Chemistry Reference
In-Depth Information
Figure 25.
Hydrogen-bond configurations of two unit cells, as viewed down the
b
-axis, used
in periodic electronic DFT calculations containing 28 and 112 waters, respectively. Hydrogens related
by symmetry and thus contributing to the same order parameter are similarly colored:
α
(yellow),
β
(green),
δ
(blue),
γ
(violet), and
(turquoise). The configuration shown here is the ground state at a
plane wave cutoff of 90 Ry or higher that corresponds to the experimentally determined proton ordered
ice XIII. (See insert for color representation of the figure.)
invariants. They differed in the number of second-order invariants, either 11 or
16. Both models predicted a transition to the ice XIII structure near 70 K [156],
although the details of the transition are different in the two models. In the follow-
ing, we focus on the results above 100 K, which are insensitive to the invariant
parametrization.
The space group symmetry of the ice V structure permits the use of five occu-
pation probabilities, or order parameters, that describe the probability of protons
being on one or the other side of a particular H-bond (Fig. 25). The order param-
eters have traditionally been designated by the first five Greek letters,
α
,
β
,
γ
,
δ
,
and
, and range from 0 to 1 in an obvious way where
1
2
indicates equal probability
of the H-bond pointing in two possible directions. As might be expected, the oc-
cupation probabilities are strongly related to some of the graph invariants. Due to
the symmetry of the ice V phase, the H-bonds described by the
parameter must
be fully disordered (i.e.,
1
=
2
).
In Fig. 26, occupation probabilities as a function of temperature are compared
with data obtained from the neutron diffraction studies of Lobban et al. [46].
From the simulations, we can calculate occupation probabilities for each model.
At temperatures above 100 K, the order parameters from the two graph invariant
schemes used to fit the energy are nearly superimposable. At high temperatures,
all order parameters take the value
1
2
, corresponding to a fully disordered struc-
ture. The
bonds, similar to experiment, remain fully disordered until the system
approaches the transition temperature below 100 K. The ground-state configura-
tion, fully ordered ice XIII, is described by the following occupation parameters:
δ
1
0. There are two symmetry-related configurations
corresponding to the ground state in which
=
1,
α
=
γ
=
2
, and
β
=
=
0or1.
