Ab initio studies on the structural and dynamical properties of ice

PHYSICAL REVIEW B VOLUME 47, NUMBER 9 1 MARCH 1993-I

Ab initio studies on the structural and dynamical properties of ice

Changyol Lee*Department ofPhysics, Haruard Uniuersity, Cambridge, Massachusetts 02138

David VanderbiltDepartment ofPhysics and Astronomy, Rutgers Uniuersity, Piscataway, New Jersey 08855-0849

Kari LaasonenInstitut Romand de Recherche Numerique en Physique des Materiaux, PHB-Ecublens, CH-1015 Lausanne, Switzerland

R. Car

and Department of Condensed Matter Physics, Uniuersity of Geneua, Geneua, CH 1211, Sw-itzerland

M. ParrinelloIBM Research Dr'Vision, Zu'rich Research Laboratory, CH-8803 Ruschlikon, Switzerland

(Received 28 September 1992)

The structural and dynamical properties of cubic H20 and D20 ice phases are studied using ab initiomolecular dynamics combined with ultrasoft pseudopotentials. Phonon frequencies are extracted fromthe velocity autocorrelation functions; contributions from diff'erent normal modes in the phonon spectraare separated and easily identified. For the low-pressure phases, the agreement with the experimentaldata is reasonable and the isotope e6'ects are well reproduced. High-pressure phases are also studied.The equations of state for cubic ice (ice I, ), and for the ice VII-VIII-X family, are calculated. It is foundthat the local-density approximation must be augmented with gradient corrections in order to obtain aproper description of the hydrogen bond. Finally, the hydrogen-bond symmetrization, which is respon-sible for the transition from ice VII-VIII to ice X, is studied and is predicted to occur at 49 GPa. Thenature of the phase transition is found to be that of a mode-softening transition. The correspondingsymmetrization is also studied in ice I„but it is found to occur at a pressure of 7 GPa at which ice I, isunstable with respect to denser phases.

I. INTRODUCTION

The physical and chemical properties of water are ofobvious interest, and have led to a huge number oftheoretical studies using semiempirical methods. Howev-er, advances in computational methodology and comput-er power hold the promise of a new generation of studiesusing ab initio quantum-mechanical approaches. Ab ini-tio quantum-chemical methods have already been appliedto water molecules and dimers' and even trimers andsmall clusters. But the most promising approach for thestudy of larger aggregates, including crystalline ice andliquid water phases, is likely to be the density-functionaltheory (DFT). To date, density-functional calculationson H20 systems have been limited to molecules and di-mers, ' but there is no fundamental obstacle to the appli-cation of these methods to larger systems. Moreover, theintroduction of the ab initio molecular dynamics by Carand Parrinello (CP) (Ref. 10) opens the possibility ofefficient studies of dynamical properties, in addition tothe calculation of static energetics.

Two technical problems have hindered the widespreadapplication of DFT methods to H20 systems. First, thepreferred approach (especially for dynamical studies) hasbeen the plane-wave pseudopotential approach, in which

forces on atoms can be calculated in a straightforwardmanner. However, the description of first-row atoms likeoxygen is problematic in plane-wave schemes, because ofthe very large number of plane-wave components neededto describe the rather localized oxygen 2p orbitals.Second, because there have been relatively few DFT cal-culations on hydrogen-bonded systems, the accuracy ofthe local-density approximation (LDA) (Ref. 11) for thisclass of systems is not well established. After all, the Vander Waals interaction, which is another (but weaker)form of attractive interaction between closed-shell sys-tems, is known to be poorly described within LDA.

In this paper we adopt the "ultrasoft" pseudopotentialsrecently introduced by Vanderbilt, ' which offer a verysmooth and high-quality description of the oxygen atomand other first-row and transition-metal elements. More-over, we augment the LDA with gradient correc-tions, ' ' and confirm that this leads to an accuratedescription of the intermolecular interaction. Theseadaptions allow us to study the properties of H20 andD20 systems within DFT, using both static energetic andCP molecular-dynamics calculations, at a modest compu-tational cost.

As the structure of the solid state is generally moreeasily described than that of the corresponding liquid, we

47 4863 1993 The American Physical Society

4864 LEE, VANDERBILT, LAASONEN, CAR, AND PARRINELLO

have decided to focus here on crystalline ice phases.There has been considerable experimental work to identi-fy the various structural modifications of ice, stable ormetastable under a variety of conditions, and to elucidatethe structures and vibrational signatures of these po-lymorphs. ' Here we investigate the structural anddynamical properties of several ice phases with cubicsymmetry. In particular, the phonon densities of statesof (low-pressure) HzO and DzO ice I, at low temperatureare calculated from first principles and compared withexperiment. The contributions from the different normalmodes in the phonon spectra are separated and easilyidentified. The isotope shifts associated with the replace-ment of protons by deuterons are well reproduced. Also,the equations of state for ice I, and the ice VII-VIII-Xsystem have been calculated. A good description of theintermolecular interaction is obtained only after gradientcorrections to LDA are included. The phase transitionsinto symmetric hydrogen-bonded ice at low pressure (iceI, ) and high pressure (ice VII-VIII~X) are studied, andidentified as mode-softening transitions.

