Proton dynamics in ice VII at high pressuresY. Finkelstein and R. Moreh Citation: J. Chem. Phys. 139, 044716 (2013); doi: 10.1063/1.4816630 View online: http://dx.doi.org/10.1063/1.4816630 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v139/i4 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 139, 044716 (2013)

Proton dynamics in ice VII at high pressuresY. Finkelstein1 and R. Moreh2,a)

1Nuclear Research Center – Negev, Beer-Sheva, Israel2Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

(Received 13 May 2013; accepted 10 July 2013; published online 31 July 2013)

We calculated the proton kinetic energies Ke(H) of ice under high pressures up to 63 GPa by assum-ing the harmonic approximation. The input measured optical frequencies of vibration, libration, andtranslation of ice VII versus pressure as well as the H2O geometry and the distances ROH necessaryfor calculating Ke(H) (at 298 K) were taken from the literature. The resulting Ke(H) values werefound to decrease gradually with increasing pressure, approaching the region where the H-atom issymmetrically hydrogen bonded between two oxygens in the OH-O system. Interestingly, the Ke(H)results were found to be consistent with those of other materials such as Rb3H(PO4)2 and KH2PO4

having similar ROH and ROO distances in the OH-O system. Similar calculations were also carriedout for D2O. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4816630]

I. INTRODUCTION

Since the first reported anomaly1 in the measured protonmean kinetic energy, Ke(H), in H2O, growing research activ-ity was focused on measuring and calculating Ke(H) in wa-ter subjected to various physico-chemical environments.2–10

In the first reported case, which involved supercooled water, avery strong temperature dependence of Ke(H) was observed,abruptly increasing from ∼131 meV in the solid phase at∼269 K to a striking 224 meV in the super-cooled fluid atT = 271.15 K, and then restoring a normal value of 143 meVin the liquid phase at 293 K.1, 4, 11

Following this puzzling behavior of Ke(H), additional re-ports, some including even more pronounced anomalies, werepublished.3, 5, 6, 12 Those results were obtained by analyzingthe radial momentum distribution function of protons mea-sured by deep inelastic neutron scattering (DINS).13 The re-sults suggest that Ke(H) can be anomalously small as low as106 meV in the case of ice confined in 14 Å diameter singlewalled carbon nanotubes (SWCNT),2, 9 or very large, up to221 meV as in water confined in 16 Å diameter double wallcarbon nanotubes (DWCNT),3 and can reach strikingly largevalues as in the case of 24 Å diameter nanoporous silica xe-rogels where Ke(H) = 295 meV.5 This latter value is nearlydouble the measured Ke(H) value in water14–16 or ice Ih2, 9, 16

at normal conditions.All DINS measurements of Ke(H) were carried out us-

ing the VESUVIO spectrometer at Rutherford Laboratory,UK.17, 18 In particular, these neutron scattering experimentsexhibited unique features revealing radial momentum distri-bution functions that contained in some cases a secondarymaximum or shoulder that has been related to a node in thetunneling direction. These secondary features were thus in-terpreted as signatures of tunneling,1, 6, 7, 13 similar to thoseobserved in the ferroelectric phase of KH2PO4.19 Accord-ingly, all anomalous Ke(H) values were explained in termsof proton tunneling through the potential barrier separating

a)Electronic mail: moreh@bgu.ac.il

two O-atoms in neighboring water molecules in liquid andsolid phases. Coherence of the proton between the two siteswas thus suggested to be the main reason for the anomalousKe(H) values.1, 7, 11

Various theoretical studies including sophisticated pathintegral calculations,13, 16, 20 and advanced quantum modelsand simulations21, 22 were applied to calculate Ke(H) in var-ious O-H containing systems. In particular, a recent abinitio calculation23 focusing on silica xerogels was made inan attempt to explain the highest Ke(H) ∼295 meV measuredin nanoporous materials. However none of the above calcu-lations could predict the anomalously large or low measuredvalues of Ke(H) obtained using the VESUVIO spectrometer.In addition none of the inelastic neutron measurements nor theoptical IR or Raman experiments could provide any evidencefor any corresponding anomaly in the OH stretch frequencywhich could explain the Ke(H) anomalies in any of the con-fining systems studied.

In this respect we decided to study the Ke(H) inice VII under high pressure (P). In this H2O phase, twokey parameters that were proposed to explain the anoma-lous behavior of Ke(H), namely confinement3 and protondelocalization1, 7, 13 due to tunneling, are practically realized.As has been inferred from path integral simulations24–26 andfrom spectroscopy,27–29 within the OH-O system, the ROO

separation in crystalline ice undergoes dramatic length re-duction under pressure, where the hydrogen bond lengthRO-H decreases and the covalent bond ROH increases. In fact,Holzapfel30 first suggested applying high pressure to controlthe lengths of H-bond and the OH covalent bond by inducingshort ROO separations down to symmetric H-bonds as in theO-H-O system. At low pressures, large inter-oxygen separa-tions, such as ROO ∼2.76 Å typical of liquid water and iceIh at ambient pressure, result in standard H bond lengths inwhich the proton is strongly localized in a highly asymmetricsingle well (SW) potential. At higher pressures and interme-diate separations, such as ROO ∼2.4 Å typical of ice VII,26, 27

the proton experiences a symmetric double-well (DW) poten-tial with a high barrier at the center between two O-atoms.

