Chemistry Reference
In-Depth Information
Each
n
-body term is obtained by adding together the energies of each unique
subset of
n
fragments within the cluster and subtracting from that the lower
order (1-body, 2-body,
1)-body) contributions. The first-order correc-
tion is merely a sum of ''monomer'' energies in the cluster:
...
,(
n
X
N
f
i
E
1
¼
E
½
½
11
i
¼
1
Note that the monomer energies (at the cluster geometry) in this summation
can be combined with monomer energies (at monomer geometry) from the
summation in Eq. [10] to obtain the energy associated with the distortion of
the monomers from their optimal structure to their geometry in the cluster
(much like the relaxation energy for CP corrections in Eq. [6]). By definition,
the contribution of
E
DIST
to the interaction energy is positive (net repulsive
effect) when the clusters are fully optimized while
E
DIST
¼
0 in the rigid mono-
mer approximation:
X
N
f
i
¼
1
ð
½
½
Þ
½
12
E
DIST
E
E
f
i
i
¼
E
int
¼
E
DIST
þ
E
2
þ
E
3
þþ
E
N
½
13
The second-order term is obtained from the energies of each of
the
2
N
¼
N
ð
N
1
Þ=
2 unique pairs of fragments (or ''dimers'') from each of
1
1-body contributions must be subtracted:
2
which
N
X
1
X
N
f
i
f
j
E
2
¼
i
2
E
½
½
14
i
¼
1
j
>
f
i
f
j
¼
f
i
f
j
ð
f
i
þ
f
j
2
E
½
E
½
E
½
E
½
Þ
½
15
One should recognize this expression for
E
2
since it is identical to the 2-body
interaction energy from Eq. [9] (i.e.,
E
2
E
2
-
body
int
). This relationship provides
a rigorous definition of the nonadditivity or cooperativity:
¼
E
2
-
body
int
E
int
¼
E
DIST
þ
þ
E
3
þþ
E
N
E
2
-
body
int
E
nonadd
¼
E
DIST
þ
þ
d
½
16
E
many
-
body
int
¼
E
DIST
þ
½
17
