  Using density functional theory (DFT) and density functional perturbation
theory (DFPT), we investigate the stability and response functions of
CsH$_2$PO$_4$, a ferroelectric material at low temperature. This material
cannot be described properly by the usual (semi-)local approximations within
DFT. The long-range e$^-$-e$^-$ correlation needs to be properly taken into
account, using, for instance, Grimme's DFT-D methods, as investigated in this
work. We find that DFT-D3(BJ) performs the best for the members of the
dihydrogenated alkali phosphate family (KH$_2$PO$_4$, RbH$_2$PO$_4$,
CsH$_2$PO$_4$), leading to experimental lattice parameters reproduced with an
average deviation of 0.5 %. With these DFT-D methods, the structural,
dielectric, vibrational and mechanical properties of CsH$_2$PO$_4$ are globally
in excellent agreement with the available experiments ($<$ 2% MAPE for
Raman-active phonons). Our study suggests the possible existence of a new
low-temperature phase for CsH$_2$PO$_4$, not yet reported experimentally.
Finally, we report the implementation of DFT-D contributions to elastic
constants within DFPT.
