Water ice is a unique material presenting intriguing physical properties, such as negative thermal expansion and anomalous volume isotope effect (VIE). They arise from the interplay between weak hydrogen bonds and nuclear quantum fluctuations, making theoretical calculations challenging. Here, we employ the stochastic self-consistent harmonic approximation to investigate how thermal and quantum fluctuations affect the physical properties of ice XI with ab initio accuracy. Regarding the anomalous VIE, our work reveals that quantum effects on hydrogen are so strong to be in a nonlinear regime: When progressively increasing the mass of hydrogen from protium to infinity (classical limit), the volume first expands and then contracts, with a maximum slightly above the mass of tritium. We observe an anharmonic renormalization of about 10% in the bending and stretching phonon frequencies probed in IR and Raman experiments. For the first time, we report an accurate comparison of the low-energy phonon dispersion with the experimental data, possible only thanks to high-level accuracy in the electronic correlation and nuclear quantum and thermal fluctuations, paving the way for the study of thermal transport in ice from first-principles and the simulation of ice under pressure.
REFERENCES
1.
T.
Bartels-Rausch
, V.
Bergeron
, J. H. E.
Cartwright
, R.
Escribano
, J. L.
Finney
, H.
Grothe
, P. J.
Gutiérrez
, J.
Haapala
, W. F.
Kuhs
, J. B. C.
Pettersson
, S. D.
Price
, C. I.
Sainz-Díaz
, D. J.
Stokes
, G.
Strazzulla
, E. S.
Thomson
, H.
Trinks
, and N.
Uras-Aytemiz
, “Ice structures, patterns, and processes: A view across the icefields
,” Rev. Mod. Phys.
84
, 885
–944
(2012
).2.
A. D.
Fortes
, I. G.
Wood
, D.
Grigoriev
, M.
Alfredsson
, S.
Kipfstuhl
, K. S.
Knight
, and R. I.
Smith
, “No evidence for large-scale proton ordering in Antarctic ice from powder neutron diffraction
,” J. Chem. Phys.
120
, 11376
–11379
(2004
).3.
A. J.
Leadbetter
, R. C.
Ward
, J. W.
Clark
, P. A.
Tucker
, T.
Matsuo
, and H.
Suga
, “The equilibrium low-temperature structure of ice
,” J. Chem. Phys.
82
, 424
–428
(1985
).4.
C.
Lobban
, J. L.
Finney
, and W. F.
Kuhs
, “The structure of a new phase of ice
,” Nature
391
, 268
–270
(1998
).5.
B.
Kamb
, “Ice. II. A proton-ordered form of ice
,” Acta Crystallogr.
17
, 1437
–1449
(1964
).6.
B.
Kamb
, A.
Prakash
, and C.
Knobler
, “Structure of ice. V
,” Acta Crystallogr.
22
, 706
–715
(1967
).7.
B.
Kamb
and A.
Prakash
, “Structure of ice III
,” Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem.
24
, 1317
–1327
(1968
).8.
B.
Kamb
and B. L.
Davis
, “Ice VII, the densest form of ice
,” Proc. Natl. Acad. Sci. U. S. A.
52
, 1433
–1439
(1964
).9.
H.
Engelhardt
and B.
Kamb
, “Structure of ice IV, a metastable high-pressure phase
,” J. Chem. Phys.
75
, 5887
–5899
(1981
).10.
L. G.
Dowell
and A. P.
Rinfret
, “Low-temperature forms of ice as studied by x-ray diffraction
,” Nature
188
, 1144
–1148
(1960
).11.
J. D.
Bernal
and R. H.
Fowler
, “A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions
,” J. Chem. Phys.
1
, 515
–548
(1933
).12.
Y.
Tajima
, T.
Matsuo
, and H.
Suga
, “Phase transition in KOH-doped hexagonal ice
,” Nature
299
, 810
–812
(1982
).13.
Y.
Tajima
, T.
Matsuo
, and H.
Suga
, “Calorimetric study of phase transition in hexagonal ice doped with alkali hydroxides
,” J. Phys. Chem. Solids
45
, 1135
–1144
(1984
).14.
S.
Kwada
, “Acceleration of dielectric relaxation by KOH-doping and phase transition in ice Ih
,” J. Phys. Chem. Solids
50
, 1177
–1184
(1989
).15.
M.
Marchi
, J. S.
Tse
, and M. L.
