Biomedical Engineering Reference
In-Depth Information
of CI methods the reader could refer to computational chemistry
textbooks [24, 100, 120].
The quite accurate Möller and Plessetmethod (MP) is based
on the Rayleigh-Schrödinger perturbation theory: for a system of
n
occupied states, the correlation potential
is treated as a perturbation of the HF Hamiltionian
electrons,
N
nuclei, and
m
(0)
H
, namely,
(0)
l
V
H
=
H
+
(8.7)
with
n
2
N
Z
(0)
i
k
HF
H
V
(8.8)
i
2
r
i
1
k
1
iki
The correlation related perturbation term
is defined in terms
of the Coulomb and the exchange terms of the HF Theory.
Thus, the exact wave function is obtained by solving the
equation:
(
V
(0)
l
V
y〉
e
y〉
(8.9)
where both the wave functions and the eigenvalues are expanded in
Taylor series of the perturbation parameter
H
+
)|
=
|
.
Each term of the wave function expansion can be represented
with the complete set of the HF eigenfunctions and the expansion
coefficients of both the eigenvalues and the wave functions can be
found by using the Slater-Condon rule. The q-order expansion of the
MP perturbation approach is denoted as MPq approach.
The MPq theory can be generalized by replacing the
single determinants with multiconfigurations states (MCSCF:
multiconfiguration self-consistent field); there are various versions
of multireference perturbation theory such as the CASTP2D or
similar [24, 100, 120].
One of the most successful methods is the Coupled Cluster (CC)
theory that has been introduced as a further development of the
above described CI method [7, 24]. The main assumption of the CC
theory is that the “true” wave function can be represented as:
l
y
=
e
T
y
(8.10)
HF
e
T
where the
operator includes all the possible excited states of
the HF determinant. It can be demonstrated that the CC scheme is
virtually equivalent to a full-CI approach. Among the different CC
schemes encountered, one of the most popular is the CCSD(T) that
includes also a singles/triples coupling term, but not all the terms
arising from triple excitations [49].
