One striking anomaly of water ice has been largely neglected and never explained. Replacing hydrogen ( 1 H) by deuterium ( 2 H) causes ice to expand, whereas the "normal" isotope effect is volume contraction with increased mass. Furthermore, the anomaly increases with temperature T , even though a normal isotope shift should decrease with T and vanish when T is high enough to use classical nuclear motions. In this study, we show that these effects are very well described by ab initio density functional theory. Our theoretical modeling explains these anomalies, and allows us to predict and to experimentally confirm a counter effect, namely that replacement of 16 =0 [15]. This "normal" isotope effect corresponds to a ∼12% zero-point expansion of 20 Ne relative to a hypothetical "classical" or "frozen" lattice [16,17]. Since H 2 O and Ne have similar molecular masses, one might expect similar effects. However, the volume of H 2 O at T = 0 is ∼0.1% smaller than that of D 2 O [12,13]. It has rarely been mentioned in the literature that this is opposite to the usual behavior, and no explanation has been offered.In this paper, we explain this effect as an interesting coupling between quantum nuclear motion and hydrogen bonding, that may be relevant also to the structure of liquid water. Our analysis shows that, despite the anomalous isotope effect, quantum ice actually has a volume 1% larger than it would have with classical nuclei. The effects are smaller than in Ne mostly because of delicate cancellations. We exploit these cancellations to make critical comparisons of: (i) quasiharmonic theory versus fully anharmonic path-integral molecular dynamics (PIMD); (ii) ab initio forces versus flexible and polarizable empirical force fields (EFF); and (iii) various flavors of ab initio density-functional theory (DFT) exchange and correlation (XC) density functionals (DF) with and without inclusion of van der Waals (vdW) interactions. We find: (i) quasiharmonic theory is satisfactory for this problem; (ii) present state of the art EFFs are not good enough to describe nuclear quantum effects in water; and (iii) all the DFs considered describe qualitatively the anomalous effects, although some versions perform better than others.Within the volume-dependent quasiharmonic approximation (QHA), the equilibrium volume V (T ) is obtained by minimizing at each T the Helmholtz free energy F (V, T ) [18,19]:where E 0 (V ) is the energy for classical (T = 0 or frozen) nuclei, at the relaxed atomic coordinates for each volume. ω k are the phonon frequencies, with k combining the branch index and the phonon wave vector within the Brillouin zone. Their volume dependence is linearized as:whereis the Grüneisen parameter of the mode, and V 0 is the equilibrium volume of E 0 (V ). ω k (V 0 ) and γ k (V 0 ) are obtained by diagonalizing the dynamical matrix, computed by finite differences from the atomic forces in a (3 × 3 × 3) supercell, at two volumes slightly below and above V 0 . As shown in the supplementary information [20] (SI), this li...

Path-integral molecular dynamics (PIMD) simulations have been carried out to study the influence of quantum dynamics of carbon atoms on the properties of a single graphene layer. Finitetemperature properties were analyzed in the range from 12 to 2000 K, by using the LCBOPII effective potential. To assess the magnitude of quantum effects in structural and thermodynamic properties of graphene, classical molecular dynamics simulations have been also performed. Particular emphasis has been laid on the atomic vibrations along the out-of-plane direction. Even though quantum effects are present in these vibrational modes, we show that at any finite temperature classical-like motion dominates over quantum delocalization, provided that the system size is large enough. Vibrational modes display an appreciable anharmonicity, as derived from a comparison between kinetic and potential energy of the carbon atoms. Nuclear quantum effects are found to be appreciable in the interatomic distance and layer area at finite temperatures. The thermal expansion coefficient resulting from PIMD simulations vanishes in the zero-temperature limit, in agreement with the third law of thermodynamics.

The Ising model in small-world networks generated from two-and three-dimensional regular lattices has been studied. Monte Carlo simulations were carried out to characterize the ferromagnetic transition appearing in these systems. In the thermodynamic limit, the phase transition has a mean-field character for any finite value of the rewiring probability p, which measures the disorder strength of a given network. For small values of p, both the transition temperature and critical energy change with p as a power law. In the limit p → 0, the heat capacity at the transition temperature diverges logarithmically in two-dimensional (2D) networks and as a power law in 3D.