Klein
, “Lattice vibrations and infrared absorption of ice Ih
,” J. Chem. Phys.
85
, 2414
–2418
(1986
).16.
J. S.
Tse
, M. L.
Klein
, and I. R.
McDonald
, “Lattice vibrations of ices Ih, VIII, and IX
,” J. Chem. Phys.
81
, 6124
–6129
(1984
).17.
P.
Bosi
, R.
Tubino
, and G.
Zerbi
, “On the problem of the vibrational spectrum and structure of ice Ih: Lattice dynamical calculations
,” J. Chem. Phys.
59
, 4578
–4586
(1973
).18.
K.
Röttger
, A.
Endriss
, J.
Ihringer
, S.
Doyle
, and W. F.
Kuhs
, “Lattice constants and thermal expansion of H2O and D2O ice Ih between 10 and 265 K
,” Acta Crystallogr., Sect. B: Struct. Sci.
50
, 644
–648
(1994
).19.
A. D.
Fortes
, “Accurate and precise lattice parameters of H2O and D2O ice Ih between 1.6 and 270 K from high-resolution time-of-flight neutron powder diffraction data
,” Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater.
74
, 196
–216
(2018
).20.
H.
Tanaka
, “Thermodynamic stability and negative thermal expansion of hexagonal and cubic ices
,” J. Chem. Phys.
108
, 4887
–4893
(1998
).21.
K.
Umemoto
, E.
Sugimura
, S.
de Gironcoli
, Y.
Nakajima
, K.
Hirose
, Y.
Ohishi
, and R. M.
Wentzcovitch
, “Nature of the volume isotope effect in ice
,” Phys. Rev. Lett.
115
, 173005
(2015
).22.
B.
Pamuk
, J. M.
Soler
, R.
Ramírez
, C. P.
Herrero
, P. W.
Stephens
, P. B.
Allen
, and M. V.
Fernández-Serra
, “Anomalous nuclear quantum effects in ice
,” Phys. Rev. Lett.
108
, 193003
(2012
).23.
M. A.
Salim
, S. Y.
Willow
, and S.
Hirata
, “Ice Ih anomalies: Thermal contraction, anomalous volume isotope effect, and pressure-induced amorphization
,” J. Chem. Phys.
144
, 204503
(2016
).24.
R.
Ramírez
, N.
Neuerburg
, M.-V.
Fernández-Serra
, and C. P.
Herrero
, “Quasi-harmonic approximation of thermodynamic properties of ice Ih, II, and III
,” J. Chem. Phys.
137
, 044502
(2012
).25.
C. P.
Herrero
and R.
Ramírez
, “Isotope effects in ice Ih: A path-integral simulation
,” J. Chem. Phys.
134
, 094510
(2011
).26.
M.
Fernandez-Serra
, B.
Pamuk
, and P. B.
Allen
, “Nuclear quantum effects in different ice phases
,” in APS March Meeting Abstracts, APS Meeting Abstracts
(APS
, 2016
), Vol. 2016, p. K47.006
.27.
J.
Wang
, G.
Román-Pérez
, J. M.
Soler
, E.
Artacho
, and M.-V.
Fernández-Serra
, “Density, structure, and dynamics of water: The effect of van der Waals interactions
,” J. Chem. Phys.
134
, 024516
(2011
).28.
B.
Pamuk
, P. B.
Allen
, and M.-V.
Fernández-Serra
, “Electronic and nuclear quantum effects on the ice XI/ice Ih phase transition
,” Phys. Rev. B
92
, 134105
(2015
).29.
M.
Fritz
, J. M.
Soler
, and M.
Fernandez-Serra
, “A general optimization method applied to a vdW-DF functional for water
,” in APS March Meeting Abstracts, APS Meeting Abstracts
(APS
, 2016
), Vol. 2016, p. C20.006
.30.
M.
Fernandez-Serra
, “van der Waals interactions in water and ice from density functional theory simulations: Improvements and challenges
,” in APS March Meeting Abstracts, APS Meeting Abstracts
(APS
, 2012
), Vol. 2012, p. H6.001
.31.
F.
Li
and J. L.
Skinner
, “Infrared and Raman line shapes for ice Ih. I. Dilute HOD in H2O and D2O
,” J. Chem. Phys.
132
, 204505
(2010
).32.
F.
Li
and J. L.
Skinner
, “Infrared and Raman line shapes for ice Ih. II. H2O and D2O
,” J. Chem. Phys.
