Betül Pamuk
Phil Allen
Marivi Fernández-Serra
Stony Brook University
Computational Condensed Matter Group
DOE Early Career grant DE-SC0003871
March Meeting 2011
Outline
 Ice Ih structure
 Zero Point Phonon Energy & Grüneisen parameter
 Motivation: Normal Isotope Shift  Zero Point
Expansion
 Experiments: Anomaly in Ice Ih  Zero Point
Contraction
 Results
 Lattice parameters
 Bulk modulus
 Dispersion curves
 Preliminary Results with Ice Ic
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Ice Ih Structure

a

c

a
Ice = O is tetrahedrally coordinated making two covalent
bonds with 2 H’s of the molecule and two H-bonds with the
nearest molecule.
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Phonon Contribution to Zero-Point
The volume dependence of energy at zero point. [1]
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Grüneisen Parameter
 The zero-point correction to the lattice parameter:
Grüneisen parameter [2]
Bulk modulus
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20Ne
22Ne
Frozen crystal
lattice constant
a0 = 4.255 Å [2]
Bolz and Mauer, 1962
Batchelder, Losee, and Simmons, 1968
Phil Allen, 1994
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Unit Cell Volume Experiments of Ice Ih
Zero Point
Contraction
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D2O
H2O
Röttger
et al.,
1994.
[3]
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Ice Ih d(O---O) = 2.75 Å
Slope: Γ negative!!!
E. Libowitzky,
1999
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[4]
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Tip4P-flexible
Tip4P-flexible does NOT show zero
point volume contraction
C. P. Herrero, R. Ramirez (2011) [5]
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rOH (Ang)
B. Pamuk et. al.
work in progress
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Electronic Energy vs. Unit Cell Volume
Experiment
10 K
DFT: SIESTA
GGA: PBE
revPBE
+ vdW
Predict water
density nicely
(Jue et. al, 2011)[6]
PBE
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PBE-vdW
revPBE
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revPBE-vdW
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Asymmetric
stretching
4 molecules/cell
Symmetric
stretching
Bending
Libration
Stretching of H-bonds
Frequency less than 0
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Ice Ih  Ice XI
 Non pure ice Ih : (KOH, defects needed) (Y. Tajima, T.
Matsuo, and H. Suga, 1982)
 Phase transition at ~ 73 K
 Angle on x-y plane different than 60o
 Vector lengths on x-y plane a≠b
 a = 4.5019 Å, b = 7.7978 Å, and c = 7.3280 Å
(A. J. Leadbetter, R. C. Ward, J. W. Clark, P. A. Tucker,
T. Matsuo and S. Suga, 1985)
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Contribution to the Shift
The sign of
the slope will
determine the
sign of Γ.
Work in
progress…
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Preliminary Results with Ice Ic
PBE
High
frequency
modes have
negative Γ!!!
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Low
frequency
modes are
different with
different
fcn’als
High
frequency
modes have
similar
behaviour
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Comparing Functionals
PBE predicts ~ 10% zero
point contraction on
H2O and 3% reverse
isotope effect on the
volume.
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PBE-vdW ~ 7% zero point
contraction on H2O and 2.5%
reverse isotope effect on the
volume. The experimental
reverse isotope effect is 0.009%.
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Conclusion
 Take into account the zero point phonon effects for
lattice parameters.
 Isotope shift in ice
 Understand the most stable structure
 Calculate volume dependency of frequencies
precisely
 Understand how different functionals explain
ice
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References
 [1] P. B. Allen, Optical and Vibrational Spectroscopies
Symposium Talk, 2010
 [2] P. B. Allen, Phil. Mag. B 70, 527 (1994)
 [3] K. Röttger et al. Acta Cryst. B50, 644-648 (1994).
 [4] E. Libowitzky, Monatshefte für Chemie (1999)
 [5] C. P. Herrero, R. Ramirez, J. Chem. Phys. 134,
094510 (2011)
 [6] J. Wang, G. Román-Pérez, J. M. Soler, E. Artacho,
and M.-V. Fernández-Serra, J. Chem. Phys. 134, 024516
(2011)
Thank You
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12 molecules/cell
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Phase Diagram
http://www.thefullwiki.org/Ice_harvesting
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Lattice Parameters
and Unit Cell Volume
Experiments of ice Ih
Röttger et al., 1994.
Lattice parameters a,
and c as a function of
temperature for ice Ih
between 10 K and 265 K.
[4]
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D2O
H 2O
Röttger et al., 1994. Unit-cell volume of ice Ih between
10 and 265 K. [4]
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Lattice Parameter
PBE
revPBE
PBE-vdW
revPBE-vdW
Measurement
(10 K) [4]
a (Å)
4.39
4.52
4.44
4.56
4.49693
c (Å)
7.17
7.39
7.24
7.43
7.32109
c/a
1.633
1.635
1.631
1.629
1.628
B0 (GPa)
17.614
12.867
15.957
14.711
12.1
I. Hamada, 2010.
Results of binding
energy, lattice
parameters, volume
per molecule and
bulk moduli;
comparing different
functionals. [6]
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