This article is organized as follows: In Sec. II, the ex-perimental background on ice is reviewed. Section IIIbrieAy discusses our theoretical method. In Sec. IV, theresults from the ab initio molecular-dynamics calcula-tions on cubic ice and ice X are presented, and Sec. Vpresents a summary and conclusion. The appendixpresents a brief discussion of the physical interpretationof the velocity autocorrelation function.

II. EXPERIMENTAL BACKGROUND

Ordinary hexagonal ice (ice Ih) is obtained when liquidwater freezes at atmospheric pressure. Roughly speak-ing, the oxygen atoms in ice I& comprise a wurtzite lat-tice (the hexagonal modification of the diamond lattice),while hydrogen atoms occupy asymmetric sites along the0—O bonds and are disordered within the Pauling icerules' (i.e., each 0 atom has two nearer and two furtherH neighbors). This ice disorder persists down to 0 K.There is a modification of ice Iz, referred to as cubic iceor ice I„which is similar except that the oxygen atomscomprise a diamond lattice. Ice I, cannot be obtained bycooling ice I&, but can be obtained by the condensation ofwater vapor below —80'C. Ice I, is metastable between—80 and —120 'C, at which temperature it becomesstable. The structure of ice I, is shown in Fig. 1. The hy-drogen atoms are also disordered within the Pauling icerules in ice I, .

The vibrations of ice Ih and ice I, crystals have beenextensively studied using infrared' ' and Ra-man' ' spectroscopy, and have been the subject oftheoretical calculations. ' Spectroscopic studies on iceI& and ice I, have shown that there is no substantialdifference in their infrared' and Raman ' spectra.This is not surprising because ice I& and I, have the samenearest-neighbor bonding configuration and the samenumber of next-nearest neighbors. (The prominentfeatures of the spectra from the experiments are summa-rized in Table II.) The assignment of observed spectralpeaks to individual modes has been greatly facilitated by

FIG. 1. Structure of ice I, . The unit cell has eight H~O mole-cules with oxygen atoms denoted by bigger circles and hydrogenatoms by smaller circles.

comparison of spectra of protonated and deuterated ice,so that a fairly complete analysis is possible. In this arti-cle, we calculate the phonon densities of states of H20and DzO ice from first principles and identify contribu-tions from the individual normal modes.

Since the stable form of a material is determined by thecondition that the Gibbs free energy is a minimum at aspecified temperature and pressure, the structures withthe lowest volume are preferred when the pressure ishigh. For the case of ice, more and more close-packedstructures are generated as the pressure is increased up toa few tens of GPa. However, all forms from ice I to iceIX have intact water molecules as their basic buildingblocks. The moderately high-pressure ice VII and VIIIphases, for example, can be thought of as consisting oftwo interpenetrating I, structures, as shown in Fig. 2. (Inthe cubic ice VII structure, the protons are again disor-dered within the Pauling ice rules. In ice VIII, on theother hand, the two sublattices are fully ordered and op-positely polarized, resulting in an antiferroelectric struc-ture that is almost cubic but with a small tetragonal dis-tortion. ) In these phases, it is known that the covalentlybonded 0—H distance increases with increasing pressure,while the hydrogen-bonded 0—0 distance naturally de-creases. At a sufFicient high pressure, the molecularpicture is expected to break down entirely, and the hy-drogens should reside at the midpoint of the 0—0 bonds.Such a speculative phase has been designated ice X.

There has been some experimental evidence suggestinga transition into ice X. Hirsch and Holzapfel found anew band above 40 GPa in the Raman spectra of ice VIIIup to 50 GPa at 100 K. Polian and Grimsditch ob-served an anomaly in the behavior of the longitudinalsound velocity at 44 GPa from Brillouin scattering stud-

47 Ab initio STUDIES ON THE STRUCTURAL AND. . . 4865

III. THEORETICAL METHOD

FIG. 2. Structure of ice VIII. The unit cell has 16 H20 mole-cules with oxygen atoms denoted by bigger circles and hydrogenatoms by smaller circles. The ice X structure would be obtainedby displacing all the hydrogen atoms into O —0 bond midpoints.The small tetragonal distortion of ice VIII is not shown.

ies up to 67 GPa. Walrafen et al. also suggested that asymmetric hydrogen-bond structure might form in iceVII at pressures of about 75+20 GPa by extrapolatingRaman spectra from lower pressures. However, these ex-periments have left many questions unanswered. Theycould not distinguish the order of the transition (first or-der vs second order), nor did they provide useful informa-tion about the structure of the high-pressure phase.Thus, the nature of the transition has been entirely specu-lative.

