Chemistry Reference
In-Depth Information
Again, a general expression for
E
n
can be obtained in the same manner from
Eq. [22]:
XX
XXX
N
f
i
n
2
f
i
n
1
f
i
n
E
n
¼
E
½
f
i
1
f
i
2
f
i
3
...
1
i
1
<
i
2
<
i
n
2
<
i
n
1
<
i
n
1
XX
N
XX
f
i
n
2
f
i
n
1
þð
1
Þ
E
½
f
i
1
f
i
2
f
i
3
...
1
i
1
<
i
2
<
i
n
2
<
i
n
1
2
XX
X
N
f
i
n
2
þð
1
Þ
½
f
i
1
f
i
2
f
i
3
...
E
1
i
1
<
i
2
<
i
n
2
þ
3
X
X
X
N
2
N
1
N
n
f
i
1
f
i
2
f
i
3
þð
1
Þ
E
½
i
1
¼
1
i
2
>
i
1
i
3
>
i
2
n
2
X
N
1
X
N
f
i
1
f
i
n
2
þð
1
Þ
E
½
i
1
¼
1
i
2
>
i
1
1
X
N
n
f
i
1
þð
1
Þ
½
½
27
E
i
1
¼
1
Application to HF Trimer, Tetramer, and Pentamer
Because the cyclic
5) used in this tutorial are sym-
metric, the number of computations required to perform a many-body decom-
position of the interaction energy is reduced dramatically. In general,
application of the decomposition procedure to a pentamer could require as
many as 25 additional calculations:
ð
HF
Þ
n
clusters (
n
¼
3
4
¼
3
¼
5
5
5 for the tetramer subsets,
10
2
¼
5
for the trimer subsets, and
10 for the dimer subsets. For (HF)
5
, however,
symmetry reduces this to 5 calculations (1 unique tetramer computation, 2
unique trimer computations, and 2 unique dimer computations).
For (HF)
3
, there is only a single unique 2-body energy since
E
½
f
1
f
2
¼
½
f
1
f
3
¼
½
f
2
f
3
while there are two such quantities for (HF)
4
and (HF)
5
E
E
(
E
).
These values are reported Table 3 for both relaxed and rigid monomers. One
finds a single unique 3-body energy (
E
½
f
1
f
2
¼
E
½
f
1
f
4
¼
E
½
f
2
f
3
¼
E
½
f
3
f
4
E
½
f
i
f
j
and
E
½
f
1
f
3
¼
E
½
f
2
f
4
E
½
f
i
f
k
f
1
f
2
f
3
¼
f
1
f
2
f
4
¼
f
1
f
3
f
4
¼
f
2
f
3
f
4
½
E
½
E
½
E
½
)
f
1
f
2
f
3
¼
f
1
f
2
f
5
¼
f
1
f
4
f
5
¼
f
2
f
3
f
4
¼
for (HF)
4
but
two for (HF)
5
(
E
½
E
½
E
½
E
½
f
3
f
4
f
5
f
i
f
j
f
k
f
1
f
2
f
4
¼
f
1
f
3
f
4
¼
f
1
f
3
f
5
¼
f
2
f
3
f
5
¼
f
2
f
4
f
5
E
½
E
½
and
E
½
E
½
E
½
E
½
E
½
f
i
f
j
f
l
E
). Of course, (HF)
5
has only one unique 4-body energy, which is given
in Table 3 along with all of the 2- and 3-body energies.
Step 5 in this tutorial is to reproduce the RHF/aug-cc-pVDZ electronic
energies in Table 3. Sample input files for several popular software packages
are available online.
92
It is worth noting that these computations could just as
easily be performed in the entire basis set of the complex, thereby yielding a
½
