  Using a generalized order parameter gradient expansion within density
functional theory, we derive a phase-field-crystal model for liquid crystals
composed by apolar particles in three spatial dimensions. Both the
translational density and the orientational direction and ordering are included
as order parameters. Different terms involving gradients in the order
parameters in the resulting free energy functional are compared to the
macroscopic Ginzburg-Landau approach as well as to the hydrodynamic description
for liquid crystals. Our approach provides microscopic expressions for all
prefactors in terms of the particle interactions. Our phase-field-crystal model
generalizes the conventional phase-field-crystal model of spherical particles
to orientational degrees of freedom and can be used as a starting point to
explore phase transitions and interfaces for various liquid-crystalline phases.
