  We present a method based on the Gruneisen formalism to calculate the thermal
expansion coefficient (TEC) tensor that is applicable to any crystal system,
where the number of phonon calculations associated with different deformations
scales linearly with the number of lattice parameters. Compared to simple
high-symmetry systems such as cubic or hexagonal systems, a proper
consideration of low-symmetry systems such as monoclinic or triclinic crystals
demands a clear distinction between the TEC tensor and the lattice-parameter
TECs along the crystallographic directions. The latter is more complicated and
it involves integrating the equations of motion for the primitive lattice
vectors, with input from the TEC tensor. A first-principles study of the TEC is
carried out for the first time on a monoclinc crystal, where we unveil high TEC
anisotropies in a recently reported monoclinic phase of niobium trisulfide
(NbS3) crystal with a relatively large primitive cell (32 atoms per cell) using
density-functional theory. We find the occurrence of a negative TEC tensor
component is largely due to the mechanical property rather than the anharmonic
effect, contrary to the common belief. Our theoretical treatment of the
monoclinic system with a single off-diagonal tensor element could be routinely
generalized to any crystal system, including the lowest-symmetry triclinic
system with three off-diagonal tensor elements.
