  The collective dynamics in liquid water is an active research topic
experimentally, theoretically and via simulations. Here, ab initio molecular
dynamics simulations are reported in heavy and ordinary water at temperature
323.15 K, or 50$^\circ$C. The simulations in heavy water were performed both
with and without dispersion corrections. We found that the dispersion
correction (DFT-D3) changes the relaxation of density-density time correlation
functions from a slow, typical of a supercooled state, to exponential decay
behaviour of regular liquids. This implies an essential reduction of the
melting point of ice in simulations with DFT-D3. Analysis of longitudinal (L)
and transverse (T) current spectral functions allowed us to estimate the
dispersions of acoustic and optic collective excitations and to observe the L-T
mixing effect. The dispersion correction shifts the L and T optic (O) modes to
lower frequencies and provides by almost thirty per cent smaller gap between
the longest-wavelength LO and TO excitations, which can be a consequence of a
larger effective high-frequency dielectric permittivity in simulations with
dispersion corrections. Simulation in ordinary water with the dispersion
correction results in frequencies of optic excitations higher than in D$_2$O,
and in a long-wavelength LO-TO gap of 24 ps$^{-1}$ (127 cm$^{-1}$).
