Chemistry Reference
In-Depth Information
on smaller unit cells, or a statistical simulation on a larger “simulation cell”. To
obtain explicit expressions required to generate invariants for enlarged unit cells,
we transform expressions like Eqs. (22)-(24) into a form that explicitly incorpo-
rates the choice of a nonprimitive unit cell. The graph invariants of Eqs. (4)-(9),
which arose in the simple example of square ice, were generated using projection
operators designed for nonprimitive unit cells.
For an
n
x
×
n
z
unit cell with basis
n
x
a
x
,n
y
a
y
,n
z
a
z
, the translation
group, denoted as
n
x
×
n
y
×
n
z
, can be written as:
n
y
×
τ
n
x
u
τ
n
y
v
τ
n
z
w
0
,
1
,...,
N
x
n
x
×
n
y
×
n
z
=
|
u
=
n
x
−
1
1
(25)
0
,
1
,...,
N
y
0
,
1
,...,
N
z
v
=
n
y
−
1
,w
=
n
z
−
(Recall that we label our cell vectors with the familiar Cartesian directions “
x, y, z
”
even though our expressions apply equally well to nonorthogonal systems.) It is
elementary to see that
is equivalent to
1
×
1
×
1
and
n
x
×
n
y
×
n
z
⊂
1
×
1
×
1
=
.
For any
n
x
×
n
y
×
n
z
,wehave
#(
n
x
×
n
y
×
n
z
)
=
#(
)
/n
x
n
y
n
z
=
N
x
N
y
N
z
/n
x
n
y
n
z
(26)
n
z
unit
cell, the value of a bond variable at a position translated by one of the members of
n
x
×
n
y
×
n
z
is equal to the bond variable at the original position:
For graphs satisfying the periodic boundary condition of an
n
x
×
n
y
×
∀
τ
i
∈
n
x
×
n
y
×
n
z
value of
τ
i
b
r
=
value of
b
r
(27)
It is crucial to realize that bond
τ
i
b
r
is distinct from bond
b
r
, even though their
values
are the same on account of periodic boundary conditions. Equation (27)
provides fewer constraints on the H-bonds than for the smaller 1
1 unit
cell. As the periodic cell is enlarged, a greater variety of H-bonding patterns is
permitted until, as the cell size approaches the thermodynamic limit, it is capable
of describing all manner of disorder in ice.
The full space group can be decomposed into cosets of the translation subgroup
n
x
×
n
y
×
n
z
. While the pure translation group for the crystal with
n
x
×
×
1
×
n
y
×
n
z
unit cell is smaller than for the 1
1 cell, the set of coset representatives is
correspondingly enlarged by a factor of
n
x
n
y
n
z
. The set of coset representatives
p
n
x
×
n
y
×
n
z
β
×
1
×
for the larger cell is given by
τ
x
τ
y
τ
z
p
β
|
=
0
,
1
,...,n
z
−
1
(28)
β
∈
G/ , u
=
0
,
1
,...,n
x
−
1
,v
=
0
,
1
,...,n
y
−
1
,w