133
, 244504
(2010
).33.
M. J.
Wójcik
, K.
Szczeponek
, and S.
Ikeda
, “Theoretical study of the OH/OD stretching regions of the vibrational spectra of ice Ih
,” J. Chem. Phys.
117
, 9850
–9857
(2002
).34.
M. S.
Bergren
and S. A.
Rice
, “An improved analysis of the OH stretching region of the vibrational spectrum of ice Ih
,” J. Chem. Phys.
77
, 583
–602
(1982
).35.
S. A.
Rice
, M. S.
Bergren
, A. C.
Belch
, and G.
Nielsen
, “A theoretical analysis of the hydroxyl stretching spectra of ice Ih, liquid water, and amorphous solid water
,” J. Phys. Chem.
87
, 4295
–4308
(1983
).36.
J. E.
Bertie
and E.
Whalley
, “Infrared spectra of ices Ih and Ic in the range 4000 to 350 cm−1
,” J. Chem. Phys.
40
, 1637
–1645
(1964
).37.
J. R.
Scherer
and R. G.
Snyder
, “Raman intensities of single crystal ice Ih
,” J. Chem. Phys.
67
, 4794
–4811
(1977
).38.
M. L.
Clapp
, D. R.
Worsnop
, and R. E.
Miller
, “Frequency-dependent optical constants of water ice obtained directly from aerosol extinction spectra
,” J. Phys. Chem.
99
, 6317
–6326
(1995
).39.
A. Y.
Zasetsky
, A. F.
Khalizov
, M. E.
Earle
, and J. J.
Sloan
, “Frequency dependent complex refractive indices of supercooled liquid water and ice determined from aerosol extinction spectra
,” J. Phys. Chem. A
109
, 2760
–2764
(2005
).40.
K.
Abe
, K.
Ishii
, M.
Nakajima
, H.
Fukuda
, and T.
Shigenari
, “Raman scattering at the proton ordering phase transition in ice crystal
,” Ferroelectrics
239
, 1
–8
(2000
).41.
K.
Abe
and T.
Shigenari
, “Raman spectra of proton ordered phase XI of ICE I. Translational vibrations below 350 cm−1
,” J. Chem. Phys.
134
, 104506
(2011
).42.
T.
Shigenari
and K.
Abe
, “Vibrational modes of hydrogens in the proton ordered phase XI of ice: Raman spectra above 400 cm−1
,” J. Chem. Phys.
136
, 174504
(2012
).43.
J.-C.
Li
, V. M.
Nield
, and S. M.
Jackson
, “Spectroscopic measurements of ice XI
,” Chem. Phys. Lett.
241
, 290
–294
(1995
).44.
M.
Arakawa
, H.
Kagi
, and H.
Fukazawa
, “Laboratory measurements of infrared absorption spectra of hydrogen-ordered ice: A step to the exploration of ice XI in space
,” Astrophys. J., Suppl. Ser.
184
, 361
–365
(2009
).45.
H.
Fukazawa
, S.
Ikeda
, and S.
Mae
, “Incoherent inelastic neutron scattering measurements on ice XI; the proton-ordered phase of ice Ih doped with KOH
,” Chem. Phys. Lett.
282
, 215
–218
(1998
).46.
H.
Itoh
, K.
Kawamura
, T.
Hondoh
, and S.
Mae
, “Molecular dynamics studies of proton ordering effects on lattice vibrations in ice Ih
,” Physica B
219–220
, 469
–472
(1996
).47.
H.
Itoh
, K.
Kawamura
, T.
Hondoh
, and S.
Mae
, “Polarized librational spectra of proton-ordered ice XI by molecular dynamics simulations
,” J. Chem. Phys.
109
, 4894
–4899
(1998
).48.
A.
Erba
, S.
Casassa
, R.
Dovesi
, L.
Maschio
, and C.
Pisani
, “Periodic density functional theory and local-MP2 study of the librational modes of ice XI
,” J. Chem. Phys.
130
, 074505
(2009
).49.
M.
Gług
, M.
Boczar
, Ł.
Boda
, and M. J.
Wójcik
, “Analysis of librational modes of ice XI studied by Car–Parrinello molecular dynamics
,” Chem. Phys.
459
, 102
–111
(2015
).50.
R.
Iftimie
and M. E.
Tuckerman
, “Decomposing total IR spectra of aqueous systems into solute and solvent contributions: A computational approach using maximally localized Wannier orbitals
,” J. Chem. Phys.
