Chemistry Reference
In-Depth Information
with {
c
lm
} denoting the CI expansion coefficients of the CSFs. With such a com-
plete many-electron basis set of CSFs, any many-electron function can be expressed
as a linear combination of these many-electron basis functions. The FCI wave
function covers therefore all electron correlation effects.
However, FCI calculations are only feasible for small molecules, since
the number of excited determinants for
m
one-electron basis functions and
N
electrons is given by
N
[
27
]. For this reason, the FCI wave function is appro-
ximated to be able to perform calculations also for larger molecules. As a
first step, the expansion of the FCI wave function is rewritten in a systematic
way by grouping the CSFs according to the degree of orbital substitution (called
excitation):
2
m
X
X
F
i
a
j
þ
;
F
i
j
;i
k
C
FCI
m
¼
c
m;
0
F
0
þ
c
m
;
a
j
;
i
j
c
m
;
a
j
;
i
j
a
k;
i
k
a
j
;a
k
þ;
ðÞ
ð
; a
k
;i
k
a
j
;i
j
a
j
;i
j
ð
Þ
(12)
with
F
i
a
j
denoting all singly excited determinants which can be obtained from
the ground-state determinant,
F
i
j
;i
k
a
j
;a
k
all doubly excited determinants, and so
forth. Obviously, the simplest approximation is obtained by truncating the expan-
sion after a certain class of terms (truncated CI). Taking all singly excited determi-
nants into account is referred to as CIS (CI-Singles), whereas incorporating also
the doubly excited determinants yields CISD (CI-Singles-Doubles), and so forth.
A major drawback is the violation of size consistency in the truncated CI approach,
i.e., the energy of two identical molecules at infinite separation is not equal to two
times the energy of a single molecule.
Different variants of truncated CI approaches exist. In truncated CI, the excited
determinants are all obtained from the ground-state determinant. If also from other
reference determinants excited determinants are produced, we arrive at the multi-
reference CI (MRCI) approach. Sometimes, it is also convenient to define a
restricted orbital space (active space), incorporate all excited determinants within
this space, and reoptimize the orbital basis which is known as complete-active-
space self-consistent-field approach (CASSCF).
There exists an approximation to the FCI wave function which overcomes the size-
consistency problem of the truncated CI approach. This is the coupled-cluster
approach. For the coupled-cluster approximation, we first define an excitation operator:
1
T
¼
T
k
;
(13)
k¼
1
which contains all possible excitations from the ground-state determinant such that
the operator
T
k
, when acting on the ground-state determinant, produces a linear
combination of all possible
k
-fold excited determinants,
X
t
i
1
;...;i
k
a
1
;...a
k
Y
i
1
;...;i
k
T
k
Y
0
¼
a
1
;...;a
k
;
(14)
ðÞ
ð
a
1
;i
1
Þ...
a
k;
i
k