0021-9606/2013/139(4)/044716/9/$30.00 © 2013 AIP Publishing LLC139, 044716-1

044716-2 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

The proton can tunnel between these two equivalent minimaalong the OH-O hydrogen bond axis, and is thus stronglydelocalized, having a bimodal distribution.26 Owing to zeropoint fluctuations, further compression to about ROO ∼2.31 Å,the central potential barrier almost vanishes and the DWtransforms into a SW.24, 26 Thus the asymmetric OH-O con-figuration goes to a symmetric O-H-O one, where the pro-ton is again localized, laying midway between the twoO-atoms,24, 26 leading to the non-molecular symmetric phaseof ice,26, 28, 30–32 denoted as ice “X.” In effect, the covalentbond length ROH increases from 0.985 Å (in ice Ih at ambientP)33 to ∼1.20 Å (in ice VII at RT and ∼62 GPa),24 while theH-bond length RO-H decreases respectively from 1.775 Å to1.20 Å and the ROO distance from 2.76 Å to ∼2.4 Å.27, 33

Ab initio path integral calculations have shown that on pres-sure increase to ∼72 GPa, at 100 K in a symmetric DW po-tential and high zero point energy (ZPE), ROO decreases to∼2.365 Å (ROH ∼1.18 Å) and bond symmetrization (ice X) isachieved.24 At still higher compression, near ∼100 GPa, thepotential barrier fully vanishes and the proton becomes cen-tered in a symmetric SW potential with ROO ∼2.32 Å (ROH

∼1.16 Å).24

Evidently, all of the above changes were found to be ac-companied with drastic decrease of the OH stretch frequency(i.e., weakening of the covalent bond) and consequently ofKe(H) as will be seen later on. In the present work we utilizethe combined measured optical and neutron inelastic vibra-tional frequencies of H2O for calculating the proton kineticenergy of water and ice at various temperatures and pressures,within the stability range of ice VII.

II. BACKGROUND

In a previous work,33 the proton kinetic energy Ke(H) inwater was calculated using an empirical approach in which

the effect of the environment was taken into account by utiliz-ing the measured frequencies as input data. It is well knownthat environmental effects in water are very strong as re-flected in the reduction of the OH stretch frequencies from ν1

= 3686 and ν3 = 3738 cm−1 in the vapor phase,34 via 3280and 3490 cm−1 in the liquid,35, 36 to 3085 and 3220 cm−1 inice Ih.37–41 Note that in ice VII at a pressure of 63 GPa, thestretch frequency, ν1 = ν3, drops steeply42 to ∼620 cm−1.This model assumes the harmonic approximation and decou-pling of the internal vibrations from the external modes of li-bration and translation; it was found to yield results which arein good agreement with experiment and to reproduce resultsof sophisticated calculations in an impressive manner. In Ref.33, the Ke(H) values of ice Ih and high density amorphous(HDA) ice at 5 K and of bulk ice at 269 K were calculatedby utilizing optical data together with the simulated libra-tional spectra.43 For ice Ih, excellent agreement with DINSexperimental results was obtained and for HDA ice the re-sult agreed within ∼3%. A less satisfactory agreement of∼10% was found for the 269 K bulk ice. For liquid water at293–300 K, however, it was necessary to assume free rotationof the water molecule in order to bring the calculated valuesto agree with experiment.33 Obviously, this is a non-physicalassumption, which when abandoned and the measured li-bration spectrum44, 45 used as an input, our previously33

calculated Ke(H) increased from ∼143 meV to 153 meV(Table I) which reveals a much better agreement with thenewly analyzed DINS data,20 as well as with recent Ke(H)values extracted from macroscopic thermodynamic data.21, 46

More recently a new, non-parametric method was also sug-gested for the analysis of DINS profiles,22 however whenutilized to reanalyze previously measured data at 271 K ofpolycrystalline ice Ih,20 a less satisfactory agreement wasobtained.22 A consideration of the newly revised results ofTable I shows that the Ke(H) values of all the phases of water

TABLE I. Comparison of previously calculated Ke(H) values (in meV units) and the new ones in water obtainedby including the measured libration spectra. The results of previous and new analysis of the DINS experimentalH2O data are also presented.

Experiment CalculatedPhase T (K) Previous analysis New analysis Ref. 32 Present

HDA ice 5 141a 145 155b

Ice Ih 5 144 152b, 155c

170 144 153b

230 146 154b

Bulk ice 269.2 131d 151.48 ± 0.03e 147 154b

130 ± 3f 154 ± 2g

133.7h 156 ± 2g

Water 293 147 ± 8d,h 143 153c

296 147d, 145h 143 153c

300 143 ± 3i 151.95 ± 0.03e,j 144 153c

aReference 2.bCalculated using Eq. (3) and the VDOS spectra of Ref. 43.cCalculated using Eq. (2) with INS data of Ref. 45.dReference 9.eReference 21.fReference 8.gReference 20.hReference 53.iReference 14.jReference 46.

044716-3 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

are almost independent of T in the range 5–300 K and that avery good agreement exists between our new values and thoseobtained by a revised analysis of the DINS data. The con-stancy of Ke(H) stems from the fact that it is mainly governedby the zero-point motion and that a decrease in the OH stretchfrequency is compensated by an increase in the frequency oflibration of the entire molecule.31, 32

All in all, our calculated Ke(H) values in all systemsstudied23, 33 were found to agree within less than 6% withother models, including the advanced path integral moleculardynamics (PIMD)16 and path integral Car-Parrinello molec-ular dynamics (PICPMD) methods,47, 48 which accounted foranisotropic motions and anharmonicity of the proton poten-tials. Such deviations in Ke(H) are relatively small comparedto the large differences of nearly a factor ∼2 between someof the recently DINS measured and calculated values in sys-tems involving OH bonds. Thus, our calculations provide in-sight into the problem and may help in tracing out the rea-son for some of the huge deviations between theory andexperiment.