Theoretically, studies based on empirical interatomicpotentials have supported the notion of a transition to theice X structure. An early theoretical calculation by Hol-zapfel ' predicted a transition to symmetric hydrogenbonding in ice VII at pressures between 35 and 80 GPa.The interaction of the hydrogen atoms with the twonearest-neighbor oxygen atoms was approximated byequivalent Morse potentials. Later, a calculation by Stil-linger and Schweizer predicted a transition at roughly60 GPa. This work included a treatment of thequantum-mechanical many-body problem for coupledproton motions along hydrogen bonds. A more recent es-timate by the same authors puts the transition at about45 GPa. Also, Stillinger and Schweizer suggested thatthe symmetrization of the hydrogen bond could beachieved in ice I, at some high pressure which is muchlower than that for ice VII-VIII ~ However, the accuracyand applicability of these semiempirical approachesremains questionable, and it is clear that ab initio calcula-tions have an important role to play in clarifying the situ-ation.

Our theoretical method consists of the ab initiomolecular-dynamics approach proposed by Car and Par-rinello' combined with the ultrasoft pseudopotentials in-troduced by Vanderbilt. ' As discussed in the introduc-tion, the latter allow for an ef5cient plane-wave represen-tation of the localized valence orbitals associated withfirst-row and transition-metal elements. A review of theVanderbilt pseudopotential formalism and some addition-al details of the pseudopotential generation procedure aregiven in Ref. 34. The pseudopotential for oxygen is gen-erated in the neutral ground-state configuration, with acutoA' radius of 1.5 a.u. for the valence electronic wavefunctions and 1.2 a.u. for the local potential. Two refer-ence energies at —1.76 and —0.67 Ry (corresponding tothe s and p eigenvalues) are used for both the s and pchannels in the nonlocal potential. Previous tests havedemonstrated that the Vanderbilt pseudopotentials gen-erated with these parameters provide a very smooth andaccurate description of oxygen atoms. For the case ofhydrogen, also generated in the neutral ground-stateconfiguration, the cutoft radii for the valence electronicwave function and the local potential are 0.8 and 0.6 a.u. ,respectively. One reference energy at —0.49 Ry is takenfor the s state of hydrogen. The charge augmentationpseudofunctions Q; (r) have a cutoff radius r;„„„=0.6a.u. for 0 and 0.5 a.u. for H, in such a way as to matchthe radial charge density and its first and second deriva-tives at r;„„„,and to preserve the appropriate moments ofthe charge density, while maximizing the smoothness us-ing a procedure similar to that used by Rappe et al.for wave functions. In these calculations, we did not im-plement the use of an L-dependent r;„„„asdiscussed inRef. 34.

In the solid-state calculations, the electronic wavefunctions are expanded in a basis of plane waves with ki-netic energy up to 20 Ry. Only the I point is used forthe Brillouin-zone summations. The program is used inthree modes of operation. We use the first-order dissipa-tive dynamics on the electrons (steepest descents) to cal-culate ground-state energies for systems with given ionicconfigurations. In some cases, we use dissipative dynam-ics simultaneously on electron and ion degrees of free-dom, in order to identify a local minimum in theconfiguration space. For the dynamical calculations, weuse the Car-Parrinello approach, i.e., second-orderNewtonian dynamics on both the fictitious electronic andreal ionic degrees of freedom. In our calculations, wetake 4.3 a.u. for the time step and 400 a.u. for the ficti-tious electron mass. The simulation is always startedfrom zero ionic velocities after a steepest-descent quenchof the electronic coordinates at the fixed startingconfiguration. Some details of the implementation, in-cluding expressions for the forces and details of themethod for evolving the constraint matrix in theNewtonian dynamics, are given in Ref. 34 (but note thatthe multiple grid scheme discussed there has not beenused in the present work).

In all cases, the Ceperley-Alder exchange-correlationenergies and potentials are used for the LDA. " Most of

4866 LEE, VANDERBILT, LAASONEN, CAR, AND PARRINELLO 47

the calculations are carried out using gradient corrections(GC) in addition, specifically using the Becke' parame-trization for the exchange and the Perdew formula' forthe correlation. For consistency, all GC solid-state calcu-lations utilize pseudopotentials that are consistently gen-erated with the same G-C scheme.

