Chemistry Reference
In-Depth Information
Bond variables
b
jα
(
j
a, b, c, d
) in Eq. (31) are labelled in Fig. 11.
The
b
jα
with the same value of
α
refer to bonds within the same 2
=
1
−
4,
α
=
×
2 subunits
of the 4
×
4 cell. Each of the first four lines of Eq. (31) are clearly recognizable
as
I
2
×
2
13
4 unit cell. Each of
the terms represent the product of bond variables for bonds that are parallel and
separated by one lattice unit in either the
x
or
y
direction, an interaction that could
be estimated by a calculation for the smaller 2
of Eq. (4) evaluated for each 2
×
2 sector of the 4
×
2 cell. Terms like
b
1
a
b
3
c
may
seem to violate this condition, since bonds 1
a
and 3
c
lie three lattice units from
each other in the
y
direction. However, the term
b
1
a
b
3
c
actually represents the
interaction of the bond 1
a
with another bond below it that lies in a neighboring
unit cell. Because of lattice periodicity, that bond has the same
value
as its periodic
image bond 3
c
. Hence, in the term
b
1
a
b
3
c
, the variable
b
3
c
represents the
value
of another bond that is its periodic image in the lattice. This example illustrates
the distinction, made immediately after Eq. (27), between bond variables and their
value. In expressions like Eq. (31), it is most convenient to replace actual bond
variables, which might be bond variables outside a primary unit cell, with other
variables within the primary cell that have the same value. Returning to Eq. (31),
we could have just as well said that the term
b
1
a
b
3
c
represents the interaction of a
bond 3
c
with another bond one lattice unit above it whose value is the same as its
periodic image, bond 1
a
.
Expression (31) is an illustration of the general formula, Eqs. (30). The terms
in the last four lines would be identical in value to those of the first four lines if
the lattice still had 2
×
2 periodicity. Put another way, if the letters were removed
from the subscripts in the last four lines, thereby enforcing 2
×
2 periodicity, the
last four lines would duplicate the first four lines. These terms are indeed part of
I
2
×
2
×
13
, but they do not appear explicitly in Eq. (4) because their
value
is identical to
terms already present in that expression. In the 4
4 setting, these terms must be
included as distinct contributions. Provided the additional invariants introduced at
the 4
×
4 level do not make significant contribution, the contribution of an invariant
like
I
4
×
4
×
1
a,
3
a
to a scalar physical property like the energy could be estimated from
ab initio
calculations for the 2
×
2 unit cell.
As discussed in Section II.C.1, invariants like
I
4
×
4
1
a,
3
a
of Eq. (31) has the physical
interpretation of counting the number of cis and trans H-bonds of square ice.
Therefore, if Bjerrum's conjecture was correct and the energetic difference between
a cis and trans H-bond was established for a 2
×
2 unit cell and the parameter
α
13
of Eq. (5) established, then for the 4
×
4 unit cell the energy would be given by
α
13
I
4
×
4
E
≈
E
0
+
(32)
13
where
α
13
is the same number as in Eq. (5) and has been established by detailed
calculations on the smaller unit cell.
Of course, an expression like Eq. (32) would
only be appropriate if Bjerrum's conjecture about cis and trans H-bonds was valid.
Therefore, an expression using additional invariants, like Eq. (10) for the 2
×
2