The harmonic decoupled model used in the present workis practically the same as that used in Ref. 15. In fact, theKe(H) value obtained by the above two approaches for iso-lated (non-bonded) H2O molecule where the vibrational fre-quencies are those of the vapor phase is ∼152 meV (seeRef. 15) at 298 K, which coincides with 152 ± 2 meV ob-tained by us for an isolated molecule.32 Similar agreementoccurs for all Ke(H) values up to 673 K.33 It should be em-phasized however that this model which was discussed inmore detail in Ref. 49 was originally suggested and appliedby our group to calculate the kinetic energy50 of isotopic15N in NaNO3 and for studying the directional atomic kineticenergies51 in a single crystal of NaNO3. It was also appliedby us earlier for studying the case of a water molecule.52

It is also of interest to compare our new results ofTable I with those derived by Ramirez-Herrero (Ref. 16). At270 K we obtained Ke(H) = 154 ± 3 meV which is close towithin ∼3% to 149.5 meV. For an isolated molecule at 270 K,however, our Ke(H) value is 150 meV, being higher by 7%than 140 meV reported in Ref. 16 at the same temperature.This deviation for an isolated molecule is too high and is veryprobably due to the assumptions made in Ref. 16 regardingthe vibrational frequencies in H2O.

III. CALCULATIONS

In the following we discuss the calculation of the pro-ton kinetic energy Ke(H) of molecular water in its threephases, vapor, liquid, and ice. We apply it to ice VII sub-jected to high pressures, between 5 and ∼63 GPa, for whichthe measured optical27, 29, 31, 42, 53–55 and inelastic neutron scat-tering (INS)44, 45 parameters necessary for such calculationsare available. Note that the present calculations do not applyto higher pressures near 70 GPa where the molecular form ofwater disappears.27, 32, 42

The proton kinetic energy Ke(H) in H2O vapor is con-tributed by three motions: translation of the whole molecule,rotation of the whole molecule, and its internal vibrations. De-coupling of the three motions is assumed as discussed above.

Thus Ke(H) of H2O in a vapor phase may be written in thefollowing form:

Ke(H ) = ST

3

2kT + SL

3

2kT

+3∑

j=1

Sj

hυj

2

(1

ehνj /kT − 1+ 1

2

), (1)

where the factors 3kT/2 represent the classical kinetic energyof translation and rotation of the entire molecule with k theBoltzmann constant, T the thermodynamic temperature, andST and SL the fractions shared by a single proton in each ofthe two motions. The third term is the kinetic energy of the in-ternal vibrations which consists of symmetric and asymmetricOH stretch and one bending HOH mode. Each mode of vibra-tion is represented by a harmonic oscillator of frequency υ j

where h is the Planck constant. The factor 1/2 is due to thefact that the kinetic part is equal to half the total energy ofeach of the 3 harmonic vibrational terms. Each H atom sharesonly a fraction Sj (j = 1–3), in the jth vibration mode. The en-ergy fraction ST of the translational motion is obtained fromthe mass ratio ST = MH/MH2O ≈ 1/18. Similarly, SL is theshare of each H atom in the rotation of H2O: it is calculatedby considering the moments of inertia along the three axespassing through its center of mass; the result is SL = 0.4752.To calculate Sj we note that it is equal to the ratio of the jthvibrational energy of one H-atom relative to that of the entire

molecule: Sj = MiA2ij /

3∑i=1

MiA2ij where Aij the amplitude of

the ith atom in the jth vibrational mode of H2O. It is usefulto check the correctness of the resulting values of Sj by usingthe sum rule which applies to each atom of H2O; it states thatthe sum of all energy fractions is equal to unity, namely: ST +SL + 1

3

3∑j=1

Sj = 1.

A. Solid H2O

Here, we focus our attention on values of Ke(H) in iceVII under high pressures, for cases where the water retains itsmolecular form. In the condensed phase, the free translationsand rotations, typical of the vapor phase, are hindered and turninto lattice vibrations and librations with frequencies νT andνL, respectively. The calculation can be carried out becausethe required optical data at elevated pressures are available inthe literature.27, 29, 31, 32, 42, 53–55 Thus Ke(H) for solid H2O isobtained by modifying Eq. (1) to yield

Ke(H ) =∑

i=T ,L

Si

hυi

2

(1

ehνi/kT − 1+ 1

2

)

+3∑

j=1

Sj

hυj

2

(1

ehνj /kT − 1+ 1

2

). (2)

In cases where the frequency distributions gT(ν) andgL(ν) of translation and libration are available from INS,44, 45

044716-4 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

3 6 9 12 15 18 21 24 272350

2400

2700

2850

3000

3150

3300

3450

30 35 40 45 50 55

1500

2000

2500

ν 1,3 (

cm-1)

P (GPa)

5 10 15 20 25 30 35 40 45 50 551000

1500

2000

2500

P (GPa)

ν 2 (cm

-1)

ν 1,ν 3

(cm

-1)

P (GPa)

FIG. 1. Internal frequencies of H2O ice-VII versus pressure at 295 K show-ing the measured Raman OH symmetric (open triangles and circles) andasymmetric (solid triangles and circles) stretch frequencies ν1 and ν3 up to27 GPa.32, 53 Lower inset: Weighted average measured IR stretch frequencyν1,3, between 31 and 55 GPa (crosses,27 asterisks55). Upper inset: measuredIR bending frequencies.27, 55 Solid and dashed lines are best fits to the data.The frequencies for calculating Ke(H) between 5 and 55 GPa were taken fromthe fitted lines.