122
, 214508
(2005
).51.
W.
Chen
, M.
Sharma
, R.
Resta
, G.
Galli
, and R.
Car
, “Role of dipolar correlations in the infrared spectra of water and ice
,” Phys. Rev. B
77
, 245114
(2008
).52.
S.
Klotz
, T.
Strässle
, A. M.
Saitta
, G.
Rousse
, G.
Hamel
, R. J.
Nelmes
, J. S.
Loveday
, and M.
Guthrie
, “In situ neutron diffraction studies of high density amorphous ice under pressure
,” J. Phys.: Condens. Matter
17
, S967
–S974
(2005
).53.
S.
Klotz
, G.
Hamel
, J. S.
Loveday
, R. J.
Nelmes
, M.
Guthrie
, and A. K.
Soper
, “Structure of high-density amorphous ice under pressure
,” Phys. Rev. Lett.
89
, 285502
(2002
).54.
S.
Klotz
, T.
Strässle
, R. J.
Nelmes
, J. S.
Loveday
, G.
Hamel
, G.
Rousse
, B.
Canny
, J. C.
Chervin
, and A. M.
Saitta
, “Nature of the polyamorphic transition in ice under pressure
,” Phys. Rev. Lett.
94
, 025506
(2005
).55.
T.
Strässle
, A. M.
Saitta
, S.
Klotz
, and M.
Braden
, “Phonon dispersion of ice under pressure
,” Phys. Rev. Lett.
93
, 225901
(2004
).56.
T.
Strässle
, S.
Klotz
, G.
Hamel
, M. M.
Koza
, and H.
Schober
, “Experimental evidence for a crossover between two distinct mechanisms of amorphization in ice Ih under pressure
,” Phys. Rev. Lett.
99
, 175501
(2007
).57.
J. M.
Besson
, S.
Klotz
, G.
Hamel
, W. G.
Marshall
, R. J.
Nelmes
, and J. S.
Loveday
, “Structural instability in ice VIII under pressure
,” Phys. Rev. Lett.
78
, 3141
–3144
(1997
).58.
R. J.
Nelmes
, J. S.
Loveday
, T.
Strässle
, C. L.
Bull
, M.
Guthrie
, G.
Hamel
, and S.
Klotz
, “Annealed high-density amorphous ice under pressure
,” Nat. Phys.
2
, 414
–418
(2006
).59.
I.
Errea
, M.
Calandra
, and F.
Mauri
, “Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides
,” Phys. Rev. B
89
, 064302
(2014
).60.
R.
Bianco
, I.
Errea
, L.
Paulatto
, M.
Calandra
, and F.
Mauri
, “Second-order structural phase transitions, free energy curvature, and temperature-dependent anharmonic phonons in the self-consistent harmonic approximation: Theory and stochastic implementation
,” Phys. Rev. B
96
, 014111
(2017
).61.
L.
Monacelli
, I.
Errea
, M.
Calandra
, and F.
Mauri
, “Pressure and stress tensor of complex anharmonic crystals within the stochastic self-consistent harmonic approximation
,” Phys. Rev. B
98
, 024106
(2018
).62.
L.
Monacelli
and F.
Mauri
, “Time-dependent self-consistent harmonic approximation: Anharmonic nuclear quantum dynamics and time correlation functions
,” Phys. Rev. B
103
, 104305
(2021
).63.
M.
Born
and R.
Oppenheimer
, “Zur quantentheorie der molekeln
,” Ann. Phys.
389
, 457
–484
(1927
).64.
B.
Cheng
, E. A.
Engel
, J.
Behler
, C.
Dellago
, and M.
Ceriotti
, “Ab initio thermodynamics of liquid and solid water
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 1110
–1115
(2019
).65.
Y.
Zhang
and W.
Yang
, “Comment on ‘Generalized gradient approximation made simple
,’” Phys. Rev. Lett.
80
, 890
(1998
).66.
C.
Adamo
and V.
Barone
, “Toward reliable density functional methods without adjustable parameters: The PBE0 model
,” J. Chem. Phys.
110
, 6158
–6170
(1999
).67.
L.
Goerigk
and S.
Grimme
, “A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions
,” Phys. Chem. Chem. Phys.
13
, 6670
(2011
).68.
S.
Grimme
, J.
Antony
, S.
Ehrlich
, and H.
