  A series of layered perovskites Sr$_2$$M$O$_4$ ($M$=Ti, V, Cr, and Mn) is
studied by $ab $ $initio$ calculations within generalized gradient
approximation (GGA) and GGA+$U$ schemes. The total energies in different
magnetic configurations, including the nonmagnetic, ferromagnetic, the layered
antiferromagnetic with alternating ferromagnetic plane and the staggered
in-plane antiferromagnetic (AFM-II) order, are calculated. It is found that
Sr$_2$TiO$_4$ is always a nonmagnetic band insulator. For Sr$_2$MnO$_4$, both
GGA and GGA+$U$ calculations show that the insulating AFM-II state has the
lowest total energy among all the considered configurations. For $M$=V and Cr,
the GGA is not enough to give out the insulating AFM-II states and including
the on-site electron-electron correlation effect $U$ is necessary and
efficient. The AFM-II state will have the lowest total energy in both cases
when $U$ is larger than a critical value. Further, the optical conductivity
spectra are calculated and compared with the experimental measurements to show
how well the ground state is described within the GGA or GGA+$U$. The results
indicate that $U$ is overestimated in Sr$_2$VO$_4$ and Sr$_2$CrO$_4$. To make
up such a deficiency of GGA+$U$, the contributions from proper changes in the
ligand field, acting cooperatively with $U$, are discussed and shown to be
efficient in Sr$_2$CrO$_4$.
