Chemistry Reference
In-Depth Information
with the expansion coefficients
t
i
1
;...;i
k
a
1
;...;
a
k
called cluster amplitudes (they can be related
to the CI coefficients). The coupled-cluster wave function is then obtained through
an exponential ansatz:
C
CC
0
¼
exp
ð
T
ÞY
0
(15)
which can be Taylor expanded:
"
#
n
1
1
1
n
ð
T
Þ¼
T
k
exp
!
n¼
0
k¼
1
1
2
1
3
T
1
þ
T
1
þ:
¼
1
þ
T
1
þ
T
2
þþ
T
1
T
2
þ
(16)
!
!
If the summations are not truncated, the coupled-cluster wave function is equal
to the FCI wave function. Even if the excitation operator
T
is truncated after single
excitations
T
T
2
(CCSD), the expan-
sion of the exponential function contains higher excitations than a corresponding
truncated CI through products of excitation operators like
T
1
T
2
, the so-called
disconnected clusters. This is also the reason why the coupled-cluster method is
size consistent even in its truncated form. Practical applications, using a truncated
excitation operator, require expressions that contain a finite number of terms. This
can be achieved using a Baker-Campell-Hausdorff expansion leading to a series of
nested commutators such that the electronic energy is calculated as:
¼
T
1
(CCS) or double excitations
T
¼
T
1
þ
F
0
1
2
1
3
1
4
E
¼
F
0
H
þ
½
H
;
T
þ
½
½
H
;
T
;
T
þ
½
½
½
H
;
T
;
T
;
T
þ
½
½
½
½
H
;
T
;
T
;
T
;
T
:
!
!
!
(17)
Because the coupled-cluster approach is nonlinear in the amplitudes as can be
deduced from (
16
), the corresponding equations for the calculations of the ampli-
tudes are solved by projection.
3.2 The Hamiltonian Operator
The Hamiltonian for a system of
M
nuclei and
N
electrons, including all one-
electron terms and two-particle interactions, is given by:
X
X
X
X
M
N
M
M
H
¼
t
n
ð
I
Þþ
t
e
ð
i
Þþ
v
nn
I
ðÞ
;
J
I¼
1
i¼
1
I¼
1
J¼Iþ
1
(18)
X
X
X
X
N
N
M
N
þ
v
ee
i
ðÞþ
;
j
v
ne
I
ðÞ
;
i
i¼
1
j¼iþ
1
I¼
1
i¼
1
