  We study the multipole interactions between $f^2$ ions with the $\Gamma_3$
non-Kramers doublet ground state under a cubic crystalline electric field. We
construct the $\Gamma_3$ doublet state of the electrons with the total angular
momentum $j=5/2$. By applying the second-order perturbation theory with respect
to the intersite hopping, we derive the multipole interactions. We obtain a
quadrupole interaction for a simple cubic lattice, an octupole interaction for
a bcc lattice, and both quadrupole and octupole interactions for an fcc
lattice. We also discuss general tendencies of the multipole interactions
depending on the lattice structure by comparing the results with those for the
$\Gamma_8$ quartet systems of $f^1$ ions.
