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We have investigated the effects of compression and quantization on atomic distribution in ice Ic and in its compressed states at 77 K and 10 K, using the path integral molecular dynamics (PIMD) simulations over wide range of volume. It has been found that the high density amorphous ice (HDA) is attained by compression but volume range to retain ice structure is wider at 10 K than 77 K. We have discovered that quantum dispersion of atoms in ice Ic at 10 K induces non-zero probability that hydrogen-bonded H
_{2}O molecular molecules are oriented nonlinearly in the crystal structure, which was believed to contain exclusively linear orientation of hydrogen-bonded molecular pairs in this ice. It has been found that for HDA there is each non-zero probability of orientational disorder of hydrogen-bonded H
_{2}O pairs, of such uniform distribution of H atoms as observed in supercritical fluids in general, and of H atoms located at the O-O midpoint. The present PIMD simulations have revealed that these observed anomalous characteristics of atomic distribution in HDA are caused by both quantization of atoms and compression of the system.

Ice Ic is known as a cubic crystal and has the lowest density among all polymorphs of ice as well as ice Ih [^{−3} (19.3 cm^{3}∙mol^{−1}) at 80 K [^{−3} (13.8 cm^{3}∙mol^{−1}) at 1 GPa and 1.17 ± 0.02 g∙cm^{−3} (15.4 cm^{3}∙mol^{−1}) at zero-pressure [

Quantum effect of H atom is considered as remarkable in low-temperature substances, and the degree of quantum dispersion is parameterized as the de Broglie thermal wavelengths_{2}O and D_{2}O [

The aim of the preset work is to investigate the detail of atomic distribution under compression of the system and quantization of atomic nuclei beyond the analysis reported so far. We have thus performed path integral molecular dynamics (PIMD) simulations of a series of H_{2}O systems, for which at the beginning of the simulations the crystalline ice Ic structure was set at molar volume 25.7 - 9.84 cm^{3}∙mol^{−1}, including the reported experimental values of ice Ic and HDA. The compression is fulfilled by varying set volume in the present study; the simulations have been done under constant- temperature and constant-volume. The temperature has been set at 77 K and 10 K; 77 K is the temperature at which the existence of HDA was reported in the experiments, while at 10 K we expect quantum effect is expected to be more enhanced than 77 K.

We have adopted a flexible model of water called the SPC/F_{2} potential [

Following the quantum-classical isomorphism based on path integral, the canonical partition function of a system of N atoms is written as

(1)

where _{2} potential, which is a function of a set of N atomic coordinates

where

are solved numerically. The mass of bead

quantum-statistical mechanical regime,

and

The equations of motion Equation (3) and MNHC have numerically been integrated following the RESPA algorithm [^{−17} s. Sixty four H_{2}O molecules have been contained in a cubic simulation cell with constant volume. The Trotter num- ber P of each atom has been taken as 140 (77 K) and 900 (10 K). Although the number of molecules in the box is not so many, the number of degrees of freedom to be computed in PIMD is even 322560 and 2073600 for^{3}∙mol^{−1} (0.700 - 1.83 g∙cm^{−3}), which includes the experimental value 19.3 cm^{3}∙mol^{−1} (0.933 g∙cm^{−3}) at 80 K [_{1}md, which is ferroelectric with all H_{2}O dipoles pointing in the same direction [

The potential energy of the system and the RDFs between atomic species have been calculated from both PIMD and classical MD simulations. In addition, we have further analyzed the density distribution of H atoms as a function of newly defined coordinates, one of which is

^{3}∙mol^{−1} of ice Ic. However, in the classical

limit, in ^{3}∙mol^{−1}). In ^{3}∙mol^{−1}. For the classical result it is seen at 17.3 cm^{3}∙mol^{−1} as well as 77 K. These results imply significant quantum effect on the relationship between energetic stability and volume of ice Ic.

_{OO}(r) and the hydrogen-hydrogen RDF g_{HH}(r) obtained from both the PIMD and classical MD calculations at 77 K and experimental density. The overall profile and the peak positions in the classical limit are in good agreement with that of ice Ic by classical MD [_{OO}(r) of PIMD has the same peak positions as that of the classical MD, the peaks are broadened because of spatial extent of beads or quantum dispersion.

In _{HH}(r) obtained from the classical MD is distinctively oscillatory, indicating the well-defined location of H atoms in ice Ic. This profile is also in good accordance with the RDFs obtained by Geiger et al.; in [_{HH}(r) of PIMD in _{HH}(r) is obviously attributed to spatial extension of the distribution of H atoms, caused by quantization of atomic nuclei. Although at a glance this profile looks to be presentation of structural disorder of H atoms, it does not mean proton disorder because the SPC/F_{2} model does not allow H atoms to jump to the other sites through breaking of the O-H covalent bond.

a couple of sharp distribution maxima at _{HH}(r) of PIMD is a smoothed version of the classical counterpart. This also supports that H atoms are not disordered but simply have broadened distribution centered around lattice points. In conclusion, for ice Ic at 77 K and experimental density, though there is significant extension of distribution of H atoms due to quantum dispersion, the crystalline structure remains in the framework of adopted model SPC/F_{2}.

