  The existence of a charge density wave (CDW) in transition metal
dichalcogenide CuS$_2$ has remained undetermined since its first experimental
synthesis nearly 50 years ago. Despite conflicting experimental literature
regarding its low temperature structure, there exists no theoretical study of
the phonon properties and lattice stability of this material. By studying the
first-principles electronic structure and phonon properties of CuS$_2$ at
various electronic temperatures, we identify temperature-sensitive soft phonon
modes which unveil a previously unreported Kohn anomaly at approximately 100K.
Variation of the electronic temperature shows the presence of two distinct
phases, characterized at low temperature by a $2\times2\times2$ periodic charge
modulation associated with the motion of the S$_2$ dimers. Investigation of the
Fermi surface presents a potential Fermi surface nesting vector related to the
location of the Kohn anomaly and observed band splittings in the unfolded
bandstructure. The combination of these results suggests a strong possibility
of CDW order in CuS$_2$. Further study of CuS$_2$ in monolayer form finds no
evidence of a CDW phase, as the identified bulk periodic distortions cannot be
realized in 2D. This behavior sets this material apart from other transition
metal dichalcogenide materials, which exhibit a charge density wave phase down
to the 2D limit. As CDW in TMDC materials is considered to compete with
superconductivity, the lack of CDW in monolayer CuS$_2$ suggests the
possibility of enhanced superconductivity relative to other transition metal
dichalcogenides. Overall, our work identifies CuS$_2$ as a previously
unrealized candidate to study interplay of superconductivity, CDW order, and
dimensionality.