IV. RESULTS

A. Tests on simple systems

We study energy-minimizing structures for H20 mono-mers and dimers. The lattice constant of the unit cell is13 0 a u. and only the I point is included in theBrillouin-zone summation. The water molecule in themonomer calculation is positioned symmetrically withrespect to the body diagonal of the cell, and the two oxy-gens in the dimer calculation are aligned on the body di-agonal. In Table I, we compare our theoretical resultsfor the 0—H bond length and H —0—H bond angle of theHzO monomer, and the 0—0 distance of the dimer, withexperiment. It is clear that gradient corrections make fora dramatic improvement in the dimer 0—0 distance, aquantity which reflects the description of the hydrogenbond. Similar results have been reported from calcula-tions utilizing conventional pseudopotentials and all-electron methods. Moreover, the underestimation ofthe 0—0 bond length by LDA is also consistent with ourcalculations of the equation of state of ice phases, to bediscussed in Sec. IV C 1. Therefore, except where noted,the gradient corrections have been adopted for all of thecalculations reported below.

B. Phonon density of states of ice I,

We carry out molecular-dynamics simulations of themotion of atoms in the crystal of ice I„and calculate thephonon density of states from the Fourier transform ofthe velocity autocorrelation function. Initially, we take alattice constant of 12.0 a.u. from experiment at—130'C. The structure of ice I, for our calculations isshown in Fig. 1. We arrange the oxygen and hydrogenatoms in a geometric configuration corresponding to iso-lated water molecules sitting in the diamond-structuresites in a cubic supercell. Since the crystal lattice of ice I,is orientationally disordered, we generate all possibleorientational configurations of the eight molecules in thesupercell which do not violate Pauling's ice rules. '

There are four physically distinct orientationalconfigurations in this eight-molecule unit cell, with de-generacy 6, 12, 24, and 48 and polarization ( 100),

( ~ —,'0), (000), and ( —,'00), respectively. We choose oneof the configurations with degeneracy 48 because it is themost disordered configuration among the four. After re-laxing the atomic degrees of freedom, we find theminimum-energy atomic configurations. The 0—H bondlength and H —0—H bond angle are found to be 1.92 a.u.and 108.5', which may be compared with 1.866 a.u. and104.7' for an isolated water molecule (see Ref. 40 orTable I), respectively. Then, we add randomly assignednormal-mode components to the minimum-energy atomicconfiguration in conformity with a target temperature100 K. We simulate the atomic dynamics in the crystalfor a duration of 320 fsec. From the series of atomic po-sitions as a function of time, we calculate the velocitiesv;(t), their Fourier transforms U, (co), and the power spec-trum ~U;(co)~ . As discussed in the appendix, the latter isdirectly related to the phonon density of states in thethermodynamic and long-time limits. (It has to be under-stood that runs of finite length will exhibit thermal fluc-tuations in the weights of peaks in the power spectrum. )In practice, the maximum entropy method ' is used toextract the power spectrum, thus providing better resolu-tion at low frequencies. The number of data is 3074 withsampling interval 0.104 fsec, and the number of poles is700. The phonon "density of states" obtained in this wayis shown in Fig. 3.

By decomposing this density of states into the contri-butions from the normal modes of a water molecule, weidentify the groups of peaks as bands arising from 0—Hstretching modes, H —0—H bending modes, librationalmodes, and translational modes, in the order of decreas-ing frequency. The corresponding projected densities ofstates for H20 and D20 ice are shown in Fig. 3. The iso-tope shifts which are evident from a comparison of theH20 and DzO ice I, results can easily be seen to agreewith expectations. In H20 ice, the 0—H stretchingmodes contribute to the peak near 3100 cm ' which isshifted to near 2250 cm ' in D20 ice. The factor ( —,

' )'~

serves to check the identification because the frequencyof a vibration is inversely proportional to the square rootof the mass of the vibrating particle. The H —0—H bend-ing modes and librational modes also show a factor of( —,')', while the translational modes show a factor of( —,", )'~ . The experimental data on the absorption spec-trum' are compared with theory in Table II. Theagreement with experiments is good.

Note that the contributions from the different normalmodes of the water molecule are well separated, con-sistent with the expectation that the intermolecular in-teraction (i.e. , the hydrogen bond) should be weak com-

TABLE I. Structural properties of H20 monomer and dimer.

Structural parameter

0—H bond length (a.u. )

H —O —H bond angle (deg. }

LDA

Monomer1.849

103.3

LDA+GC

1.866104.7

Experiment

1.809104.5

0—0 distance (a.u. )

Dimer5.10 5.50 5.63


TABLE II. Comparison between spectra from theoretical calculations and from infrared and Ramanspectra of ice I, .

Mode Theory

H20 (cm ')

Experiment Theory

D,O (cm-')

Experiment

0—H stretchingH —0—H bendingLibrationTranslation

2768-31241528

620-1024196-352

3083-33801650+30

500-105054-305

2040-22641140

376-612176-316

2283-24201210+10350-75054-294

pared to covalent or ionic chemical bonds. However, thehydrogen bond is strong enough to shift the 0—H stretchfrequency of H20 ice I, significantly from that of an iso-lated H20 molecule, roughly 3100 vs 3680 cm ', respec-tively. The hydrogen-bonding interaction must also beresponsible for the large width of the stretching-modebands which is notable in Fig. 3. This could be explainedeither as inhomogeneous broadening associated with theproton disorder of the ice crystal, or as a simple bandingresulting from the intermolecular interactions. As dis-cussed in the next paragraph, we believe it is predom-inantly the latter. However, for reasons that are not yetclear, these effects do not seem to broaden the librationalbands significantly; their frequencies remain remarkablynarrow and well defined.