_{OO}(r) calculated from PIMD simulations for each set molar volume at 77 K and 10 K. At 77 K, in ^{3}∙mol^{−1} (_{OO}(r) starts to collapse; non-zero distribution exists between the first and second peaks. The collapse of g_{OO}(r) is further promoted as the system volume is more compressed (16.2 - 9.84 cm^{3}∙mol^{−1}).

These RDFs obviously indicate the amorphousness or disorder of the distribution of oxygen atoms constituting the backbone of the whole system. Observed collapse of the RDFs is distinguished from such broadening of the distribution of quantized H atoms as observed in ice Ic in the last subsection. Rather it is certain that the present amorphization is caused by confinement of the system in small volume or compression. As seen in _{OO}(r) at 77 K becomes even flatter and is uniform in the range of 3.3 - 4.0 Å under the utmost compression, indicating that the amorphization is further promoted. _{inter} at higher density is attributed to this amorphousness of the system. The state attained for 13.8 and 15.4 cm^{3}∙mol^{−1} presently is considered as identical to the HDA which Mishima et al. discovered for the same density.

In the right column of _{OO}(r) at 10 K occurs only under the utmost compression conditions (^{3}∙mol^{−1} and more (^{3}∙mol^{−1} amorphization already occurs at 77 K (see

Thus, the volume range to retain ice Ic structure is wider (25.7 - 15.4 cm^{3}∙mol^{−1}) at 10 K than 77 K (25.7 - 19.3 cm^{3}∙mol^{−1}). This suggests that enhanced quantum dispersion at lower temperature contributes to the stabilization of ice Ic structure. It seems that the atoms with more stretched distribution due to quantum dispersion at 10 K behave as if they were more expanded cushions with which the ice structure is filled up to support wider volume. The RDFs _{HH}(r) for HDA at 10 K is extremely flat. Surprisingly, such uniform profile is similar to RDFs of supercritical fluids in general [

dispersion of atoms and the amorphous structure of the whole system.

In order to examine the distribution of H atoms, in _{OO}(r) in

been reported so far in the classical MD simulations of ice Ic. We should note that, in spite of this probability, the system is not amorphized at this density 19.3 cm^{3}∙mol^{−1} and retain the crystalline structure, as described in Section 3.2. Here let us discuss the above-described occurrence of non-linear orientation. Once a pair of hydrogen-bonded water molecules have nonlinear orientation, other adjacent molecules should be nonlinearly oriented as well to avoid rising up intermolecular potential energy. Such correlation of reorientation of water molecules can be fulfilled if the correlation is collective and concerted. In fact, a recent neutron quasielastic scattering experiment detected concerted tunneling transfer of H atoms in ice Ih and Ic at 5 K [

For HDA, on the other hand, in ^{ }Å and

tion of _{OO}(r) in

In

because the SPC/F_{2} potential employed in the present simulations does not include breaking of covalent bond of each individual H_{2}O molecule [

We have revealed significant quantum effect on static distribution of atoms in ice Ic. At first, quantized ice Ic has the lowest intermolecular potential energy at the experimental density and temperature, while the molar volume of the minimum potential energy is deviated to smaller volume in the classical limit. The distribution of H atoms in ice Ic under the experimental condition is significantly broadened with retaining the crystalline structure of ice Ic. When ice Ic is cooled down to 10 K, the quantization of atoms produces non-zero probability that hydrogen-bonded water molecules are nonlinearly oriented even in ice Ic. This is also a result of quantum effect enhanced by lowering the temperature. In addition, there is also non-zero probability that H atoms are located at the middle point of two neighboring O atoms in ice Ic at 10 K. It should be noted that this broad distribution of H atoms does not mean such proton disorder as pointed out by using neutron diffraction experiments [

In the present simulations, setting of small molar volume or compression has induced the amorphization of ice Ic. At reported experimental density by Mishima et al. (15.4 and 13.8 cm^{3}∙mol^{−1}), quantized ice Ic has also been amorphized, and the achieved state is considered as identical to the HDA which Mishima et al. discovered. For HDA at 77 and 10 K, there is non-zero distribution of H atoms spanning over _{HH}(r) under utmost compression exhibits very smooth curve and converges to unity. This feature resembles that of RDFs of supercritical fluids in general. In addition, there is also non-zero probability of H atoms at the middle point of two neighboring O atoms in HDA. These features of atomic distribution in HDA are attributed to both the quantum dispersion of atomic nuclei and the compression of the system.

Finally we should mention some perspectives. In relation to recent quasielastic neutron scattering experiments which revealed the existence of concerted motion of H atoms in ice Ih and Ic [_{2} model in which the intramolecular potential is quadratic. If a double-well-type adiabatic potential model representing proton transfer and quantum-mechanical resonance between diabatic states [

The present study was supported by JSPS Grant-in-Aid for Scientific Research No. 23654149.

Sato, N. and Kinugawa, K. (2016) Path Integral Molecular Dynamics Simulation on Atomic Distribution in Amorphized Ice Ic. Natural Science, 8, 460-474. http://dx.doi.org/10.4236/ns.2016.811048