Ke(H) takes the following form:

Ke(H ) = ST

νT∫0

gT (ν)α(ν)dν + SL

νL∫νl

gL(ν)α(ν)dν

+3∑

j=1

Sj

hυj

2

(1

ehνj /kT − 1+ 1

2

), (3)

with α(ν) = hυ2 ( 1

ehν/kT −1 + 12 ) the kinetic energy of the corre-

sponding harmonic oscillator, where νT, ν l, and νL are thelimits of integration; gT(ν) and gL(ν) are normalized so that:νT∫0

gT (ν)dν =νL∫νl

gL(ν)dν = 1.

Figures 1 and 2 show respectively plots of the experi-mental internal and lattice frequencies of ice VII at 293 Kcollected from optical measurements.27, 32, 42, 53–55 The solidand dashed lines are best fits to the measured results fromwhich the input data points of ν1, ν2, ν3, ν1,3, and νT, νL werededuced for calculating Ke(H) between 5 and ∼55 GPa us-ing Eqs. (2) and (3). Depicted in the main frame of Fig. 1are the OH stretch frequencies ν1 and ν3 (open and solidsymbols) reported up to 27 GPa by Raman studies.32, 53 Be-tween 27 and ∼55 GPa, the weighted average measured ν1,3

IR frequencies27, 55 are shown in the lower inset to Fig. 1. Inthe upper inset to Fig. 1, the IR measured bending frequencyν2 is shown.27, 55 Note that the external and internal frequen-cies of ice VII at room temperature may also be extractedfrom other studies.29, 31, 55 Typically, the differences betweenthe reported values are ∼10 cm−1, and may reach ∼200 cm−1

at the highest pressures. These differences were taken into ac-count when estimating the uncertainties of our deduced valuesof Ke(H) vs P. Figure 1 illustrates the red shift of ν1, ν3, andν13 with increasing P (at a rate of 230 cm−1/GPa), indicatinga weakening of the covalent bonds and strengthening of theH-bonds upon compression.

0 5 10 15 20 25 30 35 40 45 50 55 60 65100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

Tν T,L

(1/

cm)

P (Gpa)

L

FIG. 2. Measured IR frequencies of translation T (open squares42) and li-bration L (open circles27, 54). Solid lines T and L are best fit to the data. Thefrequencies for calculating Ke(H) between 5 and 64 GPa were taken from thefitted lines. Solid circle: weighted average frequency of ice Ih at 15 K mea-sured using INS.44 Solid square: average over measured INS frequencies of20 K ice and 253 K super cooled water.45 Note the smooth lines joining theambient and high pressure data of different phases of H2O.

At ∼63 GPa, the red shift of the stretch frequenciesgrows further with increasing P where ν13 reaches a value of∼620 cm−1 (not shown in Fig. 1).42 These changes are ac-companied by a blue shift of νT and νL as revealed in Fig. 2.By contrast, the bend frequency ν2 (Fig. 1) shows a pressure-insensitive behavior up to 40 GPa and its value was accord-ingly adopted for the calculations for the whole P range up to63 GPa. Also shown in Fig. 2 are ambient pressure measuredνL and νT using INS44, 45 which seem to comply with the ex-trapolation curves of Fig. 2, although measured for differentH2O phases (ice Ih and supercooled water).

Note that the major part of the kinetic energy of the in-ternal vibrations Sj(H) and of the libration SL(H) are takenby the H-atom, being the light partner, while the major partof the translational motion is taken by the O-atom, the heavypartner.

The calculation of SL(H) and Sj(H) of H2O in ice VIIrequired a knowledge of the molecular H2O geometry of iceVII, namely the length, ROH, and the angle � HOH (taken to beconstant:56 105.38◦ in the range 5–63 GPa in line with accu-rate quantum mechanical calculations57). ROH was obtainedfrom the empirical relation of ROH vs the stretch frequencyυOH reported by Efimov58–61 which is well confirmed by otherstudies.30, 62–64 This relation applies not only to H2O but toother systems involving standard hydrogen bonds (in systemswithout bifurcations, i.e., when a proton binds only two oxy-gen atoms).59

Table II lists all input parameters used to calculate Ke(H)and Ke(O) in ice VII vs P at 295 K together with force con-stants and energy fractions. The estimated error in the calcu-lated Ke(H) is ∼3% up to P = 55 GPa, increasing to ∼9% at63 GPa and are obtained by accounting for all uncertainties(including those of the measured frequencies).

044716-5 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

TABLE II. List of the geometrical parameters versus pressure of H2O ice VII at room temperature, used for calculating the kinetic energies of the H- andO-atoms, Ke(H) and Ke(O) (in meV units) shown in bold face in the two bottom lines bellow. The force constants (in 105dyn/cm units) are those found to bestreproduce the listed literature frequencies. Sk(H) and Sk(O) are the deduced shares of the proton and of the oxygen, respectively, in the kth mode of motion: k= T-translation, L-libration, 1-symmetric stretch (ss), 2-bending (b), 3-asymmetric stretch (ass). Literature frequencies and Sk values were used as input data inEq. (3).