We also calculate the phonon densities of states withinLDA. Preparation of the system is the same, except thatwe simulate atomic motions in all of the four distinct ice-rule configurations and that the duration of themolecular-dynamics simulations is 437 fsec. (We em-

phasize that both calculations were carried out at thesame experimental lattice constant. ) In comparison tothe calculations with gradient corrections, the phonondensities of states have almost the same features and fre-quencies. However, the frequency of the 0—H stretchingmode from tlie LDA calculations is a little lower. Wepresent in Fig. 4 the results on H20 and D20 ice for thefour physically distinct proton configurations. It is not-able that the phonon densities of states of cubic ice do notdepend strongly on the proton configurations which mayinhuence the local field. In particular, the widths of theO —H stretching bands do not correlate strongly with thedegree of proton disorder. Finally, the librational modeagain has a narrow and well-defined frequency regardlessof the proton configurations.

C. Hydrogen-bond symmetrization

l. Equation of state ofice xWe study the hydrogen-bond symmetrization with in-

creasing pressure in low- and high-pressure phases of ice.First, we calculate the equation of state for ice X. Total

CO~ ~c

6$

0)CD

M

O

~~0)

0)o

M~ ~

JD05

COCD

t5V)

O

~~V)

CDo

I

4 0

frequency (1000 cm }

FIG. 3. Phonon densities of states of H&O (left) and D20(right) ice I, . From top to bottom; total, stretching, bending, li-

brational, and transitional modes, respectively.

f I

1 2 3 4 0 1 2 3

frequency ( 1000 cm )

FIG. 4. Phonon densities of states of H&O (left) and D~O(right) ice I, within LDA. The numbers 1, 2, 3, and 4 representthe proton configurations with po1arization (100), ( 2 —,'0),(000), and ( —'00), respectively.


TABLE III. Comparisons between theoretical and experi-mental equations of state of ice X.

Latticeconstant Pressure (GPa) Pressure (GPa)

(a.u. ) from LDA from GC-LDAPressure (GPa)

from experiment

9.910.010.1

10.210.310.410.5

138.33.2

—2.1—7.5

—13—19

103979185787265

114978776716861

2. Static energy calculations

We next investigate the stability of the ice X structurewith respect to symmetry-breaking proton displacements.

-274.0

G$

-275.00)O

~~ -276.0

CDCL

~~-277.0Q)

LLI

-278.0

I I I I I

500. 1000. 1500. 2000. 2500.Volume ( a.u. )

FIG. 5. Equation of state for ice X within LDA (left) andwith gradient-corrected LDA (right). The dots are the calculat-ed data and the curve is the fitted quadratic function.

energies are calculated at lattice constants from 10.0 to12.0 a.u. in steps of 0.5 a.u. , fixing all the protons in theunit cell at the bond midpoints, using both LDA andgradient-corrected LDA. The two sets of data points arefitted to quadratic functions as shown in Fig. 5. The hy-drostatic pressure is calculated from the derivative of en-ergy with respect to volume using the thermodynamicidentity p = —BE/BV. The equation of state from thecalculations with the gradient-corrected LDA showsgood agreement with the experimental equation of stateof Hemley et al. within a few percent error, while theequation of state from LDA alone underestimates the lat-tice constant at any given pressure by more than 10%%uo.

In fact, the pressure is negative in the physically realiz-able range of lattice constants. The results are summa-rized in Table III and compared with experimental infor-mation. These results show that the gradient-correctedLDA is capable of giving a good description of thestructural properties of H2O ice, while the LDA descrip-tion is severely inadequate.

We calculate the total energies of the unit cell with latticeconstants 10.0, 10.5, and 11.0 a.u. as we displace all theprotons away from the bond midpoints by the sameamount 6 in an antiferroelectric pattern, simulating theice VIII structure, and fit the resulting data points tofourth-order polynomials. If the protons had infinitemasses, the transition into the asymmetric positionswould occur when the coefficient of the quadratic termchanges its sign, since this is the point at which thepotential-energy surface develops a double minimum.The coefficients of the quadratic terms are found to beapproximately linear in lattice constant over the range ofinterest, crossing zero at about 9.9 a.u. This correspondsto 103 GPa in our theoretical equation of state, exceedingthe experimental transition pressure by more than a fac-tor of 2.

