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We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H   $$_2$$  2 O and D   $$_2$$  2 O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density    $$\rho (T)$$  ρ ( T ) , isothermal compressibility    $$\kappa _T(T)$$  κ T ( T ) , and self-diffusion coefficients  D ( T ) of H   $$_2$$  2 O and D   $$_2$$  2 O are in excellent agreement with available experimental data; the isobaric heat capacity    $$C_P(T)$$  C P ( T ) obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H   $$_2$$  2 O and D   $$_2$$  2 O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H   $$_2$$  2 O and D   $$_2$$  2 O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H   $$_2$$  2 O, from PIMD simulations, is located at    $$P_c = 167 \pm 9$$  P c = 167 ± 9  MPa,    $$T_c = 159 \pm 6$$  T c = 159 ± 6  K, and    $$\rho _c = 1.02 \pm 0.01$$  ρ c = 1.02 ± 0.01  g/cm   $$^3$$  3 . Isotope substitution effects are important; the LLCP location in q-TIP4P/F D   $$_2$$  2 O is estimated to be    $$P_c = 176 \pm 4$$  P c = 176 ± 4  MPa,    $$T_c = 177 \pm 2$$  T c = 177 ± 2  K, and    $$\rho _c = 1.13 \pm 0.01$$  ρ c = 1.13 ± 0.01  g/cm   $$^3$$  3 . Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water,    $$P_c = 203 \pm 4$$  P c = 203 ± 4  MPa,    $$T_c = 175 \pm 2$$  T c = 175 ± 2  K, and    $$\rho _c = 1.03 \pm 0.01$$  ρ c = 1.03 ± 0.01  g/cm   $$^3$$  3 ). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of    $$T_c$$  T c for D   $$_2$$  2 O and, particularly, H   $$_2$$  2 O suggest that improved water models are needed for the study of supercooled water.

Evidence of a liquid–liquid phase transition in H$$_2$$O and D$$_2$$O from path-integral molecular dynamics simulations