Krieg
, “A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
,” J. Chem. Phys.
132
, 154104
(2010
).69.
V.
Kapil
, E.
Engel
, M.
Rossi
, and M.
Ceriotti
, “Assessment of approximate methods for anharmonic free energies
,” J. Chem. Theory Comput.
15
, 5845
–5857
(2019
).70.
J.
Matas
, J.
Bass
, Y.
Ricard
, E.
Mattern
, and M. S. T.
Bukowinski
, “On the bulk composition of the lower mantle: Predictions and limitations from generalized inversion of radial seismic profiles
,” Geophys. J. Int.
170
, 764
–780
(2007
).71.
J. C. E.
Irving
, S.
Cottaar
, and V.
Lekić
, “Seismically determined elastic parameters for Earth’s outer core
,” Sci. Adv.
4
, eaar2538
(2018
).72.
E. A.
Engel
, B.
Monserrat
, and R. J.
Needs
, “Anharmonic nuclear motion and the relative stability of hexagonal and cubic ice
,” Phys. Rev. X
5
, 021033
(2015
).73.
R.
Bianco
, I.
Errea
, M.
Calandra
, and F.
Mauri
, “High-pressure phase diagram of hydrogen and deuterium sulfides from first principles: Structural and vibrational properties including quantum and anharmonic effects
,” Phys. Rev. B
97
, 214101
(2018
).74.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
–3868
(1996
).75.
S.
Yoo
, X. C.
Zeng
, and S. S.
Xantheas
, “On the phase diagram of water with density functional theory potentials: The melting temperature of ice Ih with the Perdew–Burke–Ernzerhof and Becke–Lee–Yang–Parr functionals
,” J. Chem. Phys.
130
, 221102
(2009
).76.
A. J.
Rusnak
, E. R.
Pinnick
, C. E.
Calderon
, and F.
Wang
, “Static dielectric constants and molecular dipole distributions of liquid water and ice-Ih investigated by the PAW-PBE exchange-correlation functional
,” J. Chem. Phys.
137
, 034510
(2012
).77.
L.
Monacelli
, I.
Errea
, M.
Calandra
, and F.
Mauri
, “Black metal hydrogen above 360 GPa driven by proton quantum fluctuations
,” Nat. Phys.
17
, 63
–67
(2020
).78.
P.
Vinet
, J. R.
Smith
, J.
Ferrante
, and J. H.
Rose
, “Temperature effects on the universal equation of state of solids
,” Phys. Rev. B
35
, 1945
–1953
(1987
).79.
P.
Giannozzi
, S.
Baroni
, N.
Bonini
, M.
Calandra
, R.
Car
, C.
Cavazzoni
, D.
Ceresoli
, G. L.
Chiarotti
, M.
Cococcioni
, I.
Dabo
, A.
Dal Corso
, S.
de Gironcoli
, S.
Fabris
, G.
Fratesi
, R.
Gebauer
, U.
Gerstmann
, C.
Gougoussis
, A.
Kokalj
, M.
Lazzeri
, L.
Martin-Samos
, N.
Marzari
, F.
Mauri
, R.
Mazzarello
, S.
Paolini
, A.
Pasquarello
, L.
Paulatto
, C.
Sbraccia
, S.
Scandolo
, G.
Sclauzero
, A. P.
Seitsonen
, A.
Smogunov
, P.
Umari
, and R. M.
Wentzcovitch
, “QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials
,” J. Phys.: Condens. Matter
21
, 395502
(2009
).80.
I.
Errea
, M.
Calandra
, C. J.
Pickard
, J. R.
Nelson
, R. J.
Needs
, Y.
Li
, H.
Liu
, Y.
Zhang
, Y.
Ma
, and F.
Mauri
, “Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system
,” Nature
532
, 81
–84
(2016
).81.
L.
Paulatto
, I.
Errea
, M.
Calandra
, and F.
Mauri
, “First-principles calculations of phonon frequencies, lifetimes, and spectral functions from weak to strong anharmonicity: The example of palladium hydrides
,” Phys. Rev. B
91
, 054304
(2015
).82.
L.
Monacelli
, R.
Bianco
, M.
Cherubini
, M.
Calandra
, I.
Errea
, and F.
Mauri
, “The stochastic self-consistent harmonic approximation: Calculating vibrational properties of materials with full quantum and anharmonic effects
,” J. Phys. Condens. Matter
33
, 363001
(2021
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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