  The molecular solids $\beta^\prime$-$X$[Pd(dmit)$_2$]$_2$ (where $X$
represents a cation) are typical compounds whose electronic structures are
described by single-orbital Hubbard-type Hamiltonians with geometrical
frustration. Using the $ab$ $initio$ downfolding method, we derive the
low-energy effective Hamiltonians for $\beta^\prime$-$X$[Pd(dmit)$_2$]$_2$ with
available room- and low-temperature structures. We find that the amplitudes of
the Coulomb interactions and the anisotropy of the hopping parameters in the
effective Hamiltonians are sensitive to the changes in the lattice constants
induced by lowering the temperature. The obtained effective Hamiltonians are
analyzed using the exact diagonalization method with the boundary-condition
average. We find that a significant reduction of the antiferromagnetic ordered
moment occurs in the effective Hamiltonian of
$\beta^\prime$-EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ with the low-temperature structure.
The reduction is consistent with the quantum spin liquid behavior observed in
experiments. The comprehensive derivations of the effective Hamiltonians and
exact-diagonalization analyses of them will clarify the microscopic origins of
the exotic quantum states of matter found in
$\beta^\prime$-$X$[Pd(dmit)$_2$]$_2$ such as the quantum spin liquid behavior.
