Anomalous quantum and isotope effects in water clusters: Physical phenomenon, model artifact, or bad approximation?

Free energy differences $\Delta F:=F-F_{\text{prism}}$ are computed for several isomers of water hexamer relative to the "prism" isomer using the self-consistent phonons method. %$\Delta F:=F-F({prism})$ We consider the isotope effect defined by the quantity $\delta F_{D_2O}:=\Delta F_{\rm D_2O}-\Delta F_{\rm H_2O}$, and the quantum effect, $\delta F_{\hbar=0}:=\Delta F_{\hbar=0}-\Delta F_{\rm H_2O}$, and evaluate them using different flexible water models. While both $\delta F_{D_2O}$ and $\delta F_{\hbar=0}$ are found to be rather small for all of the potentials, they are especially small for two of the empirical models, q-TIP4P/F and TTM3-F, compared to q-SPC/Fw and the two {\it abinitio}-based models, WHBB and HBB2-pol. This qualitative difference in the properties of different water models cannot be explained by one being "more accurate" than the other. We speculate as to whether the observed anomalies are caused by the special properties of water systems, or are an artifact of either the potential energy surface form/parametrization or the numerical approximation used.