Chemistry Reference
In-Depth Information
energy (
E
2
-
body
int
) is simply the sum of the interaction energies of each pair of frag-
ments (
f
1
f
2
,
f
1
f
3
, and
f
2
f
3
) within the cluster. This RMA prescription for a tri-
mer can be extended readily to a cluster of arbitrary size,
N
, and composition in
which there are
2
¼
N
N
!=ð
2
!ð
N
2
Þ!Þ¼
N
ð
N
1
Þ=
2 unique pairwise interac-
tions:
X
X
N
1
N
E
2
-
body
int
f
i
f
j
f
i
f
j
½
f
1
f
2
...
f
N
¼
E
½
E
½
E
½
½
9
i
¼
1
j
>
i
The pairwise approximation tends to be accurate in weakly coupled sys-
tems. For example, Tauer and Sherrill demonstrated that more than 98% of
the interaction energy of various benzene tetramer structures can be recovered
by simply adding together the pairwise interactions (or ''dimers'') in the sys-
tem.
100
Despite ignoring higher order cooperative effects (3-body and 4-body
in this case),
E
2
-
body
int
differs from
E
int
by no more than 2% for the benzene tet-
ramer configurations examined in the study. Because the higher order contri-
butions account for deviations between the pairwise additive 2-body
approximation and
E
int
, they are also frequently called nonadditive or coop-
erative effects (or just the nonadditivity or cooperativity). These many-body
terms will be defined more precisely in the next section.
The nonadditivity tends to increase for more strongly coupled systems
(sometimes dramatically), and, consequently, the quality of the 2-body approx-
imation deteriorates.
101,102
In clusters of HF and/or H
2
O, the nonadditivity
can account for more than half of
E
int
, which necessarily implies that the error
associated with the 2-body approximation can exceed 50%.
59
This section of
the tutorial will use (HF)
3
,(HF)
4
, and (HF)
5
to demonstrate the procedure for
calculating these 2-body interactions as well as higher order (3-body,
...
N
-
body) contributions via a many-body decomposition of
E
int
.
Many-Body Decomposition
The most common rigorous many-body decomposition scheme for
weakly bound clusters is based upon the approach introduced by Hankins,
Moskowitz, and Stillinger in 1970.
103
Two lucid descriptions of the procedure
can be found in Ref. 104 and 105. Technically, a many-body decomposition
of
E
int
decomposes the energy of the cluster
E
½
f
1
f
2
...
f
N
into 1-body (
E
1
),
2-body (
E
2
),
...
,
N
-body (
E
N
) contributions:
X
N
E
int
¼
E
½
f
1
f
2
f
3
...
f
N
E
½
f
i
i
¼
1
X
N
¼f
E
1
þ
E
2
þ
E
3
þþ
E
N
g
E
½
f
i
½
10
i
¼1
