Chemistry Reference
In-Depth Information
Consequently, Eq. [27] and the preceding equations end up with fairly simple
forms for
ð
HF
Þ
n
,
n
¼
3
5:
E
DIST
½
HF
n
¼
n
E
RLX
½
HF
½
28
f
i
f
j
HF
E
2
½
HF
3
¼
3
E
½
6
E
½
½
29
f
i
f
j
þ
f
i
f
k
HF
E
2
½
HF
4
¼
4
E
½
2
E
½
12
E
½
½
30
f
i
f
j
þ
f
i
f
k
HF
E
2
½
HF
5
¼
5
E
½
5
E
½
20
E
½
½
31
f
i
f
j
f
k
f
f
i
f
j
þ
f
i
f
k
g þ
HF
Þ
E
3
½
HF
4
¼
4
ð
E
½
2
E
½
E
½
3
E
½
½
32
f
i
f
j
f
k
þ
f
i
f
j
f
l
f
i
f
j
þ
f
i
f
k
g þ
HF
Þ
E
3
½
HF
5
¼
5
ð
E
½
E
½
3
f
E
½
E
½
6
E
½
½
33
f
i
f
j
f
k
f
l
f
i
f
j
f
k
þ
f
i
f
j
f
k
g
E
4
½
HF
5
¼
5
ð
E
½
2
f
E
½
E
½
f
i
f
j
þ
f
i
f
k
g
HF
Þ
þ
3
f
E
½
E
½
4
E
½
½
34
Þ
n
cluster can be obtained in
different ways, the easiest of which is to subtract the lower order contributions
(
E
2
;
The full
n
-body contribution to each
ð
HF
E
n
1
) from
E
many
-
body
int
. Alternatively, Eq. [27] can be simplified in
the same manner as the lower order terms:
E
3
; ...;
f
i
f
j
þ
HF
E
3
½
HF
3
¼
E
½
HF
3
3
E
½
3
E
½
½
35
f
i
f
j
f
k
þ
f
i
f
j
þ
f
i
f
k
HF
E
4
½
HF
4
¼
E
½
HF
4
4
E
½
4
E
½
2
½
4
E
½
½
36
f
i
f
j
f
k
f
l
þ
f
i
f
j
f
k
þ
f
i
f
j
f
l
E
5
½
HF
5
¼
E
½
HF
5
5
E
½
5
E
½
5
½
f
i
f
j
f
i
f
k
þ
HF
5
E
½
5
½
5
E
½
½
37
Although every contribution to the interaction energies of these HF clus-
ters are attractive, this does not always hold. In certain cases, some of the
many-body components may actually be repulsive.
100
In step 6, you should
calculate components of
E
int
in Table 4 (
E
DIST
;
E
5
) by applying
Eqs. [12], [16], [17], [25]-[27] to the electronic energies in the Tables 1, 2,
and 3.
E
2
;
E
3
;
E
4
;
Size Consistency and Extensivity of the Energy
Energy is an extensive property. This fundamental thermodynamic prin-
ciple is introduced early in most general chemistry textbooks, and it provides
the foundation for the supermolecule description of intermolecular interac-
tions. Unfortunately, not all electronic structure techniques are size con-
sistent
106
(or more generally size extensive
107
). That is, the energy computed
by some methods does not scale properly with the number of noninteracting
fragments. Readers interested in more detail may be interested in the sections
discussing size consistency and extensivity in the review of coupled-cluster
theory by Crawford and Schaefer.
108
