  In this study, a variational method for the inverse problem of self-assembly,
i.e., a reconstruction of the interparticle interaction potential of a given
structure, is applied to three-dimensional crystals. According to the method,
the interaction potential is derived as a function that maximizes the
free-energy functional of the one- and two-particle density distribution
functions. The interaction potentials of the target crystals, including those
with face-centered cubic (fcc), body-centered cubic (bcc), and simple hexagonal
(shx) lattices, are obtained by numerical maximization of the functional. Monte
Carlo simulations for the systems of particles with these interactions were
carried out, and the self-assembly of the target crystals was confirmed for the
bcc and shx cases. However, in the many-particle system with the predicted
interaction for the fcc lattice, the fcc lattice did not spontaneously form and
was metastable.