P (GPa) 5 15 25 30 35 40 45 50 55

ROH (Å) 0.979 1.003 1.028 1.050 1.068 1.088 1.108 1.129 1.145HOH (deg) 105.38 105.38 105.38 105.38 105.38 105.38 105.38 105.38 105.38k1(105dyn/cm) 5.587 4.770 4.004 3.645 3.17 2.713 2.254 1.826 1.417k2(105dyn/cm) 0.487 0.477 0.485 0.098 0.083 0.051 0.05 0.035 0.002k3(105dyn/cm) 0.518 0.565 0.797 0.864 0.947 0.987 0.987 1.058 1.101νT (cm−1) 282 468 633 687 719 731 728 720 721νL (cm−1) 683 780 878 926 975 1 024 1 073 1 121 1 170ν1 ss(cm−1) 3 188 2 949 2 702 2 575 2 401 2 214 2 022 1 825 1 680ν2 b(cm−1) 1 574 1 585 1 623 1 653 1 690 1 722 1 740 1 740 1 570ν3 ass(cm−1) 3 302 3 079 2 855 2 575 2 401 2 214 2 022 1 825 1 680ST(H) 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056SL(H) 0.475 2 0.475 2 0.475 2 0.475 2 0.475 2 0.475 2 0.475 2 0.475 2 0.475 2S1(H) 0.462 0.461 0.460 0.466 0.467 0.468 0.471 0.485 0.485S2(H) 0.463 0.463 0.463 0.463 0.478 0.476 0.474 0.463 0.463S3(H) 0.482 0.483 0.485 0.478 0.463 0.463 0.463 0.459 0.459ST(O) 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888SL(O) 0.049 0.049 0.049 0.049 0.049 0.049 0.049 0.049 0.049S1(O) 0.077 0.078 0.081 0.068 0.067 0.063 0.059 0.03 0.03S2(O) 0.074 0.074 0.074 0.074 0.045 0.049 0.053 0.074 0.074S3(O) 0.035 0.034 0.031 0.044 0.074 0.074 0.074 0.082 0.082Ke(H) 153 ± 3 150 ± 3 148 ± 3 145 ± 3 142 ± 3 139 ± 3 136 ± 3 133 ± 3 128 ± 3Ke(O) 58 ± 2 66 ± 2 75 ± 2 79 ± 2 81 ± 2 81 ± 2 80 ± 2 79 ± 2 78 ± 2

IV. RESULTS AND DISCUSSION

A. H2O ice VII at 298 K

The calculated kinetic energies Ke(H) and Ke(O) in H2Oice VII at 298 K vs P are listed in the last two lines ofTable II and are shown graphically in Fig. 3 which also com-pares the experimental and theoretical results for related H2Ophases at ambient pressure. The first features to note in Fig. 3are the opposite behaviors of decreasing Ke(H) and increasingKe(O) upon compression. These trends may be explained bynoting that Ke(H) is mainly contributed by the internal modeswhich decrease with increasing P, while Ke(O) is mainlydetermined by translational oscillations which increase withincreasing confinement reaching a plateau at ∼40 GPa.Figure 4 illustrates these features which also show the rela-tive kinetic energy contributions to the H- and O-atoms. Ofinterest also in Fig. 4 are the opposite trends of the stretchingand librational modes with increasing pressure as was alsohinted theoretically in Ref. 13, where a trend was observedin which the momentum distributions in the directions per-pendicular to the hydrogen bond axis broadened with increas-ing confinement. This is consistent with the study of Benoitet al.24 which found that the position space variance in theperpendicular directions was shown to decrease with increas-ing pressure. The above have direct significance to the trendemerging from Fig. 4. While compression results in elonga-tion of the O–H bond which in turn induces red shift in the vi-brational frequencies, the resultant coupled hardening of thelibrational modes may be understood if one assumes that the

libration of a water molecule is the motion of the hydrogenatom perpendicular to the molecular plane.64 Thus the net ef-fect of the elongation of the covalent bond is a blue shift ofthe translational and librational frequencies.64

0 10 20 30 40 50 6050

60

70

80

90

100

110

120

130

140

150

160

P (GPa)

Ke(

H,O

) (m

eV)

O

H

FIG. 3. Calculated mean kinetic energies Ke(H) (circles) and Ke(O)(squares) vs pressure in H2O ice VII at 298 K. Solid lines are best fits tothe data points. Ambient pressure data points: (i) diamond – an average overrevised calculated and experimental values of ice Ih33, 44 and bulk water45

listed in Table I and (ii) open square – calculated by Eq. (3) utilizing INSmeasured VDOS spectra of ice Ih.44 Error bars at 35 GPa represent the typ-ical uncertainty in the calculated values between 5 and 55 GPa. The errorbars at 61.5 GPa are considerably larger due to uncertainties in the opticallymeasured frequencies. Note the smooth line joining the ambient and highpressure data of different phases of water and ice.

044716-6 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

ET(H)

EL(H)

EInt

(H)

ET(O)

EL(O)

EInt

(O)

Rel

Sha

re (

%)

P (GPa)

FIG. 4. Relative contributions of lattice external and internal modes to themean kinetic energies of the proton (solid symbols) and of the oxygen atom(open symbols) in H2O ice VII at 298 K versus pressure. Squares: translation;circles: librations; triangles: cumulative internal modes. Solid and dashedlines are best fits to the data.

Worth noting in Fig. 3 is the continuous link betweenthe experimental Ke(H) and Ke(O) values measured for otherH2O phases (ice Ih and bulk water) with those of increasinglypressurized ice VII. It illustrates the consistency of the kineticenergy values from ambient to high pressures, irrespective ofthe explicit H2O phase. Interestingly, and the most importantconclusion to be drawn from the overall data presented inFig. 3 is the noticeable emerging robustness of Ke(H). Con-sidering not only ice VII but also other phases it is clearly ev-ident that Ke(H) values in the various liquid and solid formsof H2O at ambient and high pressures simulating strong con-finement (close to the breakdown of the molecular structureof H2O) and proton coherence, are limited to an energy rangebetween ∼110 and 160 meV. The origin for this behavior isthe high zero point energy of the proton in H2O. Altogether,under strong confinement, including cases where proton co-herence is realized, no evidence of abnormal energy excess issuggested by the present calculations.