However, the protons are so light that their quantumfluctuations are not negligible. Based on the theory ofKoehler and Gillis, we estimate the size of the quantumfluctuation effects of the protons. In this theory, we con-sider a system of interacting quartic anharmonic oscilla-tors whose Hamiltonian consists of an on-site term Hoand an intercell coupling term Hi which is treated in themean-field approximation:

Ho= — +—au +yu1 d 1 2 4

2M du~ 2

H, = —yu(u ),where M is the mass of the proton, u is the displacementfrom the bond midpoint, and atomic units are used. Themean-field approximation is expected to work fairly wellhere because the interaction between water molecules is along-range dipole-dipole interaction. Ho above describesthe motion of a single proton, holding all other protonsfixed, while the calculations of Fig. 6 correspond tothinking of u as an antiferroelectric collective coordinate;thus, in the classical mean-field limit we can identifyHo+H, =

—,'(n —y)u +yu . The classical transition tothe asymmetric structure would occur at g —a =0.When quantum fluctuation effects are considered by in-

cluding the kinetic-energy term in Ho, we expect thetransition to be shifted so that it will not occur until somepositive value of g —a. Using the variational technique,one finds that the transition occurs if

(g —a) =6@/&M

We find that this condition is satisfied at lattice constant10.74 a.u. , which corresponds to 49 GPa in our equationof state shown in Fig. 5. This is now in excellent agree-ment with the experimental estimates of 42 to 47.5 GPa.

We also estimate the effect of the proton quantum fluc-tuation on our theoretical equation of state, as this couldalso affect the predicted transition pressure. Since the ox-ygens are much heavier than protons, we assume theirquantum fluctuation effects are negligible, and we treatthe oxygen sublattice as rigid. We calculate the protonground-state wave functions and energies in the fittedsingle-bond potential-energy curve, and add these


ground-state energies to the total DFT energies of theunit cell. It is found that the quantum fluctuations in-crease the pressure by only about 6 GPa in the region ofinterest. As this is a rather small correction, we have notconsidered it further.

We also calculated the transition lattice constants forother types of proton displacement patterns consistentwith the Pauling ice rules, such as a ferroelectric patternand a disordered pattern. We found no significantdifference in the transition lattice constants regardless ofthe type of displacement. Moreover, at least within themean-field theory outlined above, the shift in the transi-tion point due to quantum fluctuations should be roughlythe same for each. Therefore, our calculations do notpredict which displacement pattern is favored. However,we think the antiferroelectric displacement pattern of iceVIII is likely to require the lowest energy when couplingof the proton distortions to the tetragonal strain is takeninto account, Of course this advantage will be offsetsomewhat at higher temperatures, where the higher en-tropy of the ice VII will lower its free energy. We havenot attempted a quantitative analysis of these competingeffects.

3. Molecular-dynamics simulations

So far we have confirmed that our theory does give atransition near the experimentally observed one. In orderto understand better the nature of the transition, we tryto find a soft phonon mode at the transition by simulatingthe atomic vibrational dynamics. While the Car-Parrinello ab initio molecular dynamics assumes the va-lidity of classical mechanics' and therefore neglectsquantum fluctuation effects, this approach should stillgive important insight into the nature of the transition.We prepare the system in such a way that the ions in theunit cell deviate from the positions in ice X by a smallamount in a random fashion and the electrons are in theirground state. We let the system evolve for a duration ofabout 250 fsec. The phonon density of states is calculat-

ed from the Fourier transform of the velocity autocorre-lation function with the help of the maximum-entropymethod, as before, and the total phonon density of statesis again decomposed into the contributions from differentnormal modes of the system. In Fig. 7, we show the totalphonon density of states and the density of states project-ed onto the stretching mode (SM) of the protons, forseveral different lattice constants. Note that as the latticeconstant approaches the transition from below, the SM-projected density of states moves toward lower frequency.We further decompose the SM's of the protons into thosethat preserve the ice rules, and those that violate them.The densities of states of the ice-rule-preserving SM areshown in the right column of Fig. 7; clearly, therefore,the higher-frequency peaks in the middle column corre-spond to the ice-rule-violating SM's.

Our interpretation of Fig. 7 is that the ice-rule-preserving SM's go soft (i.e., co goes to zero) somewherebetween 10.0 and 10.5 a.u. confirming the interpretationin terms of a mode-softening transition. The theory ofthe previous parts suggests it should go soft closer to 10.0a.u. ; the fact that it seems to go at a slightly higher latticeconstant probably reAects anharmonic effects associatedwith thermal fluctuations in the molecular-dynamicssimulations, which were done at low but nonzero temper-ature. We confirm this by finite temperature mean-fieldtheory. We investigate how much the transition latticeconstant would change due to the finite temperature ofthe molecular-dynamics simulations. We measure thetemperature of the simulations to be T=647 and 569 Kat 10.0 and 10.5 a.u. , respectively. These temperaturesare somewhat higher than the relevant experiments,which are done at 100 K (Ref. 28) and 300 K (Ref. 29),but we think they are not too high to be meaningful.

