Chemistry Reference
In-Depth Information
(a)
(b)
ε/2
3
9
1
6
ε/2
4
Figure 13.
(a) An H-bond isomer of a 12-water primitive unit cell of ice Ih. Bonds representative
of the three second-order graph invariants used to fit the DFT energies are shown. All bonds used to
generate second-order invariants, used to describe energy differences for H-bond fluctuations in a
large simulation cell, lie perpendicular to the
c
-axis and are referred to as
ab
bonds. (b) An H-bond
isomer of a 48-water unit cell of ice Ih measuring 2
×
2
×
1 primitive cells on each side. Both H-bond
isomers shown are the lowest energy isomer for each unit cell in agreement with the experimentally
proposed ferroelectric, space group
Cmc
2
1
, ice XI structure. Arrows indicate direction of the relative
displacement,
/
2, of the
ab
layers that are oppositely polarized.
in Fig. 13 is assessed in Fig. 12b. There is only a slight deviation between the
prediction obtained from smaller cells and electronic DFT calculations performed
on the larger cell. As we determined later, the convergence at the small cell level
is even better than indicated in Fig. 12. The small deviation in Fig. 12b is due to
the fact that only the
point was used in the electronic DFT calculations on the
small cell. When
k
-point sampling is employed for the small cells, the predictions
for the larger cells are improved [120]. By refitting the invariant coefficients to the
large cell energies (Fig. 12c), we are able to, with Eq. (11), calculate the energy
differences arising from the various H-bond configurations in Metropolis Monte
Carlo simulations of a large ice Ih system.
Using the fit to the H-bond energetics of Fig. 12c, Monte Carlo simulations of
ice Ih were performed using an orthorhombic cell measuring 7
4 primitive
cells on each side containing 896 water molecules. Average energy as a function of
temperature (Fig. 14a) indicate that a first-order transition to the low-temperature
proton-ordered structure occurs near 98 K. The structure of the low-temperature
phase is that of the experimentally proposed ferroelectric
Cmc
2
1
structure, shown
in Fig. 2a. Entropy as a function of temperature, shown in Fig. 14b, indicates that
as ice Ih is cooled, the system loses 11% of its configurational entropy before the
transition, in agreement with pretransitional effects seen calorimetrically [8] and
in diffraction studies [14]. Only 1% of the configurational entropy for an ideal ice
phase is lost below the transition resulting in 88% of the ideal entropy lost at the
transition.
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4
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