B. D2O ice VII at 298 K

The availability of spectroscopic and neutron scatteringdata enabled us to apply our calculations to D2O ice VII also.The main purpose is to test to what extent the results conformwith the H2O data and to see if an additional insight into theanomaly problem of Ke(H) may be gained. The experimen-tal frequencies of libration and internal modes of D2O ice VIIare available up to pressures of ∼60 GPa32, 53, 55, 57 and wereused as input data in our calculations. However, the frequen-cies of translation and bending were either absent or inconclu-sive. In such cases we used the corresponding H2O frequen-cies and divided them by the empirical factor42, 58 νOH/νOD

= 1.36 based on experiment; it is an average over the literaturevalues32, 58, 65–67 between 1.31 and 1.41 and is ∼4% lower than√

2, the square root of the mass ratio. The deviation from√

2,expected for an isolated water molecule, may be understood

in the case of bonded molecule, if one accounts for complexcoupling effects imposed by the environment.

The input geometrical parameters of the D2O moleculein ice VII at 298 K were obtained by first deducing ROO fromthe empirical relations59 of νOD vs ROO where the measuredυ1,3 were taken from optical measurements.32, 53, 55, 57 In addi-tion, ROD was deduced from the empirical relation of ROD vsROO.59 The DOD angle in ice VII was taken to be identicalto that of H2O.56, 68, 69 Table III summarizes all input param-eters used in the D2O calculations together with the resultingforce constants, energy fractions, and the mean kinetic ener-gies Ke(D) and Ke(O) in D2O ice VII vs P at 298 K. Theresults are also given in graphical form in Fig. 5. It will be in-teresting to apply the new non-parametric model free methodof analysis22 to the experimental results of D-containing com-pounds to see whether the results will be in better agreementwith experiment as was the case in H-containing compounds.

By comparing Figs. 3 and 5, it is clearly evident thatKe(D) and Ke(O) in deuterated ice are less pressure sen-sitive than Ke(H) and Ke(O) in the H2O form. Otherwise,they exhibit practically similar overall behavior; both showconsistent correlation with the red shift of the stretch fre-quency vs P. In both systems relatively robust Ke(H) andKe(D) values emerge. Moreover, at the low pressure regimethey nicely comply with independent values, either measuredor simulated by others for various H2O phases under ambi-ent conditions. All in all, our calculation on pressurized iceVII provides no evidence that could support the occurrence ofanomalous Ke(H) values.

C. Comparison with Rb3H(SO4)2

It is interesting to note that in the literature a hydrogenbonded system, Rb3H(SO4)2, which is a superprotonic con-ductor, was reported70 which resembles ice VII under highpressure in the sense that ROO ∼2.4 Å in which the H-atomlay symmetrically between the two oxygens. The Ke(H) valueof this system may be deduced from the measured momentumdistribution of the H-atom along the three crystallographic di-rections at 10 K and 70 K using the DINS method.70 Thereported momentum distribution widths at 10 K were σ z

= 4.60 Å−1, σ y = 3.73 Å−1, and σ x = 4.35 Å−1 from whichthe mean kinetic energy, Ke(H) ∼113 meV, was deduced us-ing the relation: Ke(H) = 6.27 · (σ x

2 + σ y2 + σ z

2)/3; thisvalue is very close to that measured at 70 K because of theproximity of their σ z, σ y, and σ x values reported in Ref. 70.Note that this sample was more recently measured by anothergroup71; the results were σ z = 4.39 Å−1, σ y = 4.83 Å−1,and σ x = 3.37 Å−1 and seem to disagree as to the internaldirections of the crystal. However, they are in close agree-ment as to the deduced Ke(H) value. Interestingly, the aboveis close to Ke(H) = 107 ± 10 meV calculated by us for ice VIIat 61.5 GPa where ROO ∼2.4 Å,27 by using Eq. (2) togetherwith the average measured parameters of ν13, ν2, νL, and νT

reported in Refs. 27 and 42 for ice VII at 60 and 63 GPa,respectively; this average P value is below the breakdownpressure of molecular H2O which overlaps the realm of iceX. Because of the importance of this deduced Ke(H) valuewe discuss it in some detail. In ice VII two measurements

044716-7 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

TABLE III. List of the geometrical parameters versus pressure of D2O ice VII at room temperature, used for calculating the kinetic energies of the D- andO-atoms, Ke(D) and Ke(O) (in meV units) shown in bold face in the two bottom lines below. The force constants are those found to best reproduce the listedliterature frequencies. Sk(D) and Sk(O) are the deduced shares of the proton and of the oxygen, respectively, in the kth mode of motion: k = T-translation,L-libration, 1-symmetric stretch (ss), 2-bending (b), 3-asymmetric stretch (ass). Literature frequencies and Sk values were used as input data in Eq. (3) fromwhich Ke(D) and Ke(O) (in meV units) were calculated versus pressure.

