  Small displacement methods have been successfully used to calculate the
lattice dynamical properties of crystals. It involves displacing atoms by a
small amount in order to calculate the induced forces on all atoms in a
supercell for the computation of force constants. Even though these methods are
widely in use, to our knowledge, there is no systematic discussion of optimal
displacement directions from the crystal's symmetry point of view nor a
rigorous error analysis of such methods. Based on the group theory and point
group symmetry of a crystal, we propose displacement directions, with an
equivalent concept of the group of $k$, deduced directly in the Cartesian
coordinates rather than the usual fractional coordinates, that maintain the
theoretical maximum for the triple product $V$ spanned by the three
displacements to avoid possible severe roundoff errors. The proposed
displacement directions are generated from a minimal set of irreducible atomic
displacements that keep the required independent force calculations to a
minimum. We find the error in the calculated force constant explicitly depends
on the inverse of $V$ and inaccuracy of the forces. Test systems such as Si,
graphene, and orthorhombic Sb2S3 are used to illustrate the method. Our
displacement method is shown to be very robust in treating low-symmetry cells
with a large `aspect ratio' due to huge differences in lattice parameters, use
of a large vacuum height, or a very oblique unit cell due to unconventional
choice of primitive lattice vectors. It is expected that our displacement
strategy can be used to address higher-order interatomic interactions to
achieve good accuracy and efficiency.
