  We systematically derive low-energy effective Hamiltonians for molecular
solids $\beta^\prime$-$X$[Pd(dmit)$_{2}$]$_{2}$ ($X$ represents a cation) using
ab initio density functional theory calculations and clarify how the cation
controls the inter-dimer transfer integrals and the interaction parameters. The
effective models are solved using the exact diagonalization method and the
antiferromagnetic ordered moment is shown to be significantly suppressed around
the spin-liquid candidate of $X$=EtMe$_{3}$Sb, which is reported in
experiments. We also show that both the geometrical frustration and the
off-site interactions play essential roles in the suppression of
antiferromagnetic ordering. This systematic derivation and analysis of the
low-energy effective Hamiltonians offer a firm basis to clarify the nature of
the quantum spin liquid found in
$\beta^\prime$-EtMe$_{3}$Sb[Pd(dmit)$_{2}$]$_{2}$.