P (GPa) 5 15 25 30 35 40 46 52 58

ROD (Å) 0.999 1.025 1.049 1.06 1.074 1.087 1.101 1.116 1.13DOD (deg) 105.38 105.38 105.38 105.38 105.38 105.38 105.38 105.38 105.38k1 (105dyn/cm) 5.777 4.934 4.164 3.796 3.448 3.119 2.752 2.383 1.97k2 (105dyn/cm) 0.834 0.767 0.705 0.67 0.636 0.603 0.133 0.095 0.065k3 (105dyn/cm) 0.445 0.492 0.542 0.566 0.692 0.724 1.063 1.106 1.117νT (cm−1) 166 335 462 502 526 536 551 541 536νL (cm−1) 533 596 644 672 706 743 784 816 842ν1 ss(cm−1) 2 375 2 195 2 015 1 925 1 835 1 745 1 640 1 526 1 396ν2 b(cm−1) 1 176 1 176 1 176 1 176 1 243 1 243 1 335 1 328 1 304ν3 ass(cm−1) 2 494 2 319 2 145 2 058 1 970 1 883 1 640 1 526 1 396ST(D) 0.100 6 0.100 6 0.100 6 0.100 6 0.100 6 0.100 6 0.100 6 0.100 6 0.100 6SL(D) 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456S1(D) 0.432 0.432 0.432 0.432 0.432 0.432 0.431 0.431 0.431S2(D) 0.430 0.430 0.428 0.428 0.427 0.425 0.450 0.438 0.412S3(D) 0.469 0.470 0.471 0.472 0.473 0.474 0.450 0.461 0.488ST(O) 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799SL(O) 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088S1(O) 0.137 0.136 0.136 0.136 0.136 0.136 0.137 0.137 0.137S2(O) 0.140 0.140 0.144 0.144 0.146 0.150 0.101 0.124 0.176S3(O) 0.061 0.060 0.058 0.056 0.054 0.052 0.100 0.078 0.025Ke(D) 114.4 112.0 109.4 108.1 107.8 106.8 104.33 102.19 99.25Ke(O) 57.5 61.6 66.1 67.7 68.8 69.0 70.203 69.172 68.15

were reported in the literature at this high pressure regime.The first27 at P = 60 GPa, the measured frequencies ν13, ν2,νL, and νT were 908, 1600, 1236, and 165 cm−1, respectively.The second42 at P = 63 GPa and the ν13, ν2, νL, and νT valueswere 620, 1600, 1295, and 691 cm−1. Note the huge decreasein the stretch frequency ν13 caused by a 5% pressure increasewhich is indicative of the proximity to the breakdown phase

0 5 10 15 20 25 30 35 40 45 50 55 6050

60

70

80

90

100

110

120

OKe(

D,O

) m

eV

P (GPa)

D

FIG. 5. Calculated mean kinetic energies Ke(D) (solid circles) and Ke(O)(solid squares) vs pressure in D2O ice VII at 298 K. Solid lines are best fitsto the data points. The ambient pressure data points (open circle and square)were calculated using Eq. (3) and utilizing the measured INS VDOS spectraof 15 K deuterated ice Ih.44 Note the smooth line joining the ambient andhigh pressure data of different phases of D2O.

of molecular H2O. Note also the large error in the calculatedKe(H) at this high pressure region which is due to the largeuncertainties occurring in this case of the limiting propertiesof ice VII. Here, an important conclusion should be empha-sized: even though the two systems Rb3H(SO4)2 and ice VIIare different, they seem to exhibit similar properties arisingfrom the fact that their ROO distances are quite close and thecovalent bonding of the H-atom nearly disappears, and theirKe(H) values are close to each other. The good agreement be-tween the Ke(H) values provides a good confirmation of theresults of the present calculations of high pressure ice VII. Inaddition, the parameters ROO and ROH seem to represent theaverage behavior of a wide range of H bonds in widely differ-ent materials and not necessarily that of ice VII only.33, 44, 45

It is also interesting to note that H2O ice VII maintains itsmolecular character at P ∼63 GPa and ambient temperatureas revealed in its low average measured OH stretch frequencyof ν13 ∼620 cm−1.42 It is only at higher pressures that theH-bonding becomes dominant and the molecular structure ofH2O disappears.28, 30, 32

D. Comparison with KH2PO4

Another case with which we could compare our calcu-lations is (KH2PO4) which is a paraelectric-ferroelectric ma-terial, whose ROO is 2.497 Å at 102 K (ferroelectric phase)and 2.483 Å at room temperature (paraelectric phase).72 Thiscomparison is possible because the Ke(H) value is known andwas deduced from the results of a DINS study73 where theproton momentum distribution was measured along two axes

044716-8 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

at 90 K (ferroelectric phase): σ z = 5.69 Å−1, σ x = 4.14 Å−1

from which Ke(H) = 139.3 meV is obtained using the rela-tion: Ke(H) = 6.27 · (2σ x

2 + σ z2)/3 meV; this is very close

to 138.6 meV measured at 130 K (paraelectric phase).73 Themain point to be emphasized here is that this value is in verygood agreement with Ke(H) = 139 ± 3 meV deduced fromour calculated values in ice VII at P = 40 GPa. It turns outthat at such a pressure, the value of ROO for ice VII, being2.47 ± 0.03 Å, conforms with ROO = 2.497 Å of KH2PO4.The distance ROO of ice VII at 40 GPa was deduced by usingthe two empirical relations59 of Efimov: the first, connectingthe measured stretch frequency ν13 and ROH and the secondrelating ROH to ROO. The good agreement between the Ke(H)values of two entirely different systems which happen to haveclose values of ROO supports the validity of the results of thepresent calculated values of Ke(H).

