Chemistry Reference
In-Depth Information
2.2.1 Pitfall in Transforming Trio Strength Between Lattice-Gas
and Ising Models
Many researchers recast the lattice-gas model into an Ising model before doing
computations. This seemingly innocent procedure invites pitfalls when multisite
interactions are involved. Here we offer a particular example: what seems like a
relatively modest three-spin interaction can correspond to an unphysically large trio
interaction in the lattice-gas Hamiltonian. Our example involves a square lattice
with
M
2 (first and second neighbor pair interactions) and the right-triangle trio
E
RT
corresponding to two
E
1
legs at right angles and an
E
2
hypotenuse (or
E
112
for
short, in a general formal notation [
3
]).
The mapping to spin language, with
s
i
=
=±
1, is
n
i
=
(
1
+
s
i
)/
2. We see
E
1
4
+
E
2
4
+
E
RT
2
E
RT
4
E
RT
8
H
=
s
i
s
j
+
s
i
s
j
+
s
i
s
j
s
k
(2.3)
ij
ij
ijk
1
2
RT
In Ising or spin language, the three coefficients are called
−
J
1
,
−
J
2
, and
−
J
RT
,
respectively. For the pair interactions, we easily see that
J
2
J
1
=
E
2
/
E
1
+
E
RT
/
E
1
E
2
E
1
E
RT
|
∼
≈
for
|
4
|
E
1
|
(2.4)
1
+
2
E
RT
/
E
1
One might then naively - but incorrectly - expect that
J
RT
/
J
1
≈
E
RT
/
E
1
. Instead
J
RT
J
1
=
E
RT
/
E
1
E
RT
E
1
=
2
J
RT
/
J
1
⇔
(2.5)
2
+
4
E
RT
/
E
1
1
−
4
J
RT
/
J
1
J
1
∼
This highly non-linear relation leads to a surprising result: For
J
RT
/
1
/
4,
E
1
becomes large and negative. A similar effect would occur in the opposite direction
for
E
RT
/
E
RT
/
E
1
J
RT
/
J
1
. Furthermore, if
J
RT
/
J
1
increases to slightly above 1/4,
E
RT
/
2, a larger magnitude than one is likely to encounter.
Finally, it is unfortunate (to say the least) that some researchers (including promi-
nent ones) denote the lattice-gas energy parameters as
J
. Even in the simplest case
of just a nearest-neighbor interaction, the lattice-gas energy differs from the Ising
energy by a factor of 4; thus, misinterpretation and numerical inaccuracies are very
likely.
E
1
≈−
1
/
2.2.2 Effect on Phase Boundaries: Asymmetries Not Inevitable
A trio interaction, which breaks particle-hole symmetry (or up-down symmetry in
the Ising viewpoint), is generally expected to lead inevitably to substantial asymme-
tries in phase diagrams about half-monolayer coverage. A rather surprising finding
of numerical [Monte Carlo] calculations is that a single type of trio interaction need
