Chemistry Reference
In-Depth Information
potentials with sufficient accuracy to describe the energetics of H-bond rearrange-
ments in ice, it is what would be needed to, predict proton-ordering phase transi-
tions in ice. A link between H-bond topology and the energy, if such a correlation
did exist, would provide an inexpensive route to the energies needed to predict
H-bond order-disorder phenomena in ice. Even if a suitable empirical potential
was available (and we are optimistic that one will be developed), a description in
terms of H-bond topology would provide a concise language for analyzing and
understanding H-bond fluctuations in ice. In this section, we review a method to
link physical quantities with H-bond topology. The cases considered in Sections III
and V will provide evidence that physical quantities can be described conveniently
and concisely in terms of the H-bond topology.
The key idea is that since energy is a scalar, invariant to symmetry operations, it
must depend on combinations of H-bonds that are likewise invariant to symmetry
operations. Hence, we write the energy in terms of such combinations, which we
have called
graph invariants
. Alternatively, local site displacements transform as
vectors. Hence, the site displacements that arise from local variations of H-bond
topology must depend on combinations of H-bond variables that transform as a
first-rank tensor. While there is an exceedingly large number of H-bond configura-
tions available in even a small simulation cell, the number of invariant combinations
of H-bonds needed to describe the energy or the number of vectorial combinations
of H-bonds needed to describe site displacements will turn out to be quite small.
Furthermore, they can be parametrized by first-principles calculations (Section
II.B.2) on small unit cells.
The idea of linking the H-bond topology to energetics in ice has been around
since the work of Bjerrum [98] and Pitzer and Polissar [99], who postulated that
trans H-bonds, those where the nonbonded hydrogens lie on opposite sides of the
bond, are lower energy than cis H-bonds [100]. (See Fig. 2 for an illustration of cis
and trans H-bonds.) However, such past attempts were rather
ad hoc
in nature and
not sufficiently general to provide accurate predictions. Consider how the scheme
would work for the energy if one accepted Bjerrum's conjecture about cis and trans
H-bonds: The energy difference between a cis and a trans H-bond in ice-Ih could
be established by
ab initio
calculations on small unit cells, for which this type of
detailed calculation is feasible. Such
ab initio
calculations are not possible for the
many H-bond arrangements in large simulation cells, but it is certainly possible
to either enumerate all the H-bond topologies (Section IV) or perform a Monte
Carlo simulation for large cells and assign an energy to each configuration based
on the number of cis and trans H-bonds. The energies
E
i
needed in Eq. (3) for the
billions of topologies possible for large unit cells would be given, relative to an
all-trans configuration, by counting the number of cis bonds in each topology and
multiplying by the cis-trans energy difference. This illustrates the two ingredients
needed in our scheme. First, parameters are derived from detailed calculations (like
ab initio
) on small unit cells. Second, it also illustrates that an invariant for small