It should however be pointed out that the DINSresult73 for Ke(D) in the corresponding deuterated compound,KD2PO4 (DKDP) is ∼71 meV which is far lower than theexpected value. Note that the Ke(D) values usually conformwith the expectation when compared to that of H2O. In fact avalue of Ke(D) = 108.3 meV at 269 K and 118 K was respec-tively extracted for solid D2O from macroscopic thermody-namic data,21 and from elastic electron scattering10 measure-ment. This is to be compared with Ke(D) = 113 ± 3 meVcalculated by the present empirical method when utilizing themeasured INS vibrational density of states (VDOS) spectra44

of 15 K deuterated ice Ih. The origin for the huge deviation isnot clear, and will hopefully induce further experimental andtheoretical efforts for D-containing compounds.

E. Difficulties in understanding the anomalies

As mentioned earlier there are several reported anoma-lies in the measured values of Ke(H) in systems involving theOH-O configuration. We start with the Ke(H) anomalies firstreported in the case of supercooled water1 at temperaturesaround 273 K. In this particular case, several calculationswere made,13, 21 none of which could explain the anomalies.In this respect we mention the calculations of Ref. 16 whocarried out PIMD simulations which could reproduce the T-dependence of the density of water at ∼273 K but predicted asmooth variation of Ke(H) between 220 K and 300 K, in con-trast to the huge increase of Ke(H) of Ref. 1. In addition, inthe advanced PICPMD study of Ref. 13, no clear indicationsof secondary peaks or nodes in the momentum distributionwere found which could be associated with tunneling as sug-gested in Refs. 1, 6, 7, and 19. Also, no excess kinetic energycompared to ambient liquid water was apparent. It is very in-teresting however to find out whether a renewed analysis ofthe supercooled water data along the same lines carried out inRef. 22 would yield lower Ke(H) values in better agreementwith the results of the present model.

The measured Ke(H) value at 271 K was reported1, 11 tobe higher than that at 300 K by ∼50% and was attributedto proton coherent delocalization between two neighboringO-atoms. However, the measured momentum distribution ofRef. 1 implies a narrow proton wave function with ROO

∼2.7 Å which is inconsistent16, 74 with an increased proton de-

localization. This distance is smaller than ROO ∼2.76 Å (fornormal water) but is not small enough to fulfill the coherentdelocalization conditions of Ref. 75 (for an asymmetric DWpotential representing the OH-O interaction) requiring: ROO

∼2.50 Å and the elevation of the proton to the second excitedstate of the OH stretch vibration. Thus the assumption of co-herent delocalization in this case is quite problematic and maynot be correct to use it for explaining cases where the averageproton momentum distribution is anomalously high. It wasalso pointed out in Ref. 23 that it was not possible to explainthe huge increase of ∼50% in Ke(H) of water at ∼271 K be-cause no jump in the OH stretch frequency was observed76–78

in the above T range using optical IR and Raman measure-ments.

It should also be noted that in the case of ice confinedin SWCNTs of 14.1 Å diameter (at 5 K), a steep drop inKe(H) to 106 meV was reported.2 This behavior has beensuggested in Ref. 13 to be analogous to that of ice VII un-der high pressure and that it is caused by weakening the cova-lent bond and strengthening the H-bond. While this analogybetween the two systems13 seems to be plausible, it is how-ever contrasted by the results of two inelastic neutron scat-tering measurements: The value of ROO in water confinedin SWCNTs was reported79 to be ROO = 2.92 Å comparedwith ∼2.4 Å for compressed ice VII.27 Second, the stretchand bend energies in nanotube (NT) water were found to belarger by 4% than that of ice Ih at 5 K, implying a calculatedvalue33: Ke(H) = 146 meV being far higher than the mea-sured 106 meV of Ref. 2. Thus while the measured Ke(H) inice Ih confined in NT is reported to be nearly the same as thatcalculated for ice VII under high pressure, the results of INSmeasurements are in striking contrast with the suggestion ofRef. 13. Note also that the trend of decrease of Ke(H) withincreasing P seem to work in an opposite manner to that ofwater confined in DWCNTs where a huge increase3 in Ke(H)was reported. Nevertheless, as pointed out in Ref. 13 thereis a strong analogy between the high pressure ice VII phaseand that of the superprotonic conductor Rb3H(SO4)2 in thesense that both are hydrogen bonded systems revealing quiteclose ROO and Ke(H) values.

V. CONCLUDING REMARKS

The Ke(H) values of ice VII at high pressures up to 63GPa was calculated by assuming the harmonic approximationand utilizing the measured optical frequencies of vibration,libration, and translation and the water geometry of ice VII(at 298 K) versus pressure. Calculations were performed forboth protonated and deuterated ice VII. The resulting Ke(H)values were found to decrease gradually with increasing pres-sure reaching a region where the H-atom is symmetricallyhydrogen bonded between two oxygens in the OH-O sys-tem. It is of interest to note the interplay between the weak-ening stretch modes and the strengthening librational modeswith increasing pressure. In both H and D systems, relativelyrobust Ke(H) and Ke(D) values emerge. At the low pres-sure regime the results comply with independent experimen-tal values measured for various H2O phases under ambient

044716-9 Y. Finkelstein and R. Moreh J. Chem. Phys. 139, 044716 (2013)

conditions. Our calculation on pressurized ice VII providesno evidence that could support the occurrence of anomalousKe(H) values. Finally, the Ke(H) results were found to beconsistent with those of other materials such as Rb3H(PO4)2

and KH2PO4 having similar ROO distances in the OH-Osystem.

ACKNOWLEDGMENTS

One of us (R.M.) would like to thank Dr. D. Nemirovskyfor useful remarks.

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