  To determine how electron correlations open a gap in two-dimensional massless
Dirac electrons in the organic compounds $\alpha$-(BEDT-TTF)$_2$I$_3$
[$\alpha$-(ET)$_2$I$_3$] and $\alpha$-(BEDT-TSeF)$_2$I$_3$
[$\alpha$-(BETS)$_2$I$_3$], we derive and analyze $ab$ $initio$ low-energy
effective Hamiltonians for these two compounds. We find that the horizontal
stripe charge ordering opens a gap in the massless Dirac electrons in
$\alpha$-(ET)$_2$I$_3$, while an insulating phase without explicit symmetry
breaking appears in $\alpha$-(BETS)$_2$I$_3$. We clarify that the combination
of the anisotropic transfer integrals and the electron correlations induces a
dimensional reduction in the spin correlations, i.e., one-dimensional spin
correlations develop in $\alpha$-(BETS)$_2$I$_3$. We show that the
one-dimensional spin correlations open a gap in the massless Dirac electrons.
Our finding paves the way for opening gaps for massless Dirac electrons using
strong electronic correlations.
