Chemistry Reference
In-Depth Information
series sometimes become oscillatory. Moreover, higher orders of MP perturbation
theory can even diverge [
5
].
Another popular post-HF technique is the CC method [
4
-
6
]. The CC method
solves the size consistency problem of CI by forming a wave function where the
excitation operators are exponentiated:
ð T
j i¼
CC
exp
Þ
j ;
HF
(9)
where
T
¼ T
1
þ T
2
þ T
3
þ :::
(10)
and
T
n
is a linear combination of n-type excitations, for example,
ð T
1
Þ
j
CCS
i ¼
exp
j ;
HF
(11)
ð T
1
þ T
2
Þ
j
CCSD
i¼
exp
j ;
HF
(12)
ð T
1
þ T
2
þ T
3
Þ
j
CCSDT
i ¼
exp
j i
HF
(13)
and so on. Thus, at the |CCS
level, all possible single excitations are included in the
cluster operator, and at the |CCSD
i
the double excitations are also taken into
consideration, etc. At each level of CC theory one includes through the exponential
parametrisation of (9) all possible determinants that can be generated within a given
orbital basis. These are all the determinants that enter the full CI wave function in
the same orbital basis. Thus, the improvement in the sequence |CCSD
i
,
and so on does not occur because more determinants are included but because of an
improved representation of their expansion coefficients. Owing to the presence of
the disconnected clusters, CC wave functions truncated at a given excitation level
also contain contributions from determinants corresponding to higher-order excita-
tions. The terms that are missing relative to full CI represent higher-order connected
clusters and the associated disconnected clusters. By contrast, CI wave functions
truncated at the same level contain contributions from determinants only up to this
level.
The CC model is not variational. With a large enough basis set CCSD typically
recovers 95% of the correlation energy for a molecule at equilibrium geometry, while
the inclusion of triple excitations give rise to a further five- to tenfold reduction in
error. The CCSD(T) method has become the method of choice for accurate small-
molecule calculations.
In recent years there has been a growing interest in numerical techniques which
can speed up quantum chemical computations. Various methods are available to
approximate the four-index electron repulsion integrals as products of three-index
intermediates. These methods are called density fitting (DF) or resolution of the
identity (RI), and Cholesky decomposition (CD) techniques. A general comparison
i
, |CCSDT
i
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