Chemistry Reference
In-Depth Information
over 1, 2, 3 to give
i
a
1
¼
3,
i
a
2
¼
5, and
i
a
3
¼
7.
f
i
n
1
f
i
n
n
E
½
f
i
1
f
i
2
f
i
3
...
f
i
n
1
f
i
n
¼
E
½
f
i
1
f
i
2
f
i
3
...
X
n
f
i
a
E
½
a
¼
1
n
X
1
X
n
f
i
a
1
f
i
a
2
a
1
2
E
½
a
2
>
a
1
¼
1
X
X
X
n
2
n
1
n
f
i
a
1
f
i
a
2
f
i
a
3
a
2
3
E
½
a
1
¼
1
a
2
>
a
1
a
3
>
XX
n
X
f
i
a
n
2
a
n
2
n
2
E
½
f
i
a
1
f
i
a
2
f
i
a
3
...
1
a
1
<
a
2
<
XX
XX
n
f
i
a
n
2
f
i
n
1
a
n
1
1
E
½
f
i
a
1
f
i
a
2
f
i
a
3
...
½
23
n
1
a
1
<
a
2
<
a
n
2
<
n
E
, the expressions for the components of
the cluster energy (and therefore interaction energy via Eq. [10] and [13]) can
be simplified. In the following form, it is easier to see a connection between
this many-body decomposition and the inclusion-exclusion principle (also
known as the sieve principle) from combinatorial mathematics:
By expanding
;
; ...;
2
E
3
E
N
X
1
X
N
f
i
f
j
E
2
¼
E
½
i
¼
1
j
>
i
i
þ
j
Þ
ð
E
½
E
½
½
24
X
X
X
N
2
N
1
N
f
i
f
j
f
k
E
3
¼
E
½
i
¼
1
j
>
i
k
>
j
f
i
f
j
þ
f
i
f
k
þ
f
j
f
k
Þ
ð
E
½
E
½
E
½
f
i
þ
f
j
þ
f
k
Þ
þð
E
½
E
½
E
½
½
25
X
X
X
X
N
3
N
2
N
1
N
f
i
f
j
f
k
f
l
E
4
¼
E
½
i
¼1
j
>
i
k
>
j
l
>
k
f
i
f
j
f
k
þ
f
i
f
j
f
l
þ
f
i
f
k
f
l
þ
f
j
f
k
f
l
ð
E
½
E
½
E
½
E
½
Þ
f
i
f
j
þ
f
i
f
k
þ
f
i
f
l
þ
f
j
f
k
þ
f
j
f
l
þ
f
k
f
l
þð
E
½
E
½
E
½
E
½
E
½
E
½
Þ
f
i
þ
f
j
þ
f
k
þ
f
l
ð
½
½
½
½
Þ
½
26
E
E
E
E
