-type
antiferromagnetic, in agreement with the experiments. Reasonable Curie-Weiss
and $\mathrm{N\acute{e}el}$ temperatures compared to the experiments are given
by mean-field approximation theory. Our results give a complete explanation of
its electronic structure, magnetic and orbital order, and help to further
comprehend the behaviors of Jahn-Teller active perovskite-type fluoride.
