Chemistry Reference
In-Depth Information
3
¼
N
Continuing to the third-order expression gives
N
ð
N
1
Þð
N
2
Þ=
6
2
2-body and
1
1-body contributions that
3
3
unique ''trimers.'' There are
must be removed from each trimer energy:
X
2
X
1
X
N
N
N
f
i
f
j
f
k
E
3
¼
j
3
E
½
½
18
i
¼
1
j
>
i
k
>
f
i
f
j
f
k
¼
f
i
f
j
f
k
ð
3
E
½
E
½
f
i
f
j
þ
f
i
f
k
þ
f
j
f
k
Þð
f
i
þ
f
j
þ
f
k
Þ ½
½
½
½
½
½
½
19
2
E
2
E
2
E
E
E
E
4
''tetramer'' energies from each of
N
For the 4-body contribution, there are
3
3-body,
2
2-body, and
1
1-body terms are subtracted:
4
4
4
which
X
X
X
X
N
3
N
2
N
1
N
l
>
k
4
E
f
i
f
j
f
k
f
l
E
4
¼
½
½
20
>
i
¼
1
j
i
k
>
j
f
i
f
j
f
k
f
l
¼
f
i
f
j
f
k
f
l
ð
3
E
f
i
f
j
f
k
þ
3
E
f
i
f
j
f
l
þ
3
E
f
i
f
k
f
l
4
E
½
E
½
½
½
½
f
j
f
k
f
l
f
i
f
j
þ
2
E
f
i
f
k
þ
2
E
f
i
f
l
f
j
f
k
þ
3
E
½
Þð
2
E
½
½
½
2
E
½
f
j
f
l
þ
2
E
f
k
f
l
f
i
þ
f
j
þ
f
k
þþ
f
l
þ
2
E
½
½
Þð
E
½
E
½
E
½
E
½
Þ
½
21
A new indexing notation is introduced to help provide a generalized
expression for the
n
-body contribution to the (interaction) energy of a cluster
with
N
components. The indices
i
;
;
; ...
are replaced with
i
1
;
;
; ...;
;
i
n
to emphasize that this
n
-body component,
E
n
, contains
n
nested summations
giving rise to
j
k
i
2
i
3
i
n
1
n
¼
N
N
!=ð
n
!ð
N
n
Þ!
terms:
X
X
X
X
N
n
þ
1
N
n
þ
2
N
1
N
f
i
n
1
f
i
n
E
n
¼
i
1
i
n
1
n
E
½
f
i
1
f
i
2
f
i
3
...
½
22
i
1
¼
1
i
2
>
i
n
1
>
i
n
2
i
n
>
Again, each
n
E
term is obtained by removing the lower order contributions
from the
electronic
energy of
the
n
-mers
composed of
fragments
1
1-body terms,
2
2-body
n
n
f
i
1
f
i
2
f
i
3
...
f
i
n
1
f
i
n
. That is, one must subtract
1
ð
n
f
i
n
1
f
i
n
terms,
...
,
n
1
Þ
-body terms from
E
½
f
i
1
f
i
2
f
i
3
...
. To denote these
n
terms; the indices
a
1
;
a
2
; ...;
a
n
1
are used to run over the values of a particular
set of fragment indices,
S
¼f
i
1
;
i
2
;
i
3
; ...;
i
n
1
;
i
n
g
. Note that the index
a
j
cor-
responds to the
j
th element in
and, therefore, fragment
f
i
j
. It does not neces-
sarily have anything to do with fragment
f
j
. This is a subtle but important
distinction. For example,
f
1
does not appear in
S
3
E
½
f
3
f
5
f
7
even though
a
loops