9.0 a.u.

EOG$

OQ)

CO

05

6$

CO0)

CO

O

CO

0)

9.5 a.u.

10.0 a.u.

10.5 a.u.

-1.0 0. 1.00 1 2 3 4 0 1 2 3 4 0 1 2 3 4

frequency (lOOO cm )

8 (a.u. )

FIG. 6. Born-Oppenheimer potential-energy surfaces of anantiferroelectric collective mode in the ice-X structure. Theupper and lower curves correspond to lattice constants of 10.0and 11.0 a.u. , respectively.

FIG. 7. Total densities of states of ice X at several latticeconstants (left), and the densities of states projected onto all0—H stretching modes (middle), and only ice-rule preservingstretching modes (right) of the protons. Vertical axis is scaledby a factor of 2.4 in middle and right panels.

4870 LEE, VANDERBILT, LAASONEN, CAR, AND PARRINELLO

Even in the classical case, the expectation value of theproton displacement from the bond midpoint (u ) doesnot become nonzero when the potential develops a doubleminimum. The critical temperature and the critical in-tercell coupling g are related as follows, again from sim-ple mean-field theory:

k~ T=y& u'&,

where

I u exp[ —V(u ) /kii T ]du&u'&= jexp [ —V( u ) /k' T ]du

and

V(u)= —,'au +Pu

Calculating (u ) numerically at the temperature of themolecular-dynamics simulations, we get the critical g.Comparing the critical g to the actual y, we find that it isin the symmetric regime at 10.0 a.u. (i.e., actual y issmaller than the critical g) and in the asymmetric regimeat 10.5 a.u. (i.e., actual y is larger than the critical y). So,the classical transition at finite temperature should be be-tween 10.0 and 10.5 a.u. It is difficult to predict the tran-sition lattice constant with confidence, but assuming thatthe actual and critical y can be smoothly interpolatedover the range of lattice constants between 10.0 and 10.5a.u. , we estimate that the transition should occur at about10.37 a.u. Therefore we think the finite temperature isresponsible for the discrepancy between static and dy-namic estimates of the location of the classical transition.

4. Hydrogen-bond symmetrization in ice I,

We have thus far studied the hydrogen-bond symmetri-zation in the high-pressure phases ice VII-VIII. Therehas been a prediction that the hydrogen bond in the low-pressure cubic ice phase should also become sym-metrized, but at a much lower pressure. Here, we carry32

over our previous analysis to the case of ice I„calculat-ing the equation of state and attempting to predict thecritical pressure for the transition into symmetricallyhydrogen-bonded cubic ice. We use the ice I, supercellhaving the most disordered proton configuration, withpolarization ( —,'00). The total energies of the unit cellare calculated with gradient-corrected LDA at latticeconstants from 10.0 to 12.0 a.u. in steps of 0.5 a.u. , allow-ing the atomic configurations to relax at each lattice con-stant. The calculations were also repeated for the sym-metric structure, in which the protons are fixed at the0—0 "bond" midpoints, and the 0 atoms lying on a dia-mond lattice. The data points and the fitted function areshown in Fig. 8. The equilibrium lattice constants for thetwo structures are found to be 11.58 a.u. for ice I, and11.01 a.u. for symmetrically hydrogen-bonded cubic ice,respectively. (However, the latter figure has limitedsignificance, as the symmetric phase is unstable to the I,phase except at zero pressure. ) The calculated equilibri-

um volume of 11.58 a.u. for ice I, is about 3% less thanthe experimental value of 12.0 a.u. adopted in Sec. IVBfor the phonon calculations. However, this smalldiscrepancy would most likely be reduced further if theeffect of the H quantum fluctuations on the equation ofstate were considered, as discussed above in Sec. IV C 1.

The hydrostatic pressure is calculated from the equa-tion of state in the same way as for ice X, and the samemean-field analysis of the collective and single-bond Hpotentials is carried out. We find that the transition be-tween ice I, and symmetrically hydrogen-bonded cubicice occurs at a lattice constant is 10.80 a.u. , which corre-sponds to 7 GPa using our calculated equation of state.This is well above the pressure of about 0.213 GPa atwhich ice II, transforms into other denser phases such asice II or ice III at —35 C, ' and ice I, is metastable withrespect to ice Ih at lower pressure. (The rather low tran-sition pressure out of ice I, and I& can easily be under-stood because these phases are characterized by a largeamount of empty space in the crystal structure. ) Thus, itis not surprising that symmetric hydrogen-bonded cubicice has not been observed experimentally.

