  We numerically investigate dynamical properties of $\kappa$-type molecular
conductors in their antiferromagnetic Mott insuating state. By studying the
extended Hubbard model on the two-dimensional $\kappa$-type lattice within the
Lanzcos exact diagonalization method, the one-particle spectral function
$A(\boldsymbol{k}, \omega)$ and the optical absorption spectra are calculated
taking advantage of twisted boundary conditions. We find a spin splitting in
$A(\boldsymbol{k}, \omega)$ predicted by a previous Hartree-Fock study [M. Naka
et al., Nat. Commun. 10, 4305 (2019)]; namely, their up-spin and down-spin
components become different in the general $\boldsymbol{k}$-points of the
Brillouin zone, even without the spin-orbit coupling. Furthermore, we
demonstrate the variation in the optical properties near the Mott gap by the
presence or absence of the antiferromagnetic order, tuned by a small staggered
magnetic field.
