Anomalous Nuclear Quantum Effects in Ice

Abstract

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...

Simulation and understanding of atomic and molecular quantum crystals

“…Interestingly, an anomalous thermal expansion isotopic effect has been observed in ice; the volume of solid D 2 O is larger than that of solid H 2 O (Röttger et al, 1994), in contrast to what occurs in other crystals upon substitution with heavier species. This quantum nuclear effect has been rationalised recently by Pamuk et al (2012) with ab initio calculations based on the quasi-harmonic approximation.…”

“…For instance, narrowing (widening) of the peaks in g(r) may be caused by the presence of heavier (lighter) species. Kinetic isotopic effects can also manifest in the thermal expansion of quantum solids (Pamuk et al, 2012;Herrero and Ramírez, 2011a) and corresponding P − T boundaries delimiting the thermodynamic stability regions of different phases (Lorenzana, Silvera, and Goettel, 1990;Goncharov, Hemley, and Mao, 2011).…”

“…In some cases it has been actually demonstrated that the combined description of molecular interactions and ionic effects at the quantum level is necessary for reproducing correctly the experimental findings in ice. Examples include the anomalous volume expansion observed in ice isotopes (Pamuk et al, 2012) and the interpretation of measured x-ray absorption spectra (Kong, Wu, and Car, 2012;Kang et al, 2013).…”

Temperature dependence of the zero point kinetic energy in ice and water above room temperature

“…These isotope effects are most evident in their temperature dependence, in that the volume difference between heavy and light ice increases with temperature [36]. Moreh and Nemirovski [24] calculated hE K ðTÞi of the proton in ice Ih between 5 K and 269 K, using optical vibration frequencies from the literature, assuming the harmonic approximation and decoupling between the degrees of freedom of translation, rotation-libration, and internal vibrations.…”

“…In the solid phase, a recent study assessed the anomalous isotope effects on the thermal expansion of ice [36], which were shown to originate on competing anharmonicities in the vibrational modes of the system. These isotope effects are most evident in their temperature dependence, in that the volume difference between heavy and light ice increases with temperature [36].…”