V. CONCLUSION

We have presented results from a study of the structur-al and dynamical properties of various phases of ice car-ried out using the ab initio molecular-dynamics methodcombined with ultrasoft pseudopotentials. We havefound that the LDA must be augmented with gradientcorrections for a proper description of the hydrogenbonding. The phonon density of states of cubic ice hasbeen extracted from the molecular-dynamics simulations.We have calculated the equations of states for ice I„symmetrically hydrogen-bonded cubic ice, and the iceVII-VIII-X system. We have also calculated the criticalpressure of the transition into the symmetricallyhydrogen-bonded phases. The transition from ice VIII-

-13?.8

(5-13?.9

Q)0~ -~38.0—CL

CD~ -138.1Q)

-138.2

1000. 3 200. 1400. 1600. 1800.

Volume ( a.u. )

FIG. 8. Equations of state for ice I, and symmetricallyhydrogen-bonded cubic ice. The dots are calculated data andthe curves are the fitted quadratic functions.

Ab initio STUDIES ON THE STRUCTURAL AND. . . 4871

VII to ice X is predicted to occur at about 49 GPa whenthe eAects of the quantum Auctuations of the protons areincluded within mean-field theory. While the peaks inthe phonon density of states corresponding to most othermodes shift only moderately lower in frequency with in-creasing pressure, the modes corresponding to 0—Hstretch combinations which preserve the ice rules be-comes soft (co ~0) at the transition. The phase transi-tion between ice I, and syrnrnetric hydrogen-bonded cu-bic ice is predicted to occur at 7 GPa. The theoretical re-sults are in good agreement with experiments.

This work also helps to demonstrate the accuracy andefficiency of calculations using the recently introduced ul-trasoft pseudopotentials. Calculations on a rather com-plex system containing oxygen atoms were carried out atmodest computational cost. We hope this calculationalscheme will provide many future opportunities to carryout first-principles calculations of the structural and elec-tronic properties of materials that include first-row andtransition-metal elements. Other properties of ice, suchas the ionic and orientational defects in ice crystals, thehydration geometry around an anion (CI, etc. ) or a cat-ion (Ni +, Ca +, Mn +, etc. ), '" and the structural anddynamical properties of bulk water, are interestingproblems which are now open to treatment using similarmethods. Studies of transition-metal systems are alsocoming within reach, as evidenced by a recent calculationon liquid Cu by Pasquarello et al. Thus, the approachused here should be very useful in future studies of a wide

range of systems that have previously been difficult tostudy using plane-wave-based DFT methods.

ACKNOWLEDGMENTS

We thank Xiao-Ping Li, Dominic King-Smith, and Al-fredo Pasquarello for helpful discussions. This work wassupported by NSF Grant No. DMR-91-15342 (C.L. andD.V.) and by the Swiss NSF Grant No. 21-31144,91(K.L. and R.C.). Supercomputer time was provided bythe National Center for Supercomputing Applicationsunder Grant No. DMR920003N.

APPENDIX: DENSITY OF STATESFROM VELOCITY AUTOCORRELATIGN FUNCTION

We discuss how the Fourier transform of the velocityautocorrelation function is related to the phonon densityof states. The immediate output from the molecular-dynamics simulations is the time trace u(tI, ) withk =1, . . . , X. According to the definition of the densityof states, we are required to count frequencies which areequal to cu:

g(co)= +5(co—cu (k)) .jk

(A 1)

Using second quantized notation, the ath Cartesian com-ponent of the velocity of the wth atom in the lth unit cell,u I,(t), is given by

u I (t)=1/2

1g ej„(k)[—ice (k)]exp[ i o,r(k—)t+ik R(]jk ~J

1/2

[a.(k)+a (—k)], (A2)

where e, (k) is the polarization vector, JV is the number of atoms, M, is the mass of ath atom, and a (k), a (k) are theannihilation and creation operators. Using the commutation relations between the annihilation and creation operators,

[a (k), a,"(k')]=5 5(k —k'),

[a (k), a (k')] =[a (k), aj'(k')]=0,it follows that

(A3)

(A4)

gM (u I (t)u, „(0))=—alit.

(2m )+exp[ in) (k)t](A—co)(k)[n (kJ)+ —,']) .

(A5)

In the thermodynamic limit, the quantity in the bracket in the right-hand side is kz'r. (This does not necessarily meanthat the energy in each mode is equal in a finite run, as in Figs. 3 or 4.) By taking the Fourier transform of both sides ofEq. (A5), it follows that

al x.

(2vr )k~ T g 5(co—co, (k)),

jk(A6)

where the left-hand side is the density of states within a constant factor.


*Present address: Laboratory of Atomic and Solid State Phys-ics, Cornell University, Ithaca, New York 14853-2501.

~Present address: IBM Research Division, Zurich ResearchLaboratory, CH-8803 Riischlikon, Switzerland.

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Ab initio studies on the structural and dynamical properties of ice